# Patent

## Title
Optical structure on an optical fiber and method of fabrication

## Abstract

An optical structure on an optical fiber and a method of fabrication is provided. The optical structure includes an end of an optical fiber and a layer formed on the end of the optical fiber. The layer comprises one or more first portions having a first optical pathlength in a direction perpendicular to the layer and one or more second portions having a second optical pathlength in the direction perpendicular to the layer, the second optical pathlength different from the first optical pathlength.

## Background
CLAIM OF PRIORITY
This application is a divisional from U.S. patent application Ser. No. 11/971,714, filed Jan. 9, 2008, incorporated in its entirety by reference herein, which claims the benefit of U.S. Provisional Patent Appl. No. 60/879,465, filed Jan. 9, 2007, which is incorporated in its entirety by reference herein, and U.S. Provisional Patent Appl. No. 61/000,003, filed Oct. 23, 2007, which is incorporated in its entirety by reference herein.


BACKGROUND OF THE INVENTION
1. Field of the Invention
This application relates generally to sensor systems, and more particularly to optical-fiber-compatible sensor systems.
2. Description of the Related Art
Fiber-optic acoustic sensors have been extensively researched since their first emergence in the 1970s (see, e.g., J. H. Cole, R. L. Johnson, and P. G. Bhuta, “Fiber-optic detection of sound,” J. Acoust. Soc. Am. Vol. 62, 1136 (1977); J. A. Bucaro, H. D. Dardy, and E. F. Carome, “Fiber-optic hydrophone,” J. Acoust. Soc. Am. Vol. 62, 1302 (1977)), largely because of their multiple advantages over conventional acoustic sensors. These advantages include small size and weight, ability to operate in chemically and/or electrically harsh environments, ease of multiplexing large numbers of sensors together, and compatibility with fiber-optic networks for data transport and processing.
Various forms of biological, chemical, and mechanical sensors (such as acoustic or pressure sensors) that can be addressed optically at the end of an optical fiber can be useful for medical and security applications. The very small size (e.g., 125 μm diameter) of these sensors, can be used for example to penetrate tissue or veins, or to deploy in places where small size is crucial. Also, optical devices such as filters, mirrors, and polarizers at a fiber end can be very useful in fiber communication applications.
In several key applications, such as undersea oil exploration and smart wells, the demand for more sensitive and more compact fiber sensors has been a strong drive behind recent research efforts. A current limitation of acoustic fiber sensors is that in order to be highly sensitive, they require a long length of fiber, which makes them bulky and poorly to non-responsive to frequencies above a few hundred Hz.
Various fiber optic sensor systems have been previously disclosed that provide acoustic pressure measurements based on the relative displacements of the two mirrors of a Fabry-Perot interferometric cavity. See, e.g., M. Yu et al., “Acoustic Measurements Using a Fiber Optic Sensor System,” J. Intelligent Mat'l Systems and Structures, vol. 14, pages 409-414 (July 2003); K. Totsu et al., “Ultra-Miniature Fiber-Optic Pressure Sensor Using White Light Interferometry,” J. Micromech. Microeng., vol. 15, pages 71-75 (2005); W. B. Spillman, Jr. et al., “Moving Fiber-Optic Hydrophone,” Optics Lett., vol. 5, no. 1, pages 30-31 (January 1980); K. Kardirvel et al., “Design and Characterization of MEMS Optical Microphone for Aeroacoustic Measurement,” 42nd AIAA Aerospace Sciences Meeting and Exhibit, 5-8 Jan. 2004, Reno, Nev.; J. A. Bucaro et al., “Miniature, High Performance, Low-Cost Fiber Optic Microphone,” J. Acoust. Soc. Am., vol. 118, no. 3, part 1, pages 1406-1413 (September 2005); T. K. Gangopadhyay et al., “Modeling and Analysis of an Extrinsic Fabry-Perot Interferometer Cavity,” Appl. Optics, vol. 44, no. 16, pages 312-3196 (1 Jun. 2005); and P. J. Kuzmenko, “Experimental Performance of a Miniature Fabry-Perot Fiber Optic Hydrophone,” Proceedings of 8th Optical Fiber Sensors Conference, Monterey, Calif., Jan. 29-31, 1992, pages 354-357.
Photonic crystal slabs (PCSs) are photonic crystal structures having a spatially periodically varying refractive index. A PCS exhibits guided resonance optical modes that are strongly confined within the PCS, but are coupled to incident radiation through a phase matching mechanism due to the periodically varying refractive index. These guided resonance modes are typically manifest in transmission or reflection spectra as sharp Fano lineshapes superimposed on a smoothly varying background. See, e.g., M. Kanskar et al., “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett., vol. 70, page 1438 (1997); V. N. Astratov et al., “Resonant coupling of near-infrared radiation to photonic band structure waveguides,” J. Lightwave Technol., vol. 17, page 2050 (1999); and S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B, vol. 65, page 235112 (2002). Such guided resonance modes have been used previously as optical filters or mirrors in light emitting diodes and lasers.
SUMMARY OF THE INVENTION
In certain embodiments, an acoustic sensor is provided. The acoustic sensor comprises at least one photonic crystal structure. The acoustic sensor further comprises an optical fiber having an end optically coupled to the at least one photonic crystal structure. The acoustic sensor further comprises a structural portion mechanically coupled to the at least one photonic crystal structure and to the optical fiber. The at least one photonic crystal structure, the optical fiber, and the structural portion substantially bound a region having a volume such that a frequency response of the acoustic sensor is generally flat in a range of acoustic frequencies.
In certain embodiments, a method of fabricating an acoustic sensor is provided. The method comprises providing at least one photonic crystal structure. The method further comprises optically coupling an end of an optical fiber to the at least one photonic crystal structure. The method further comprises providing a structural portion and mechanically coupling the structural portion to the at least one photonic crystal structure and to the optical fiber. The at least one photonic crystal structure, the optical fiber, and the structural portion substantially bound a region having a volume such that a frequency response of the acoustic sensor is generally flat within a range of acoustic frequencies.
In certain embodiments, an acoustic sensor is provided. The acoustic sensor comprises at least one photonic crystal structure. The acoustic sensor further comprises an optical fiber having an end optically coupled to the at least one photonic crystal structure. The acoustic sensor further comprises a structural portion mechanically coupled to the at least one photonic crystal structure and to the optical fiber. The at least one photonic crystal structure, the optical fiber, and the structural portion substantially bound a region having a volume such that a sensitivity of the acoustic sensor is better than about 100 μPa/Hz1/2 in a range of acoustic frequencies below 20 kHz.
In certain embodiments, a method of fabricating an acoustic sensor is provided. The method comprises providing at least one photonic crystal structure. The method further comprises optically coupling an end of an optical fiber to the at least one photonic crystal structure. The method further comprises providing a structural portion and mechanically coupling the structural portion to the at least one photonic crystal structure and to the optical fiber. The at least one photonic crystal structure, the optical fiber, and the structural portion substantially bound a region having a volume such that a sensitivity of the acoustic sensor is better than about 100 μPa/Hz1/2 in a range of acoustic frequencies below 20 kHz.
In certain embodiments, a method of fabricating an acoustic sensor is provided. The method comprises providing at least one photonic crystal structure. The method further comprises optically coupling an end of an optical fiber to the at least one photonic crystal structure. The method further comprises providing a structural portion and mechanically coupling the structural portion to the at least one photonic crystal structure and to the optical fiber. A distance between the at least one photonic crystal structure and the end of the optical fiber is selected to improve a temperature stability of the acoustic sensor.
In certain embodiments, a method of fabricating an optical structure on an optical fiber is provided. The method comprises preparing an end of an optical fiber. The method further comprises forming a layer onto the fiber end. The method further comprises patterning the layer to form the optical structure.
In certain embodiments, a method of fabricating a photonic crystal structure is provided. The method comprises forming a first dielectric layer on an end of an optical fiber. The method further comprises forming a sacrificial layer over at least a portion of the first dielectric layer. The method further comprises forming a second dielectric layer over the sacrificial layer. The method further comprises patterning the first dielectric layer to form a first photonic crystal slab. The method further comprises patterning the second dielectric layer to form a second photonic crystal slab. The method further comprises removing the sacrificial layer.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 schematically illustrates an example acoustic sensor compatible with certain embodiments described herein.
FIG. 2A schematically illustrates an example photonic crystal slab (PCS) having a substantially square array of substantially circular holes extending completely through the slab.
FIG. 2B illustrates a scanning electron microscope micrograph of portions of an example PCS.
FIG. 2C schematically illustrates another example PCS having a substantially square array of substantially circular holes extending only partly through the PCS.
FIG. 2D schematically illustrates another example PCS having a substantially square distribution of protrusions.
FIGS. 2E and 2F schematically illustrate cross-sectional views of other example PCSs having a plurality of elongated regions with a substantially one-dimensionally-periodic distribution.
FIGS. 3A-3C schematically illustrates an example PCS exhibiting an optical resonance in the simulated transmitted optical power spectrum for light incident in a direction substantially perpendicular to the PCS.
FIG. 4A shows the reflectivity of an example PCS over a wavelength range of 1300 nanometers to 1600 nanometers
FIG. 4B schematically illustrates the measured resonance wavelength shift for substantially perpendicularly incident light on an example PCS as a function of temperature.
FIG. 5 illustrates the resonance wavelength shift for substantially perpendicularly incident light on an example PCS as a function of mechanical forces applied to the PCS.
FIG. 6 schematically illustrates an experimental configuration of a 1-centimeter long PCS in proximity to an acoustic speaker.
FIGS. 7A and 7B schematically illustrate an example acoustic sensor having a single PCS.
FIG. 8 schematically illustrates an example photonic crystal structure comprising a first PCS and a second PCS substantially parallel to the first PCS.
FIG. 9 is a plot of various normalized transmission spectra measured from a photonic crystal structure comprising a pair of PCSs.
FIGS. 10A-10C schematically illustrate the dependence of the resonance frequencies of a photonic crystal structure comprising a first PCS and a second PCS.
FIG. 11 schematically illustrates the transmission spectra of two PCSs coupled in the near-field configuration when the PCSs are laterally displaced from one another.
FIG. 12 illustrates the measured transmission spectra corresponding to TE polarized light incident on the PCS at various incidence angles.
FIGS. 13A-13D schematically illustrate example PCS structures having at least one photonic crystal defect.
FIGS. 14A and 14B schematically illustrate an example implementation for mirror-symmetry breaking in a PCS structure compatible with certain embodiments described herein.
FIG. 15 schematically illustrates several example hole structures which break or remove one or more of the mirror symmetries of the PCS unit cell.
FIG. 16A schematically illustrates a unit cell of a PCS having circularly symmetric holes on a periodic square lattice distribution.
FIGS. 16B-16E schematically illustrate the dot products of various resonance modes of the PCS with plane waves polarized in the horizontal direction (x-polarization) and with plane waves polarized in the vertical direction (y-polarization).
FIG. 17A schematically illustrates an example unit cell of a PCS having holes on a periodic square lattice distribution, in which each hole comprises a small region to one side of the hole.
FIGS. 17B and 17C schematically illustrate an asymmetric resonance mode of the PCS of FIG. 17A.
FIG. 17D schematically illustrates the dot product of the odd-symmetric resonance mode with an incident plane wave with y-polarization.
FIG. 18A schematically illustrates a PCS unit cell with the circularly symmetric hole of FIG. 16A having four mirror symmetry axes.
FIG. 18B schematically illustrates two doubly degenerate resonances and four non-degenerate resonances of the PCS structure of FIG. 18A.
FIG. 18C schematically illustrates x-polarized and y-polarized incident plane waves and the corresponding electric fields.
FIG. 18D schematically illustrates a PCS unit cell with an asymmetric hole that is missing a mirror symmetry about the horizontal axis.
FIG. 18E schematically illustrates a PCS unit cell with a rotationally-asymmetric hole.
FIGS. 19A and 19B show finite-difference time-domain simulations (FDTD) of transmission spectra for the three different hole shapes of FIGS. 18A, 18D, and 18E for polarizations perpendicular and parallel, respectively, to the hole elongations.
FIGS. 20A and 20B shows FDTD simulations of transmission spectra for incident light with polarizations perpendicular and parallel, respectively, to the hole elongations.
FIGS. 21A-21C are scanning-electron microscopy images of PCS structures with circularly-symmetric holes, minor-asymmetric holes, and rotationally-asymmetric holes, respectively.
FIGS. 21D-21F are scanning-electron microscopy images of the circularly-symmetric holes, minor-asymmetric holes, and rotationally-asymmetric holes, respectively.
FIGS. 22A and 22B show experimental measurements of the transmission spectrum for the three different PCS structures for polarizations perpendicular and parallel, respectively, to the hole elongations.
FIG. 23 illustrates the transmission spectra for the perpendicular polarization case of FIG. 22A on a larger wavelength range.
FIG. 24 schematically illustrates an example acoustic sensor system having a housing compatible with certain embodiments described herein.
FIG. 25 schematically illustrates an example acoustic sensor system having a secondary housing compatible with certain embodiments described herein.
FIG. 26 schematically illustrates another example acoustic sensor system having a secondary housing compatible with certain embodiments described herein.
FIG. 27 schematically illustrates an example acoustic sensor system having a metal layer on the optical fiber and a single PCS compatible with certain embodiments described herein.
FIG. 28 schematically illustrates an example acoustic sensor system having a fiber Bragg grating and a single PCS compatible with certain embodiments described herein.
FIG. 29 schematically illustrates a perspective view of an example configuration of an acoustic sensor system coupled to one end of an optical fiber.
FIGS. 30A-30Q schematically illustrate an example fabrication process flow compatible with certain embodiments described herein for the components of the acoustic sensor system.
FIG. 31 schematically illustrates an example configuration of a movable reflective element (e.g., a membrane) and an optical fiber.
FIG. 32 is a graph of an optical resonance as a function of wavelength.
FIG. 33 shows the impedance of an example acoustic sensor at a frequency of 30 kHz with a diaphragm diameter of 125 microns.
FIGS. 34A and 34B schematically illustrate the resonant wavelength changes for two different cavity lengths, respectively.
FIGS. 35A and 35B schematically illustrate a standard Fabry-Perot interferometer and an example fiber Fabry-Perot interferometer, respectively.
FIG. 36 is a plot of the resonances of a standard Fabry-Perot interferometer and of a fiber Fabry-Perot interferometer using an SMF-28 single-mode fiber.
FIG. 37 is a plot of the measured transmission spectrum from an example fiber Fabry-Perot interferometer.
FIG. 38A schematically illustrates an experimental set-up for characterizing a fiber Fabry-Perot acoustic sensor.
FIG. 38B schematically illustrates an example PCS-based acoustic sensor.
FIGS. 39A and 39B shows the experimental and theoretical response of an fiber Fabry-Perot acoustic sensor for different incident pressures at a frequency of 30 kHz.
FIG. 39C schematically illustrates an example weak-signal measurement experimental set-up.
FIG. 39D shows the frequency response of the example fiber FP acoustic sensor of FIGS. 39A and 39B.
FIG. 40 schematically illustrates an example configuration of a fiber Fabry-Perot acoustic sensor for analysis of thermal noise.
FIG. 41A schematically illustrates an equivalent circuit for an example fiber Fabry-Perot acoustic sensor and FIG. 41B schematically illustrates the equivalent circuit of FIG. 41A overlaid onto a schematic of the fiber FP acoustic sensor.
FIG. 42 schematically illustrates a noise equivalent circuit with no incident pressure, and with noise sources in series with the dissipative elements.
FIGS. 43A and 43B show the sensor response and the noise, respectively, of the sensor for an air medium.
FIGS. 44A and 44B show the sensor response and noise for an air medium, respectively, for an array of N=20×20 holes.
FIGS. 45A and 45B show the sensor response and noise for an air medium, respectively, for holes all around the diaphragm but with an increased cavity size.
FIGS. 46A and 46B show the sensor response and the noise, respectively, of the sensor for a water medium.
FIGS. 47A and 47B show the sensor response and noise for a water medium, respectively, for an array of N=20×20 holes.
FIGS. 48A and 48B show the sensor response and noise for a water medium, respectively, for holes all around the diaphragm but with an increased cavity size.
FIGS. 48C and 48D show the sensor response and the noise level of FIGS. 48A and 48B, respectively, in the lower frequency range.
FIGS. 49A and 49B schematically illustrate two example embodiments of the acoustic sensor having an increased cavity volume.
FIG. 50 schematically illustrates an example structure of a photonic crystal structure compatible with certain embodiments described herein.
FIG. 51 shows a pair of scanning electron micrographs of an example photonic crystal structure fabricated on a single-mode silica fiber with a 350-nm-thick amorphous-silicon layer at the fiber end in accordance with certain embodiments described herein.
FIG. 52 is a flow diagram of an example method of fabricating a photonic crystal structure on a fiber end in accordance with certain embodiments described herein.
FIG. 53A schematically illustrates an example device as an interface between the optical fiber and free space in accordance with certain embodiments described herein.
FIG. 53B schematically illustrates an example device spliced to another cleaved or polished fiber in accordance with certain embodiments described herein.
FIG. 53C schematically illustrates an example device having an end within a surrounding medium in accordance with certain embodiments described herein.
FIG. 54 is a flow diagram of an example method 900 of fabricating a photonic crystal structure in accordance with certain embodiments described herein.
FIGS. 55A-55F schematically illustrate a series of example fabrication steps for forming a near-field coupled photonic crystal device at the end of an optical fiber in accordance with certain embodiments described herein.
FIG. 56A shows three example holographic waveplates in accordance with certain embodiments described herein.
FIG. 56B shows two scanning electron micrographs of cross-sections of example sub-wavelength waveplates in accordance with certain embodiments described herein.
FIG. 57 schematically illustrates a wire-grid polarizer in accordance with certain embodiments described herein.



DETAILED DESCRIPTION OF EXAMPLE EMBODIMENTS
Present-day optical resonators which have sufficient quality factors to achieve sensitivities comparable to those of piezoelectric transducers are typically large and impractical to fabricate, install, align, and operate. In contrast, certain embodiments described herein comprise an acoustic sensor based on optical resonators formed by photonic crystal slab (PCS) structures with apertures which are orders of magnitude smaller than those of traditional optical cavities. The small size of certain such embodiments provides a sensitivity comparable to that of piezoelectric and capacitive displacement sensors for frequencies larger than about 10 kHz. Photonic crystal structures comprising a pair of PCSs can be used to provide notch and bandpass transmission and reflection filters, and such structures can be utilized in acoustic sensor systems compatible with various applications (e.g., oil exploration, undersea acoustic wave detection).
Certain embodiments described herein provide a novel fiber acoustic sensor based on an external fiber Fabry-Perot (FP) interferometer formed by placing a photonic-crystal slab (PCS) in proximity to the metal- or dielectric-coated reflective end of a single-mode fiber. Certain such embodiments provide an acoustic sensor that is extremely small (e.g., about 0.5 millimeter×about 0.5 millimeter×about 0.1 millimeter), operates up to high frequencies (e.g., 100 kHz and higher), and has a very high sensitivity. The fiber FP acoustic sensor is interrogated in certain embodiments with a single-frequency laser. When an acoustic wave is incident on the PCS, the PCS deflects, which modulates the spacing between the PCS and the mirrored fiber end, at the acoustic frequency. This spacing modulation is detected as a change in the laser power reflected by the FP interferometer. Employing a PCS as the external mirror in certain embodiments provides a lightweight membrane that moves easily in response to an incident acoustic wave, either in a gas (e.g., air) or in a liquid (e.g., water). Certain embodiments also provide the high reflectivity advantageously used for high sensitivity, as detailed herein. Certain such a fiber FP acoustic sensor can be advantageously more sensitive than other reported low-finesse miniature fiber FP sensors (see, e.g., K. Totsu, Y. Haga, and M. Esashi, “Ultra-miniature fiber-optic pressure sensor using white light interferometry,” J. Micromech. Microeng. Vol. 15, 71-75 (2005)). Compared to FP sensors based on other reflecting media, certain embodiments utilizing a PCS combine the advantages of being extremely thin and mass producible, as well as offering a high reflectivity (e.g., greater than 99%) and hence high sensitivity.
PCS structures have been used previously as filters and mirrors, in the same way as multi-layer dielectric stacks. However, PCS structures have several advantages over multi-layer mirrors, including but not limited to, being a single dielectric layer, being compatible with microelectromechanical systems (MEMS), and having unique properties that are difficult or impossible to achieve with multilayer stacks and that can be controlled through geometrical parameters. For example, PCS structures can have a high reflectivity over a broad range of wavelengths (e.g., an observed extinction in transmission of over 99% in a range of wavelengths greater than about 30 nanometers), and can be used as efficient filters at telecom wavelengths (e.g., 1540 nanometers) with sharp resonances observed to have Q of about 5000. In addition, a PCS structure can be used as a circular polarization beam-splitter separating plane-polarized light into its spin-polarized components. Also, though an introduction of a small form birefringence, a PCS structure can act as a dual quarter-wave retarder-based polarizing beam splitter, which separates an incoming wave equally into two orthogonal polarizations through reflection and transmission.
FIG. 1 schematically illustrates an example acoustic sensor 10 compatible with certain embodiments described herein. The acoustic sensor 10 comprises at least one photonic crystal structure 20 having at least one optical resonance with a resonance frequency and a resonance lineshape. The acoustic sensor 10 further comprises a housing 30 substantially surrounding the at least one photonic crystal structure 20 and mechanically coupled to the at least one photonic crystal structure 20. At least one of the resonance frequency and the resonance lineshape of the at least one photonic crystal structure 20 is responsive to acoustic waves 40 incident upon the housing 30. As illustrated by FIG. 1, in certain embodiments, the acoustic sensor 10 further comprises an optical fiber 50 optically coupled to the at least one photonic crystal structure 20.
Single PCS Structures
In certain embodiments, the at least one photonic crystal structure 20 comprises a PCS 70, an example of which is schematically illustrated by FIG. 2A. The PCS 70 comprises a first material 72 and an array of regions 74 within the PCS 70. The regions 74 comprise a second material 76 having a refractive index different from a refractive index of the first material 72. The PCS 70 of FIG. 2A has a thickness T and a substantially planar configuration.
In certain embodiments, the first material 72 comprises a solid dielectric material, examples of which include but are not limited to, silicon, silica, silicon nitride, ceramics, and plastics. In certain embodiments, the first material 72 comprises a solid semiconductor material, examples of which include but are not limited to, silicon, germanium, indium phosphide, gallium arsenide, or other III-V semiconductor materials. In certain embodiments, the second material 76 comprises a gas (e.g., air). In certain embodiments, the second material 76 comprises a fluid, examples of which include but are not limited to, water, isopropanol, ethanol, methanol, and other alcohols.
In certain embodiments, the thickness T of the PCS 70 is in a range between about 100 nanometers and about 1000 nanometers. In certain embodiments, the PCS 70 has a substantially square shape, while in other embodiments, the PCS 70 has a substantially circular, rectangular, hexagonal, elliptical, or other shape.
In certain embodiments, the regions 74 have a maximum width along a direction substantially parallel to the PCS 70 in a range between about 100 nanometers and about 1500 nanometers. In certain embodiments, the regions 74 have a substantially circular shape, while in certain other embodiments, the regions 74 have a substantially elliptical, oval, square, rectangular, triangular, pentagonal, hexagonal, semicircular, or other shape.
In certain embodiments, the array of regions 74 has a substantially two-dimensionally-periodic distribution. The periodicities of the distribution in two different directions generally parallel to the PCS 70 are substantially the same in certain embodiments, while in certain other embodiments, the periodicities are different. In certain embodiments, the center-to-center distance between nearest-neighboring regions 74 is in a range between about 100 nanometers and about 1550 nanometers. In certain embodiments, the substantially two-dimensionally-periodic distribution of the array of regions 74 is square, while in certain other embodiments, the substantially two-dimensionally-periodic distribution is rectangular, triangular, square, rhombic, oblique, or hexagonal. Other substantially two-dimensionally-periodic distributions are also compatible with certain embodiments described herein.
In certain embodiments, the regions 74 comprise a plurality of holes extending at least partially through the thickness of the PCS 70, containing the second material 76, and having a substantially two-dimensionally-periodic distribution within the PCS 70. For example, FIG. 2A schematically illustrates an example PCS 70 having an array of regions 74 comprising substantially circular holes extending completely through the thickness of the PCS 70 and having a substantially square distribution, in accordance with certain embodiments described herein. FIG. 2B illustrates a scanning electron microscope micrograph of portions of such an example PCS 70 fabricated on 450-nanometer thick silicon. FIG. 2C schematically illustrates another example PCS 70 having a substantially square array of regions 74 comprising substantially circular holes extending only partly through the thickness T of the PCS 70, thereby having a depth D less than the thickness T of the PCS 70, in accordance with certain other embodiments described herein.
FIG. 2D schematically illustrates another example PCS 70 having a substantially square distribution of protrusions 78 (e.g., pillars) having a substantially circular cross-section in a plane substantially parallel to the PCS 70, in accordance with certain other embodiments described herein. The protrusions 78 have a height H above the PCS 70 in a range between about 100 nanometers and about 1000 nanometers. In certain embodiments, the height H is greater than the thickness T, while in certain other embodiments, the height H is less than or equal to the thickness T. In certain embodiments, the protrusions 78 comprise the same material as does the underlying portions of the PCS 70, while in certain other embodiments, the protrusions 78 comprise a different material (e.g., the PCS 70 comprises silicon oxide while the protrusions 78 comprise silicon). In certain embodiments, the PCS 70 comprises a dielectric material (e.g., silicon, silica, silicon nitride, ceramics, plastics) or a semiconductor material (e.g., silicon, germanium, indium phosphide, gallium arsenide, or other III-V semiconductor). In certain embodiments, the protrusions 78 comprises a dielectric material (e.g., silicon, silica, silicion nitride, ceramics, plastics) or a semiconductor material (e.g., silicon, germanium, indium phosphide, gallium arsenide, or other III-V semiconductor). Other shapes, sizes, and distributions of the protrusions 78 are also compatible with certain embodiments described herein.
FIGS. 2E and 2F schematically illustrate cross-sectional views of other example slabs 70 having a plurality of elongated regions 74 with a substantially one-dimensionally-periodic distribution (e.g., a one-dimensional grating). In FIGS. 2E and 2F, the regions 74 extend in a direction substantially perpendicular to the cross-sectional view. In certain embodiments, the spacing between adjacent regions 74 is in a range between about 100 nanometers and about 1550 nanometers. In certain embodiments, the widths of the regions 74 are in a range between about 100 nanometers and about 1500 nanometers. In certain embodiments, the center-to-center spacing between adjacent regions 74 is in a range between about 100 nanometers and about 1550 nanometers.
As schematically illustrated by FIG. 2E, in certain embodiments, the PCS 70 comprises a first material (e.g., a dielectric material such as silica, silicon oxide, or silicon nitride) with regions 74 comprising troughs or grooves 80 within the PCS 70 containing the second material 76 (e.g., air or water). In certain embodiments, the grooves 80 extend completely through the thickness T of the PCS 70, while in certain other embodiments, the grooves 80 extend only partly through the thickness T of the PCS 70. The depth D of the grooves 80 is in a range between about 10 nanometers and about 1000 nanometers. In certain embodiments, the grooves 80 have a generally square, trapezoidal, curved or “U”-shaped, or triangular cross-section in a plane substantially perpendicular to the PCS 70. Other shapes and sizes of the grooves 80 are also compatible with certain embodiments described herein.
In certain other embodiments, as schematically illustrated by FIG. 2F, the regions 74 comprise protrusions 82 having a height H above the PCS 70 in a range between about 10 nanometers and about 1000 nanometers. The protrusions 82 of certain embodiments comprise the same material as does the underlying portions of the PCS 70, while in certain other embodiments, the protrusions 82 comprises a different material from the first material 72 (e.g., the PCS 70 comprises silicon oxide while the protrusions 82 comprise silicon). In certain embodiments, the PCS 70 comprises a dielectric material (e.g., silicon, silica, silicon nitride, ceramics, plastics) or a semiconductor material (e.g., silicon, germanium, indium phosphide, gallium arsenide, or other III-V semiconductor). In certain embodiments, the protrusions 82 comprises a dielectric material (e.g., silicon, silica, silicion nitride, ceramics, plastics) or a semiconductor material (e.g., silicon, germanium, indium phosphide, gallium arsenide, or other III-V semiconductor). In certain embodiments, the protrusions 82 have a generally square, trapezoidal, curved or “U”-shaped, or triangular cross-section in a plane substantially perpendicular to the PCS 70. Other shapes and sizes of the protrusions 82 are also compatible with certain embodiments described herein.
In certain embodiments, the at least one photonic crystal structure 20 comprises a single PCS 70 that exhibits at least one optical resonance having a resonance frequency and a resonance lineshape. FIGS. 3A-3C schematically illustrates an example PCS 70 (shown on the left) exhibiting an optical resonance in the simulated transmitted optical power spectrum (shown on the right) for light incident in a direction substantially perpendicular to the PCS 70. In FIGS. 3A-3C, the optical resonance is shown as a dip in the transmitted optical power spectrum. The horizontal axes of the simulated transmitted optical power spectra of FIGS. 3A-3C are in units of (c/a), where c is the speed of light in vacuum and a is the lattice constant of the PCS 70 (e.g., the center-to-center spacing of the holes). FIG. 3A illustrates the PCS 70 with no forces applied, FIG. 3B illustrates the PCS 70 with a compressive force applied, and FIG. 3C illustrates the PCS 70 with an expansive or stretching force applied. The compressive force shifts the frequency of the optical resonance towards higher frequencies, as shown by a comparison of FIGS. 3A and 3B. The expansive force shifts the frequency of the optical resonance towards lower frequencies, as shown by a comparison of FIGS. 3A and 3C.
FIG. 4A shows the reflectivity of an example PCS 70 over a wavelength range of 1300 nanometers to 1600 nanometers. The reflectivity is greater than 99% over a bandwidth of about 120 nanometers. Such measurements show that a PCS can be used as an efficient polarization independent mirror. FIG. 4B schematically illustrates the measured resonance wavelength shift for substantially perpendicularly incident light on an example PCS 70 as a function of temperature. For temperature T0 of about 25° C., the resonance wavelength is about 1431 nanometers, for temperature T1 of about 450° C., the resonance wavelength is about 1434 nanometers, and for temperature T2 of about 800° C., the resonance wavelength is about 1436 nanometers. By changing the temperature of the PCS 70, both the geometry is changed via thermal expansion and the dielectric constant is changed, both of which contribute to the shift of the resonance wavelength.
FIG. 5 illustrates the resonance wavelength shift for substantially perpendicularly incident light on an example PCS 70 as a function of mechanical forces applied to the PCS 70. For the measurements illustrated by FIG. 5, one end of an example PCS 70 was fixedly mounted to a stationary position and the other end of the PCS 70 was mounted to a piezoelectric oscillator which was oscillated at 4.7 kHz using a 4-volt peak-to-peak voltage. The relative sensitivity of the change in optical power with respect to different optical wavelengths for a constant acoustic power generally follows the slope of the optical transmission spectrum of the PCS 70.
Similar behavior was observed for a PCS 70 in the experimental apparatus schematically illustrated by FIG. 6. As shown by FIG. 6, one end 82 of a 1-centimeter long PCS 70 was fixedly mounted (e.g., by epoxy) to a stationary position, and the other end 84 was fixedly mounted (e.g., by epoxy) to one end of a movable cantilever which was used to reduce the frequency of the PCS structure. An audio speaker 86 facing the cantilever and spaced about 3 centimeters from the cantilever was oscillated at about 500 Hz using a 10-volt peak-to-peak voltage.
FIGS. 7A and 7B schematically illustrate an example acoustic sensor 10 having photonic crystal structure 20 comprising a single PCS 70 in accordance with certain embodiments described herein. The PCS 70 is mounted with a first end 92 fixedly mounted to a stationary position and a second end 94 fixedly mounted to a movable membrane 96. In certain embodiments, the membrane 96 is a portion of the housing 30. An optical fiber 50 is positioned to irradiate the PCS 70 with light in a direction substantially perpendicular to the PCS 70. In certain embodiments, light reflected by the PCS 70 re-enters the optical fiber 50 and is detected by an optical sensor (not shown), while in certain other embodiments, light transmitted through the PCS 70 is detected by an optical sensor (not shown). In certain embodiments, acoustic waves 40 incident on the membrane 96 induce forces (e.g., strain) in the plane of the PCS 70 (e.g., by stretching and compressing the PCS 70), thereby shifting at least one of the resonance frequency and the resonance lineshape of the PCS 70, as detected by either the reflection spectrum, the transmission spectrum, or both. In certain other embodiments, the PCS 70 is mounted to the membrane 96 such that acoustic waves 40 incident on the membrane 96 induce strain in the PCS 70 by bending the PCS 70. In certain such embodiments, the measured Q for the resonance is about 2500 to 3000. In certain such embodiments, the corresponding sensitivity of the acoustic sensor 10 is about 1 micropascal/Hz1/2, and the dynamic range is limited by the yield strength of the PCS 70 to be about 50 decibels. In an example embodiment, a theoretical strain of about 1×10−5 applied to a PCS 70 yields a 103 change in the transmitted power at a wavelength of about 1550 nanometers.
Dual PCS Structures
In certain embodiments, the photonic crystal structure 20 comprises a first PCS 100 and a second PCS 102 substantially parallel to the first PCS 100, as schematically illustrated by FIG. 8. Each of these PCSs 100, 102 can have physical parameters (e.g., thicknesses, region sizes, materials, periodicities, distributions) as described above for the single PCS structure.
In certain embodiments, there is no physical contact between the first PCS 100 and the second PCS 102. The first and second PCSs 100, 102 can undergo displacements relative to one another in response to incident acoustic waves 40. In certain embodiments, light is incident on the first and second PCSs 100, 102 in a direction substantially perpendicular to the PCSs 100, 102. In certain embodiments, the light is provided by an optical fiber 50, as schematically illustrated by FIG. 8, while in certain other embodiments, the light is collimated prior to irradiating the PCSs 100, 102.
FIG. 9 is a plot of various normalized transmission spectra measured from a photonic crystal structure 20 comprising a pair of PCSs (e.g., as shown in FIG. 8), each transmission spectrum corresponding to a different manual displacement between the two PCSs. The measured transmission spectra of FIG. 9 were obtained by using two PCSs in proximity to one another and a micron-actuator to manually vary the displacement between the two slabs. As can be seen from FIG. 9, the pair of PCSs exhibits optical resonances each having a resonance frequency and a resonance lineshape, and that both the resonance frequency and the resonance lineshape are responsive to changes of the relative position between the two PCSs. As shown in FIG. 9, one example resonance of the pair of PCSs has a tuning bandwidth of about 50 nanometers at a center wavelength of about 1377 nanometers. This resonance is sufficiently sharp (e.g., about 0.5 THz with a peak-to-floor ratio of 25 dB) to be used in an acoustic sensor system. Theoretical calculations can be used to design PCS structures with sharper resonances, to be used in acoustic sensor systems with even higher sensitivities.
The resonance frequency and the resonance lineshape of the pair of PCSs are both dependent on changes of the perpendicular distance between the two PCSs and on changes of the lateral relative positions of the two PCSs. The two PCSs exhibit optical behavior similar to that of a single PCS, and through the relative displacements, the geometry and optical properties of the photonic crystal structure can be tuned. U.S. Patent Application Publication No. US 2004/0080726 A1, which is incorporated in its entirety by reference herein, discloses calculations (e.g., temporal coupled-mode theory calculations and finite-difference time-domain simulations) of the transmission spectrum for a pair of PCSs as functions of the frequency of the incident light and of the displacement between the two PCSs. These calculations replicate the behavior shown in FIG. 9.
In certain embodiments, the two PCSs are brought sufficiently close to one another that they are optically coupled in the near-field to one another (referred to herein as a near-field configuration). In certain embodiments, the two PCSs are spaced apart from one another such that the PCSs are not optically coupled to one another, but form a cavity (referred to herein as a Fabry-Perot configuration). In either the Fabry-Perot configuration or the near-field configuration, the optical resonances shift in frequency (or wavelength) with changing displacement between the two PCSs. Thus, the amount of displacement between the two PCSs can be detected by measuring the transmitted power (or the reflected power) at a predetermined frequency (or wavelength). In general, the near-field configuration generates a larger shift of frequency (or wavelength) than does the Fabry-Perot configuration, such that the near-field configuration has a higher sensitivity to displacements than does the Fabry-Perot configuration.
In certain embodiments in which the two PCSs are optically coupled together in the near-field configuration, the optical resonances are split into two resonances. The amount of splitting varies with the displacement between the two PCSs which, in certain embodiments, provides a measure of the displacement. FIGS. 10A-10C schematically illustrate the dependence of the resonance frequencies of the photonic crystal structure 20 comprising a first PCS 100 and a second PCS 102. In FIG. 10A, a single PCS 70 is schematically shown with its transmission spectrum having a single optical resonance mode. In FIG. 10B, a pair of PCSs 100, 102 coupled in the near-field configuration are schematically shown and the transmission spectrum has a pair of optical resonance modes having frequencies that are split from one another. In FIG. 10C, one or both of the PCSs are displaced in a direction substantially perpendicular to the PCSs such that the distance between the two PCSs 100, 102 is decreased, thereby shifting the frequencies of the two modes such that the splitting between the frequencies of the two modes increases.
In certain embodiments in which the two PCSs are coupled in the near-field configuration, additional resonances appear in the transmission spectra when the PCSs are laterally displaced relative to one other in a direction substantially parallel to the PCSs, as schematically illustrated by FIG. 11. As discussed more fully below, these resonances are generated by breaking the mirror symmetry of the double PCS structure, which allows incident light to couple to non-degenerate resonances. These additional resonances shift in frequency (or wavelength) as a function of the perpendicular displacement between the two PCSs. These additional resonances shift in frequency (or wavelength) and their lineshapes (e.g., linewidths) also change as a function of the lateral displacement parallel to the two PCSs. In certain embodiments, by optically coupling the two PCSs, the linewidth and the frequency of these additional resonances can advantageously be tuned dynamically by displacements between the two PCSs perpendicular to the PCSs and parallel to the PCSs. In certain embodiments, a sub-Angstrom displacement (either perpendicular or parallel to the PCSs) between the two PCSs introduces a detectable change in the transmitted or reflected power at a sensitive resonance wavelength. In certain embodiments, electrical actuation can be used to shift the PCSs in a direction generally parallel to the PCSs and using resonance frequency shifts due to acoustic-wave-induced displacements between the PCSs in a direction generally perpendicular to the PCSs. Certain such embodiments are advantageously used in acoustic sensor systems.
Fiber Compatibility
The sharp resonances of typical optical resonators or filters are sensitive to the incident angle of the light. Typically, to avoid this sensitivity to the incident angle, the incident light is collimated so as to approximate a plane wave. When using an optical fiber as the light source, the light emitted by the optical fiber possesses a certain angular distribution which is typically collimated for present-day optical resonators using additional collimation optics and additional fiber-to-fiber coupling hardware.
In contrast, certain embodiments described herein have one or more resonances which are substantially independent of the incidence angle of the optical beam over a range of incidence angles. In certain such embodiments, the light emitted by the optical fiber has an angular distribution such that a substantial fraction (e.g., more than 50%) of the light incident on the PCS is within the range of incidence angles for which the resonance frequency of such resonances does not change. For such resonances, the linewidth of the resonance is also essentially independent of the incidence angle. Such an angular insensitivity implies that the resonances do not have to be excited by a collimated beam (e.g., by light which approximates a plane wave).
In certain embodiments in which the resonance is insensitive to the incidence angle, the various angular components of the light emitted by the optical fiber are all affected by the PCS structure in the same way, so the acoustic sensor behaves in much the same way as if the light was collimated. In certain such embodiments, since the resonance is insensitive to the incidence angle, the light from the optical fiber directly impinges the PCS structure without intervening collimation optics between the optical fiber and the PCS structure. Certain such embodiments advantageously avoid using complicated collimation or coupling components, thereby simplifying integration and packaging and lowering cost.
The fiber-compatibility of the PCS structure advantageously permits certain embodiments described herein to be easily incorporated into already-present and widely-used fiber-based acoustic sensor systems. In addition, the angular insensitivity of the PCS structure advantageously facilitates incorporating several types of filters into fiber-based optical communication networks.
In an example embodiment, a silicon-nitride PCS illuminated by transverse-electric (TE) polarized light has a resonance mode with a wavelength of about 695 nanometers. FIG. 12 illustrates the measured transmission spectra corresponding to TE polarized light incident on the PCS at various incidence angles. As shown in FIG. 12, the transmission spectra have various features which are dependent on the incidence angle, but the resonance mode at about 695 nanometers is substantially insensitive to the incidence angle of the TE polarized light. In another example embodiment, the silicon-nitride PCS is illuminated by transverse-magnetic (TM) polarized light, and exhibits a resonance mode with a wavelength of about 770 nanometers, and this resonance is substantially insensitive to the incidence angle of the TM polarized light.
In certain embodiments in which the acoustic sensor 10 further comprises an optical fiber 50 optically coupled to the at least one photonic crystal structure 20 (e.g., as schematically illustrated by FIG. 1), the light emitted from the optical fiber 50 is incident to the at least one photonic crystal structure 20 in a range of incidence angles within about 10 degrees from a direction perpendicular to the at least one photonic crystal structure 20. In certain such embodiments, the light is not collimated between being emitted from the optical fiber 50 and reaching the at least one photonic crystal structure 20.
Tailoring the Optical Resonance
Certain eigenmodes in a PCS possess infinite lifetimes, hence are uncoupled to outside radiation at normal incidence. Therefore, in present-day optical resonator systems utilizing photonic crystals, it is generally not possible to couple to certain resonances (referred to herein as non-degenerate resonances) with normally-incident plane waves due to a symmetry mismatch between the resonance mode and the incident wave. This effect was observed experimentally by Pacradouni et al., “Photonic band structure of dielectric membranes periodically textured in two dimensions,” Phys. Rev. B, vol. 62, page 4204 (2000), and discussed theoretically by Paddon and Young, “Two-dimensional vector-coupled-mode theory for textured planar waveguides,” Phys. Rev. B, vol. 61, page 2090 (2000). Using group theoretical arguments, Ochiai and Sakoda, in “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B, vol. 63, page 125107 (2001), showed that these resonances are uncoupled due to a symmetry mismatch with outside radiation.
However, measurements and group theory calculations show that it is possible to couple to these non-degenerate resonances in a PCS lacking mirror symmetry. As described more fully below, simulations and experimental results show that such non-degenerate resonances can indeed be excited by breaking the mirror symmetry of the PCS structure, either by breaking the periodicity of the lattice array or by breaking the mirror symmetry of the unit cells (e.g., in a square lattice array). In addition, it is possible to control the sharpness (e.g., linewidth, quality factor) of such resonances by adjusting the degree of asymmetry (e.g., the size of the non-symmetric region of the holes of the PCS structure). In certain embodiments, the quality factor of these resonances can be tuned from a finite minimum to infinity. Resonances sharper than, the spectral linewidth of the source are generally practically useless, so in certain embodiments, the tuning is, done from a finite minimum to a finite maximum (as determined by the linewidth of the incident light).
Such PCS structures are expected to have applications for mode selection and linewidth control in lasers, and will find use in acoustic sensor applications by advantageously improving and controlling the sensitivity of the acoustic sensor system. Certain embodiments described herein advantageously improve the sensitivity of the acoustic sensor system up to a limit imposed by other factors, such that the PCS structure is not the limiting element. In certain embodiments in which a lower sensitivity is desirable (e.g., to improve the dynamic range), the sensitivity of the acoustic sensor system is lowered such that the PCS structure is the limiting element. In certain embodiments, the lack of mirror symmetry is implemented for a PCS structure with a triangular lattice array or any other lattice array geometry, or in general, for any kind of an optical resonator system.
In certain embodiments, the non-degenerate resonances of a PCS with a symmetric structure that are uncoupled to normally-incident plane waves are excited in a mirror-symmetry-lacking PCS structure. In certain embodiments, one or more of the mirror symmetries of the PCS structure is advantageously broken or removed to allow coupling to the non-degenerate resonances. In certain embodiments, the coupling to these non-degenerate resonances is advantageously controlled by selecting the degree of asymmetry. In certain embodiments, the at least one photonic crystal structure has a symmetry axis and the light incident normal to the at least one photonic crystal structure is polarized in a direction substantially perpendicular to the symmetry axis. In certain other embodiments, the normally-incident light is polarized in a direction substantially parallel to the symmetry axis.
In certain embodiments, the asymmetry of the PCS structure is generated by an asymmetry in the substantially periodic distribution of holes. FIGS. 13A-13D schematically illustrate example PCS structures having at least one photonic crystal defect in the substantially periodic distribution. The PCS structure of FIG. 13A has a photonic crystal defect comprising a missing hole, and such a photonic crystal defect possesses mirror symmetry with respect to the horizontal and vertical axes. In certain embodiments, the PCS structure comprises at least one hole with a reduced size or an increased size as compared to the other holes of the PCS structure. In certain embodiments, this reduced-size or increased-size hole is at an expected lattice position of the substantially periodic distribution, while in other embodiments, it is displaced from the expected lattice position. In certain other embodiments, this reduced-size or increased-size hole is in proximity to the position of a missing hole. For example, FIG. 13B schematically illustrates a PCS structure with a hole having a reduced size and adjacent to the missing hole position. FIG. 13C shows a hole adjacent to the missing hole position to be slightly shifted from its expected lattice position of the substantially periodic distribution. FIG. 13D shows a hole which itself lacks a mirror symmetry acting as the defect. In certain other embodiments, the dielectric constant of a portion of the PCS structure is reduced or increased to break the mirror symmetry. For example, at least one of the holes of the PCS structure can contain a third material having a refractive index different from the refractive indices of the first material or the second material. The photonic crystal defects of FIGS. 13B, 13C, and 13D lack mirror symmetry with respect to the horizontal axis. Various possibilities to break the mirror symmetry, not limited to those schematically illustrated by FIGS. 13A-13D, are compatible with embodiments described herein. While FIGS. 13A-13D have been described in terms of a PCS structure comprising a plurality of holes, persons skilled in the art recognize that a PCS structure comprising a plurality of protrusions would exhibit similar behavior.
FIGS. 14A and 14B schematically illustrate an example implementation for mirror-symmetry breaking in a PCS structure compatible with certain embodiments described herein. The PCS structure shown in FIG. 14A possesses mirror symmetry with respect to both the horizontal and vertical axes. The PCS structure shown in FIG. 14B lacks mirror symmetry with respect to the horizontal axis.
FIG. 15 schematically illustrates several example hole structures which break or remove one or more of the mirror symmetries of the PCS unit cell. Each of the structures schematically illustrated by FIG. 15 lack mirror symmetry with respect to the horizontal axis, while possessing mirror symmetry with respect to the vertical axis. Besides the structures schematically illustrated by FIG. 15, there is an infinite number of hole shapes compatible with embodiments described herein.
FIG. 16A schematically illustrates a unit cell 150 of a PCS having circularly symmetric holes 152 on a periodic square lattice distribution. The dashed lines of FIG. 16A denote various mirror symmetry axes 154 of the PCS. FIGS. 16B-16E schematically illustrate the dot products of various resonance modes of the PCS with plane waves polarized in the horizontal direction (x-polarization) and with plane waves polarized in the vertical direction (y-polarization). The dot products schematically illustrated by FIGS. 16B and 16C are not equal to zero, so these two resonance modes couple to incident plane wave. However, the dot products schematically illustrated by FIGS. 16D and 16E equal zero, so this resonance mode does not couple to incident plane waves, and is a non-degenerate resonance.
In certain embodiments, one or more of the mirror symmetries of the PCS structure is broken or removed. In certain such embodiments, one or more of the mirror symmetries of the unit cell of the periodic array of holes in the PCS is removed. FIG. 17A schematically illustrates an example unit cell 160 of a PCS having holes 162 on a periodic square lattice distribution, in which each hole 162 comprises a small region 163 to one side of the hole 162. The region 163 of FIG. 17A has a generally square shape, while in certain other embodiments, the region 163 has another shape (e.g., triangular, rectangular, irregular). As shown in FIG. 17A, the hole 162 does not have a mirror symmetry about the horizontal axis 164, as denoted by the horizontal dashed line marked by an “X,” but the hole 162 maintains the mirror symmetry about the vertical axis 165. The region 163 removes one of the mirror symmetries of the unit cell 160, as compared to the circularly symmetric hole 150 of FIG. 16A, thereby changing the symmetry of the non-degenerate resonances. As schematically illustrated by FIGS. 17B and 17C, the region 163 modifies the resonance mode schematically illustrated by FIGS. 16D and 16E to be an asymmetric resonance mode, which can be equated to the sum of an even-symmetric resonance mode and an odd-symmetric resonance mode. As schematically illustrated by FIG. 17D, the dot product of this odd-symmetric resonance mode with an incident plane wave with y-polarization is non-zero, indicating that this odd-symmetric resonance mode can couple to incident plane waves. Thus, the change of the symmetry of the resonance modes by the asymmetric hole 162 makes coupling to the non-degenerate resonances possible using normally-incident plane waves.
FIG. 18A schematically illustrates a PCS unit cell 150 with the circularly symmetric hole 152 of FIG. 16A having four mirror symmetry axes 154. FIG. 18B schematically illustrates two doubly degenerate resonances (E(1) and E(2)) and four non-degenerate resonances (A1, A2, B1, B2) of the PCS structure, and FIG. 18C schematically illustrates x-polarized (ex) and y-polarized (ey) incident plane waves and the corresponding electric fields. The hole 152 of FIG. 18A has a substantially symmetric shape possessing mirror symmetry with respect to a first axis (e.g., {circumflex over (σ)}x) along the PCS 70 and with respect to a second axis (e.g., {circumflex over (σ)}y) along the PCS 70, the second axis substantially perpendicular to the first axis. The dot products E(1)·ey and E(2)·ex are non-zero, indicating that these doubly degenerate resonances of FIG. 18B couple to y-polarized and x-polarized incident plane waves, respectively. The dot products A1ex, A2ex, B1ex, B2ex, A1 ey, A2ey, B1ey, and B2ey are each equal to zero, indicating that these non-degenerate resonances of FIG. 18B are not coupled to either x-polarized or y-polarized incident plane waves.
In certain embodiments, the coupling to the non-degenerate resonances can be controlled by advantageously selecting the degree of asymmetry of the hole. FIG. 18D schematically illustrates a PCS unit cell 160 with the asymmetric hole 162 with a region 163 to one side. The asymmetric hole 162 has a substantially asymmetric shape lacking mirror symmetry with respect to one axis along the PCS 70. For example, as shown in FIG. 18D, the hole 162 has the mirror symmetry about the horizontal axis broken and has the rotational symmetry broken, possesses mirror symmetry with respect to the vertical axis 165 along the PCS 70, the vertical axis 165 substantially perpendicular to the horizontal axis. FIG. 18E schematically illustrates a PCS unit cell 170 with a hole 172 having two similar regions 173 positioned to maintain the two mirror symmetry axes 174, while the rotational symmetry remains broken. The PCS structure corresponding to FIG. 18E can be used to demonstrate that it is the breaking of the mirror symmetry that is responsible for the excitation of the sharp non-degenerate resonances. As described more fully below, for PCS structures where only the rotational symmetry is broken (e.g., for elliptical holes), the non-degenerate resonances remain uncoupled to the normally-incident plane waves.
FIGS. 19A and 19B show finite-difference time-domain simulations (FDTD) of transmission spectra for these three, different hole shapes for polarizations perpendicular and parallel, respectively, to the hole elongations. The transmission spectra of FIGS. 19A and 19B correspond to normal incidence transmission through a PCS structure with circular holes, minor-asymmetric holes, and rotationally-asymmetric holes.
The simulations were done for a dielectric constant of 12, corresponding roughly to the dielectric constant of Si or GaAs at optical frequencies. The PCS thickness was chosen to be 0.75a, where a is the lattice constant of the periodic structure. The radius of the circular portion of the hole was chosen to be 0.4a and the width of the square-shaped regions was chosen to be 0.025a. As can be seen in FIGS. 19A and 19B, additional sharp features (denoted by arrows) due to non-degenerate resonances are present only in the PCS structure lacking mirror symmetry. Each of these additional resonances appears only for one polarization and not for the other, thereby demonstrating the non-degenerate nature of these resonances.
In certain embodiments, the magnitude of the asymmetry of the holes is selected to provide a desired amount of coupling to normally-incident plane waves. FIGS. 20A and 20B shows FDTD simulations of transmission spectra for incident light with polarizations perpendicular and parallel, respectively, to the hole elongations. To show that the quality factor of these resonances can be controlled, the size of the elongations was increased by 100% to 0.05a. As shown by a comparison of FIGS. 20A and 20B with FIGS. 19A and 19B, the strength and linewidths of the non-degenerate resonances have increased with the increase in asymmetry. This behavior has also been measured from PCS structures with increasing asymmetry.
To demonstrate that the results of the analysis and simulations can be observed in a real structure, the three PCS structures generally corresponding to FIGS. 18A, 18D, and 18E were fabricated on free-standing silicon membranes. FIGS. 21A-21C are scanning-electron microscopy images of PCS structures with circularly-symmetric holes, mirror-asymmetric holes, and rotationally-asymmetric holes, respectively. FIGS. 21D-21F are scanning-electron microscopy images of the circularly-symmetric holes, mirror-asymmetric holes, and rotationally-asymmetric holes, respectively. The circular line overlayed on these SEM images facilitates seeing the small hole elongations of these PCS structures that produce the asymmetries. The material of the PCS was silicon, the thickness of the PCS was about 450 nanometers, the period of the lattice array was about 1000 nanometers, and the diameter of the holes was about 450 nanometers.
FIGS. 22A and 22B show experimental measurements of the transmission spectrum for the three different PCS structures for polarizations perpendicular and parallel, respectively, to the hole elongations. Sharp doubly-degenerate modes are observed for both polarizations, as denoted with arrows (labeled as DD) in all three of the PCS structures. There are also broader doubly-degenerate resonances present which are not denoted by arrows. As shown in FIG. 22A, there is an additional, relatively sharp resonance for the mirror-asymmetric PCS structure (corresponding to FIG. 21B and FIG. 21E) and this resonance is only present for one polarization (perpendicular to the hole elongation), showing its non-degeneracy (labeled as ND). There is a small difference in the transmission spectra for the two polarizations even for the case of the symmetric PCS structure (corresponding to FIG. 21A and FIG. 21D). This small difference is due to a small elongation of the lattice array in one direction due to the electron-beam exposure and subsequent fabrication steps used to form the PCS structure. However, this situation is not essential for the observation of the non-degenerate resonances.
The measured sharp resonances shown in FIGS. 22A and 22B do not vary over as large a transmission range as do the idealized calculations (which vary between 0 and 100% transmission in a range of one linewidth) due to the deterioration of the resonances through fabrication-related disorders. The measurements described herein were for a relatively large lattice array of size 100 microns by 100 microns, where disorder effects can play a significant role for sharp resonances. The angular content of the incident light with finite spot-size is another effect that can deteriorate sharp resonances. For a single defect cavity, such as one for a laser, the non-degenerate resonances can be much more dominant (e.g., they can vary from 0 to 100%).
To illustrate that the non-degenerate resonance appears only in the mirror-asymmetric PCS structure (corresponding to FIG. 21B and FIG. 21E), FIG. 23 illustrates the transmission spectra for the perpendicular polarization case of FIG. 22A on a larger wavelength range. The non-degenerate nature of these resonances, combined with the fact that their inherently high quality factor can be tuned through a simple geometrical parameter that can be controlled lithographically enable a variety of applications including acoustic sensing systems and devices for mode selection and linewidth control in lasers. Such structures will also find use as very sharp filters in sensor applications.
Acoustic Sensor Systems
FIG. 24 schematically illustrates an example acoustic sensor system 200 compatible with certain embodiments described herein. In certain embodiments, the acoustic sensor system 200 comprises at least one photonic crystal structure 20 having at least one optical resonance with a resonance frequency and a resonance lineshape. The acoustic sensor system 200 further comprises a housing 30 substantially surrounding the at least one photonic crystal structure 20 and mechanically coupled to the at least one photonic crystal structure 20. At least one of the resonance frequency and the resonance lineshape of the at least one photonic crystal structure 20 is responsive to acoustic waves 40 incident upon the housing 30. As illustrated by FIG. 24, in certain embodiments, the acoustic sensor system 200 further comprises an optical fiber 50 optically coupled to the at least one photonic crystal structure 20.
In certain embodiments, the acoustic sensor system 200 is compatible with operation in a liquid (e.g., seawater) or other media. As schematically illustrated in FIG. 24, an acoustic wave 40 impinges on, and is detected by, the acoustic sensor system 200.
In the embodiment schematically illustrated by FIG. 24, the at least one photonic crystal structure 20 comprises two PCSs 70a, 70b optically coupled to one another and in close proximity to one another (referred to herein as a double-PCS structure). In certain embodiments the two PCSs 70a, 70b are substantially parallel to one another. In certain embodiments, the two PCSs 70a, 70b are optically coupled to each other in the near-field configuration. In certain other embodiments, the two PCSs 70a, 70b are placed further apart so that they are not optically coupled in the near-field configuration, but form a simple Fabry-Perot cavity (i.e., the Fabry-Perot configuration). In certain embodiments, the resonances of the photonic crystal structure 20 shift in frequency (and in the corresponding wavelength) when the vertical distance between the two PCSs 70a, 70b is changed. Example photonic crystal structures 20 compatible with certain embodiments described herein are described in “Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs,” W. Suh et al., Appl. Phys. Lett. vol. 82, No. 13, pages 1999-2001 (1999), and U.S. Patent Publication No. 2004/0080726 A1 which is incorporated in its entirety by reference herein.
In certain embodiments, the PCSs 70a, 70b undergo movement relative to one another (e.g., one movable PCS 70b moves relative to a non-moving PCS 70a) in response to forces applied to the at least one photonic crystal structure 20. In the embodiment schematically illustrated by FIG. 24, the PCSs 70a, 70b of the photonic crystal structure 20 are illuminated by light emitted from the fiber core 52 of the optical fiber 50. When the PCSs 70a, 70b move vertically with respect to one another, the frequency (and the corresponding wavelength) of the sharp optical resonances supported by the photonic crystal structure 20 shift due to the changed optical coupling between the guided resonances of the individual PCSs 70a, 70b. This shift results in a change of the intensity or the phase of the light reflected from or transmitted through the photonic crystal structure 20 and provides an observable quantity to measure the relative displacement between the two PCSs 70a, 70b. 
In certain embodiments, the housing 30 comprises a structure 210 comprising one or more supports 212 and a movable portion 220. The housing 30 further comprises a coupler 230 configured to be coupled to the optical fiber 50. The movable portion 220 is mechanically coupled to the coupler 230 by the one or more supports 212. The optical fiber 50 of certain embodiments passes through an opening in the coupler 230 and the fiber core 52 is in proximity to and is optically coupled with the photonic crystal structure 20.
Example materials for the structure 210, the movable portion 220, and the supports 212 include, but are not limited to, crystalline silicon, polysilicon, silica, silicon nitride, ceramics, plastics, amorphous diamond, germanium, indium phosphide, gallium arsenide, and metals and metal alloys. Example materials for the coupler 230 include, but are not limited to, crystalline silicon, Pyrex glass, quartz, polysilicon, silica, silicon nitride, ceramics, plastics, amorphous diamond, germanium, indium phosphide, gallium arsenide, and metals and metal alloys.
In certain embodiments, the coupler 230 comprises an optically transmissive portion 232 (e.g., a hole, a window, an optically transmissive membrane) through which the optical fiber 50 emits light to irradiate the photonic crystal structure 20. The optically transmissive portion 232 allows light emitted by the fiber core 52 to irradiate the photonic crystal structure 20, and allows light reflected by the photonic crystal structure 20 to be received by the fiber core 52.
The movable portion 220 is configured to move (e.g., as denoted by the double-headed arrow in FIG. 24) in response to the pressure modulations of an acoustic wave 40 incident on the movable portion 220. In the embodiment schematically illustrated by FIG. 24, one PCS 70a (e.g., the PCS closer to the optical fiber 50) is generally stationary, while the other PCS 70b (e.g., the PCS farther from the optical fiber 50) is attached to the movable portion 220 of the structure 210. In certain other embodiments, the PCS 70b is generally stationary while the PCS 70a is attached to the movable portion 220.
In certain embodiments, the movement of the PCS 70b has a component in a direction substantially perpendicular to the PCS 70a, wherein the movement changes a distance between the PCSs 70a, 70b. In the embodiment schematically illustrated by FIG. 24, the PCS 70b attached to the structure 210 will simultaneously move in response to an incident acoustic wave 40, such that the acoustic wave 40 modulates the distance between the two PCSs 70a, 70b. In this way, the reflectivity (e.g., the power of the reflected light) and/or the transmissivity (e.g., the power of the transmitted light) of the photonic crystal structure 20 is modulated by the incident acoustic wave 40. The optical signal reflected from the photonic crystal structure 20 is transmitted back to the optical fiber 50 and directed to a detector (not shown), which measures the reflected signal power. In certain embodiments, the phase of the reflected light is measured instead of the power of the reflected light. In certain embodiments, the movement of the PCS 70b has a component in a direction substantially parallel to the PCS 70a. 
In certain embodiments, the sensitivity (e.g., the change of the detected reflected power per unit of incident acoustic pressure) of the photonic crystal structure 20 is advantageously increased by utilizing a signal having a frequency (or wavelength) offset slightly from one of the resonance frequencies (or wavelengths) of the double-PCS photonic crystal structure 20. In certain embodiments utilizing extremely high sensitivities, the PCSs 70a, 70b are designed to have extremely sharp resonances, e.g., by breaking a mirror symmetry of at least one of the PCSs 70a, 70b, as described herein.
In certain embodiments, the mechanical properties of the acoustic sensor structure 200 (e.g., mechanical resonance frequency, spring constant) are dependent on both the movable portion 220 of the structure 210 and the one or more supports 212. In certain embodiments, the movable portion 220 serves as the mechanical spring by providing a restoring force in response to displacements of the movable portion 220 by acoustic waves 40. In certain other embodiments, the supports 212 serve as the mechanical spring by providing the restoring force in response to displacements of the movable portion 220 by acoustic waves 40. Other embodiments utilizing other spring designs for the structure 210 or the supports 212 are also compatible with embodiments described herein.
In certain embodiments, the acoustic sensor system 200 is insensitive to static pressure variations in the medium (e.g., seawater) in which it operates. As an example, the acoustic sensor system 200 of certain embodiments is operable close to the surface of seawater, or several feet below the surface of seawater. In certain embodiments, the housing 30 comprises at least one pressure conduit 240 between an inner region 250 within the housing 30 and an outer region 260 outside the housing 30. In certain embodiments, the at least one pressure conduit 240 comprises the movable portion 220 of the housing 30. In certain such embodiments, the movable portion 220 comprises an elastic membrane that is responsive to a pressure differential between the inner region 250 and the outer region 260 by moving to remove the pressure differential. In certain embodiments, the supports 210 provide the restoring force to the movable portion 220 and are responsive to a pressure differential across the movable portion by moving the movable portion 220 to reduce the pressure differential. The at least one pressure conduit 240 of certain embodiments serves as low-pass filters that equalize the static pressure between the inner region 250 and the outer region 260.
In certain embodiments, the at least one pressure conduit 240 comprises a hole through the housing 30, with the hole fluidly coupling the inner region 250 with the outer region 260. In certain such embodiments, the inner region 250 is filled with the same medium (e.g., seawater) of the acoustic waves 40 as is the outer region 260, and the medium is free to flow between the inner region 250 and the outer region 260. In certain embodiments, the at least one pressure conduit 240 comprises a hole through the housing 30 and an elastic membrane that seals the at least one pressure conduit 240 to fluidly isolate the inner region 250 from the outer region 260. The membrane of certain embodiments is responsive to a pressure differential between the inner region 250 and the outer region 260 by moving to reduce the pressure differential, thereby still acting as a low-pass filter equalizing the pressure inside and outside the acoustic sensor system 200, while keeping the medium (e.g., seawater) from entering the acoustic sensor system 200. In certain such embodiments in which it is desirable to not expose the photonic crystal structure 20 or other internal components of the acoustic sensor system 200 to the medium (e.g., seawater) which can be corrosive and dirty, the membrane advantageously keeps the medium of the acoustic waves 40 from entering the inner region 250 within the housing 30. Example materials for the membrane include, but are not limited to, silicon nitride or rubber.
In certain embodiments, the acoustic sensor system 200 includes other structural components for better performance and reliability. These other structural components are not crucial for the operation of the acoustic sensor system 200. In certain embodiments, the acoustic sensor system 200 comprises one or more spacers 270 positioned to avoid contact between the two PCSs 70a, 70b in response to a large-magnitude pressure wave incident on the acoustic sensor system 200, thereby advantageously avoiding stiction between the two PCSs 70a, 70b. The spacers 270 of certain embodiments serve as safety structures which define a minimum separation between the two PCSs 70a, 70b, thereby preventing the two PCSs 70a, 70b from contacting and sticking to each other. Example materials for the spacers 270 include, but are not limited to, crystalline silicon, polysilicon, silicon nitride, silicon oxide, amorphous diamond, ceramics, plastics, germanium, indium phosphide, gallium arsenide, and metals and metal alloys. In certain embodiments, amorphous diamond is used because it is hydrophobic which facilitates the prevention of sticking of the two PCSs 70a, 70b. 
Due to the sensitivity of the optical properties of the photonic crystal structure 20 on the medium surrounding the PCSs 70a, 70b, in certain embodiments, the medium in which the acoustic sensor system 200 is placed (e.g., water) is advantageously restricted from the region 280 within the acoustic sensor system 200. In certain such embodiments, the PCSs 70a, 70b of the photonic crystal structure 20 operate within a gas (e.g., air). In certain embodiments, the housing 30 defines a region (e.g., inner region 250) comprising a liquid and external to the at least one photonic crystal structure 20 and defines the region 280 containing the at least one photonic crystal structure 20 and that is substantially free of the liquid. While liquid may be able to intrude into the region 280 through the opening under the spacers 270, in certain embodiments, both the pressure of the gas inside the region 280 and the small size of the openings under the spacers 270 are selected to advantageously prevent intrusion of the liquid into the region 280, which could otherwise degrade the operation of the acoustic sensor system 200. Certain embodiments advantageously improve the liquid expulsion out of the region 280 further by providing at least a portion of the photonic crystal structure 20 with a hydrophobic surface configured to restrict the liquid from the region 280.
FIG. 25 schematically illustrates an example acoustic sensor system 200 comprising a secondary housing 310. The secondary housing 310 of certain embodiments is mechanically coupled to the housing 30 and contains a non-corrosive liquid or gas, including but not limited to, deionized water, isopropanol, or air. Certain such embodiments advantageously protect various components of the acoustic sensor system 200 from corrosion or other damage from the medium (e.g., seawater) in which the acoustic waves 40 are being measured.
In certain embodiments, the secondary housing 310 is sufficiently elastic to equalize the pressure outside and inside the secondary housing 310 such that pressure modulations due to the incident acoustic wave 40 are translated into the medium (e.g., gas or fluid) within the secondary housing 310. In certain such embodiments, the secondary housing 310 comprises a balloon. In certain other embodiments, the secondary housing 310 comprises a rigid portion and an elastic membrane.
FIG. 26 schematically illustrates another example acoustic sensor system 200 having a secondary housing 310 which protects the photonic crystal structure 20 within the secondary housing 310. In certain embodiments, the photonic crystal structure 20 is sealed within the secondary housing 310 with a clean, non-corrosive, and non-damaging liquid or gas in the inner region 250 and in the outer region 260. In certain such embodiments, the movable PCS 70b of the photonic crystal structure 20 is directly on the movable portion 220 of the housing 30.
FIG. 27 schematically illustrates an example acoustic sensor system 200 comprising a photonic crystal structure 20 comprising a single PCS 70. The acoustic sensor system 200 further comprises a metal layer 320 that is at least partially transmissive and at least partially reflective to light emitted by the optical fiber 50. In certain embodiments, the metal layer 320 is a metal coating on the end of the optical fiber 50. In certain embodiments, the PCS 70 and the metal layer 320 form a Fabry-Perot interferometric cavity that is sensitive to displacements of the PCS 70 relative to the metal layer 320. In certain embodiments, the metal layer 320 comprises a thin adhesion layer (e.g., chromium or titanium layer with a thickness of about 4 nanometers) on the optical fiber 50, and a gold or silver layer on the adhesion layer and having a thickness in a range between about 5 nanometers and about 50 nanometers. In certain other embodiments, the metal layer 320 comprises an aluminum layer on the optical fiber 50 and having a thickness in a range between about 5 nanometers and about 50 nanometers. In certain other embodiments, other metals and metal alloys can be used. In certain embodiments, utilizing the metal layer 320 simplifies the fabrication process of the device.
FIG. 28 schematically illustrates an example acoustic sensor system 200 comprising a photonic crystal structure 20 comprising a single PCS 70. The acoustic sensor system 200 further comprises a Bragg grating at or near the end of the optical fiber 50. In certain embodiments, the Bragg grating comprises a grating deposited at or near the end of the optical fiber 50 and that is a few micrometers thick. In certain other embodiments, as schematically illustrated by FIG. 28, the Bragg grating comprises a fiber Bragg grating 330 which is part of the optical fiber 50. The fiber Bragg grating 330 is at least partially transmissive and at least partially reflective to light emitted by the optical fiber 50. In certain embodiments, the PCS 70 and the fiber Bragg grating 330 form a Fabry-Perot interferometric cavity that is sensitive to displacements of the PCS 70 relative to the fiber Bragg grating 330. Typically, fiber Bragg gratings have a pitch of several hundred nanometers and a total length ranging from several hundred micrometers to several millimeters. The fiber Bragg grating of certain embodiments provides a reflectivity from a few percent up to almost 100% in a wavelength bandwidth ranging from picometers up to several nanometers. The optical properties of such combinations of a single PCS 70 and a fiber Bragg grating 330 are described more fully below. Fiber Bragg gratings 330 compatible with certain embodiments described herein are commercially available and use of such fiber Bragg gratings can simplify fabrication of the acoustic sensor system 200.
FIG. 29 schematically illustrates a perspective view of an example configuration of an acoustic sensor system 200 coupled to one end of an optical fiber 50. The acoustic sensor system 200 comprises a housing 30 having a structure 210 with a movable portion 220 and pressure conduits 240 (e.g., holes) and a coupler 230. Other configurations of the acoustic sensor system and the optical fiber are also compatible with embodiments described herein.
Certain embodiments of the acoustic sensor system 200 described herein provide various advantages over standard fiber-based sensor systems. In certain embodiments, the acoustic sensor system 200 advantageously achieves higher frequency operation due to the flexibility provided by MEMS fabrication technology. In certain such embodiments, the acoustic sensor system 200 is designed to operate at frequencies larger than 10 kHz, a range that is inaccessible for present-day acoustic fiber sensor systems, and in certain embodiments, can operate at frequencies up to about 50 kHz. In certain embodiments, the PCS-based acoustic sensor system described herein is advantageously more sensitive at higher frequencies than are present-day acoustic fiber sensor systems. In certain embodiments, the acoustic sensor system 200 advantageously provides high sensitivity (e.g., sensitive to less than 30 micropascals/Hz1/2). In certain embodiments, the acoustic sensor system 200 comprises a photonic crystal structure 20 that can be fabricated on substrates (e.g., chips) using lithography techniques (as described more fully below), thereby facilitating mass production and low cost, and that is fiber-compatible. In certain embodiments, utilizing MEMS fabrication technology to fabricate the acoustic sensor system 200 advantageously results in acoustic sensor systems that are small in size, light, and compact. In certain embodiments, the compactness of the PCS-based acoustic sensor systems described herein advantageously facilitates their deployment. In certain embodiments, the PCS-based acoustic sensor systems described herein can be advantageously designed to be insensitive to the polarization of the incident light, thereby eliminating the need for compensation for polarization-induced signal fading.
Fabrication of an Acoustic Sensor
In certain embodiments, surface micromachining techniques and bulk micromachining techniques are used in the fabrication process flow to form various components of the acoustic sensor system 200. Lithography techniques compatible with embodiments described herein include, but are not limited to, optical lithography, electron-beam lithography, nano-imprinting techniques, and other techniques generally compatible with microelectromechanical system (MEMS) fabrication. Surface micromachining techniques compatible with embodiments described herein include, but are not limited to, film deposition, dry etching, wet etching, epitaxial growth, wafer bonding, and sacrificial releasing. Bulk micromachining techniques compatible with embodiments described herein include, but are not limited to, anisotropic or isotropic deep reactive ion etching, anisotropic wet etching using KOH (potassium hydroxide) or TMAH (tetramethylammonium hydroxide), and isotropic wet etching.
FIGS. 30A-30Q schematically illustrate an example fabrication process flow compatible with certain embodiments described herein for the components of the acoustic sensor system 200. Many other fabrication process flows, with different process steps, number of process steps, and/or order of process steps are also compatible with certain embodiments described herein, and the choice of which process flow to use is typically dependent on the types of equipment that are available for use. As schematically illustrated by FIG. 30A, the starting material for fabrication is a silicon-on-insulator (SOI) wafer 500 having a substrate 510 with a (100) crystal orientation and a thickness of about 500 microns, an oxide layer 520 over the substrate 510 with a thickness of about 1 micron, and a silicon layer 530 over the oxide layer 510 with a thickness of about 10 microns. Other materials for the wafer 500 are also compatible with certain embodiments described herein.
As schematically illustrated by FIG. 30B, the SOI wafer 500 is oxidized to form an oxide layer 540 over the silicon layer 530 and having a thickness of about 1 micron. As schematically illustrated by FIG. 30C, the oxide layer 540 is patterned by etching the oxide layer 540 down to the silicon layer 530 (e.g., by using a first mask) to isolate various portions of the oxide layer 540 from one another. As schematically illustrated by FIG. 30D, portions of the oxide layer 540 are further etched (e.g., by using a second mask) by about 500 nanometers.
As schematically illustrated by FIG. 30E, the silicon layer 520 is etched down to the oxide layer 510. As schematically illustrated by FIG. 30F, the oxide layer 530 is etched down by about 500 nanometers, thereby removing portions of the oxide layer 540. As schematically illustrated by FIG. 30G, portions of the silicon layer 530 are etched down by about 5 microns. As schematically illustrated by FIG. 30H, the oxide layer 540 is removed.
As schematically illustrated by FIG. 30I, a silicon wafer 600 having an oxide layer 610 on at least one side is bonded to the SOI wafer 500 with the oxide layer 610 in contact with the silicon layer 530. In certain embodiments, the oxide layer 610 has a thickness of about 10 microns. In certain embodiments, the side of the silicon wafer 600 that is not in contact with the silicon layer 530 is polished or grinded down to produce a silicon layer 620 having a thickness of about 10 microns on top of the oxide layer 610.
As schematically illustrated by FIG. 30J, the silicon layer 620 is patterned to make alignment marks visible and to form MEMS structures. In certain embodiments, this patterning includes using a third mask and etching the silicon layer 620 down to the oxide layer 610.
As schematically illustrated by FIG. 30K, an oxide layer 630 is formed on the silicon layer 620 (e.g., deposited and patterned using a fourth mask and etching with hydrogen fluoride) and an oxide layer 632 is formed on the silicon layer 510. In certain embodiments, each of the oxide layer 630 and the oxide layer 632 has a thickness of about 2 microns. As schematically illustrated by FIG. 30L, another oxide layer 640 is formed on the oxide layer 630 and on the exposed portions of the silicon layer 620 and another oxide layer 642 is formed on the oxide layer 632. In certain embodiments, each of the oxide layer 640 and the oxide layer 642 has a thickness of about 2 microns.
As schematically illustrated by FIG. 30M, the SOI wafer 500 is patterned (e.g., using a fifth mask) by etching an aperture 650 through the oxide layer 642, the oxide layer 632, and the silicon layer 510, stopping at the oxide layer 520. As schematically illustrated by FIG. 30N, the aperture is extended by etching away a portion of the oxide layer 520, stopping at the silicon layer 530, and the oxide layer 640 is etched away. In certain embodiments, the etching of the oxide layer 642, the oxide layer 632, the silicon layer 532, the oxide layer 520, and the oxide layer 640 are performed during the same etching step. In certain embodiments, the resultant structure is separated into individual chips, and the subsequent process steps are performed on the chip scale.
As schematically illustrated by FIG. 30O, a controlled etch of a portion of the silicon layer 530 through the aperture 650 is performed (e.g., the aperture 650 self-aligns and masks the silicon layer 530) and a controlled etch of a portion of the silicon layer 620 through a portion of the oxide layer 630 is performed. In certain embodiments, the remaining portion 660 of the silicon layer 530 has a thickness of about 450 nanometers and the remaining portion 670 of the silicon layer 620 has a thickness of about 450 nanometers. These remaining portions 660, 670 serve as the silicon substrates for the photonic crystal slabs 70a, 70b of the acoustic sensor system 200. In certain embodiments, the oxide layer 632 is removed.
As schematically illustrated by FIG. 30P, the lattice of the photonic crystal structure 20 is formed by patterning (e.g., by PMMA coating, electron-beam exposure, etching, and stripping resist) to form the two photonic crystal slabs 70a, 70b and the oxide layer 610 is removed, as schematically illustrated by FIG. 30Q. In certain embodiments, the two PCSs 70a, 70b are self-aligned with the same geometrical parameters. To avoid detrimental stress effects due to the oxide layer 610 underneath the portion 670 resulting from the silicon layer 530, in certain embodiments, hydrofluoric acid can be used to remove the oxide layer 610 from portions of the membrane before the lattice is patterned. For defining the lattice, a Raith150 electron-beam lithography tool can be used. In certain embodiments, the primary masking material for transferring the photonic crystal lattice onto the membrane is a monolayer of 496,000 relative molecular mass polymethylmethacrylate (PMMA), a high resolution, high current positive resist. The exposed patterns are developed in a 1:2 solution of methyl isobutyl ketone:isopropyl alcohol and then anisotropically etched with a plasma etcher, using a plasma of SF6 and CHClF2, resulting in near 90° sidewalls. A single masking material gives reproducible and well-resolved structures. In certain embodiments, the size of the individual photonic crystal slabs 70a, 70b is about 100 microns×100 microns. A similar fabrication method can be adapted for other materials such as silicon nitride or silicon oxide.
In certain embodiments, to create 100 micron×100 micron free-standing silicon PCSs 70a, 70b, 808-micron-wide square apertures 650 are formed on the back of the SOI wafer 500 using anisotropic etching to etch through the 500-micron-thick substrate 510. Using an anisotropic etchant of 30% KOH in water with 1% isopropyl alcohol to reduce surface tension, well-defined structures with smooth etched surfaces can be achieved.
Analysis of the Mechanics of a Diaphragm
The mechanics of the movable portion 220 and of the elastic portions (e.g., the secondary housing 310) of the acoustic sensor system 200 affect the performance of various embodiments described herein. These mechanics are analyzed below for various configurations of the acoustic sensor system 200. While the calculations below provide some insight into the operation of various embodiments described herein, but are not intended to be limiting.
A. Free Vibration of a Diaphragm
The equation of motion for the transverse displacement u of a stretched diaphragm with thickness h, and density ρ can be expressed as:




  
    
      
        
          
            (
            
              
                h
                ⁢
                
                  
                
                ⁢
                ρ
                ⁢
                
                  
                    ∂
                    2
                  
                  
                    ∂
                    
                      t
                      2
                    
                  
                
              
              +
              
                D
                ⁢
                
                  
                
                ⁢
                
                  
                    ∇
                    4
                  
                  ⁢
                  
                    -
                    h
                  
                
                ⁢
                
                  
                
                ⁢
                σ
                ⁢
                
                  
                
                ⁢
                
                  ∇
                  2
                
              
            
            )
          
          ⁢
          u
        
        =
        0
      
    
    
      
        (
        1
        )
      
    
  




(See, e.g., I. Ladabaum et al., “Surface micromachined capacitive ultrasonic transducers,” Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions, vol. 45, issue 3, pages 678-690 (May 1998); and M. Yu, “Fiber-Optic Sensor Systems for Acoustic Measurements,” Ph.D. Dissertation, University of Maryland, College Park, Md.) Here σ is the residual stress and D is the flexural rigidity, defined as:





  
    
      
        D
        =
        
          
            Eh
            3
          
          
            12
            ⁢
            
              (
              
                1
                -
                
                  v
                  2
                
              
              )
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




where E is the Young's modulus, and ν is Poisson's ratio. It should be noted that equation (1) is only applicable for small transverse displacements. When the deflection is large, the equation becomes non-linear.

For a clamped circular diaphragm with radius a, assuming a solution u(r,θ,t)=u(r,θ)ejωt, equation (1) becomes:

D∇4u−hσ∇2u=hρω2u  (3)

which has a solution of the form:

u(r,θ)=[AJm(αr)+BIm(βr)] cos(mθ)  (4)

where Jm( ) is the Bessel function of the first kind of order m, and Im( ) is the modified Bessel function of the first kind of order m, with





  
    
      
        
          
            α
            2
          
          =
          
            
              
                
                  
                    
                      h
                      2
                    
                    ⁢
                    
                      σ
                      2
                    
                  
                  +
                  
                    4
                    ⁢
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    hD
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ω
                      2
                    
                  
                
              
              -
              
                h
                ⁢
                
                  
                
                ⁢
                σ
              
            
            
              2
              ⁢
              D
            
          
        
        ⁢
        
          
        
        ⁢
        
          

        
        ⁢
        
          
            and
            ⁢
            
              
            
            ⁢
            
              β
              2
            
          
          =
          
            
              
                
                  
                    
                      
                        h
                        2
                      
                      ⁢
                      
                        σ
                        2
                      
                    
                    +
                    
                      4
                      ⁢
                      
                        
                      
                      ⁢
                      ρ
                      ⁢
                      
                        
                      
                      ⁢
                      hD
                      ⁢
                      
                        
                      
                      ⁢
                      
                        ω
                        2
                      
                    
                  
                
                +
                
                  h
                  ⁢
                  
                    
                  
                  ⁢
                  σ
                
              
              
                2
                ⁢
                D
              
            
            .
          
        
      
    
    
      
        (
        5
        )
      
    
  




The boundary conditions state that u(a, θ)=0, and





  
    
      ∂
      
        ∂
        r
      
    
    ⁢
    
      u
      ⁡
      
        (
        
          a
          ,
          θ
        
        )
      
    
  
  =
  0.




These conditions reduce to the eigenvalue equation:





  
    
      
        
          
            
              
                J
                m
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  a
                
                )
              
            
            ⁢
            
              d
              
                d
                r
              
            
            ⁢
            
              
                I
                m
              
              ⁡
              
                (
                
                  β
                  ⁢
                  
                    
                  
                  ⁢
                  a
                
                )
              
            
          
          -
          
            
              
                I
                m
              
              ⁡
              
                (
                
                  β
                  ⁢
                  
                    
                  
                  ⁢
                  a
                
                )
              
            
            ⁢
            
              d
              
                d
                r
              
            
            ⁢
            
              
                J
                m
              
              ⁡
              
                (
                
                  α
                  ⁢
                  
                    
                  
                  ⁢
                  a
                
                )
              
            
          
        
        =
        0
      
    
    
      
        (
        6
        )
      
    
  




that can be solved together with equations (5), which can be summarized as:

(βa)2−(αa)2=κ2  (7)

where κ is the useful “tension parameter” defined as κ=a√{square root over (hσ/D)}.

The solutions to equations (6) and (7) for each m=0, 1, 2, . . . can be denoted as αmn and βmn, where n=1, 2, 3 . . . denotes the nth root. The boundary conditions give the eigenmodes as:




  
    
      
        
          
            u
            mn
          
          ⁡
          
            (
            
              r
              ,
              θ
            
            )
          
        
        =
        
          
            C
            ⁡
            
              [
              
                
                  
                    J
                    m
                  
                  ⁡
                  
                    (
                    
                      
                        α
                        mn
                      
                      ⁢
                      r
                    
                    )
                  
                
                -
                
                  
                    
                      
                        J
                        m
                      
                      ⁡
                      
                        (
                        
                          
                            α
                            mn
                          
                          ⁢
                          a
                        
                        )
                      
                    
                    
                      
                        I
                        m
                      
                      ⁡
                      
                        (
                        
                          
                            β
                            mn
                          
                          ⁢
                          a
                        
                        )
                      
                    
                  
                  ⁢
                  
                    
                      I
                      m
                    
                    ⁡
                    
                      (
                      
                        
                          β
                          mn
                        
                        ⁢
                        r
                      
                      )
                    
                  
                
              
              ]
            
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            (
            
              m
              ⁢
              
                
              
              ⁢
              θ
            
            )
          
        
      
    
    
      
        (
        8
        )
      
    
  




where the eigenfrequency of a mode is found through equations (5) as:





  
    
      
        
          ω
          mn
        
        =
        
          
            α
            mn
          
          ⁢
          
            β
            mn
          
          ⁢
          
            
              D
              
                h
                ⁢
                
                  
                
                ⁢
                ρ
              
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




B. Forced Oscillation of a Diaphragm
For a forced and damped diaphragm, the equation of motion becomes:




  
    
      
        
          
            (
            
              
                h
                ⁢
                
                  
                
                ⁢
                ρ
                ⁢
                
                  
                    ∂
                    2
                  
                  
                    ∂
                    
                      t
                      2
                    
                  
                
              
              +
              
                b
                ⁢
                
                  ∂
                  
                    ∂
                    t
                  
                
              
              +
              
                D
                ⁢
                
                  
                
                ⁢
                
                  
                    ∇
                    4
                  
                  ⁢
                  
                    -
                    h
                  
                
                ⁢
                
                  
                
                ⁢
                σ
                ⁢
                
                  
                
                ⁢
                
                  ∇
                  2
                
              
            
            )
          
          ⁢
          u
        
        =
        
          P
          ⁡
          
            (
            
              r
              ,
              θ
              ,
              t
            
            )
          
        
      
    
    
      
        (
        10
        )
      
    
  




where b is the damping coefficient, and P is the pressure applied on the diaphragm surface. In the case of a miniature microphone, where λacoustic>>a, the pressure wave will be a plane wave, hence P(r,θ,t)=P(t)=P0ejωt.

Due to the similarity to the free vibration problem, we expect a solution of the form:




  
    
      
        
          u
          ⁡
          
            (
            
              r
              ,
              θ
              ,
              t
            
            )
          
        
        =
        
          
            ∑
            
              m
              ,
              n
            
          
          ⁢
          
            
              
                A
                mn
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              
                u
                mn
              
              ⁡
              
                (
                
                  r
                  ,
                  θ
                
                )
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




where umn are the modes from the free vibration problem, and Amn are modal participation factors. Putting equation (11) into equation (10) provides the following:





  
    
      
        
          
            ∑
            
              m
              ,
              n
            
          
          ⁢
          
            [
            
              
                
                  u
                  mn
                
                (
                
                  
                    h
                    ⁢
                    
                      
                    
                    ⁢
                    ρ
                    ⁢
                    
                      
                        
                          ∂
                          2
                        
                        ⁢
                        
                          A
                          mn
                        
                      
                      
                        ∂
                        
                          t
                          2
                        
                      
                    
                  
                  +
                  
                    b
                    ⁢
                    
                      
                        ∂
                        
                          A
                          mn
                        
                      
                      
                        ∂
                        t
                      
                    
                  
                
                )
              
              +
              
                
                  A
                  mn
                
                ⁡
                
                  (
                  
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          ∇
                          4
                        
                        ⁢
                        
                          u
                          mn
                        
                      
                    
                    -
                    
                      h
                      ⁢
                      
                        
                      
                      ⁢
                      σ
                      ⁢
                      
                        
                          ∇
                          2
                        
                        ⁢
                        
                          u
                          mn
                        
                      
                    
                  
                  )
                
              
            
            ]
          
        
        =
        
          
            P
            0
          
          ⁢
          
            e
            
              jω
              ⁢
              
                
              
              ⁢
              t
            
          
        
      
    
    
      
        (
        12
        )
      
    
  




The second term on the left-hand-side is given by equation (3). Hence, equation (12) becomes:





  
    
      
        
          
            ∑
            
              m
              ,
              n
            
          
          ⁢
          
            
              (
              
                
                  h
                  ⁢
                  
                    
                  
                  ⁢
                  ρ
                  ⁢
                  
                    
                      
                        ∂
                        2
                      
                      ⁢
                      
                        A
                        mn
                      
                    
                    
                      ∂
                      
                        t
                        2
                      
                    
                  
                
                +
                
                  b
                  ⁢
                  
                    
                      ∂
                      
                        A
                        mn
                      
                    
                    
                      ∂
                      t
                    
                  
                
                +
                
                  h
                  ⁢
                  
                    
                  
                  ⁢
                  ρ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ω
                    mn
                    2
                  
                  ⁢
                  
                    A
                    mn
                  
                
              
              )
            
            ⁢
            
              u
              mn
            
          
        
        =
        
          
            P
            0
          
          ⁢
          
            e
            
              j
              ⁢
              
                
              
              ⁢
              ω
              ⁢
              
                
              
              ⁢
              t
            
          
        
      
    
    
      
        (
        13
        )
      
    
  




To solve this equation, the orthogonality of the eigenmodes can be exploited, which is:





  
    
      
        
          
            ∫
            0
            
              2
              ⁢
              π
            
          
          ⁢
          
            
              ∫
              0
              a
            
            ⁢
            
              
                u
                mn
              
              ⁢
              
                u
                kl
              
              ⁢
              r
              ⁢
              
                
              
              ⁢
              
                d
                r
              
              ⁢
              
                
              
              ⁢
              
                d
                θ
              
            
          
        
        =
        
          
            δ
            mk
          
          ⁢
          
            δ
            nl
          
          ⁢
          
            
              ∫
              0
              
                2
                ⁢
                π
              
            
            ⁢
            
              
                ∫
                0
                a
              
              ⁢
              
                
                  u
                  mn
                  2
                
                ⁢
                r
                ⁢
                
                  
                
                ⁢
                
                  d
                  r
                
                ⁢
                
                  
                
                ⁢
                
                  d
                  θ
                
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




Using the orthogonality, the left-hand-side in equation (13) becomes:





  
    (
    
      
        h
        ⁢
        
          
        
        ⁢
        ρ
        ⁢
        
          
            
              ∂
              2
            
            ⁢
            
              A
              mn
            
          
          
            ∂
            
              t
              2
            
          
        
      
      +
      
        b
        ⁢
        
          
            ∂
            
              A
              mn
            
          
          
            ∂
            t
          
        
      
      +
      
        h
        ⁢
        
          
        
        ⁢
        ρ
        ⁢
        
          
        
        ⁢
        
          ω
          mn
          2
        
        ⁢
        
          A
          mn
        
      
    
    )
  
  ⁢
  
    
      ∫
      0
      
        2
        ⁢
        π
      
    
    ⁢
    
      
        ∫
        0
        a
      
      ⁢
      
        
          u
          mn
          2
        
        ⁢
        
          
        
        ⁢
        r
        ⁢
        
          d
          r
        
        ⁢
        
          
        
        ⁢
        
          d
          θ
        
      
    
  




while the right-hand-side becomes:





  
    
      P
      0
    
    ⁢
    
      e
      
        jω
        ⁢
        
          
        
        ⁢
        t
      
    
    ⁢
    
      
        ∫
        0
        
          2
          ⁢
          π
        
      
      ⁢
      
        
          ∫
          0
          a
        
        ⁢
        
          
            
              u
              mn
            
            ⁡
            
              (
              
                r
                ,
                θ
              
              )
            
          
          ⁢
          r
          ⁢
          
            
          
          ⁢
          
            d
            r
          
          ⁢
          
            
          
          ⁢
          
            d
            θ
          
        
      
    
  
  =
  
    
      P
      0
    
    ⁢
    
      e
      
        jω
        ⁢
        
          
        
        ⁢
        t
      
    
    ⁢
    
      
        ∫
        0
        a
      
      ⁢
      
        
          
            u
            mn
          
          ⁡
          
            (
            r
            )
          
        
        ⁢
        
          
        
        ⁢
        r
        ⁢
        
          d
          r
        
        ⁢
        
          
            ∫
            0
            
              2
              ⁢
              π
            
          
          ⁢
          
            
              cos
              ⁡
              
                (
                
                  m
                  ⁢
                  
                    
                  
                  ⁢
                  θ
                
                )
              
            
            ⁢
            
              
            
            ⁢
            
              d
              θ
            
          
        
      
    
  




Since





  
    
      ∫
      0
      
        2
        ⁢
        π
      
    
    ⁢
    
      
        cos
        ⁡
        
          (
          
            m
            ⁢
            
              
            
            ⁢
            θ
          
          )
        
      
      ⁢
      
        
      
      ⁢
      
        d
        θ
      
    
  
  =
  
    {
    
      
        
          
            
              2
              ⁢
              π
            
            ,
          
        
        
          
            m
            =
            0
          
        
      
      
        
          
            0
            ,
          
        
        
          
            
              m
              ≠
              0
            
            ,
          
        
      
    
  




the incident pressure wave only couples to modes with m=0, the modes that have only radial nodes (no polar nodes). Therefore, the index m can be dropped, so that only the index n is used.

In this case, the eigenvalue equation (6) reduces to:




  
    
      
        
          
            α
            n
          
          
            β
            n
          
        
        =
        
          
            -
            
              
                
                  J
                  0
                
                ⁡
                
                  (
                  
                    
                      α
                      n
                    
                    ⁢
                    a
                  
                  )
                
              
              
                
                  J
                  1
                
                ⁡
                
                  (
                  
                    
                      α
                      n
                    
                    ⁢
                    a
                  
                  )
                
              
            
          
          ·
          
            
              
                I
                1
              
              ⁡
              
                (
                
                  
                    β
                    n
                  
                  ⁢
                  a
                
                )
              
            
            
              
                I
                0
              
              ⁡
              
                (
                
                  
                    β
                    n
                  
                  ⁢
                  a
                
                )
              
            
          
        
      
    
    
      
        (
        15
        )
      
    
  




And the eigenmodes in equation (8) become:





  
    
      
        
          
            u
            n
          
          ⁡
          
            (
            r
            )
          
        
        =
        
          C
          ⁡
          
            [
            
              
                
                  J
                  0
                
                ⁡
                
                  (
                  
                    
                      α
                      n
                    
                    ⁢
                    r
                  
                  )
                
              
              -
              
                
                  
                    
                      J
                      0
                    
                    ⁡
                    
                      (
                      
                        
                          α
                          n
                        
                        ⁢
                        a
                      
                      )
                    
                  
                  
                    
                      I
                      0
                    
                    ⁡
                    
                      (
                      
                        
                          β
                          n
                        
                        ⁢
                        a
                      
                      )
                    
                  
                
                ⁢
                
                  
                    I
                    0
                  
                  ⁡
                  
                    (
                    
                      
                        β
                        n
                      
                      ⁢
                      r
                    
                    )
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        16
        )
      
    
  




The equation for the modal participation factor An becomes then:





  
    
      
        
          
            
              
                ∂
                2
              
              ⁢
              
                A
                n
              
            
            
              ∂
              
                t
                2
              
            
          
          +
          
            
              
                ω
                n
              
              
                Q
                n
              
            
            ⁢
            
              
                ∂
                
                  A
                  n
                
              
              
                ∂
                t
              
            
          
          +
          
            
              ω
              n
              2
            
            ⁢
            
              A
              n
            
          
        
        =
        
          
            
              U
              n
            
            
              h
              ⁢
              
                
              
              ⁢
              ρ
            
          
          ⁢
          
            P
            0
          
          ⁢
          
            e
            
              jω
              ⁢
              
                
              
              ⁢
              t
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




where Qn=hρωn/b is the quality factor of the nth mode, and the constant Un is:





  
    U
    n
  
  =
  
    
      
        ∫
        0
        a
      
      ⁢
      
        
          u
          n
        
        ⁢
        r
        ⁢
        
          
        
        ⁢
        
          d
          r
        
      
    
    
      
        ∫
        0
        a
      
      ⁢
      
        
          u
          n
          2
        
        ⁢
        r
        ⁢
        
          
        
        ⁢
        
          d
          r
        
      
    
  




Assuming a solution of the form An(t)=Anej(ωt+φn), equation (17) gives:




  
    
      
        
          
            A
            n
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              U
              n
            
            ⁢
            
              P
              0
            
            ⁢
            
              e
              
                jω
                ⁢
                
                  
                
                ⁢
                t
              
            
          
          
            h
            ⁢
            
              
            
            ⁢
            
              ρ
              ⁡
              
                (
                
                  
                    ω
                    n
                    2
                  
                  -
                  
                    ω
                    2
                  
                  +
                  
                    
                      jω
                      n
                    
                    ⁢
                    
                      ω
                      /
                      
                        Q
                        n
                      
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        18
        )
      
    
  




Hence, we get the displacement as:





  
    
      
        
          u
          ⁡
          
            (
            
              r
              ,
              t
            
            )
          
        
        =
        
          
            
              ∑
              n
            
            ⁢
            
              
                
                  A
                  n
                
                ⁡
                
                  (
                  t
                  )
                
              
              ⁢
              
                
                  u
                  n
                
                ⁡
                
                  (
                  r
                  )
                
              
            
          
          =
          
            
              P
              0
            
            ⁢
            
              e
              
                jω
                ⁢
                
                  
                
                ⁢
                t
              
            
            ⁢
            
              
                ∑
                n
              
              ⁢
              
                
                  
                    U
                    n
                  
                  ⁢
                  
                    
                      u
                      n
                    
                    ⁡
                    
                      (
                      r
                      )
                    
                  
                
                
                  h
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ρ
                    ⁡
                    
                      (
                      
                        
                          ω
                          n
                          2
                        
                        -
                        
                          ω
                          2
                        
                        +
                        
                          
                            jω
                            n
                          
                          ⁢
                          
                            ω
                            /
                            
                              Q
                              n
                            
                          
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        19
        )
      
    
  




This is the general solution for any frequency. For low frequencies, such that ω<<ωn:





  
    
      
        
          u
          ⁡
          
            (
            
              r
              ,
              t
            
            )
          
        
        =
        
          
            P
            0
          
          ⁢
          
            e
            
              jω
              ⁢
              
                
              
              ⁢
              t
            
          
          ⁢
          
            
              ∑
              n
            
            ⁢
            
              
                
                  U
                  n
                
                ⁢
                
                  
                    u
                    n
                  
                  ⁡
                  
                    (
                    r
                    )
                  
                
              
              
                h
                ⁢
                
                  
                
                ⁢
                ρ
                ⁢
                
                  
                
                ⁢
                
                  
                    ω
                    n
                    2
                  
                  ⁡
                  
                    (
                    
                      1
                      +
                      
                        j
                        ⁢
                        
                          1
                          
                            Q
                            n
                          
                        
                        ⁢
                        
                          ω
                          
                            ω
                            n
                          
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        20
        )
      
    
  




This is a general solution for the transverse displacement of a stretched diaphragm that is vibrated by a pressure plane wave at a frequency below resonance.

C. Solutions for the Special Cases of Membrane and Plate
Two different kinds of structures, which are diaphragms made of silicon-nitride and crystalline-silicon, are of interest. Due to the mechanical properties of these two materials, the diaphragm solutions have closed forms as is discussed below.
C.1 Membrane Solution
A membrane is a diaphragm where the residual stress is dominant, e.g. κ→∞. The membrane structure is a good approximation for κ>20, which is the case for a silicon-nitride diaphragm that usually has a high residual stress. In this case, since βn→κ/a→∞, the eigenvalue equation (15) becomes simply J0(αna)=0. For notational simplicity, αna=zn, where zn denotes the nth zero of J0(x).
Also, the eigenmodes in equation (16) become un(r)=CJ0(znr/a), so that:




  
    
      U
      n
    
    ⁢
    
      
        u
        n
      
      ⁡
      
        (
        r
        )
      
    
  
  =
  
    
      
        
          
            ∫
            0
            a
          
          ⁢
          
            
              
                J
                0
              
              ⁡
              
                (
                
                  
                    z
                    n
                  
                  ⁢
                  
                    r
                    /
                    a
                  
                
                )
              
            
            ⁢
            r
            ⁢
            
              
            
            ⁢
            
              d
              r
            
          
        
        
          
            ∫
            0
            a
          
          ⁢
          
            
              
                J
                0
                2
              
              ⁡
              
                (
                
                  
                    z
                    n
                  
                  ⁢
                  
                    r
                    /
                    a
                  
                
                )
              
            
            ⁢
            r
            ⁢
            
              
            
            ⁢
            
              d
              r
            
          
        
      
      ⁢
      
        
          J
          0
        
        ⁡
        
          (
          
            
              z
              n
            
            ⁢
            
              r
              /
              a
            
          
          )
        
      
    
    =
    
      
        2
        
          z
          n
        
      
      ·
      
        
          
            J
            0
          
          ⁡
          
            (
            
              
                z
                n
              
              ⁢
              
                r
                /
                a
              
            
            )
          
        
        
          
            J
            1
          
          ⁡
          
            (
            
              z
              n
            
            )
          
        
      
    
  




The eigenfrequencies in equation (9), on the other hand, become:





  
    ω
    n
  
  =
  
    
      
        z
        n
      
      a
    
    ⁢
    
      
        σ
        ρ
      
    
  




Using these in the general solution of equation (20):





  
    
      
        
          u
          ⁡
          
            (
            
              r
              ,
              t
            
            )
          
        
        =
        
          
            P
            0
          
          ⁢
          
            e
            
              jω
              ⁢
              
                
              
              ⁢
              t
            
          
          ⁢
          
            
              2
              ⁢
              
                a
                2
              
            
            
              h
              ⁢
              
                
              
              ⁢
              σ
            
          
          ⁢
          
            
              ∑
              n
            
            ⁢
            
              
                
                  1
                  
                    z
                    n
                    3
                  
                
                ·
                
                  
                    
                      J
                      0
                    
                    ⁡
                    
                      (
                      
                        
                          z
                          n
                        
                        ⁢
                        
                          r
                          /
                          a
                        
                      
                      )
                    
                  
                  
                    
                      J
                      1
                    
                    ⁡
                    
                      (
                      
                        z
                        n
                      
                      )
                    
                  
                
              
              ⁢
              
                
                  (
                  
                    1
                    +
                    
                      j
                      ⁢
                      
                        
                          
                            ω
                            ⁢
                            
                              
                            
                            ⁢
                            
                              ba
                              2
                            
                          
                          
                            h
                            ⁢
                            
                              
                            
                            ⁢
                            σ
                          
                        
                        ·
                        
                          1
                          
                            z
                            n
                            2
                          
                        
                      
                    
                  
                  )
                
                
                  -
                  1
                
              
            
          
        
      
    
    
      
        (
        21
        )
      
    
  




To find a closed form of this expression, two different damping conditions, which are b=0, and b→∞ will be considered.
C.1.a Membrane Solution—Negligible Damping Case
For b=0, the displacement in equation (21) becomes:




  
    u
    ⁡
    
      (
      
        r
        ,
        t
      
      )
    
  
  =
  
    
      P
      0
    
    ⁢
    
      e
      
        jω
        ⁢
        
          
        
        ⁢
        t
      
    
    ⁢
    
      
        2
        ⁢
        
          a
          2
        
      
      
        h
        ⁢
        
          
        
        ⁢
        σ
      
    
    ⁢
    
      
        ∑
        n
      
      ⁢
      
        
          1
          
            z
            n
            3
          
        
        ·
        
          
            
              J
              0
            
            ⁡
            
              (
              
                
                  z
                  n
                
                ⁢
                
                  r
                  /
                  a
                
              
              )
            
          
          
            
              J
              1
            
            ⁡
            
              (
              
                z
                n
              
              )
            
          
        
      
    
  




which can be recognized as a Fourier-Bessel series. A function in the interval x=(0,1) can be expanded in a Fourier-Bessel series as:





  
    f
    ⁡
    
      (
      x
      )
    
  
  =
  
    
      ∑
      n
    
    ⁢
    
      
        C
        n
      
      ⁢
      
        
          J
          0
        
        ⁡
        
          (
          
            
              z
              n
            
            ⁢
            x
          
          )
        
      
    
  




where the coefficients Cn are given as:





  
    C
    n
  
  =
  
    
      2
      
        
          J
          1
          2
        
        ⁡
        
          (
          
            z
            n
          
          )
        
      
    
    ⁢
    
      
        ∫
        0
        1
      
      ⁢
      
        
          f
          ⁡
          
            (
            x
            )
          
        
        ⁢
        
          
            J
            0
          
          ⁡
          
            (
            
              
                z
                n
              
              ⁢
              x
            
            )
          
        
        ⁢
        x
        ⁢
        
          
        
        ⁢
        
          d
          x
        
      
    
  




Considering the integral




  
    
      
        ∫
        0
        1
      
      ⁢
      
        
          (
          
            1
            -
            
              x
              2
            
          
          )
        
        ⁢
        
          
            J
            0
          
          ⁡
          
            (
            
              
                z
                n
              
              ⁢
              x
            
            )
          
        
        ⁢
        x
        ⁢
        
          
        
        ⁢
        
          d
          x
        
      
    
    =
    
      
        4
        ⁢
        
          
            J
            1
          
          ⁡
          
            (
            
              z
              n
            
            )
          
        
      
      
        z
        n
        3
      
    
  
  ,




the displacement for negligible damping in a closed form can be expressed as:





  
    
      
        
          u
          ⁡
          
            (
            
              r
              ,
              t
            
            )
          
        
        =
        
          
            P
            0
          
          ⁢
          
            e
            
              jω
              ⁢
              
                
              
              ⁢
              t
            
          
          ⁢
          
            
              a
              2
            
            
              4
              ⁢
              h
              ⁢
              
                
              
              ⁢
              σ
            
          
          ⁢
          
            (
            
              1
              -
              
                
                  r
                  2
                
                
                  a
                  2
                
              
            
            )
          
        
      
    
    
      
        (
        22
        )
      
    
  




This solution is also consistent with other reports. See, e.g., W. P. Eaton et al., “A new analytical solution for diaphragm deflection and its application to a surface micromachined pressure sensor,” Int'l Conf. on Modeling and Simulation of Microsystems, 1999. Note that equation (22) is an exact solution applicable to the whole range of r=(0,a).

C.1.b Membrane Solution—Strong Damping Case
For b→∞, the displacement in equation (21) becomes:




  
    u
    ⁡
    
      (
      
        r
        ,
        t
      
      )
    
  
  =
  
    
      P
      0
    
    ⁢
    
      e
      
        jω
        ⁢
        
          
        
        ⁢
        t
      
    
    ⁢
    
      2
      
        j
        ⁢
        
          
        
        ⁢
        ω
        ⁢
        
          
        
        ⁢
        b
      
    
    ⁢
    
      
        ∑
        n
      
      ⁢
      
        
          1
          
            z
            n
          
        
        ·
        
          
            
              J
              0
            
            ⁡
            
              (
              
                
                  z
                  n
                
                ⁢
                
                  r
                  /
                  a
                
              
              )
            
          
          
            
              J
              1
            
            ⁡
            
              (
              
                z
                n
              
              )
            
          
        
      
    
  




Considering the integral




  
    
      
        ∫
        0
        1
      
      ⁢
      
        
          
            J
            0
          
          ⁡
          
            (
            
              
                z
                n
              
              ⁢
              x
            
            )
          
        
        ⁢
        x
        ⁢
        
          
        
        ⁢
        
          d
          x
        
      
    
    =
    
      
        
          J
          1
        
        ⁡
        
          (
          
            z
            n
          
          )
        
      
      
        z
        n
      
    
  
  ,




the displacement for strong damping in a closed form can be expressed as:





  
    
      
        
          u
          ⁡
          
            (
            
              r
              ,
              t
            
            )
          
        
        =
        
          
            P
            0
          
          ⁢
          
            e
            
              jω
              ⁢
              
                
              
              ⁢
              t
            
          
          ⁢
          
            1
            
              j
              ⁢
              
                
              
              ⁢
              ω
              ⁢
              
                
              
              ⁢
              b
            
          
        
      
    
    
      
        (
        23
        )
      
    
  




Therefore, when the damping is very strong, the membrane tends to move as a whole without a noticeable bending.

C.2 Plate Solution
A plate is a diaphragm where the bending stiffness is dominant, e.g. κ=0. The plate structure is a good approximation for κ<2, which is the case for a crystalline-silicon diaphragm that usually has very low residual stress.
In this case, since βn=αn, the eigenvalue equation (15) becomes:




  
    
      
        J
        0
      
      ⁡
      
        (
        
          
            α
            n
          
          ⁢
          a
        
        )
      
    
    +
    
      
        
          
            J
            1
          
          ⁡
          
            (
            
              
                α
                n
              
              ⁢
              a
            
            )
          
        
        
          
            I
            1
          
          ⁡
          
            (
            
              
                α
                n
              
              ⁢
              a
            
            )
          
        
      
      ⁢
      
        
          I
          0
        
        ⁡
        
          (
          
            
              α
              n
            
            ⁢
            a
          
          )
        
      
    
  
  =
  0




For notational simplicity, αna=zn, where zn denotes the nth zero of the function I0(x) that is defined as:





  
    
            0
    
    ⁢
    
      (
      
        
          z
          n
        
        ⁢
        
          r
          /
          a
        
      
      )
    
  
  =
  
    
      
        J
        0
      
      ⁡
      
        (
        
          
            z
            n
          
          ⁢
          
            r
            /
            a
          
        
        )
      
    
    +
    
      
        
          
            J
            1
          
          ⁡
          
            (
            
              z
              n
            
            )
          
        
        
          
            I
            1
          
          ⁡
          
            (
            
              z
              n
            
            )
          
        
      
      ⁢
      
        
          I
          0
        
        ⁡
        
          (
          
            
              z
              n
            
            ⁢
            
              r
              /
              a
            
          
          )
        
      
    
  




Whence, the eigenmodes in equation (16) become un(r)=CI0(znr/a), so that:





  
    
      
        
          
            U
            n
          
          ⁢
          
            
              u
              n
            
            ⁡
            
              (
              r
              )
            
          
        
        =
        
          
            
              
                
                  
                    ∫
                    0
                    a
                  
                  ⁢
                  
                    
                                            0
                    
                    ⁢
                    
                      (
                      
                        
                          z
                          n
                        
                        ⁢
                        
                          r
                          /
                          a
                        
                      
                      )
                    
                    ⁢
                    r
                    ⁢
                    
                      d
                      r
                    
                  
                
                ⁢
                
                  
                
              
              
                
                  ∫
                  0
                  a
                
                ⁢
                
                  
                                        0
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        z
                        n
                      
                      ⁢
                      
                        r
                        /
                        a
                      
                    
                    )
                  
                  ⁢
                  
                    
                  
                  ⁢
                  r
                  ⁢
                  
                    d
                    r
                  
                
              
            
            ⁢
            
                            0
            
            ⁢
            
              (
              
                
                  z
                  n
                
                ⁢
                
                  r
                  /
                  a
                
              
              )
            
          
          =
          
            
              
                2
                
                  z
                  n
                
              
              ·
              
                
                  
                    J
                    1
                  
                  ⁡
                  
                    (
                    
                      z
                      n
                    
                    )
                  
                
                
                  
                    J
                    0
                    2
                  
                  ⁡
                  
                    (
                    
                      z
                      n
                    
                    )
                  
                
              
            
            ⁢
            
                            0
            
            ⁢
            
              (
              
                
                  z
                  n
                
                ⁢
                
                  r
                  /
                  a
                
              
              )
            
          
        
      
    
    
      
        
      
    
  




The eigenfrequencies in equation (9), on the other hand, become:




  
    ω
    n
  
  =
  
    
      
        z
        n
        2
      
      
        a
        2
      
    
    ⁢
    
      
        D
        
          h
          ⁢
          
            
          
          ⁢
          ρ
        
      
    
  




Using these in the general solution of equation (20):





  
    
      
        
          u
          ⁡
          
            (
            
              r
              ,
              t
            
            )
          
        
        =
        
          
            P
            0
          
          ⁢
          
            e
            
              jω
              ⁢
              
                
              
              ⁢
              t
            
          
          ⁢
          
            
              2
              ⁢
              
                a
                4
              
            
            D
          
          ⁢
          
            
              ∑
              n
            
            ⁢
            
              
                
                  1
                  
                    z
                    n
                    5
                  
                
                ·
                
                  
                    
                      J
                      1
                    
                    ⁡
                    
                      (
                      
                        z
                        n
                      
                      )
                    
                  
                  
                    
                      J
                      0
                      2
                    
                    ⁡
                    
                      (
                      
                        z
                        n
                      
                      )
                    
                  
                
              
              ⁢
              
                                0
              
              ⁢
              
                (
                
                  
                    z
                    n
                  
                  ⁢
                  
                    r
                    /
                    a
                  
                
                )
              
              ⁢
              
                
                  (
                  
                    1
                    +
                    
                      j
                      ⁢
                      
                        
                          
                            ω
                            ⁢
                            
                              
                            
                            ⁢
                            
                              ba
                              4
                            
                          
                          D
                        
                        ·
                        
                          1
                          
                            z
                            n
                            4
                          
                        
                      
                    
                  
                  )
                
                
                  -
                  1
                
              
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




To find a closed form of this expression, two different damping conditions, which are b=0, and b→∞ are considered.

C.2.a Plate Solution—Negligible Damping Case
For b=0, the displacement in equation (24) becomes:




  
    u
    ⁡
    
      (
      
        r
        ,
        t
      
      )
    
  
  =
  
    
      P
      0
    
    ⁢
    
      e
      
        jω
        ⁢
        
          
        
        ⁢
        t
      
    
    ⁢
    
      
        2
        ⁢
        
          a
          4
        
      
      D
    
    ⁢
    
      
        ∑
        n
      
      ⁢
      
        
          
            1
            
              z
              n
              5
            
          
          ·
          
            
              
                J
                1
              
              ⁡
              
                (
                
                  z
                  n
                
                )
              
            
            
              
                J
                0
                2
              
              ⁡
              
                (
                
                  z
                  n
                
                )
              
            
          
        
        ⁢
        
                    0
        
        ⁢
        
          (
          
            
              z
              n
            
            ⁢
            
              r
              /
              a
            
          
          )
        
      
    
  




It is possible to define a generalized Fourier-Bessel series for the function I0(x), using the orthogonality of I0(znx), which is:





  
    
      ∫
      0
      1
    
    ⁢
    
      
                0
      
      ⁢
      
        (
        
          
            z
            n
          
          ⁢
          x
        
        )
      
      ⁢
      
                0
      
      ⁢
      
        (
        
          
            z
            m
          
          ⁢
          x
        
        )
      
      ⁢
      x
      ⁢
      
        
      
      ⁢
      
        d
        x
      
    
  
  =
  
    
      δ
      nm
    
    ⁢
    
      
        J
        0
        2
      
      ⁡
      
        (
        
          z
          n
        
        )
      
    
  




Using this orthogonality, a function in the interval x=(0,1) can be expanded as:





  
    f
    ⁡
    
      (
      x
      )
    
  
  =
  
    
      ∑
      n
    
    ⁢
    
      
        C
        n
      
      ⁢
      
                0
      
      ⁢
      
        (
        
          
            z
            n
          
          ⁢
          x
        
        )
      
    
  




Where the coefficients Cn are given in this case as:





  
    C
    n
  
  =
  
    
      1
      
        
          J
          0
          2
        
        ⁡
        
          (
          
            z
            n
          
          )
        
      
    
    ⁢
    
      
        ∫
        0
        1
      
      ⁢
      
        
          f
          ⁡
          
            (
            x
            )
          
        
        ⁢
        
                    0
        
        ⁢
        
          (
          
            
              z
              n
            
            ⁢
            x
          
          )
        
        ⁢
        
          
        
        ⁢
        x
        ⁢
        
          d
          x
        
      
    
  




Calculation shows that





  
    
      ∫
      0
      1
    
    ⁢
    
      
        
          (
          
            1
            -
            
              x
              2
            
          
          )
        
        2
      
      ⁢
      
                0
      
      ⁢
      
        (
        
          
            z
            n
          
          ⁢
          x
        
        )
      
      ⁢
      x
      ⁢
      
        
      
      ⁢
      
        d
        x
      
    
  
  =
  
    64
    ⁢
    
      
        2
        ⁢
        
          
            J
            1
          
          ⁡
          
            (
            
              z
              n
            
            )
          
        
      
      
        z
        n
        5
      
    
  




Hence, the displacement for negligible damping in a closed form can be expressed as:




  
    
      
        
          u
          ⁡
          
            (
            
              r
              ,
              t
            
            )
          
        
        =
        
          
            P
            0
          
          ⁢
          
            e
            
              j
              ⁢
              
                
              
              ⁢
              ω
              ⁢
              
                
              
              ⁢
              t
            
          
          ⁢
          
            
              a
              4
            
            
              64
              ⁢
              D
            
          
          ⁢
          
            
              (
              
                1
                -
                
                  
                    r
                    2
                  
                  
                    a
                    2
                  
                
              
              )
            
            2
          
        
      
    
    
      
        (
        25
        )
      
    
  




Note that this is an exact solution applicable to the whole range of r=(0,a). This solution is also consistent with other reports. See, e.g., W. P. Eaton et al., “A new analytical solution for diaphragm deflection and its application to a surface micromachined pressure sensor,” Int'l Conf. on Modeling and Simulation of Microsystems, 1999. Also note that the decay from r=0 to r=a is more rapid compared to the membrane case.

C.2.b Plate Solution—Strong Damping Case
For b→∞, the displacement in equation (21) becomes:




  
    
      
        
          
            u
            ⁡
            
              (
              
                r
                ,
                t
              
              )
            
          
          =
          
            
              P
              0
            
            ⁢
            
              e
              
                jω
                ⁢
                
                  
                
                ⁢
                t
              
            
            ⁢
            
              2
              
                j
                ⁢
                
                  
                
                ⁢
                ω
                ⁢
                
                  
                
                ⁢
                b
              
            
            ⁢
            
              
                ∑
                n
              
              ⁢
              
                1
                
                  z
                  n
                
              
            
          
        
        ⁣
        
          
            ·
            
              
                
                  J
                  1
                
                ⁡
                
                  (
                  
                    z
                    n
                  
                  )
                
              
              
                
                  J
                  0
                  2
                
                ⁡
                
                  (
                  
                    z
                    n
                  
                  )
                
              
            
          
          ⁢
          
                        0
          
          ⁢
          
            (
            
              
                z
                n
              
              ⁢
              
                r
                /
                a
              
            
            )
          
        
      
    
    
      
        
      
    
  




Calculation shows that





  
    
      ∫
      0
      1
    
    ⁢
    
      
                0
      
      ⁢
      
        (
        
          
            z
            n
          
          ⁢
          x
        
        )
      
      ⁢
      x
      ⁢
      
        
      
      ⁢
      
        d
        x
      
    
  
  =
  
    
      2
      ⁢
      
        
          J
          1
        
        ⁡
        
          (
          
            z
            n
          
          )
        
      
    
    
      z
      n
    
  




Hence, the displacement for strong damping in a closed form can be expressed as:





  
    
      
        
          u
          ⁡
          
            (
            
              r
              ,
              t
            
            )
          
        
        =
        
          
            P
            0
          
          ⁢
          
            e
            
              jω
              ⁢
              
                
              
              ⁢
              t
            
          
          ⁢
          
            1
            
              jω
              ⁢
              
                
              
              ⁢
              b
            
          
        
      
    
    
      
        (
        26
        )
      
    
  




Therefore, as in the membrane case, when the damping is very strong, the plate tends to move as a whole without a noticeable bending.

D. Mechanical Impedance Effects of the Surrounding Medium
Calculations of mechanical impedances can facilitate understanding what effect the surrounding medium (such as air or water), and the damping, will have on the displacement of the diaphragm. The mechanical impedance Z is defined as the ratio of pressure to speed, namely Z=P/υ. In the case discussed here, υ(r)=jωu(r). To calculate the impedance of the diaphragm, the lumped speed is used, which is:




  
    υ
    _
  
  =
  
    
      
        1
        
          π
          ⁢
          
            
          
          ⁢
          
            a
            2
          
        
      
      ⁢
      
        
          ∫
          0
          a
        
        ⁢
        
          
            υ
            ⁡
            
              (
              r
              )
            
          
          ⁢
          2
          ⁢
          π
          ⁢
          
            
          
          ⁢
          r
          ⁢
          
            
          
          ⁢
          
            d
            r
          
        
      
    
    =
    
      
        
          2
          ⁢
          jω
        
        
          a
          2
        
      
      ⁢
      
        
          ∫
          0
          a
        
        ⁢
        
          
            u
            ⁡
            
              (
              r
              )
            
          
          ⁢
          r
          ⁢
          
            
          
          ⁢
          
            d
            r
          
        
      
    
  




so that,





  Z
  =
  
    
      P
      υ
    
    =
    
      
        
          
            P
            0
          
          ⁢
          
            e
            
              jω
              ⁢
              
                
              
              ⁢
              t
            
          
        
        
          υ
          _
        
      
      .
    
  




Using equation (22), the impedance of a diaphragm that can be approximated by a membrane is then:




  
    
      
        
          Z
          d
        
        =
        
          
            
              
                P
                0
              
              ⁢
              
                e
                
                  jω
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            
              
                
                  2
                  ⁢
                  jω
                
                
                  a
                  2
                
              
              ⁢
              
                (
                
                  
                    P
                    0
                  
                  ⁢
                  
                    e
                    
                      jω
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                  ⁢
                  
                    
                      a
                      4
                    
                    
                      16
                      ⁢
                      
                        
                      
                      ⁢
                      h
                      ⁢
                      
                        
                      
                      ⁢
                      σ
                    
                  
                
                )
              
            
          
          =
          
            
              -
              j
            
            ⁢
            
              
                8
                ⁢
                h
                ⁢
                
                  
                
                ⁢
                σ
              
              
                ω
                ⁢
                
                  
                
                ⁢
                
                  a
                  2
                
              
            
          
        
      
    
    
      
        (
        27
        )
      
    
  




And similarly, using equation (25), the impedance of a diaphragm that can be approximated by a plate is then:





  
    
      
        
          Z
          d
        
        =
        
          
            
              
                P
                0
              
              ⁢
              
                e
                
                  jω
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            
              
                
                  2
                  ⁢
                  jω
                
                
                  a
                  2
                
              
              ⁢
              
                (
                
                  
                    P
                    0
                  
                  ⁢
                  
                    e
                    
                      jω
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                  ⁢
                  
                    
                      a
                      6
                    
                    
                      384
                      ⁢
                      D
                    
                  
                
                )
              
            
          
          =
          
            
              -
              j
            
            ⁢
            
              
                192
                ⁢
                D
              
              
                ω
                ⁢
                
                  
                
                ⁢
                
                  a
                  4
                
              
            
          
        
      
    
    
      
        (
        28
        )
      
    
  




The impedance due to the damping will be just Zb=b, as can be verified using equation (23) or equation (26). The fact that the damping impedance is a real number means that it is responsible for dissipation loss in the system.

To calculate the impedance of the surrounding medium, the expression for the displacement of particles in an acoustic wave can be used:




  
    
      
        
          u
          a
        
        =
        
          
            1
            
              ω
              ⁢
              
                
              
              ⁢
              
                ρ
                a
              
              ⁢
              
                υ
                a
              
            
          
          ⁢
          
            P
            0
          
          ⁢
          
            e
            
              jω
              ⁢
              
                
              
              ⁢
              t
            
          
        
      
    
    
      
        (
        29
        )
      
    
  




where ρa is the density of the medium, and υn is the speed of the acoustic wave (not to be confused with the speed of the particles that are displaced in the medium). Using equation (29), the impedance of the surrounding medium can be expressed as:





  
    
      
        
          Z
          s
        
        =
        
          
            
              
                P
                0
              
              ⁢
              
                e
                
                  jω
                  ⁢
                  
                    
                  
                  ⁢
                  t
                
              
            
            
              j
              ⁢
              
                
              
              ⁢
              
                ω
                ⁡
                
                  (
                  
                    
                      P
                      0
                    
                    ⁢
                    
                      e
                      
                        jω
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                    
                    ⁢
                    
                      1
                      
                        ω
                        ⁢
                        
                          
                        
                        ⁢
                        
                          ρ
                          a
                        
                        ⁢
                        
                          υ
                          a
                        
                      
                    
                  
                  )
                
              
            
          
          =
          
            
              -
              j
            
            ⁢
            
              
            
            ⁢
            
              ρ
              a
            
            ⁢
            
              υ
              a
            
          
        
      
    
    
      
        (
        30
        )
      
    
  




The total impedance of the system will then be the sum of the impedance of the diaphragm, the damping impedance, and the impedance of the surrounding medium:

Ztotal=Zd+Zb+Zs 

The total displacement of the diaphragm will depend on the value of this total impedance. If one of the impedances is much larger than the others, the diaphragm displacement will be dominated by it. For example, if the membrane impedance is dominant, i.e. Zd>>Zb,Zs, the displacement will be just as in equation (22) or equation (25), the diaphragm displacements under negligible damping. If the damping impedance is dominant, i.e. Zb>>Zd,Zs, the displacement will be just as in equation (23) or equation (26), the diaphragm displacements under large damping conditions. And, if the impedance of the surrounding medium is dominant, i.e. Zs>>Zd,Zb, the displacement will be just as in equation (29), which is the displacement of the particles in the acoustic wave.

E. Numerical Values for the Impedances
As an example system, a circular diaphragm made of either silicon-nitride or crystalline-silicon has the radius of a typical SMF-28 singlemode fiber (e.g., 62.5 microns), and is separated by a distance d from the end of the fiber. Table 1 gives values of various parameters and constants to be used in the calculations.







TABLE 1





Parameters and constants
















Diaphragm parameters



radius of diaphragm:
a = 62.5 microns


thickness of diaphragm:
h = 0.5 micron


gap length between diaphragm and fiber:
d = 1 micron


operation frequency:
ω = 2π × 30 kHz


Silicon-nitride constants


density:
ρSiN = 3270 kg/m3


estimates residual stress in high stress nitride film:
σSiN = 300 MPa


Young's modulus:
ESiN = 320 GPa


Poisson's ratio:
νSiN = 0.26


Crystalline-silicon constants


density:
ρSi = 2330 kg/m3


residual stress (estimated to be insignificant):
σSi ≈ 0 MPa


Young's modulus:
ESi = 150 GPa


Poisson's ratio:
νSi = 0.2


Air constants


density (of dry air at 1 atm pressure and 20° C.):
ρair = 1.21 kg/m3


speed of sound (at 20° C.):
υair = 343 m/s


dynamic viscosity (at 20° C.):
μair = 1.82 × 10−5 kg/m/s


Water constants


density (of pure water at 20° C.):
ρwater = 998 kg/m3


speed of sound (in pure water at 20° C.):
υwater = 1482 m/s


dynamic viscosity (at 20° C.):
μwater = 9.77 × 10−4 kg/m/s


density (of sea water with 3.5% salinity at 20° C.):
ρsea-water = 1025 kg/m3


speed of sound (in sea water with 3.5% salinity at 20° C.):
υsea-water = 1522 m/s









Other than the formulas given in the previous sections, an expression can be used to calculate the damping. The calculation of damping is usually complex, and has also a strong dependence on the overall geometry. However, an estimate of the damping can still be made. Because of the close proximity of the diaphragm and the fiber end, the dominant damping will be the squeeze film damping which can estimated as:




  b
  ≈
  
    
      1
      
        π
        ⁢
        
          
        
        ⁢
        
          a
          2
        
      
    
    ⁢
    
      (
      
        0.42
        ⁢
        
          
            μ
            ⁢
            
              
            
            ⁢
            
              A
              2
            
          
          
            d
            3
          
        
      
      )
    
  




where μ is the dynamic viscosity of the medium, A=πa2 is the area of the plates, and d is the gap length (see, e.g., M. Andrews et al., “A comparison of squeeze-film theory with measurements on a microstructure, Sensors and Actuators A, vol. 36, pages 79-87 (1993)).

Using the values in Table 1:
bair=9.38×104 kg/m2/s, damping in air
bwater=5.04×106 kg/m2/s, damping in water
Similarly, using the values in Table 1 in the impedance formulas equations (27), (28), and (30):
|ZSiN|=1.62×106 kg/m2/s, impedance of a silicon-nitride membrane
|ZSi|=1.09×105 kg/m2/s, impedance of a silicon plate
|Zair|=415 kg/m2/s, impedance of air
|Zwater|=1.48×106 kg/m2/s, impedance of water
|Zsea-water|=1.56×106 kg/m2/s, impedance of sea-water
These results show that for the given dimensions, the impedances, of the membranes are comparable to the damping and water impedances. FIG. 33 shows the impedance of an example acoustic sensor at a frequency of 30 kHz with a diaphragm diameter of 125 microns. The damping is generally inversely proportional to the cube of the diaphragm-fiber gap. The damping is larger for smaller distances, and is significant for distances smaller than about 3 microns. In addition, a larger diaphragm diameter would yield more advantageous results. A diaphragm radius more than 50% larger would make the silicon-nitride calculations more accurate, since in this case, the tension parameter value of κ≈13 is insufficient to model the nitride diaphragm as a membrane. Also, the damping impedance can be reduced through careful design, such as the size and position of the pressure equalizing holes.
These results show that the initial assumptions about the displacement of the diaphragm in air and water were inaccurate, and these calculations can be used to make a more optimal sensor design, either for air or water operation.
Fiber Fabry-Perot-Based Acoustic Sensor Designs
The expressions previously derived can be used to find optimal parameters for the acoustic sensor systems. FIG. 31 schematically illustrates an example configuration of a movable reflective element (e.g., a membrane) and an optical fiber The main parameters to be optimized, shown schematically in FIG. 31, are the cavity length (L), the radius of the membrane (a), and the reflectivities of the fiber end (Rf) and the membrane mirror (Rm).
As a first step, the limitations or ranges of these parameters can be calculated. The membrane radius a is preferably equal to at least the radius of the fiber, which is about 62.5 microns, so that the impedance of the membrane does not becomes excessively large so that it limits the pressure sensitivity of the sensor. The size of the membrane is preferably sufficiently small to provide a compact and robust sensor. Therefore, the membrane diameter advantageously does not exceed about 300 microns, above which it becomes large and fragile.
For reasons shown below, the reflectivity of the membrane mirror Rm is preferably as large as possible (e.g., Rm ̃1), which is achieved in certain embodiments with a photonic crystal mirror. The reflectivity of the fiber end (Rf) is preferably as small as possible. The reasons for this are discussed more fully below. Also, a small reflectivity on the fiber end is preferably used, since it is usually difficult to deposit a highly reflective mirror, either metal or dielectric, on the end of a fiber. Also, if a fiber Bragg mirror is used, its length is preferably reduced by requiring a small reflectivity, as it was explained in previous texts.
To calculate limitations on the cavity length L, several factors are considered. From the mechanical point of view, it is advantageous to have a large length, since this results in a smaller damping. However, when the optical properties are considered, there are certain limitations on the cavity length, as calculated below.
The contrast of the resonance tends to decrease with increasing mirror reflectivities, so that for very high reflectivities, it appears that there is almost no visible resonance. This effect can be avoided by balancing the mirrors of the Fabry-Perot. In fact, as calculated, full contrast can be restored by having:

Rf=Rm2−L/z0 

where z0=πw02nc/λ=kw02/2 is the Rayleigh range, a characteristic length over which the beam does not diverge significantly.

Also, the maximum sensitivity to displacement occurs at the point where the overall reflection is RP=PrPi=1⁄2, on the steeper side of the asymmetric resonance. At that point, the sensitivity is equal to the sensitivity of a regular Fabry-Perot that has an effective reflectivity of:

Reff=√{square root over (RfRm)}=Rm2−L/2z0 

The sensitivity to displacement σ=∂RP/∂L of a regular Fabry-Perot at the point RP=1⁄2 is calculated as:





  σ
  =
  
    
      π
      λ
    
    ⁢
    
      K
    
  




where K=4Reff/(1−Reff)2.

The above relations can be used to calculate the maximum L. This calculated value is dependent on the minimum reflectivity Reff that is used to achieve the required sensitivity. The minimum required reflectivity for the best case scenario corresponds to the noise level being in the shot-noise limit, and the impedance of water being dominant, so that the membrane displaces with the water particles.
The relations between pressure and water displacement is expressed as:




  
    Δ
    ⁢
    
      
    
    ⁢
    L
  
  =
  
    
      (
      
        1
        
          ω
          ⁢
          
            
          
          ⁢
          υ
          ⁢
          
            
          
          ⁢
          ρ
        
      
      )
    
    ·
    P
  




Using the values υwater=1482 m/s, ρwater=998 kg/m3, and the numbers wanted for the sensor ω=2π×30 kHz, and P=30 μPa/√{square root over (Hz)}:

ΔL=1.08×10−7 nm/√{square root over (Hz)}

When the noise level is at the shot-noise limit, then the minimum detectable displacement is:





  
    Δ
    ⁢
    
      
    
    ⁢
    
      L
      min
    
  
  =
  
    
      
        2
      
      π
    
    ⁢
    λ
    ⁢
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          fhv
        
        
          K
          ⁢
          
            
          
          ⁢
          η
          ⁢
          
            
          
          ⁢
          
            P
            i
          
        
      
    
  




Substituting ΔLmin with the above number, and using the values, Pi=1 mW, λ=1550 nm, η=0.9, and solving for Reff:

Reff=0.954

This is the minimum effective reflectivity to achieve the desired sensitivity under the best conditions. This value can be used to calculate the maximum cavity length. Using the above expression Reff=Rm2−L/2z0, and requiring that Rm ̃1:




  
    L
    max
  
  =
  
    
      
        
          2
          ⁢
          
            z
            0
          
        
        
          ln
          ⁢
          
            
          
          ⁢
          2
        
      
      ⁢
      
        ln
        ⁡
        
          (
          
            
              R
              m
            
            
              R
              eff
            
          
          )
        
      
    
    =
    
      9.48
      ⁢
      
        
      
      ⁢
      
        μm
        .
      
    
  




This is the maximum cavity length in water, a limitation imposed by the optical properties of the fiber Fabry-Perot. This number is smaller (7.21 microns) in air, due to the difference in the Rayleigh range, calculated for a regular SMF-28 fiber.

The other constraint on the cavity length is that it is a multiple of half the operation wavelength, which in certain embodiments is λ=1550 nm.
With this knowledge of the limitations for the main parameters: the cavity length (L), the radius of the membrane (a), and the reflectivities of the fiber end (Rf) and the membrane mirror (Rm), the values can be optimized.
To optimize these parameters, the mechanical properties of the device are considered. In the mechanics calculations, the following impedance values were found for the parameters a=62.5 μm (=radius of an SMF-28 fiber) and L=1 μm:
Zwater=1.48×106 kg/m2/s, impedance of water
bwater=5.04×106 kg/m2/s, damping in water
ZSi=1.09×105 kg/m2/s, impedance of a silicon plate
The three impedances are on the same order, which means that the displacement of the membrane will be reduced by a factor f with respect to the displacement of water particles, where:




  
    f
    ≈
    
      
        Z
        water
      
      
        
          Z
          water
        
        +
        
          b
          water
        
        +
        
          Z
          Si
        
      
    
  
  =
  0.22




With these impedance values, the membrane will displace only by about 22% of the displacement of the water particles. This number is advantageously closer to about 90% for a sensitive device. To achieve this result, the damping in water, and also possibly, the impedance of the silicon plate are advantageously reduced to have:

bwater+ZSi≈1.64×105 kg/m2s

The expressions we need to consider are:



  
    
      
        b
        water
      
      ≈
      
        
          μ
          ⁢
          
            
          
          ⁢
          
            a
            2
          
        
        
          2
          ⁢
          
            L
            3
          
        
      
    
    ,
    
      

    
    ⁢
    and
  




  
    
      Z
      Si
    
    =
    
      
        192
        ⁢
        
          
        
        ⁢
        D
      
      
        ω
        ⁢
        
          
        
        ⁢
        
          a
          4
        
      
    
  



To reduce the overall impedance, L can be increased without changing a, since bwater, has a larger contribution. In such a case, ZSi will remain unchanged, so that advantageously:

bwater≈5.50×104 kg/m2/s

Hence, the length is advantageously:





  L
  =
  
    
      1
      ⁢
      
        
      
      ⁢
      μm
      ×
      
        
          (
          
            
              5.04
              ×
              
                10
                6
              
              ⁢
              
                
              
              ⁢
              kg
              ⁢
              
                /
              
              ⁢
              
                m
                2
              
              ⁢
              
                /
              
              ⁢
              s
            
            
              5.50
              ×
              
                10
                4
              
              ⁢
              
                
              
              ⁢
              kg
              ⁢
              
                /
              
              ⁢
              
                m
                2
              
              ⁢
              
                /
              
              ⁢
              s
            
          
          )
        
        
          1
          /
          3
        
      
    
    =
    
      4.51
      ⁢
      
        
      
      ⁢
      μm
    
  




Since the cavity length is a multiple of half the operation wavelength, the closest number to this is 6×1.55 μm/2=4.65 μm, which is still within the range of Lmax=9.48 μm. Using the cavity length L=4.65 μm, the reduction factor is f=0.9=90%. Since a=62.5 μm remained unchanged in this calculation, the other two design parameters Rf and Rm remain to be optimized.
The displacement of the membrane will be:

ΔL≈f×ΔLwater=0.9×1.08×10−7 nm/√{square root over (Hz)}=9.72×10−8 nm/√{square root over (Hz)}

which results in Reff=0.958 and Rm=Reff2L/z0=0.980, and Rf=Rm2−L/z0=0.936.

Therefore, a set of parameters for an example design that provides a sensitivity of 30 μPa/√{square root over (Hz)} at 30 kHz are: a=62.5 microns, L=4.65 microns, Rm=0.980, and Rf=0.936. Other values are also compatible with certain embodiments described herein.
Fabry-Perot-Based Acoustic Sensor Compared to a Fiber Bragg Grating
A simple Fabry-Perot structure to be employed as an acoustic sensor can be constructed with a photonic crystal mirror and a fiber Bragg grating, (e.g., as shown in FIG. 28), or with just two photonic crystal mirrors, as described herein. The sensitivity of such a sensor can be calculated from the basic Fabry-Perot equations. (See, e.g., Thomson et al., “A Fabry-Perot acoustic surface vibration detector-application to acoustic holography,” J. Phys. D: Appl. Phys., Vol. 6, page 677 (1973).) In certain embodiments, both of the mirrors forming the Fabry-Perot cavity have a high reflectivity R. Then, for K=4R/(1−R)2 and φ=2πL/λ, with L being the mirror spacing, the relation between the reflected power Pr and the incident power Pi can be calculated as:




  
    
      
        
          
            P
            r
          
          
            P
            i
          
        
        =
        
          1
          -
          
            
              (
              
                1
                +
                
                  K
                  ⁢
                  
                    
                  
                  ⁢
                  
                    sin
                    2
                  
                  ⁢
                  φ
                
              
              )
            
            
              -
              1
            
          
        
      
    
    
      
        (
        31
        )
      
    
  




The sensitivity σ to the displacement L will then be:





  
    
      
        σ
        =
        
          
            d
            
              d
              L
            
          
          ⁢
          
            (
            
              
                P
                r
              
              
                P
                i
              
            
            )
          
        
      
    
    
      
        (
        32
        )
      
    
  




To find the maximum sensitivity, equation (32) is solved for dσ/dL=0, which is satisfied for φ=(3K)−1/2+mπ, keeping in mind that K>>1. This result is expected, stating that the highest sensitivity is at the side of a resonance. Using this value, the maximum sensitivity can be expressed as:




  
    
      
        
          σ
          max
        
        =
        
          
            
              3
              ⁢
              
                3
              
              ⁢
              π
            
            
              4
              ⁢
              
                
              
              ⁢
              λ
            
          
          ⁢
          
            K
          
        
      
    
    
      
        (
        33
        )
      
    
  




Therefore, the maximum sensitivity only depends on the reflectivity of the mirrors, provided that the linewidth of the laser is much smaller than the linewidth of the Fabry-Perot resonance. This condition is satisfied if L<<c/Δνlaserπ√{square root over (K)}, where Δνlaser is the linewidth of a single-mode laser (or the frequency spread of a multi-mode laser). Thus, for a sensitive measurement, the linewidth of the laser Δνlaser, is advantageously selected to be much smaller than the linewidth of the Fabry-Perot resonance ΔνF-P=c/Lπ√{square root over (K)}, which is dependent on the cavity length L. Thus, equation (33) for the maximum sensitivity imposes a limit on the maximum cavity length on the Fabry-Perot cavity depending on the laser linewidth.

For a sufficiently large laser power such as 1 milliwatt, the dominant noise will be the photodiode shot current. The mean current in the photodiode measuring the reflected power will be I0=Preη/hν, where η is the quantum efficiency of the photodiode.
At the maximum sensitivity point, calculated from equation (31), Pr=Pi/4. This current gives rise to a shot noise current:




  
    
      
        
          I
          N
        
        =
        
          
            
              2
              ⁢
              
                eI
                0
              
              ⁢
              Δ
              ⁢
              
                
              
              ⁢
              f
            
          
          =
          
            
              
                
                  
                    P
                    i
                  
                  ⁢
                  
                    e
                    2
                  
                  ⁢
                  η
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  f
                
                
                  2
                  ⁢
                  hv
                
              
              ,
            
          
        
      
    
    
      
        (
        34
        )
      
    
  




where Δf is the electronic system bandwidth.

For a small mirror displacement with peak amplitude ΔL, the signal current in the photodiode will be:




  
    
      
        
          I
          S
        
        =
        
          
            1
            
              2
            
          
          ⁢
          
            
              Δ
              ⁢
              
                
              
              ⁢
              
                P
                r
              
              ⁢
              e
              ⁢
              
                
              
              ⁢
              η
            
            hv
          
        
      
    
    
      
        (
        35
        )
      
    
  




and ΔPr, can be calculated using equation (2) to be:





  
    
      
        
          σ
          =
          
            
              1
              
                P
                i
              
            
            ⁢
            
              (
              
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    r
                  
                
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  L
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        36
        )
      
    
  




hence ΔPr=σPiΔL

Operating at maximum sensitivity given in equation (33), the power signal of equation (36) inside the signal current expression of equation (35):




  
    I
    S
  
  =
  
    
      
        3
        ⁢
        
          3
        
        ⁢
        π
      
      
        4
        ⁢
        
          2
        
      
    
    ⁢
    
      (
      
        
          Δ
          ⁢
          
            
          
          ⁢
          L
        
        λ
      
      )
    
    ⁢
    
      
        
          K
        
        ⁢
        e
        ⁢
        
          
        
        ⁢
        η
      
      hv
    
    ⁢
    
      P
      i
    
  




From which the signal-to-noise ratio can be expressed as:





  
    
      
        
          S
          N
        
        =
        
          
            
              I
              S
              2
            
            
              I
              N
              2
            
          
          =
          
            
              
                27
                ⁢
                
                  
                
                ⁢
                
                  π
                  2
                
              
              16
            
            ⁢
            
              
                K
                ⁢
                
                  
                
                ⁢
                η
                ⁢
                
                  
                
                ⁢
                
                  P
                  i
                
              
              
                Δ
                ⁢
                
                  
                
                ⁢
                fhv
              
            
            ⁢
            
              
                (
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    L
                  
                  λ
                
                )
              
              2
            
          
        
      
    
    
      
        (
        38
        )
      
    
  




For a unity signal-to-noise ratio, the detection sensitivity of the system will then be:





  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          
            L
            min
          
        
        =
        
          
            4
            
              3
              ⁢
              
                3
              
              ⁢
              π
            
          
          ⁢
          λ
          ⁢
          
            
              
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  fhv
                
                
                  K
                  ⁢
                  
                    
                  
                  ⁢
                  η
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    i
                  
                
              
            
            .
          
        
      
    
    
      
        (
        39
        )
      
    
  




Using the values, Pi=1mW, λ=1500 nm, η=0.9, and a modest reflectivity R=0.99, the value of ΔLmin=2.25·10−8 nm/√{square root over (Hz)} is obtained. The sensitivity can be increased through the reflectivity of the mirrors. For example, a reflectivity of R=0.999 provides about 10 times better sensitivity. Throughout the calculations below, the value of ΔLmin=10−6 nm/√{square root over (Hz)} is used, since the experimental values have previously been reported to be about an order of magnitude worse than the theoretical limit.

The sensitivity given in equation (33) is only dependent on the mirror reflectivity. It may be expected that the length of the cavity would play a crucial role in the sensitivity, so that a much smaller sensitivity would be provided by a long cavity. If equation (33) is written in terms of the cavity length L, and the resonance linewidth Δν1/2:




  
    
      
        
          σ
          max
        
        =
        
          
            
              3
              ⁢
              
                3
              
            
            
              4
              ⁢
              
                
              
              ⁢
              λ
            
          
          ⁢
          
            (
            
              c
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  v
                  
                    1
                    /
                    2
                  
                
                ⁢
                L
              
            
            )
          
        
      
    
    
      
        (
        40
        )
      
    
  




which is an expected result. The sensitivity drops with increasing length. Also, as expected, the sensitivity drops with increasing linewidth, since the resonances become less steep. However, in a Fabry-Perot cavity with high reflectivity mirrors, the resonance linewidth is dependent on L, so that the resonances become sharper for longer cavity lengths:





  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          
            v
            
              1
              /
              2
            
          
        
        =
        
          
            c
            
              π
              ⁢
              
                K
              
            
          
          ·
          
            1
            L
          
        
      
    
    
      
        (
        41
        )
      
    
  




(See, e.g., P. Yeh, “Optical Waves in Layered Media,” (Wiley, New York, 1988).) Therefore, the dependence on L in the sensitivity equation (40) cancels out, so that it is the mirror reflectivity provides the dominant contribution (as long as it is high). In certain such embodiments, the important criterion is therefore that the laser linewidth should be much smaller than the Fabry-Perot resonance linewidth.

To calculate the dynamic range, the minimum detectable length is known, and therefore the maximum length shift is to be calculated. For a Fabry-Perot cavity, L=constant·λ, hence:




  
    
      
        
          
            Δ
            ⁢
            
              
            
            ⁢
            L
          
          L
        
        =
        
          
            Δ
            ⁢
            
              
            
            ⁢
            λ
          
          λ
        
      
    
    
      
        (
        42
        )
      
    
  




Therefore, the maximum shift is ΔLmax=(L/λ) Δλmax. The maximum wavelength shift one can detect is one-quarter the linewidth of the Fabry-Perot resonance. Therefore, the maximum detectable cavity length change is, using equation (41):





  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          
            L
            max
          
        
        =
        
          
            
              L
              λ
            
            ⁢
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                
                  λ
                  
                    1
                    /
                    2
                  
                
              
              4
            
          
          =
          
            λ
            
              4
              ⁢
              π
              ⁢
              
                K
              
            
          
        
      
    
    
      
        (
        43
        )
      
    
  




Hence, the dynamic range is:





  
    
      
        
          D
          ⁢
          
            
          
          ⁢
          R
        
        =
        
          
            
              Δ
              ⁢
              
                
              
              ⁢
              
                L
                max
              
            
            
              Δ
              ⁢
              
                
              
              ⁢
              
                L
                min
              
            
          
          =
          
            
              
                3
                ⁢
                
                  3
                
              
              16
            
            ⁢
            
              
                
                  η
                  ⁢
                  
                    
                  
                  ⁢
                  
                    P
                    i
                  
                
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  fhv
                
              
            
          
        
      
    
    
      
        (
        44
        )
      
    
  




which shows that the dynamic range is independent of the Fabry-Perot parameters such as the reflectivity or the cavity length. For the values used above, a dynamic range about 130 dB (20 log) results. Again, assuming an order of magnitude less sensitivity (10−6 nm/√{square root over (Hz)}) than that predicted, the dynamic range is then around 110 dB. Although this dynamic range is for the displacement measurements, it also applies for pressure, since the displacement is proportional to the pressure.

To compare these results to a single fiber Bragg grating, it is desirable to know if it is possible to get the same values by stretching a piece of such a fiber. FIG. 32 is a graph of an optical resonance as a function of wavelength. As a first step, the sensitivity for a general sharp resonance is calculated, which is shown in FIG. 32. From simple geometrics, we obtain the relation:




  
    
      
        
          
            
              Δ
              ⁡
              
                (
                
                  
                    P
                    r
                  
                  /
                  
                    P
                    i
                  
                
                )
              
            
            
              Δ
              ⁢
              
                
              
              ⁢
              λ
            
          
          ≈
          
            
              1
              /
              2
            
            
              Δ
              ⁢
              
                
              
              ⁢
              
                
                  λ
                  
                    1
                    /
                    2
                  
                
                /
                2
              
            
          
        
        =
        
          1
          
            Δ
            ⁢
            
              
            
            ⁢
            
              λ
              
                1
                /
                2
              
            
          
        
      
    
    
      
        (
        45
        )
      
    
  




If the resonance wavelength and distance is proportional to each other, as in a Fabry-Perot cavity, so that equation (42) is valid, the sensitivity can be expressed as:





  
    
      
        σ
        =
        
          
            
              Δ
              ⁡
              
                (
                
                  
                    P
                    r
                  
                  /
                  
                    P
                    i
                  
                
                )
              
            
            
              Δ
              ⁢
              
                
              
              ⁢
              L
            
          
          =
          
            
              λ
              /
              L
            
            
              Δ
              ⁢
              
                
              
              ⁢
              
                λ
                
                  1
                  /
                  2
                
              
            
          
        
      
    
    
      
        (
        46
        )
      
    
  




To verify this expression, the expressions for a Fabry-Perot cavity can be used, to get:





  
    
      
        
          σ
          =
          
            
              π
              λ
            
            ⁢
            
              K
            
          
        
        ,
      
    
    
      
        (
        47
        )
      
    
  




which is very close to equation (33), thereby verifying equations (45) and (46).

Having general expressions for the sensitivity, the sensitivity for a fiber Bragg grating can be calculated. The resonance wavelength of such a structure is:




  
    
      
        λ
        =
        
          2
          ⁢
          
            n
            eff
          
          ⁢
          
            L
            N
          
        
      
    
    
      
        (
        48
        )
      
    
  




where neff is the effective refractive index, L the total length of the grating, and N the number of layers. (See, e.g., Kersey et al., “Fiber grating sensors,” J. Lightwave Technol., vol. 15, no. 8, page 1442 (1997).) When such a structure is stretched by ΔL, the wavelength shifts by:





  
    
      
        
          
            Δ
            ⁢
            
              
            
            ⁢
            λ
          
          =
          
            2
            ⁢
            
              
                n
                eff
              
              ⁡
              
                (
                0.78
                )
              
            
            ⁢
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                L
              
              N
            
          
        
        ,
      
    
    
      
        (
        49
        )
      
    
  




where the factor 0.78 comes from changes in the fiber index due to photo-elastic effects. Therefore:





  
    
      
        
          
            Δ
            ⁢
            
              
            
            ⁢
            L
          
          L
        
        =
        
          
            1
            0.78
          
          ⁢
          
            
              Δ
              ⁢
              
                
              
              ⁢
              λ
            
            λ
          
        
      
    
    
      
        (
        50
        )
      
    
  




which shows that equation (42) is valid to that order, meaning equation (46) is also valid to that order. Thus, the sensitivity of a Fabry-Perot cavity and a fiber Bragg grating are on the same order for a given wavelength, provided that L·Δλ1/2 of equation (46) is the same.

For example, a commercially available fiber Bragg gratings operating at around 1500 nanometers, a linewidth of 0.02 picometer for a grating around 5 centimeters long, the structure gives L·Δλ1/2=103 nm2. For a Fabry-Perot cavity on the other hand, using equation (11):




  
    
      
        
          
            L
            ·
            Δ
          
          ⁢
          
            
          
          ⁢
          
            λ
            
              1
              /
              2
            
          
        
        =
        
          
            
              λ
              2
            
            
              π
              ⁢
              
                K
              
            
          
          =
          
            
              
                
                  λ
                  2
                
                π
              
              ·
              
                
                  1
                  -
                  R
                
                
                  R
                
              
            
            ≈
            
              
                
                  λ
                  2
                
                π
              
              ·
              
                (
                
                  1
                  -
                  R
                
                )
              
            
          
        
      
    
    
      
        (
        51
        )
      
    
  




To get a similar number (e.g., L·Δλ1/2=103 nm2) for a Fabry-Perot cavity, a reflectivity of R≈0.998 would be used. Hence, such a commercially available fiber Bragg grating seems to have the same sensitivity of a Fabry-Perot cavity with high reflectivity mirrors.

In this analysis of the Fabry-Perot cavity, it was assumed that the linewidth of the laser is much smaller than the linewidth of the Fabry-Perot resonance. The linewidth of the laser should be one to two orders of magnitude smaller than the resonance linewidth, so that the laser does not average over less sensitive regions. When a small cavity length is used, the Fabry-Perot resonance linewidth is large, hence the laser does not have to be very narrow. When the cavity length is large however, the Fabry-Perot resonance becomes sharper, hence a narrower laser is used to achieve the same sensitivity achieved in a short Fabry-Perot cavity. The main problem arises at this point when the laser has to be extremely narrow.
Consider the above case for the 0.02 picometer linewidth, for example. To achieve the calculated sensitivity, a laser as narrow as 10−3 to 10−4 picometer would be used. When a laser is that narrow, other noise sources become dominant over the shot-noise. One of the most important noises for such narrow lasers is the free running frequency noise. In fact, by reducing this noise using a pre-stabilized laser, it was previously experimentally shown that a sensitivity of 10−5/√{square root over (Hz)} can be obtained for a greater than 25 millimeter long Fabry-Perot formed by two fiber Bragg gratings. (See, Chow et al., “Ultra resolution fiber sensor using a pre-stabilized diode laser,” page CPDA9, Post-deadline CLEO 2005 (2005).) This reported value is just about an order of magnitude worse than the fundamental shot-noise limited sensitivity for the same structure. Therefore, it is harder to get high sensitivities with long cavity lengths, since in that case a very good laser is used. However, these results should be encouraging for the fiber Bragg grating structure shown in FIG. 28, as well as for a Fabry-Perot sensor using two thin photonic crystal slabs as the mirrors.
FIGS. 34A and 34B schematically illustrate the resonant wavelength changes for two different cavity lengths, respectively. FIG. 34A corresponds to a cavity length Lshort of about 0.75 microns, and FIG. 34B corresponds to a cavity length Llong=20Lshort (e.g., about 15 microns). For the same displacement between the two reflectors of the Fabry Perot resonator, the change of power for the cavity of FIG. 34A is equal to that of the cavity of FIG. 34B. The temperature sensitivity of a Fabry-Perot cavity having a length L is given by ΔL=(a·ΔT)·L, where ΔL is the change of length, a is the thermal expansion coefficient, and ΔT is the change of temperature. Thus, while the sensitivity to displacements is independent of cavity length, the longer cavities are more sensitive to temperature. Therefore, in certain embodiments, a shorter cavity is advantageously used for higher temperature stability. In addition, a shorter cavity is advantageously used since the resonance is wider, thereby avoiding using a narrowband laser in certain embodiments. In certain various embodiments, the cavity length is less than about 15 microns, less than about 10 microns, less than about 5 microns, less than about 1 micron, or less than about 0.5 micron.
Fiber Fabry-Perot Interferometer Compared to Standard Fabry-Perot Interferometer
FIG. 35A schematically illustrates a standard Fabry-Perot (FP) interferometer comprising infinitely large mirrors with field reflection coefficients rf and rm. If a plane wave with a field E, is incident on the FP interferometer, the total field reflected by the FP interferometer is Er=rfEi+tfEc−, where Ec− is the field traveling backward in the cavity and tf is the mirror's field transmission coefficient. In most cases, rf can be taken to be real and tf=j√{square root over (1−rf2)}. Summing all the fields reflected between the two mirrors yields the classic field reflection coefficient of a standard FP interferometer:




  
    r
    SFP
  
  =
  
    
      
        E
        r
      
      
        E
        i
      
    
    =
    
      
        
          r
          f
        
        -
        
          
            
              
                (
                
                  1
                  -
                  
                    r
                    f
                  
                
                )
              
              2
            
            
              r
              f
            
          
          ⁢
          
            
              ∑
              
                p
                =
                1
              
              ∞
            
            ⁢
            
              
                
                  (
                  
                    
                      r
                      f
                    
                    ⁢
                    
                      r
                      m
                    
                  
                  )
                
                p
              
              ⁢
              
                e
                
                  
                    -
                    j
                  
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  kLp
                
              
            
          
        
      
      =
      
        
          
            
              
                r
                f
              
              ⁢
              
                e
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  kL
                
              
            
            -
            
              r
              m
            
          
          
            
              e
              
                j
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                kL
              
            
            -
            
              
                r
                f
              
              ⁢
              
                r
                m
              
            
          
        
        .
      
    
  




The reflectivity of a fiber FP interferometer is quite different due to the mode-selective properties of the fiber. As an example, FIG. 35B schematically illustrates a fiber FP interferometer comprising a metallic coating deposited on the end of the fiber (reflectivity rf) facing a thin PCS (reflectivity rm). The fiber FP interferometer is probed with single-frequency laser diode light coupled into the fiber FP interferometer through the fiber core. For simplicity, the field entering the FP from the core can be taken to have a Gaussian profile E(z), which is a good approximation for a single-mode fiber. The backward-traveling field in the cavity is then the sum of the p fields reflected by the PCS mirror:




  
    E
    c
    -
  
  =
  
    
      
        t
        f
      
      
        r
        f
      
    
    ⁢
    
      
        ∑
        
          p
          =
          1
        
        ∞
      
      ⁢
      
        
          
            (
            
              
                r
                f
              
              ⁢
              
                r
                m
              
            
            )
          
          p
        
        ⁢
        
          
            E
            ⁡
            
              (
              
                p
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                L
              
              )
            
          
          .
        
      
    
  




The detected field is the portion of the total field reflected by the fiber FP interferometer that is coupled to the fiber fundamental mode, or Er=rfEi+ai·tfEi, where ai is the coupling coefficient of the free-space field to the fiber mode. So the only difference with a standard FP interferometer is the factor ai, which eliminates the fraction of the total reflected field that does not have the proper amplitude profile and phase front to be coupled back into the fiber (this fraction is coupled to cladding modes, and/or reflected). The coupling coefficient can be calculated as:





  
    
      a
      i
    
    =
    
      
        
          〈
          
            
              E
              i
            
            |
            
              E
              c
              -
            
          
          〉
        
        
          〈
          
            
              E
              i
            
            |
            
              E
              i
            
          
          〉
        
      
      =
      
        
          
            
              t
              f
            
            
              r
              f
            
          
          ⁢
          
            
              ∑
              
                p
                =
                1
              
              ∞
            
            ⁢
            
              
                
                  (
                  
                    
                      r
                      f
                    
                    ⁢
                    
                      r
                      m
                    
                  
                  )
                
                p
              
              ⁢
              
                
                  〈
                  
                    
                      E
                      i
                    
                    |
                    
                      E
                      ⁡
                      
                        (
                        
                          p
                          ⁢
                          
                            
                          
                          ⁢
                          2
                          ⁢
                          L
                        
                        )
                      
                    
                  
                  〉
                
                
                  〈
                  
                    
                      E
                      i
                    
                    |
                    
                      E
                      i
                    
                  
                  〉
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        =
        
          
            
              t
              f
            
            
              r
              f
            
          
          ⁢
          
            
              ∑
              
                p
                =
                1
              
              ∞
            
            ⁢
            
              
                
                  
                    (
                    
                      
                        r
                        f
                      
                      ⁢
                      
                        r
                        m
                      
                    
                    )
                  
                  p
                
                
                  1
                  +
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      pL
                      /
                      
                        z
                        0
                      
                    
                  
                
              
              ⁢
              
                e
                
                  
                    -
                    j
                  
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  kLp
                
              
            
          
        
      
    
  
  ,




where z0 is the Rayleigh range of the Gaussian beam inside the fiber FP interferometer. Using this result, the field reflection coefficient for the fiber Fabry-Perot interferometer can be expressed as:





  
    r
    FFP
  
  =
  
    
      
        E
        r
      
      
        E
        i
      
    
    =
    
      
        
          r
          f
        
        +
        
          
            a
            i
          
          ·
          
            t
            f
          
        
      
      =
      
        
          r
          f
        
        -
        
          
            
              
                (
                
                  1
                  -
                  
                    r
                    f
                  
                
                )
              
              2
            
            
              r
              f
            
          
          ⁢
          
            
              ∑
              
                p
                =
                1
              
              ∞
            
            ⁢
            
              
                
                  
                    (
                    
                      
                        r
                        f
                      
                      ⁢
                      
                        r
                        m
                      
                    
                    )
                  
                  p
                
                
                  1
                  +
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      pL
                      /
                      
                        z
                        0
                      
                    
                  
                
              
              ⁢
              
                
                  e
                  
                    
                      -
                      j
                    
                    ⁢
                    
                      
                    
                    ⁢
                    2
                    ⁢
                    kLp
                  
                
                .
              
            
          
        
      
    
  




Comparing this result to the formula for a standard FP interferometer shows that the spatial filtering performed by the single-mode fiber adds a factor of) 1/(1+jpL/z0) in each term of the series. In contrast to a standard FP interferometer, the summation expression for the fiber FP interferometer does not have a simple analytical closed form. However, a Lerch transcendent can be used to convert it into the integral form:




  
    r
    FFP
  
  =
  
    
      ∫
      0
      1
    
    ⁢
    
      
        
          
            
              r
              f
            
            ⁢
            
              e
              
                jθ
                ⁡
                
                  (
                  u
                  )
                
              
            
          
          -
          
            r
            m
          
        
        
          
            e
            
              jθ
              ⁡
              
                (
                u
                )
              
            
          
          -
          
            
              r
              f
            
            ⁢
            
              r
              m
            
          
        
      
      ⁢
      
        
      
      ⁢
      
        d
        u
      
    
  




where





  
    θ
    ⁡
    
      (
      u
      )
    
  
  =
  
    
      2
      ⁢
      kL
    
    -
    
      
        L
        
          z
          0
        
      
      ⁢
      ln
      ⁢
      
        
      
      ⁢
      
        u
        .
      
    
  




Note that the reflectivity of a standard FP interferometer can also be expressed in a similar form:





  
    r
    SFP
  
  =
  
    
      
        
          r
          f
        
        ⁢
        
          e
          jθ
        
      
      -
      
        r
        m
      
    
    
      
        e
        jθ
      
      -
      
        
          r
          f
        
        ⁢
        
          r
          m
        
      
    
  




where θ=2kL. The integral form for rSFP above is more efficient in numerical calculations than the series form which requires computing a very large number of terms when the reflectivities are high.

The reflection spectrum of both a fiber FP interferometer and a standard FP interferometer are plotted in FIG. 36 for typical reflectivities (Rm=|rm|2=0.99, Rf=|rf|2=0.99) and a mirror spacing L=15.5 microns. In the fiber FP interferometer, the fundamental mode 1/e2 power radius was taken to be 5.1 microns. Diffraction of the laser beam inside the fiber FP interferometer, combined to the spatial filtering of the fiber, introduces a strong asymmetry in the resonances, as well as a loss of contrast, as shown in FIG. 36. Specifically, the resonance lineshape is unchanged on the short-wavelength side (other than the reduced contrast), but it is considerably broadened on the long-wavelength side. This asymmetry can clearly find very useful applications in fiber sensors. In particular, it offers the unique potential of probing the same fiber FP interferometer at two wavelengths, one centered on a steeper short-wavelength slope, the other on a less steep long-wavelength slope, and thus enhance the sensor's dynamic range.
The external-mirror fiber FP reflection amplitude can be expressed as rEFFP=∫0∞rFP(θn)e−udu, where rFP=rFP(θ) is the standard FP reflection amplitude with θ=2kL and θn=θ+(L/z0)u. The fiber FP reflection amplitude is then a weighted average of the standard FP reflection amplitude and reflection amplitudes with larger effective lengths. Thus the resonance lineshape is asymmetric and elongated towards increasing wavelength. The contrast is reduced, but can be maximized when Rf≈Rm2−L/z0, where Rf is the reflectivity of the fiber mirror and Rm is the reflectivity of the external mirror. Therefore, for high reflectivities of the external mirror, the contrast can be very close to unity, and the fiber FP interferometer is still asymmetric, though not as strongly, to provide high contrast and high sensitivity. Note that since z0 is generally larger than L (e.g., 53 microns versus 15.5 microns), this condition is generally achieved when Rm is close to unity. However, high contrast and high sensitivity are generally not achievable using large cavity lengths. In addition, since mirror thickness contributes to the cavity length, a thin external mirror is advantageously used in certain embodiments. The use of a PCS in certain embodiments advantageously provides not only the high reflectivity used in an FP interferometer for high sensitivity, but also the high reflectivity required to maximize the contrast in a fiber FP interferometer.
The transmission spectrum from an example fiber FP interferometer was measured to confirm the predicted asymmetry. One of the FP mirrors was formed by the cleaved end of an SMF-28 fiber, which was coated with 12 nanometers of gold on a 4-nanometer chrome adhesion layer. The calculated reflectivity of the mirror formed on the fiber end was 80%, and the measured reflectivity of this mirror was 65%. The other mirror was a PCS fabricated by standard techniques on a 450-nanometer-thick silicon membrane with a square lattice of 644-nanometer diameter circular holes spaced by 805 nanometers. The calculated reflectivity of the PCS mirror at 1.55 microns was greater than 99%, although due to less-than-ideal fabrication conditions, its actual reflectivity was only 65%, a value actually better matched to the fiber mirror reflectivity. The measured transmission spectrum of this fiber FP interferometer is shown in FIG. 37. As expected, the resonances are strongly asymmetric, and less steep on the long-wavelength side.
FIG. 38A schematically illustrates an experimental set-up for characterizing a fiber FP acoustic sensor, and FIG. 38B schematically illustrates an example PCS-based FP acoustic sensor. To interrogate the FP acoustic sensor, a single-frequency tunable laser diode (e.g., tunable in a range from 1520 nanometers to 1570 nanometers) is tuned to the steepest slope of one of the FP acoustic sensor's resonances. An acoustic tone of frequency f in the 10-50 kHz range, generated by an audio speaker, is launched onto the PCS. The modulated power reflected by the FP acoustic sensor is tapped with a 3-dB coupler, detected with a fast detector, and analyzed with a spectrum analyzer, which provides the magnitude of the detected power at f. The acoustic pressure incident on the fiber FP acoustic sensor is also measured with a calibrated microphone (e.g., Bruel & Kjaer Model No. 4191).
FIG. 39A shows the dependence of the detected optical power of the fiber FP acoustic sensor in air on acoustic pressure at 30 kHz in a spectrum analyzer bandwidth of 1 kHz. FIG. 39B shows the detected optical power over a wider range of pressures. The two noise levels of FIG. 39A correspond to the 1-kHz bandwidth of the spectrum analyzer, and the noise level for a 1-Hz bandwidth detection system (e.g., a lock-in amplifier), an example of which is schematically illustrated by FIG. 39C. As expected, the fiber FP acoustic sensor response is linear up to the maximum tested pressure of 250 mPa. The departure from linearity at low pressures is believed to originate from the proximity of the instrument noise floor (−85 dBm at a 1-kHz bandwidth, top noise-floor curve in FIG. 39A). The fiber FP acoustic sensor was also found to exhibit a relatively flat response (better than 10 dB) over the 10 kHz-50 kHz range. In a 1-kHz bandwidth, the minimum detectable pressure of the sensor operated in air at a high frequency of 30 kHz is therefore 560 μPa. The measured response is in good agreement (within a few dB) with the response predicted theoretically from the mechanical response of the PBS and the FP parameters, as shown in FIGS. 39A and 39B. The discrepancy between the theoretical and experimental results may be due to the mechanical calculations of the sensor response, which assumes that the presence of the PCS holes does not affect the silicon membrane response.
By reducing the bandwidth to 1 Hz (−115 dBm, the lower noise-floor curve in FIG. 39A), which is a standard value for sensors, the minimum detectable pressure would be reduced to about 18 μPa in a 1-Hz bandwidth, corresponding to a sensitivity of about 18 μPa/Hz1/2 (corresponding to a minimum detectable displacement of 1×10−14 meters). This value was confirmed experimentally by testing the fiber FP acoustic sensor at very weak acoustic signal levels and using a lock-in amplifier detection. The noise in all these measurements was dominated by the thermal noise of the detector, which is higher than the shot noise, even at the 2.5-mW detected power in the measurements. With a shot-noise-limited detector, the minimum detectable pressure should be reduced to 1 μPa/Hz1/2. In certain embodiments, the sensitivity is better than about 100 μPa/Hz1/2 at frequencies below 20 kHz. In certain embodiments, the sensitivity is better than about 10 μPa/Hz1/2 at frequencies greater than about 50 kHz with a relatively flat frequency response. In certain other embodiments, the sensitivity can be improved to be less than about 10 nPa/Hz1/2.
FIG. 39D shows the frequency response of the example fiber FP acoustic sensor of FIGS. 39A and 39B. The frequency response is generally flat to about 10 dB over a range of frequencies between about 10 kHz and about 40 kHz, and is expected to extend to about 100 kHz, or even greater. Most of the ripples shown in FIG. 39C are due to standing sound waves between the sensor and the source.
For this fiber FP acoustic sensor, the reflectivity of both the fiber and the PCS was around 65%. In certain other embodiments, this number can be easily increased to above 90% to improve the sensitivity by more than 10 dB. Also, the diameter or width of the PCS silicon membrane was around 150 microns, and in certain other embodiments, this diameter or width can be larger. In certain various embodiments, the PCS has a diameter greater than about 50 microns, greater than about 100 microns, greater than about 125 microns, greater than about 150 microns, or greater than about 200 microns. For a given applied pressure, the membrane displacement increases as the fourth power of this diameter or width. Therefore, even a small increase in membrane area will greatly improve the sensitivity (e.g., 25 dB for two-fold increase in radius). With at least one of these two improvements, both of which can be implemented with straightforward fabrication steps, the sensitivity of certain other embodiments is better than about 1 μPa/Hz1/2, and in certain other embodiments, is better than the 10 nPa/Hz1/2 level in a shot-noise-limited detection.
Analysis of Thermal Mechanical Noise
To be able to calculate the thermal mechanical noise associated with a fiber FP sensor, it is helpful to shortly review the sources of thermal noise. Thermal noise is associated with the random motion of the diaphragm, which has an equation of motion for the transverse displacement u without any incident pressure of:




  
    
      (
      
        
          h
          ⁢
          
            
          
          ⁢
          ρ
          ⁢
          
            
              ∂
              2
            
            
              ∂
              
                t
                2
              
            
          
        
        +
        
          B
          ⁢
          
            ∂
            
              ∂
              t
            
          
        
        +
        
          D
          ⁢
          
            
          
          ⁢
          
            
              ∇
              4
            
            ⁢
            
              -
              h
            
          
          ⁢
          
            
          
          ⁢
          σ
          ⁢
          
            ∇
            2
          
        
      
      )
    
    ⁢
    u
  
  =
  0.




The first term is the mass term, the second term is the damping term, the third term is the bending term, and the fourth term is the stress term. The damping term implies that any oscillation in the diaphragm will continue to diminish with no limit. Even the small thermal vibrations of the diaphragm would decay, leaving the diaphragm at a temperature below that of the surrounding, so the equation is flawed thermodynamically. To allow the diaphragm to be in thermal equilibrium, a pressure term is included that moves the diaphragm, which makes up for the loss due to the damping. Hence, the thermodynamically proper equation for no incident pressure is:





  
    
      (
      
        
          h
          ⁢
          
            
          
          ⁢
          ρ
          ⁢
          
            
              ∂
              2
            
            
              ∂
              
                t
                2
              
            
          
        
        +
        
          B
          ⁢
          
            ∂
            
              ∂
              t
            
          
        
        +
        
          D
          ⁢
          
            
          
          ⁢
          
            
              ∇
              4
            
            ⁢
            
              -
              h
            
          
          ⁢
          
            
          
          ⁢
          σ
          ⁢
          
            ∇
            2
          
        
      
      )
    
    ⁢
    u
  
  =
  
    P
    N
  




The pressure term PN has its origin in spontaneous thermal fluctuations which are a function of the damping B, which is included to make up for the damping loss. Thus, while the damping allows the system to transfer energy to the surrounding, it also allows the system to gather energy from the surrounding. This behavior is the basis of the fluctuation-dissipation theorem (see, e.g., H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. Vol. 83, 34-40 (1951)), which states that for every mechanism of dissipation, there is always a fluctuating force in the system. This equation can be complicated to solve, but simplifications can be made in certain embodiments in which the wavelength of the acoustic pressure wave is much larger Than the radius of the diaphragm and the frequency of the acoustic pressure wave is much less than the resonant frequency of the sensor. Under such conditions, it is possible to calculate displacements, resonances, and impedances analytically.

A generalized form of the Nyquist relation (see, e.g., C. Kittel, Elementary Statistical Physics, New York, Wiley (1958)) can be used to calculate the spectral density of the fluctuating pressure, which will be the thermal noise level of the acoustic sensor. Analogously to the regular Johnson noise in a resistor VN=√{square root over (4kBTR)}[V/√{square root over (Hz)}], the noise in a pressure sensor can be expressed as:

PN=PN2=√{square root over (4kBTRtotal)}[Pa/√{square root over (Hz)}],

where Rtotal=B/S is the acoustic resistance related to the damping B in the system, with S=πa2 the area of the diaphragm (see, e.g., T. B. Gabrielson, “Mechanical thermal noise in micromachined acoustic and vibration sensors,” IEEE Trans. Electron Devices Vol. 40, 903-909 (1993)). The individual contributions from viscous resistance (Rvsc), and radiation resistance (Rrad) to the total acoustic resistance Rtotal=Rvsc+Rrad are discussed below, along with a more complete model of the sensor with a more accurate expression for the effective dissipation affecting the diaphragm.

The origin of viscous resistance Rvsc is Brownian motion, which is due to thermal agitation. When the diaphragm is moved through a fluid, it will experience a drag. This drag is due to the random collisions of the fluid molecules. Therefore, there is a damping term associated with this dissipation, referred to as the viscous resistance. Hence, when the diaphragm is at rest, it will still experience this random force that will contribute to the thermal noise.
The source of radiation resistance Rrad is the random pressure fluctuation in the fluid due to thermal agitation. An oscillating diaphragm will gradually lose energy through the generation of pressure waves. Therefore, there is a damping term associated with this dissipation, referred to as the radiation resistance. Hence, the diaphragm is capable of picking up the fluctuating pressure of the ambient environment, which will contribute to the thermal noise.
A.1 Uniform Diaphragm Approximation
FIG. 40 schematically illustrates an example configuration of a fiber Fabry-Perot acoustic sensor for analysis of thermal noise. The configuration of FIG. 40 has a circular diaphragm with radius a, thickness h, where rh is the radius of the holes through it, and Λ is the pitch of the hole array. The diaphragm gap is d, and the dynamic viscosity of the medium is μ. Example values for these dimensions are: a=62.5 μm, h=0.45 μm, rh=0.32 μm, Λ=0.88 μm, d=5 μm, with an operation frequency of ω=2π×30 kHz, and an ambient temperature of T=20° C.=293 K. In the calculation below, the holes are assumed to be covering the whole diaphragm.
For simplicity, any effects of the holes are first ignored, thereby assuming that the diaphragm in FIG. 40 is uniform. The major contribution to the viscous resistance in certain such embodiments will be the horizontal flow of the squeezed fluid in the gap, called squeeze-film damping. The acoustic resistance due to the squeeze-film damping in the case when all the fluid is forced out of the edges is:




  
    
      R
      edge
    
    =
    
      
        3
        ⁢
        μ
      
      
        2
        ⁢
        π
        ⁢
        
          
        
        ⁢
        
          d
          3
        
      
    
  
  ,




so for this calculation, Rvsc=Redge.

The radiation resistance of the diaphragm can be calculated for ωa/υa>>1 by:




  
    
      R
      rad
    
    =
    
      
        
          ω
          2
        
        ⁢
        
          ρ
          a
        
      
      
        2
        ⁢
        π
        ⁢
        
          
        
        ⁢
        
          υ
          a
        
      
    
  
  ,




where ρa is the density of the fluid, and Va is the speed of sound in the fluid (see, e.g., L. E. Kinder, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals of Acoustics, 3rd. ed., New York, Wiley (1982)). The contribution of the radiation resistance to the thermal noise for this case will be small. However, its contribution will be important for the cases when it is desired to decrease the overall damping for a lower thermal noise level.

The thermal noise levels for the case where we have no holes is calculated through the above expressions as Pair=33.5 μPa/Hz1/2 for air, and Pwater=246 μPa/Hz1/2 for water. These noise levels are almost an order of magnitude higher than generally desired. Thus, a more complete model that includes the effects of the holes is desired to analyze whether the presence of the holes can reduce the thermal noise level.
A.2 Diaphragm with Perforations
The presence of holes in the diaphragm creates an additional path for the viscous loss in parallel to the loss from the edges. By including the effect of the N number of holes, there will be an additional N number of parallel paths for the fluid to get out of the gap. Therefore, the viscous resistance in this case will be:




  
    1
    
      R
      vsc
    
  
  =
  
    
      1
      
        R
        edge
      
    
    +
    
      N
      
        R
        hole
      
    
  




where N=(a2/rh2) is the total number of holes,  is the fill-factor, defined as the ratio of the open area to the whole area (=π(rh/Λ)2 for a square lattice and





    =
  
    
      
        2
        ⁢
        π
      
      
        3
      
    
    ⁢
    
      
        (
        
          
            r
            h
          
          /
          Λ
        
        )
      
      2
    
  




for a triangular lattice).

The hole resistance will have two contributions, which are due to the horizontal flow of the fluid from the surroundings of the hole (squeeze-film flow), and the vertical flow of the fluid through the hole (Poiseuille flow). The horizontal flow contribution from each hole will be (see, e.g., Z. {hacek over (S)}kvor, “On acoustical resistance due to viscous losses in the air gap of electrostatic transducers,” Acustica, Vol. 19, 295-297 (1967-1968)):




  
    
      R
      vsc
      ↔
    
    =
    
      
        
          12
          ⁢
          μ
        
        
          π
          ⁢
          
            
          
          ⁢
          
            d
            3
          
        
      
      ⁢
      
        G
        ⁡
        
          (
                    )
        
      
    
  
  ,




where





  
    G
    ⁡
    
      (
            )
    
  
  =
  
    
      
        1
        2
      
      ⁢
          
    -
    
      
        1
        8
      
      ⁢
      
                2
      
    
    -
    
      
        1
        4
      
      ⁢
      ln
      ⁢
      
        
      
      ⁢
          
    -
    
      
        3
        8
      
      .
    
  




The vertical flow contribution from each hole, on the other hand, will be (see, e.g., D. Homentcovschi and R. N. Miles, “Modeling of Viscous Damping of Perforated Planar Microstructures. Applications in Acoustics,” Journal of the Acoustical Society of America, Vol. 116, 2939-2947 (2004)):





  
    R
    vsc
    ↕
  
  =
  
    
      8
      ⁢
      μ
      ⁢
      
        
      
      ⁢
      
        h
        eff
      
    
    
      π
      ⁢
      
        
      
      ⁢
      
        r
        h
        4
      
    
  




where an effective thickness





  
    h
    eff
  
  =
  
    h
    +
    
      
        
          3
          ⁢
          π
        
        8
      
      ⁢
      
        r
        h
      
    
  




is used to make corrections for the effect of the hole-end when the radius rh and the thickness h is comparable, as in the present case. Hence, the total hole resistance can be expressed as:





  
    R
    hold
  
  =
  
    
      
        R
        vsc
        ↔
      
      +
      
        R
        vsc
        ↕
      
    
    =
    
      
        
          
            12
            ⁢
            μ
          
          
            π
            ⁢
            
              
            
            ⁢
            
              d
              3
            
          
        
        ⁢
        
          G
          ⁡
          
            (
                        )
          
        
      
      +
      
        
          
            8
            ⁢
            
              
            
            ⁢
            μ
            ⁢
            
              
            
            ⁢
            
              h
              eff
            
          
          
            π
            ⁢
            
              
            
            ⁢
            
              r
              h
              4
            
          
        
        .
      
    
  




Therefore, if we include the contribution of the holes to the squeeze-film damping, the thermal noise level becomes: Pair=29.4 μPa/Hz1/2 for air, and Pwater=216 μPa/Hz1/2 for water. Therefore, the holes do not help much to reduce the noise level, mainly due to the high viscous resistance of the vertical flow through the small holes. For a closed cavity with no flow out of the edges, the damping will be even higher so that the thermal noise becomes: Pair=61.1 μPa/Hz1/2 for air, and Pwater=448 μPa/Hz1/2 for water.
These relatively simple calculations illustrate that to obtain a low self-noise for the fiber FP acoustic sensor, the design should be carefully considered. A more accurate model of various aspects of the sensor can be used to calculate the frequency dependent response of the sensor to an incident pressure that includes the effects of the ambient medium. Although the thermal noise in the ambient is essentially white noise, it will not transfer to the diaphragm as simple as assumed in the calculations above. The acoustic impedances of the diaphragm and the dissipative elements may not match perfectly, so that the noise transferred to the diaphragm can be more complex. Furthermore, since the impedances are in general frequency dependent, the noise transferred to the diaphragm will change with frequency. These effects are addressed in the discussion below.
B. Lumped-Element Equivalent-Circuit Modeling of the Sensor
To calculate the response and noise characteristics of the sensor, it is advantageous to have a model that describes the operation of the sensor in a simple way. One way of constructing such a model is by drawing an equivalent circuit of the sensor that includes the acoustic impedance of various elements of the sensor. In the circuit, instead of electrical resistance and capacitance, acoustic resistance R corresponding to dissipation and acoustic compliance C corresponding to the storing of potential energy are used. Likewise, instead of electrical inductance, an acoustic mass M corresponding to the storing of kinetic energy is used.
An equivalent circuit for an example fiber FP acoustic sensor is schematically illustrated by FIG. 41A. The same circuit is redrawn in FIG. 41B overlaid onto a schematic of the fiber FP acoustic sensor to better understand the modeling. The relation between the incident pressure Pin and the pressure acting on the diaphragm Pd is described by the transfer function Hsensor=Pd/Pin, Through simple circuit analysis, the pressure on the diaphragm can be calculated, so that the sensor transfer function is written in terms of the following three transfer functions Hd, Hcav, and Hleak, which are reciprocals of band-pass filter functions. The choice of indices for the functions is due to their respective relations to the energy passed to the oscillating diaphragm (Hd), the energy passed through the holes to the oscillating cavity (Hcav), and to the reduction of the diaphragm response due to the leaking through the holes (Hleak). These transfer functions can be expressed as:

Hd(ω)=1−ω2Cd(Mrad+Md)+jωCdRrad 

Hcav(ω)=1−ω2CcavMhole+jωCcavRvsc 

Hleak(ω)=1−ω2CdMhole+jωCdRvsc 

The transfer function for the sensor can then be expressed as:





  
    H
    sensor
  
  =
  
    
      
        P
        d
      
      
        P
        
          i
          ⁢
          
            
          
          ⁢
          n
        
      
    
    =
    
      
        
          
            H
            cav
          
          -
          1
        
        
          
            
              H
              d
            
            ⁢
            
              H
              cav
            
          
          +
          
            H
            leak
          
          -
          1
        
      
      .
    
  




To calculate the noise, a noise equivalent circuit can be used, as schematically illustrated by FIG. 42, with no incident pressure, and with noise sources in series with the dissipative elements, the viscous resistance Rvsc, and the radiation resistance Rrad. Due to impedance mismatch, the noise transferred to the diaphragm will be more complex than assumed in the previous section. For the configuration of FIG. 40, the impedances at 30 kHz (in units of 1012 Pa-s/m3 are:
Acoustic resistances: Rvsc=12; Rrad=0.0038
Acoustic mass: ωMd=0.029; ωMrad=0.81; ωMhole=0.031
Acoustic compliance: (1/ωCd)=6.5; (1/ωCcav)=190,000.
In certain embodiments, the high impedance of the Ccav term will dominate the system.
To calculate the effect of these noise sources on the diaphragm, the transfer functions for these sources can be calculated just in the same way as Hsensor, yielding:



  
    
      H
      vsc
    
    =
    
      
        
          P
          d
        
        
          P
          vsc
        
      
      =
      
        
          (
          
            1
            
              
                H
                cav
              
              -
              1
            
          
          )
        
        ⁢
        
          H
          sensor
        
      
    
  




  
    
      H
      rad
    
    =
    
      
        
          P
          d
        
        
          P
          rad
        
      
      =
      
        
          (
          
            
              H
              cav
            
            
              
                H
                cav
              
              -
              1
            
          
          )
        
        ⁢
        
          H
          sensor
        
      
    
  



The power spectral density of the fluctuating pressure on the diaphragm will then be:

PN2=|Hvsc|2Pvsc2+|Hrad|Prad2.

From the Nyquist theorem: Pvsc2=4kBTRvsc, and Prad2=4kBTRrad and using the expressions for the transfer functions Hvsc and Hrad:





  
    P
    N
    2
  
  =
  
    4
    ⁢
    
      k
      B
    
    ⁢
    T
    ⁢
    
      
        
          
          
            H
            sensor
          
          
        
        2
      
      
        
          
          
            
              H
              cav
            
            -
            1
          
          
        
        2
      
    
    ⁢
    
      
        (
        
          
            R
            vsc
          
          +
          
            
              
                
                
                  H
                  cav
                
                
              
              2
            
            ⁢
            
              R
              rad
            
          
        
        )
      
      .
    
  




Therefore, the signal-to-noise ratio (SNR) on the diaphragm can be expressed as:





  SNR
  =
  
    
      
        P
        d
      
      
        P
        N
      
    
    =
    
      
        
          
            
            
              
                H
                cav
              
              -
              1
            
            
          
          ⁢
          
            P
            in
          
        
        
          
            4
            ⁢
            
              k
              B
            
            ⁢
            
              T
              ⁡
              
                (
                
                  
                    R
                    vsc
                  
                  +
                  
                    
                      
                        
                        
                          H
                          cav
                        
                        
                      
                      2
                    
                    ⁢
                    
                      R
                      rad
                    
                  
                
                )
              
            
          
        
      
      .
    
  




Thus, to design a sensor with a minimum detectable pressure of Pin=MDP [Pa/√{square root over (Hz)}], the design formula will be, using SNR≧1:





  
    
      
        R
        vsc
      
      +
      
        
          
            
            
              H
              cav
            
            
          
          2
        
        ⁢
        
          R
          rad
        
      
    
    
      
        
        
          
            H
            cav
          
          -
          1
        
        
      
      2
    
  
  ≤
  
    
      MDP
      2
    
    
      4
      ⁢
      
        k
        B
      
      ⁢
      T
    
  




To achieve an acceptable low noise level, it is advantageous to choose the design parameters to have low Rvsc and Rrad. It is also advantageous to have a steep frequency response for the filter function Hcav, so that the noise is only amplified at very low frequencies, and not inside the bandwidth in which the sensor is to operate. This expression also gives the interesting result that the self-noise of the sensor cannot be improved by changing the stiffness of the diaphragm.
C. Spectral Behavior of the Sensor
To do calculations based on the equivalent-circuit model, it is advantageous to be able to calculate the individual acoustic impedances. The model includes two dissipative elements, the viscous resistance Rvsc, and the radiation resistance Rrad. The effective viscous resistance of the N number of holes can be expressed as:




  
    
      R
      vsc
    
    =
    
      
        
          R
          vsc
          ↕
        
        +
        
          R
          vsc
          ↔
        
      
      =
      
        
          
            1
            N
          
          ⁢
          
            
              8
              ⁢
              
                
              
              ⁢
              μ
              ⁢
              
                
              
              ⁢
              
                h
                eff
              
            
            
              π
              ⁢
              
                
              
              ⁢
              
                r
                h
                4
              
            
          
        
        +
        
          
            1
            N
          
          ⁢
          
            
              12
              ⁢
              
                
              
              ⁢
              μ
            
            
              π
              ⁢
              
                
              
              ⁢
              
                d
                3
              
            
          
          ⁢
          
            G
            ⁡
            
              (
              ℘
              )
            
          
        
      
    
  
  ,




where





  
    G
    ⁡
    
      (
      ℘
      )
    
  
  =
  
    
      
        1
        2
      
      ⁢
      ℘
    
    -
    
      
        1
        8
      
      ⁢
      
        ℘
        2
      
    
    -
    
      
        1
        4
      
      ⁢
      ln
      ⁢
      
        
      
      ⁢
      ℘
    
    -
    
      3
      8
    
  




(see, e.g., D. Homentcovschi et al. cited above). The

radiation resistance can be expressed as





  
    R
    rad
  
  =
  
    
      
        ω
        2
      
      ⁢
      
        ρ
        a
      
    
    
      2
      ⁢
      π
      ⁢
      
        
      
      ⁢
      
        υ
        a
      
    
  




(see, e.g., L. E. Kinder et al., cited above; W. Kuhnel and G. Hess, “A silicon condenser microphone with structured back plate and silicon nitride membrane,” Sensors and Actuators A, Vol. 30, 251-258 (1992)).

Two acoustic compliances, the diaphragm compliance Cd and the cavity compliance Ccav, are considered in the model. The diaphragm compliance for a plate can be expressed as




  
    C
    d
  
  =
  
    
      π
      ⁢
      
        
      
      ⁢
      
        a
        6
      
    
    
      192
      ⁢
      
        
      
      ⁢
      D
    
  




(see, e.g., M. Rossi, Acoustics and Electroacoustics, Artech House, Inc., 1st ed., (1988)) and the compliance of a cavity with volume Vcav can be expressed as





  
    C
    cav
  
  =
  
    
      
        V
        cav
      
      
        
          ρ
          a
        
        ⁢
        
          υ
          a
          2
        
      
    
    =
    
      
        π
        ⁢
        
          
        
        ⁢
        
          a
          2
        
        ⁢
        d
      
      
        
          ρ
          a
        
        ⁢
        
          υ
          a
          2
        
      
    
  




(see, e.g., W. Kuhnel et al., cited above).

Three acoustic masses are considered in the model: the diaphragm mass Md, the radiation mass Mrad, and the hole mass Mhole. The diaphragm mass can be expressed as




  
    M
    d
  
  =
  
    
      9
      ⁢
      
        
      
      ⁢
      ρ
      ⁢
      
        
      
      ⁢
      h
    
    
      5
      ⁢
      
        
      
      ⁢
      π
      ⁢
      
        
      
      ⁢
      
        a
        2
      
    
  




(see, e.g., M. Rossi, cited above), the radiation mass as





  
    M
    rad
  
  =
  
    
      8
      ⁢
      
        
      
      ⁢
      
        ρ
        a
      
    
    
      3
      ⁢
      
        
      
      ⁢
      
        π
        2
      
      ⁢
      a
    
  




(see, e.g., M. Rossi, cited above and W. Kuhnel et al., cited above), and the effective mass of the N number of holes as





  
    M
    hole
  
  =
  
    
      1
      N
    
    ⁢
    
      
        
          ρ
          a
        
        ⁢
        
          h
          eff
        
      
      
        π
        ⁢
        
          
        
        ⁢
        
          r
          h
          2
        
      
    
  




(see, e.g., M. Zollner and E. Zwicker, Elektroakustik, Berlin, Springer Verlag, (1987)).

As discussed more fully below, calculations can be made using the model to predict the sensor performance in air and in water. In certain embodiments, optimum sensor performance is achievable by having (1) a close to unity response at a broad range, and (2) a noise level close to the ambient noise level.
C.1 Sensor in an Air Medium
FIG. 43A shows the sensor response and FIG. 43B shows the noise of the sensor for an air medium. As shown in FIG. 43A, the sensor response has a slope, which is due to the high leakage through the holes. While it is advantageous to filter out the DC pressure variations, it is not desired to filter out the higher frequencies. Certain embodiments can obtain a flatter response by reducing the number of holes.
In FIG. 43B, the upper solid line corresponds to the noise calculated with the model, the dashed line is the noise level calculated with the more simple model described above, and the lower solid line is the ambient noise level, which is the noise due to radiation resistance only. The radiation resistance is an inherent property of the medium, so it is always present. The sensor noise level is less than about 65 μPa/Hz1/2 for frequencies less than about 100 kHz, and is well above the ambient noise level.
In FIGS. 43A and 43B, the holes are distributed over substantially the whole diaphragm. In certain other embodiments, the number of holes can be reduced, so that they are distributed within a central portion of the diaphragm and do not cover the whole diaphragm. For example, FIGS. 44A and 44B show the sensor response and noise for an air medium, respectively, for an array of N=20×20 holes. FIG. 44A shows that by reducing the leakage through the holes, the response saturates at higher frequencies. However, although the response becomes flat, it is not close to unity. The reason for this can be understood by looking at the equivalent circuit model of FIG. 41A. Since part of the pressure is dropping over the cavity, the cavity acts as a spring that reduces the response of the diaphragm. To get rid of this effect, the stiffness of the cavity can be reduced in certain embodiments. In other words, the acoustic compliance of the cavity can be increased, so that the impedance of the cavity becomes insignificant and most of the pressure drops over the diaphragm. To increase the cavity compliance Ccav, the model indicates that a larger cavity volume can be used. FIG. 44B shows that the sensor noise has improved by reducing the number of holes.
FIGS. 45A and 45B show the sensor response and noise for an air medium, respectively, for holes all around the diaphragm but with an increased cavity size (e.g., Vcav=(0.3 mm)3) FIG. 45A shows that by just increasing the cavity volume (e.g., to be greater than about (0.1 mm)3), a unity response that is very flat is achieved. The rapid convergence is also much better than the effect of reducing the number of holes (compare with FIG. 44A). FIG. 45B shows that the noise has also improved significantly, so that the sensor noise level is very close to the ambient noise level within the air medium. The sensor noise level is less than about 10 μPa/Hz1/2 for frequencies between about 25 kHz and about 100 kHz. Therefore, in certain embodiments, the volume of the cavity is advantageously increased to improve both the response and the noise of the fiber FP acoustic sensor. In certain various embodiments, the cavity volume is greater than (0.1 mm)3, greater than (0.3 mm)3, greater than (0.5 mm)3, greater than (1 mm)3, or greater than (2 mm)3. In certain various embodiments, the cavity volume is in a range between (0.1 mm)3 and (3 mm)3, in a range between (0.2 mm)3 and (3 mm)3, or in a range between (0.3 mm)3 and (3 mm)3.
C.2. Sensor in a Water Medium
FIGS. 46A and 46B show the sensor response and the noise, respectively, of the sensor for a water medium. The sensor response is drastically reduced when the ambient medium is water, with a response down to 3.5×10−5 (−90 dB). The noise level shown in FIG. 46B is quite bad considering the very low response, so that not much of the noise is picked up. At low frequencies, the response drops to zero due to venting through the holes, with a high-pass cutoff at 15 kHz. At higher frequencies, Pd<<Pin since the water in the small cavity pushes back. Thus, while certain embodiments compatible with the configuration of FIG. 40 can be used in air, such configurations do not work as well in water, since the added stiffness of the water trapped in the cavity reduces the response by about 90 dB.
FIGS. 47A and 47B show the sensor response and noise for a water medium, respectively, for an array of N=20×20 holes. FIG. 47A shows that although the convergence to a flat band has improved, the maximum response is still the same very low value. Fewer holes corresponds to less venting, hence a lower cutoff frequency (e.g., 25 Hz). The noise level shown in FIG. 47B seems to have improved, but again does not mean much when the response is very low.
FIGS. 48A and 48B show the sensor response and noise for a water medium, respectively, for holes all around the diaphragm but with an increased cavity size (e.g., greater than about (0.1 mm)3). FIG. 48A shows that the response for an acoustic sensor having a cavity volume of Vcav=(0.3 mm)3 is unity over a wide band (e.g., up to about 70 kHz) and is generally flat (e.g., to about 10 dB) with Pd≈Pin. At higher frequencies, a resonance is observed, which is well away from the frequencies of interest. For the smaller cavity volume configuration, this resonance had been at higher frequencies, but is pulled to lower frequencies for the larger cavity volume configuration due to reduced overall stiffness. FIG. 48B shows that the noise level is comparable to the ambient noise level (e.g., for 10 kHz-70 kHz). Therefore, as for the air medium, the cavity volume can advantageously be increased. In certain various embodiments, the cavity volume is greater than (0.1 mm)3, greater than (0.3 mm)3, greater than (0.5 mm)3, greater than (1 mm)3, or greater than (2 mm)3. In certain various embodiments, the cavity volume is in a range between (0.1 mm)3 and (3 mm)3, in a range between (0.2 mm)3 and (3 mm)3, or in a range between (0.3 mm)3 and (3 mm)3.
FIG. 48C shows the sensor response of FIG. 48A in the lower frequency range, having a high-pass cutoff at about 1 kHz, so that the frequency response of the acoustic sensor is generally flat in the range between about 1 kHz and about 70 kHz. FIG. 48D shows the noise level of FIG. 48B in the lower frequency range, in which the noise level is close to the ambient noise level for frequencies greater than about 10 kHz. The sensor noise level is less than about 30 μPa/Hz1/2 for frequencies between about 15 kHz and about 50 kHz. Therefore, in certain embodiments, a larger cavity volume is advantageously used for an acoustic sensor in water.
FIGS. 49A and 49B schematically illustrate two example embodiments of the acoustic sensor having an increased cavity volume. In FIG. 49A, a larger cavity has a plurality of sensors (e.g., two fiber-PCS pairs) and a region between the sensors providing additional volume to the cavity. Certain such embodiments will provide advantages in fabrication and in aligning of the fibers. In FIG. 49B, a larger cavity is used with a single sensor (e.g., a single fiber-PCS pair) and walls of the cavity extending generally away from the sensor and the optical fiber. In certain embodiments, the cavity volume is increased by a factor of about 1000. The cavity can be sealed in certain embodiments by a high-viscosity epoxy coupled to the optical fiber and the surrounding structure, as schematically illustrated by FIG. 49B.
D. Effect of the Holes on the Mechanics of the Diaphragm
In certain embodiments, the effect of the perforations on the stiffness of the diaphragm can be advantageously adjusted. The equation of motion discussed above for a uniform diaphragm was:




  
    
      (
      
        
          h
          ⁢
          
            
          
          ⁢
          ρ
          ⁢
          
            
              ∂
              2
            
            
              ∂
              
                t
                2
              
            
          
        
        +
        
          B
          ⁢
          
            ∂
            
              ∂
              t
            
          
        
        +
        
          D
          ⁢
          
            
              ∇
              4
            
            ⁢
            
              -
              h
            
          
          ⁢
          
            
          
          ⁢
          σ
          ⁢
          
            ∇
            2
          
        
      
      )
    
    ⁢
    u
  
  =
  
    P
    ⁡
    
      (
      
        r
        ,
        θ
        ,
        t
      
      )
    
  




The holes will make the elasticity of the diaphragm highly anisotropic (see, e.g., M. Pedersen, W. Olthuis, and P. Bergveld, “On the mechanical behaviour of thin perforated plates and their application in silicon condenser microphones,” Sens. Actuators A, Vol. 54, 499-504 (1996)). However, the structure can be approximated as being uniform-homogenous with effective elastic parameters: Deff=D(1−√{square root over ()}); ρeff=ρ(1−√{square root over ()}); and σeff=σ√{square root over (1−)}. The resonance frequencies of a membrane (e.g., SiN) will change as:





  
    
      ω
      n
    
    =
    
      
        
          z
          n
        
        a
      
      ⁢
      
        
          σ
          ρ
        
      
    
  
  ,
  
    
      
        =
      
      ⁢
      
        >
      
      ⁢
      
        
      
      ⁢
      
        ω
        n
        ′
      
    
    =
    
      
        ω
        n
      
      
        
          (
          
            1
            -
            ℘
          
          )
        
        
          1
          /
          4
        
      
    
  
  ,




which would yield, e.g., for =0.4, ωn′=1.14ωn, a 14% larger resonance frequency. The resonance frequencies of a plate (e.g., Si) on the other hand will change as:





  
    
      ω
      n
    
    =
    
      
        
          z
          n
          2
        
        
          a
          2
        
      
      ⁢
      
        
          D
          
            h
            ⁢
            
              
            
            ⁢
            ρ
          
        
      
    
  
  ,
  
    
      
        =
      
      ⁢
      
        >
      
      ⁢
      
        
      
      ⁢
      
        ω
        n
        ′
      
    
    =
    
      
        ω
        n
      
      
        
          (
          
            1
            +
            
              ℘
            
          
          )
        
        
          1
          /
          2
        
      
    
  
  ,




which would yield, e.g., for =0.4, ωn′=0.783ωn, a 22% smaller resonance frequency. The response to pressure of a membrane (e.g., SiN) will change as:





  
    
      u
      ⁡
      
        (
        
          r
          ,
          t
        
        )
      
    
    =
    
      
        P
        0
      
      ⁢
      
        e
        
          j
          ⁢
          
            
          
          ⁢
          ω
          ⁢
          
            
          
          ⁢
          t
        
      
      ⁢
      
        
          a
          2
        
        
          4
          ⁢
          
            
          
          ⁢
          h
          ⁢
          
            
          
          ⁢
          σ
        
      
      ⁢
      
        (
        
          1
          -
          
            
              r
              2
            
            
              a
              2
            
          
        
        )
      
    
  
  ,
  
    
      
        =
      
      ⁢
      
        >
      
      ⁢
      
        
      
      ⁢
      
        
          u
          ′
        
        ⁡
        
          (
          
            r
            ,
            t
          
          )
        
      
    
    =
    
      
        u
        ⁡
        
          (
          
            r
            ,
            t
          
          )
        
      
      
        
          1
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which would yield, e.g., for =0.4, u′=1.29u, a 29% larger response. The response to pressure of a plate (e.g., Si) will change as:





  
    
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which would yield, e.g., for =0.4, u′=2.72u, a 172% larger response.

Photonic Crystal Structure on a Fiber End

Certain embodiments described herein provide a photonic crystal structure formed on a fiber tip or end. While in certain embodiments, such a structure can be used as one of the mirrors of the fiber FP acoustic sensor, in certain other embodiments, such a structure can be used a disposable index sensor, for example to detect biological agents in blood. Certain embodiments can be used as a filter, mirror, or polarizer at the end of the fiber for applications that employ free-space to fiber interfaces. In certain embodiments, the structure can be also spliced (e.g., with fusion splicing) to another cleaved or polished fiber, such that the photonic crystal remains in between two fibers. Thus the photonic crystal can be used as an in-line reflector, filter, or polarizer. One advantage provided by certain embodiments over e.g. fiber Bragg grating based filters would be the temperature stability of such a filter due to its extremely small thickness (e.g., <500 nanometers).
Fabricating such devices can be challenging, e.g., to deposit materials such as silicon at the end of a fiber, and use lithography techniques to pattern small structures on it. The high temperatures involved (e.g., several 100° C.) in depositing materials such as silicon, and especially etching patterns into it with a hot plasma, can damage the polymeric coating of the optical fibers. Also, the extremely large aspect ratio of a piece of fiber makes it difficult to pattern, due to space limitations for putting such a structure into most micro-fabrication equipments.
Certain embodiments described herein utilize an optical structure (e.g., a photonic crystal structure) formed on an end of the fiber. In certain such embodiments, a method of fabrication is used that overcomes temperature issues and other physical constraints, and certain such embodiments can be used to fabricate photonic crystal structures on silicon on the end of a single-mode fiber. FIG. 50 schematically illustrates an example device 700 comprising a photonic crystal structure 710 formed in a dielectric layer 720 on an end of a fiber 730 in accordance with certain embodiments described herein. FIG. 51 shows a pair of scanning electron micrographs of an example photonic crystal structure 710 fabricated in a 350-nm-thick amorphous-silicon layer 720 at the fiber end of a single-mode silica fiber 730 in accordance with certain embodiments described herein.
FIG. 52 is a flow diagram of an example method 800 of fabricating an optical structure on a fiber end in accordance with certain embodiments described herein. In certain embodiments, the method 800 comprises preparing the fiber end in an operational block 810. Examples of fiber end preparation include, but are not limited to, cleaving the fiber, or polishing the end of the fiber, so that there is a smooth surface at the end of the fiber. In certain embodiments, a curved fiber surface might be used. In certain embodiments, the fiber can be single-mode or multi-mode.
In certain embodiments, the method 800 further comprises depositing a dielectric layer 720 onto the prepared fiber end in an operational block 820. In certain embodiments, the dielectric layer 720 is evaporated onto the fiber end by e-beam evaporation. For example, amorphous silicon (or other high-refractive-index dielectrics such as germanium) can be evaporated onto the fiber end using an e-beam evaporator. In certain other embodiments, the dielectric layer 720 can be deposited by other methods including but not limited to sputtering or chemical vapor deposition. In certain other embodiments, silicon dioxide, silicon nitride, magnesium fluoride, or silicon monoxide can be used as the dielectric layer 720.
In certain embodiments, the method further comprises patterning the dielectric layer 720 in an operational block 830. For example, in certain embodiments, the dielectric layer 720 is patterned by ion-milling using a focused ion beam which patterns a photonic crystal lattice 710 directly onto the dielectric layer 720 at the end of the fiber 730. The photonic crystal of certain embodiments comprises a periodic array of holes that extend either partially or completely through the dielectric layer 720. In certain embodiments, the method 800 further comprises covering the dielectric layer 720 with a thin layer of gold (e.g., thickness of about 10 nanometers) prior to the patterning process to reduce charging effects during the patterning process. In certain other embodiments, the dielectric layer 720 can be patterned using lithography techniques (e.g., covering the fiber end with a photo-resist material, exposing it with an optical or electron-beam lithography tool, developing the photo-resist, etching through the dielectric layer 720, and stripping the photo-resist). In certain embodiments, the periodic array of holes of the photonic structure 710 covers at least part of the area of the core region of the fiber 730, and in certain such embodiments, the periodic array of holes covers the whole core region of the fiber 730. The nature of the e-beam evaporation and ion-milling does not pose any temperature issues or problems regarding the length of the fiber structures.
The holes of the photonic crystal pattern in certain embodiments can have a variety of shapes, including but not limited to, circle, square, rectangle, and keyhole. These holes can be laid out on different types of two dimensional lattices including but not limited to a square lattice, a triangular lattice, and a hexagonal lattice. The patterns can also have non-uniformities, such as hole sizes being different in the center of the pattern as compared to the outer parts of the pattern. Similarly, the pattern can also have defects, such as missing holes, or individual holes with different sizes compared with the rest of the pattern.
FIG. 53A schematically illustrates an example device 700 as an interface between the optical fiber 730 and free space. The photonic crystal 710 covers the core region 732 of the optical fiber 730. In certain such embodiments, the device 700 can be used for several applications where the photonic crystal 710 acts as a reflector or a filter. As described above, such a device 700 can be used in a diaphragm sensor as a mirror in a Fabry-Perot interferometer to detect, for example acoustic signals. Also, the device 700 can serve as a filter or polarizer for the light that comes out of the fiber 730, or that is coupled into the fiber 730. Similarly, an active photonic crystal that is used, e.g. as a laser, could be pumped by the fiber in this configuration.
In certain embodiments, the device 700 can be spliced (e.g., with fusion splicing) to another cleaved or polished fiber 740, such that the photonic crystal 710 remains in between two fibers 730, 740, as depicted in FIG. 53B. In certain such embodiments, most of the light will remain in-line, inside the fibers 730, 740. Thus, the photonic crystal 710 can be used as an in-line reflector, filter or polarizer. Similarly, certain embodiments can be used as a in-line polarizing mirror, that can be very useful in applications based on fibers, such as fiber-optic gyroscopes. Certain embodiments can be used a fiber laser similar to the one described above, with a photonic crystal 710 patterned out of an active material.
In certain embodiments, the modes in the photonic crystal 710 can be coupled directly to optical near-fields, and the device 700 can be used as a compact near-field sensor. The optical properties of the photonic crystal 710 can change drastically when the index of refraction of the surrounding medium 750 changes. In certain such embodiments, the device 700 can be used as an index sensor. For example as schematically illustrated by FIG. 53C, a single-mode fiber (e.g., having a diameter of 125 microns) can have medical applications, since its size is compatible for penetrating veins and tissue. Because of its potential low cost, certain such embodiments can be used as a disposable sensor. In certain embodiments, the sensitivity of such a sensor can be increased by employing holes that lack a mirror symmetry, such as a circular holes with a little protrusion on one side, so that the holes look like keyholes.
FIG. 54 is a flow diagram of an example method 900 of fabricating a photonic crystal structure (e.g., a near-field coupled photonic crystal device) in accordance with certain embodiments described herein. FIGS. 55A-55F schematically illustrate a series of example fabrication steps for forming a near-field coupled photonic crystal device 1000 at the end of an optical fiber 1010 in accordance with certain embodiments described herein.
The method 900 comprises forming a first dielectric layer 1020 on an end of an optical fiber 1010 in an operational block 910. In FIG. 55A, a first dielectric layer 1020 comprising a 450-nanometer amorphous silicon layer is evaporated onto a cleaved or polished end of the optical fiber 1010.
The method 900 further comprises forming a sacrificial layer 1030 over at least a portion of the first dielectric layer 1020 in an operational block 920. In certain embodiments, forming the sacrificial layer 1030 comprises forming a 100-nanometer chrome layer 1032 over the first dielectric layer 1020 (e.g., by e-beam evaporation), and forming a 500-nanometer gold layer 1034 over the chrome layer 1032 (e.g., by e-beam evaporation). In certain embodiments, the first dielectric layer 1020, the chrome layer 1032, and the gold layer 1034 are formed subsequently to one another in the same e-beam evaporation system. As shown in FIG. 55B, a portion of the chrome layer 1032 is removed, e.g., by dipping the structure in a wet chrome etchant at room temperature. As shown in FIG. 55C, the gold layer is removed, e.g., by dipping the structure in a wet gold etchant at room temperature.
The method 900 further comprises forming a second dielectric layer 1040 over the sacrificial layer 1030 in an operational block 930. For example, as shown in FIG. 55D, the structure is returned to the e-beam evaporator system and a second dielectric layer 1040 comprising a second 450-nanometer silicon layer is evaporated onto the structure.
The method 900 further comprises patterning the first dielectric layer 1020 to form a first photonic crystal slab 1050 in an operational block 940, and patterning the second dielectric layer 1040 to form a second photonic crystal slab 1060 in an operational block 950. For example, as shown in FIG. 55E, the first photonic crystal slab 1050 and the second photonic crystal slab 1060 are patterned into the first and second silicon layers in one ion-milling step.
The method 900 further comprises removing the sacrificial layer 1030 in an operational block 960. For example, as shown in FIG. 55F, the remaining portion of the chrome layer 1032 is removed by dipping the structure in a wet chrome etchant. In certain other embodiments, various other patterns can be introduced to the structure besides the photonic crystal lattices. For example, in an index sensing application, bigger holes can be milled through the second dielectric layer 1040, so that channels leading to the gap between the two dielectric layers are created, thereby allowing a liquid to easily penetrate the gap between the two photonic crystal slabs.
In certain embodiments, the fabrication methods described above can be used to fabricate other optical devices besides photonic crystals at the end of an optical fiber. For example, for several applications including telecommunication and fiber-based physical devices such as gyroscopes, it is desirable to have one or more sub-wavelength gratings which can manipulate the phase distribution, electric-field amplitude distribution, and polarization distribution in an optical mode.
By patterning structures such as holographic waveplates (or computer-generated holograms) on the amorphous silicon layer at the fiber tip, certain embodiments described herein can be used to convert the fundamental Gaussian mode of a single-mode fiber into higher order Laguerre-Gaussian modes. Such modes include, but are not limited to, doughnut-shaped modes and even more complex modes that can be employed in applications such as optical trapping and manipulation of microscopic particles. For some applications such as confocal microscopy, a doughnut-shaped focal point is desirable at the front focal plane of high numerical-aperture objectives. Holographic waveplates also allow the creation of modes with unconventional space-variant polarizations (conventional here refers to either linearly, circularly, or elliptically polarized), which are usually radial and azimuthal polarizations. By using these space-variant polarizations states, in certain embodiments, it is even possible to focus light to a spot smaller than the diffraction limit (see, e.g., R. Dorn et al, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett., Vol. 91, 233901 (2003)). FIG. 55A shows three example holographic waveplates (from U. Levy et al, “Engineering space-variant inhomogeneous media for polarization control,” Opt. Lett., Vol. 29, 1718 (2004)) in accordance with certain embodiments described herein. FIG. 55B shows two scanning electron micrographs of cross-sections of example sub-wavelength waveplates formed in GaAs in accordance with certain embodiments described herein (from Levy et al.).
In certain embodiments described herein, a fiber can be formed with a holographic waveplate at its end, that has an output mode with either a useful intensity distribution such as a doughnut mode or a polarization distribution such as radial polarization. Such a mode can be used to trap and apply torque to microscopic particles in various embodiments. A doughnut-shaped mode can also increase the trapping force to trap particles with lower refractive index than the surrounding fluid. One advantage of certain embodiments described herein over standard optical trapping and manipulation methods is that the optics are all integrated into the fiber, so that it is easy to dip it into a solution, target a specific area or particle group, and move around both the fiber and the trapped particles.
In certain embodiments, one or more polarizers can be created by fabricating sub-wavelength gratings at the fiber end. One method in accordance with certain embodiments described herein can be used to obtain a conventional, linear polarizer by creating a regular one-dimensional grating pattern on the amorphous silicon layer at the fiber end. For certain applications and wavelength ranges, a metal layer can be added to the silicon layer, or the silicon layer can be completely substituted by a metal layer. These types of polarizers are usually referred to as wire-grid polarizers, schematically illustrated by FIG. 56. More complex polarization filters can also be made by this same process by depositing a pattern of metal wires with more complex shapes. For example, a pattern of concentric circles centered on the core will act as a filter that differentiates between the TE and the TM modes: one mode is attenuated while the other one is not. A pattern of radial metal lines spreading out from the center of the core will create a polarizer with just the opposite properties. There are also other methods to create polarizers, such as embedding elliptical metal particles into a glass layer to make polarizers at the end of a fiber (see, e.g., Kazutaka Baba, “Theoretical characteristics of optical polarizing films using oblique metal island films,” Proceedings of SPIE, Vol. 6116, 611605-1 (2006)).
These fibers, which are patterned with a sub-wavelength structure at the end, can also be combined in certain embodiments with a second fiber, such as through splicing, or using a lower temperature gluing method. Then, the sub-wavelength structures will remain in-line in between the fibers. Certain embodiments can, for example, interface two different types of fibers with a holographic waveplate, and facilitate the coupling between the fibers. Also, certain embodiments can create a fiber polarizer, by embedding a sub-wavelength polarizer in between two fibers.
Various embodiments have been described above. Although the invention has been described with reference to these specific embodiments, the descriptions are intended to be illustrative of the invention and are not intended to be limiting. Various modifications and applications may occur to those skilled in the art without departing from the true spirit and scope of the invention as defined in the appended claims.

## Claims

1. A method of fabricating an optical structure on an optical fiber, the method comprising:
preparing an end of an optical fiber;
forming a layer onto the fiber end; and
subsequent to the forming, patterning the layer to form the optical structure, wherein the optical structure comprises a photonic crystal structure.

2. The method of claim 1, wherein preparing the end comprises cleaving the optical fiber to form the end.

3. The method of claim 1, wherein forming the layer comprises depositing the layer onto the fiber end by e-beam evaporation, sputtering, or chemical vapor deposition.

4. The method of claim 1, wherein patterning the layer comprises ion-milling the layer using a focused ion beam.

5. The method of claim 1, wherein the optical structure comprises a periodic array of holes that extend either partially or completely through the layer.

6. The method of claim 1, wherein the layer comprises a dielectric material and the method further comprising covering the dielectric material with one or more metal layers prior to patterning the layer.

7. The method of claim 1, wherein the optical structure covers at least part of the area of a core region of the optical fiber.

8. The method of claim 1, wherein the optical structure comprises a sub-wavelength grating.

9. A method of fabricating a photonic crystal structure, the method comprising:
forming a first dielectric layer on an end of an optical fiber;
forming a sacrificial layer over at least a portion of the first dielectric layer;
forming a second dielectric layer over the sacrificial layer;
subsequent to the forming of the first dielectric layer, patterning the first dielectric layer to form a first photonic crystal slab;
patterning the second dielectric layer to form a second photonic crystal slab; and
removing the sacrificial layer.

10. The method of claim 9, wherein the sacrificial layer comprises chrome and forming the sacrificial layer comprises:
forming a chrome layer over at least a portion of the first dielectric layer;
forming a gold layer over the chrome layer;
removing a portion of the chrome layer; and
removing the gold layer.

11. The method of claim 9, wherein patterning the first dielectric layer and patterning the second dielectric layer are performed concurrently.

12. An optical apparatus comprising:
an end of an optical fiber; and
a layer comprising material deposited onto the end of the optical fiber, the layer comprising one or more first portions having a first optical pathlength in a direction perpendicular to the layer and one or more second portions having a second optical pathlength in the direction perpendicular to the layer, the second optical pathlength different from the first optical pathlength, wherein the layer comprises a photonic crystal structure.

13. The optical apparatus of claim 12, wherein the one or more first portions have a first refractive index and the one or more second portions have a second refractive index different from the first refractive index.

14. The optical apparatus of claim 12, wherein the one or more first portions have a first thickness in the direction perpendicular to the layer and the one or more second portions have a second thickness in the direction perpendicular to the layer, the second thickness different from the first thickness.

15. The optical apparatus of claim 12, wherein the end of the optical fiber is cleaved.

16. The optical apparatus of claim 12, wherein the layer comprises amorphous silicon, silicon dioxide, silicon nitride, magnesium fluoride, or silicon monoxide.

17. The optical apparatus of claim 12, wherein the layer comprises metal.

18. The optical apparatus of claim 12, wherein the one or more first portions comprise a dielectric material and the one or more second portions comprise a periodic array of holes that extend at least partially through the layer.

19. The optical apparatus of claim 18, wherein the periodic array of holes covers at least part of the area of a core region of the optical fiber.

20. The optical apparatus of claim 12, wherein the photonic crystal structure forms one end of a fiber Fabry-Perot cavity.

21. The optical apparatus of claim 12, wherein the optical fiber is coupled to another optical fiber such that the layer is between the two optical fibers.

22. The optical apparatus of claim 12, wherein the layer comprises a subwavelength grating.

23. The optical apparatus of claim 12, wherein the layer comprises a holographic waveplate.

24. The optical apparatus of claim 12, wherein the layer comprises a wire-grid polarizer.

25. The method of claim 9, wherein the patterning the second dielectric layer is subsequent to the forming of the second dielectric layer over the sacrificial layer.