# Patent

## Title
Method of measuring a physical function using a composite function which includes the physical function and an arbitrary reference function

## Abstract

A method measures a nolinearity profile of a sample with at least one sample surface and having a sample nonlinearity profile along a sample line through a predetermined point on the sample surface. The sample line is oriented perpendicularly to the sample surface. The method includes measuring a Fourier transform of the sample nonlinearity profile and obtaining a reference nonlinearity profile from a reference material. The method includes forming a first composite sample having a first composite nonlinearity profile and forming a second composite sample having a second composite nonlinearity profile inequivalent to the first composite nonlinearity profile. The method further includes measuring a Fourier transform of the first composite nonlinearity profile and measuring a Fourier transform of the second composite nonlinearity profile. The method further includes calculating the sample nonlinearity profile using the Fourier transforms of the sample nonlinearity profile, the reference nonlinearity profile, the first composite nonlinearity profile, and the second composite nonlinearity profile.

## Background
CLAIM OF PRIORITY
This application is a continuation-in-part of, and claims priority to, U.S. patent application Ser. No. 10/357,275, filed Jan. 31, 2003, which is incorporated by reference in its entirety herein. U.S. patent application Ser. No. 10/357,275 claims priority to U.S. Provisional Application No. 60/405,405, filed Aug. 21, 2002, which is also incorporated by reference in its entirety herein.


BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates generally to the field of measuring attributes of a physical system, and more particularly to methods of measuring a non-symmetric physical function.
2. Description of the Related Art
Various optical devices are based on induced second-order susceptibilities in silica-based glass waveguides (e.g., electro-optic modulators, switches, parametric amplifiers). For example, G. Bonfrate et al. describe optical parametric oscillators useful for the study of quantum interference and quantum cryptography, and for metrology applications in Parametric Fluorescence in Periodically Poled Silica Fibers, Applied Physics Letters, Vol. 75, No. 16, 1999, pages 2356-2358, which is incorporated in its entirety by reference herein. Second-order susceptibility can be induced in a several-microns-thick region of fused silica (a material that is not normally centro-symmetric, and therefore normally does not exhibit a second-order susceptibility) by poling at elevated temperatures. This phenomenon has been described by R. A. Myers et al. in Large Second-Order Nonlinearity in Poled Fused Silica, Optics Letters, Vol. 16, No. 22, 1991, pages 1732-1734, which is incorporated in its entirety by reference herein.
FIGS. 1A and 1B schematically illustrate the poling of a silica wafer 1. As schematically illustrated in FIG. 1A, poling typically comprises using an anode electrode 2 placed proximate to one surface 3 of the wafer 1 and a cathode electrode 4 placed proximate to the opposite surface 5 of the wafer 1. A voltage is applied across the wafer 1 for a period of time, resulting in a second-order optical nonlinearity profile having a thickness and localized beneath the surface 3 where the anode electrode was placed, as schematically illustrated in FIG. 1B. As used herein, the term “anodic surface” refers to the surface which is placed proximate to the anode electrode, and the term “cathodic surface” refers to the surface which is placed proximate to the cathode electrode. Such a poling procedure is described in more detail by Thomas G. Alley et al. in Space Charge Dynamics in Thermally Poled Fused Silica, Journal of Non-Crystalline Solids, Vol. 242, 1998, pages 165-176, which is incorporated herein in its entirety.
The field of poled silica has suffered from the lack of a common method to reliably measure the second-order optical nonlinearity profile of poled samples. This absence of a reliable procedure for measuring nonlinearity profiles may be the reason, at least in part, for wide discrepancies in the measured magnitudes and the positions of the nonlinearity profiles of various poled systems as reported in the literature. The Maker fringe (MF) technique is the most common method currently used to investigate the nonlinearity profile of poled silica. The MF technique comprises focusing a pulsed laser beam of intensity I1 (known as the fundamental signal) onto a sample at an incident angle θ and measuring the intensity I2 of the second harmonic (SH) signal generated within the nonlinear region as a function of the incident angle θ. For a transverse magnetic (TM) polarized fundamental laser beam, the conversion efficiency ηTM(θ) is given by: 



  
    
      
        
          
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where

    
    
        d33(z) is the nonlinear coefficient (which is proportional to the second-order susceptibility X(2));
        z is the direction normal to the sample surface (i.e., parallel to the poling field);
        n1 and n2 are the refractive indices at the fundamental and SH frequencies, respectively;
        Δk=k2−2k1, where k1 and k2 are the fundamental and SH wave numbers, respectively, and
        ƒ(θ, n1, n2) is a well-defined function of the incident angle θ (relative to the surface normal direction) and refractive indices n1 and n2.
    
    


The function ƒ(θ, n1, n2) accounts for both the power loss due to reflection suffered by the fundamental and the SH beams, and the projection of the input electric field along the appropriate direction. In general, ƒ(θ, n1, n2) depends on both the polarization of the input fundamental wave and the geometry of the second harmonic generation configuration. The exact formula of ƒ(θ, n1, n2) is given by D. Pureur, et al. in Absolute Measurement of the Second-Order Nonlinearity Profile in Poled Silica, Optics Letters, Vol. 23, 1998, pages 588-590, which is incorporated in its entirety by reference herein. This phenomenon is also described by P. D. Maker et al. in Effects of Dispersion and Focusing on the Production of Optical Harmonics, Physics Review Letters, Vol. 8, No. 1, 1962, pages 21-22, which is incorporated in its entirety by reference herein.
The conversion efficiency ηTM(θ) is obtained experimentally by rotating the sample with respect to the incident laser beam and measuring the power of the SH signal as a function of the incident angle θ. Due to dispersion of the laser beam, Δk is finite and ηTM(θ) exhibits oscillations (called the Maker fringes) which pass through several maxima and minima. The objective of this measurement is to retrieve the second-order nonlinearity profile d33(z). The absolute value of the integral in Equation 1 is the amplitude of the Fourier transform of d33(z). In principle, if one knew both the amplitude and the phase of a Fourier transform, one can readily infer the argument of the Fourier transform (in this case d33(z)) by taking the inverse Fourier transform of the Fourier transform. However, the measured Maker fringes provide only the magnitude of the Fourier transform, not its phase. Consequently, for an arbitrary and unknown nonlinearity profile, the MF measurement alone is not sufficient to determine a unique solution for d33(z). Even if the phase information were available, it would provide the shape of d33(z) but it would still not provide the location of this shape beneath the surface of the sample (i.e., where the nonlinearity profile starts beneath the surface).
Previous efforts to determine d33(z) have involved fitting various trial profiles to the measured MF data. Examples of such efforts are described by M. Qiu et al. in Double Fitting of Maker Fringes to Characterize Near-Surface and Bulk Second-Order Nonlinearities in Poled Silica, Applied Physics Letters, Vol. 76, No. 23, 2000, pages 3346-3348; Y. Quiquempois et al. in Localisation of the Induced Second-Order Non-Linearity Within Infrasil and Suprasil Thermally Poled Glasses, Optics Communications, Vol. 176, 2000, pages 479-487; and D. Faccio et al. in Dynamics of the Second-Order Nonlinearity in Thermally Poled Silica Glass, Applied Physics Letters, Vol. 79, No. 17, 2001, pages 2687-2689. These references are incorporated in their entirety by reference herein.
However, the previous methods do not produce a unique solution for d33(z). Two rather different trial profiles can provide almost equally good fits to the measured MF data. This aspect of using fitting routines to determine d33(z) is described in more detail by Alice C. Liu et al. in Advances in the Measurement of the Poled Silica Nonlinear Profile, SPIE Conference on Doped Fiber Devices II, Boston, Mass., November 1998, pages 115-119, which is incorporated in its entirety by reference herein.
SUMMARY OF THE INVENTION
According to one aspect of the present invention, a method measures a second-order optical nonlinearity profile of a sample. The method comprises providing the sample having the second-order optical nonlinearity profile. The method further comprises placing a surface of the sample in proximity to a surface of a supplemental sample. The sample and supplemental sample form a composite sample having a symmetric second-order optical nonlinearity profile. The method further comprises measuring a Fourier transform of the symmetric second-order optical nonlinearity profile. The method further comprises calculating an inverse Fourier transform of the measured Fourier transform. The calculated inverse Fourier transform provides information regarding the second-order optical nonlinearity profile.
In another aspect of the present invention, a method measures a physical function. The method comprises forming a symmetric composite function by superimposing the physical function with a substantially identical physical function. The method further comprises obtaining a Fourier transform of the symmetric composite function. The method further comprises calculating an inverse Fourier transform of the obtained Fourier transform. The calculated inverse Fourier transform provides information regarding the physical function.
In still another aspect of the present invention, a method determines the temporal waveform of a laser pulse. The method comprises providing a laser pulse having a temporal waveform. The method further comprises providing a time-reversed pulse having a time-reversed temporal waveform which corresponds to the temporal waveform of the laser pulse after being time-reversed. The method further comprises forming a symmetric composite waveform by combining the temporal waveform of the laser pulse and the time-reversed temporal waveform of the time-reversed pulse. The method further comprises obtaining a Fourier transform of the symmetric composite waveform. The method further comprises calculating an inverse Fourier transform of the obtained Fourier transform. The calculated inverse Fourier transform provides information regarding the temporal waveform of the laser pulse.
In still another aspect of the present invention, a method measures a nolinearity profile of a sample. The method comprises providing a sample having at least one sample surface and having a sample nonlinearity profile along a sample line through a predetermined point on the sample surface. The sample line is oriented perpendicularly to the sample surface. The method further comprises measuring a Fourier transform of the sample nonlinearity profile. The method further comprises providing a reference material having at least one reference surface and having a reference nonlinearity profile along a reference line through a predetermined point on the reference surface. The reference line is oriented perpendicularly to the reference surface. The method further comprises obtaining a Fourier transform of the reference nonlinearity profile. The method further comprises forming a first composite sample having a first composite nonlinearity profile by placing the sample and the reference material proximate to one another in a first configuration with the sample line substantially collinear with the reference line. The method further comprises measuring a Fourier transform of the first composite nonlinearity profile. The method further comprises forming a second composite sample having a second composite nonlinearity profile which is inequivalent to the first composite nonlinearity profile by placing the sample and the reference material proximate to one another in a second configuration with the sample line substantially collinear with the reference line. The method further comprises measuring a Fourier transform of the second composite nonlinearity profile. The method further comprises calculating the sample nonlinearity profile using the Fourier transforms of the sample nonlinearity profile, the reference nonlinearity profile, the first composite nonlinearity profile, and the second composite nonlinearity profile.
In still another aspect of the present invention, a method measures a nolinearity profile of a sample. The method comprises providing a sample having at least one sample surface and having a sample nonlinearity profile along a sample line through a predetermined point on the sample surface. The sample line is oriented perpendicularly to the sample surface. The method further comprises measuring a Fourier transform of the sample nonlinearity profile. The method further comprises providing a reference material having at least one reference surface and having a reference nonlinearity profile along a reference line through a predetermined point on the reference surface. The reference line is oriented perpendicularly to the reference surface. The method further comprises obtaining a Fourier transform of the reference nonlinearity profile. The method further comprises forming a first composite sample having a first composite nonlinearity profile by placing the sample and the reference material proximate to one another in a first configuration with the sample line substantially collinear with the reference line. The method further comprises measuring a Fourier transform of the first composite nonlinearity profile. The method further comprises forming a second composite sample having a second composite nonlinearity profile which is inequivalent to the first composite nonlinearity profile by placing the sample and the reference material proximate to one another in a second configuration with the sample line substantially collinear with the reference line. The method further comprises measuring a Fourier transform of the second composite nonlinearity profile. The method further comprises using the Fourier transforms of the sample nonlinearity profile, the reference nonlinearity profile, the first composite nonlinearity profile, and the second composite nonlinearity profile to calculate the sample nonlinearity profile.
In still another aspect of the present invention, a method measures a sample temporal waveform of a sample optical pulse. The method comprises providing a sample optical pulse having a sample temporal waveform. The method further comprises measuring a Fourier transform of the sample temporal waveform. The method further comprises providing a reference optical pulse having a reference temporal waveform. The method further comprises obtaining a Fourier transform of the reference temporal waveform. The method further comprises forming a first composite optical pulse comprising the sample optical pulse followed by the reference optical pulse. The first composite optical pulse has a first composite temporal waveform. The method further comprises measuring the Fourier transform of the first composite temporal waveform. The method further comprises providing a time-reversed pulse having a time-reversed temporal waveform corresponding to the reference temporal waveform after being time-reversed. The method further comprises forming a second composite optical pulse comprising the sample optical pulse followed by the time-reversed optical pulse. The method further comprises measuring the Fourier transform of the second composite temporal waveform. The method further comprises calculating the sample temporal waveform using the Fourier transform of the sample temporal waveform, the Fourier transform of the reference temporal waveform, the Fourier transform of the first composite temporal waveform, and the Fourier transform of the second composite temporal waveform.
For purposes of summarizing the invention, certain aspects, advantages and novel features of the invention have been described herein above. It is to be understood, however, that not necessarily all such advantages may be achieved in accordance with any particular embodiment of the invention. Thus, the invention may be embodied or carried out in a manner that achieves or optimizes one advantage or group of advantages as taught herein without necessarily achieving other advantages as may be taught or suggested herein.



BRIEF DESCRIPTION OF THE DRAWINGS
FIGS. 1A and 1B schematically illustrate the poling of a silica wafer.
FIG. 2 is a flow diagram of a method of measuring a second-order optical nonlinearity profile of a sample in accordance with embodiments of the present invention.
FIG. 3A schematically illustrates a composite sample having a symmetric second-order optical nonlinearity profile.
FIGS. 3B and 3C schematically illustrate two measurement configurations in accordance with embodiments of the present invention.
FIGS. 4A-4D schematically illustrate various embodiments of the supplemental sample in accordance with embodiments of the present invention.
FIG. 5 illustrates exemplary Maker fringe profiles (in units of 10−19 m2/W) measured from a common sample (shown as open circles) and a composite sample (shown as crosses).
FIG. 6A illustrates the second-order nonlinearity profile d33(z) (in units of 10−13 m/V) of the composite sample obtained by calculating the inverse Fourier transform of the measured fringe profile from the composite sample as illustrated in FIG. 5.
FIG. 6B illustrates the original nonlinearity profile d33(z) (in units of 10−13 m/V) of the sample corresponding to the z>0 portion of the nonlinearity profile of FIG. 6A.
FIG. 7 illustrates the charge distribution ρ(z) (in units of C/m3) derived from the nonlinearity profile of FIG. 6B.
FIG. 8 illustrates the comparison of the measured fringe profile (in units of 10−19 m2/W) from the original sample (open circles) with a theoretical fringe profile calculated from d33(z) (as shown in FIG. 6B) obtained from the measured SH signal from the composite sample (solid line).
FIG. 9 is a flow diagram of a method of measuring a sample nonlinearity profile in accordance with embodiments of the present invention.
FIGS. 10A and 10B schematically illustrate single-pass configurations for measuring the Fourier transforms of the sample nonlinearity profile and the reference nonlinearity profile, respectively.
FIGS. 11A-11D schematically illustrate various configurations for forming composite samples in accordance with embodiments of the present invention.
FIGS. 12A and 12B schematically illustrate two symmetric nonlinearity profiles.
FIG. 13 is a flowchart of a method in accordance with embodiments of the present invention for measuring physical functions.
FIG. 14 is a flow diagram of a method of determining the temporal waveform of a laser pulse in accordance with embodiments of the present invention.
FIG. 15 schematically illustrates four-wave mixing (FWM) with pulsed pumps for providing the time-reversed pulse in accordance with embodiments of the present invention.
FIGS. 16A and 16B schematically illustrate femtosecond spectral holography for providing the time-reversed pulse in accordance with embodiments of the present invention.
FIG. 17 schematically illustrates one embodiment of a classical intensity correlator utilizing a Michelson interferometer.
FIG. 18 schematically illustrates a general configuration for converting a periodically repeated sequence of pulses into a periodically repeated sequence of symmetric pulses using a movable phase conjugative mirror.
FIG. 19 illustrates an exemplary temporal waveform (in units of W/m2) of an asymmetric input pulse compatible with embodiments of the present invention.
FIG. 20 illustrates the magnitude of the Fourier transform (unitless) of the autocorrelation function corresponding to the temporal waveform of FIG. 19.
FIG. 21 illustrates the symmetric composite waveform (in units of W/m2) corresponding to the temporal waveform of FIG. 19.
FIG. 22 illustrates the magnitude of the Fourier transform (unitless) of the autocorrelation function of the symmetric composite waveform of FIG. 21.
FIG. 23A illustrates the recovered symmetric temporal waveform (in units of W/m2).
FIG. 23B illustrates the difference (in units of W/m2) between the recovered symmetric temporal waveform and the temporal waveform of the original pulse.
FIG. 24 schematically illustrates a system for another embodiment for determining the temporal waveform of a laser pulse.
FIGS. 25A and 25B illustrate the magnitude (in units of W/m2) and phase of an arbitrary asymmetric complex envelope function to be characterized.
FIG. 26 illustrates the intensity profile (in units of W/m2) for the asymmetric complex envelope function of FIG. 25A with the carrier frequency oscillations.
FIG. 27 illustrates the symmetric temporal waveform (in units of W/m2) with the carrier frequency corresponding to the asymmetric complex envelope function of FIG. 26.
FIGS. 28A and 28B illustrate the detected intensity (in units of W/m2) on the CCD imaging device for the symmetric pulse of FIG. 27, and the single pulse of FIG. 26, respectively.
FIGS. 29A and 29B illustrate the recovered symmetric temporal waveform and the original temporal waveform (in units of W/m2), respectively, including the carrier frequencies.
FIG. 30 illustrates the waveforms of FIGS. 29A and 29B (in units of W/m2) overlaid with one another in an expanded time frame for comparison purposes.
FIG. 31 is a flow diagram of a method of measuring a sample temporal waveform of a sample optical pulse in accordance with embodiments of the present invention.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
FIG. 2 is a flow diagram of a method 100 of measuring a second-order optical nonlinearity profile of a sample 10 in accordance with embodiments of the present invention. While the flow diagrams herein illustrate particular embodiments with steps in a particular order, other embodiments with different orders of steps are also compatible with the present invention.
In the method 100, the sample 10 is provided in an operational block 110. The sample 10 has the second-order optical nonlinearity profile 15 to be measured. In an operational block 120, a surface of the sample 10 is placed proximate to a surface of a supplemental sample 20, as schematically illustrated in FIGS. 3A-3C and 4A-4D. As schematically illustrated in FIG. 3A, the sample 10 and supplemental sample 20 form a composite sample 30 having a symmetric second-order optical nonlinearity profile 35. In an operational block 130, a Fourier transform of the symmetric second-order optical nonlinearity profile 35 is measured. In an operational block 140, an inverse Fourier transform of the measured Fourier transform is calculated. The calculated inverse Fourier transform provides information regarding the second-order optical nonlinearity profile 15.
In certain embodiments, the information comprises the magnitude and sign of the second-order optical nonlinearity profile 15 as a function of depth below the surface of the sample 10. In other embodiments, information comprises the position of the second-order optical nonlinearity profile 15 below the surface of the sample 10.
In certain embodiments, the second-order optical nonlinearity profile 15 is non-symmetric, while in other embodiments, the second-order optical nonlinearity profile 15 is symmetric. Other existing methods are useful for measuring symmetric second-order optical nonlinearity profiles, but such existing methods are not compatible with non-symmetric profiles. Embodiments described herein are compatible with both symmetric and non-symmetric second-order optical nonlinearity profiles. Since it is not generally known whether the second-order optical nonlinearity profile is symmetric or non-symmetric prior to measurement, embodiments described herein provide a general method of measurement independent of the symmetry of the profile to be determined. In certain embodiments, the information comprises whether the second-order optical nonlinearity profile is symmetric or non-symmetric.
Embodiments of the present invention can be used to determine the nonlinearity profile of any optically nonlinear material (e.g., crystalline, amorphous, organic, or inorganic) in a bulk form or in a film form. In certain embodiments, the nonlinear material comprises an organic material, such as polymers and solids doped with dye molecules. In other embodiments, the nonlinear material comprises inorganic materials such as crystalline lithium niobate or amorphous materials (e.g., oxide-based, fluoride-based, or sulfide-based glasses).
In certain embodiments, the sample 10 comprises silica glass that has been poled so as to induce a second-order optical nonlinearity profile in the sample 10. For example, a fused silica wafer comprised of INFRASEL quartz glass and measuring 1′′×1′′×0.03′′ (i.e., 2.54 cm×2.54 cm×7.62 mm) can be poled at approximately 270° C. in air by using an anode electrode placed proximate to one surface of the wafer (i.e., the anodic surface) and a cathode electrode placed proximate to the opposite surface of the wafer (i.e., the cathodic surface) to apply approximately 5 kV across the wafer for approximately 15 minutes. This procedure results in a second-order optical nonlinearity profile approximately 20-30 μm thick and localized beneath the anodic surface as described in more detail by Thomas G. Alley et al. in Space Charge Dynamics in Thermally Poled Fused Silica, Journal of Non-Crystalline Solids, Vol. 242, 1998, pages 165-176, which is incorporated herein in its entirety. Other materials and objects (e.g., optical fibers) with inherent or induced second-order optical nonlinearity profiles, and a wide range of poling conditions, are compatible with embodiments of the present invention.
FIGS. 3B and 3C schematically illustrate two measurement configurations in accordance with embodiments of the present invention. In such embodiments, the composite sample 30 has a surface normal direction n. A laser 40 produces a laser beam 42 of light with a fundamental frequency ν. The laser beam 42 is incident on the composite sample 30 at an angle θ relative to the surface normal direction n. In the embodiment of FIG. 3A, a detector 50 is positioned on the opposite side of the composite sample 30 so as to detect the second harmonic (SH) signal 52 (having a frequency 2ν) from the composite sample 30. In certain embodiments, appropriate optical filters (not shown) are placed between the composite sample 30 and the detector 50 to reduce the laser power transmitted by the composite sample 30 to be well below the weaker power level of the SH signal. The embodiment of FIG. 3B can be considered to be a transmission configuration. In the embodiment of FIG. 3C, the detector 50 is positioned on the same side of the composite sample 30 as the laser 40 so as to detect the SH signal 52. The embodiment of FIG. 3C can be considered to be a reflection configuration. As used herein, the term “double pass” refers to configurations in which Maker fringe data are obtained from two wafers proximate to one another (e.g., as shown in FIGS. 3B and 3C). As used herein the term “single pass” refers to configurations in which Maker fringe data are obtained from a single wafer.
In certain embodiments, the composite sample 30 is positioned so that the laser beam 42 is first incident on the sample 10, while in other embodiments, the laser beam 42 is first incident on the supplemental sample 20. In certain embodiments, the SH signal 52 is measured as a function of the incident angle θ of the laser beam 42 by rotating the composite sample 30 relative to the laser beam 42.
FIGS. 4A-4D schematically illustrate various embodiments of the supplemental sample 20 in accordance with embodiments of the present invention. In certain embodiments, the supplemental sample 20 has a second-order optical nonlinearity profile substantially identical to that of the sample 10. In the embodiment schematically illustrated in FIG. 4A, the sample 10 and the supplemental sample 20 comprise two portions of a common sample 60. For example, a fused silica wafer which serves as the common sample 60 can be poled as described above, resulting in a poled region 62 beneath the anodic surface 63 of the wafer 60. The wafer can then be cut in half, producing two portions 64, 65, which can serve as the sample 10 and the supplemental sample 20. The sample 10 and the supplemental sample 20 can then be sandwiched together, thereby forming the composite sample 30. In certain embodiments, the sample 10 and the supplemental sample 20 are sandwiched together with the poled regions of the portions 64, 65 proximate to one another (referred to herein as an anode-to-anode configuration). In certain such embodiments, the two halves of the surface 63 are in contact with one another, while in other embodiments, there is empty space or a spacer material between the two portions 64, 65. In other embodiments, the sample 10 and the supplemental sample 20 are sandwiched together with the poled regions of the portions 64, 65 on the outer sides of the composite sample 30 (referred to herein as a cathode-to-cathode configuration). In certain such embodiments, the two portions 64, 65 are in contact with one another, while in other embodiments, there is empty space or a spacer material between the two portions 64, 65.
In the embodiment schematically illustrated in FIG. 4B, the supplemental sample 20 is prepared using substantially identical conditions as those used to prepare the sample 10. For example, two substantially identical fused silica wafers 70, 74 can be poled sequentially or simultaneously as described above using substantially identical conditions, resulting in corresponding poled regions 71, 75 beneath corresponding surfaces 72, 76. In this way, one wafer 70 serves as the sample 10, and the other wafer 74 serves as the supplemental sample 20. The sample 10 and the supplemental sample 20 can then be sandwiched together, thereby forming the composite sample 30. In certain embodiments, the sample 10 and the supplemental sample 20 are sandwiched together with the poled regions 71, 75 proximate to one another in the anode-to-anode configuration. In certain such embodiments, the two surfaces 72, 76 are in contact with one another, while in other embodiments, there is empty space, or a spacer material, between the two wafers 70, 74. In other embodiments, the sample 10 and the supplemental sample 20 are sandwiched together with the poled regions 71, 75 on opposite sides of the composite sample 30 in the cathode-to-cathode configuration. In certain such embodiments, the two wafers 70, 74 are in contact with one another, while in other embodiments, there is empty space or a spacer material between the two wafers 70, 74.
In the embodiment schematically illustrated in FIG. 4C, the supplemental sample 20 comprises a reflector 80 with a reflecting surface 81. The sample 10 of such embodiments can comprise a wafer 82 with a pooled region 83 beneath a surface 84 of the wafer 82. The sample 10 and the supplemental sample 20 can then be sandwiched together, thereby forming the composite sample 30. In certain embodiments, the sample 10 and the supplemental sample 20 are sandwiched together with the pooled region 83 proximate to the reflecting surface 81. In certain such embodiments, the reflecting surface 81 and the surface 84 are in contact with one another, while other embodiments have an empty space or a spacer material between the two surfaces 81, 84. When detecting the SH signal 52 in the reflection configuration, the reflector 80 of such embodiments provides an image of the second-order nonlinearity profile of the sample 10 substantially identical to the second-order nonlinearity profile of the sample 10.
In alternative embodiments, as schematically illustrated in FIG. 4D, the supplemental sample 20 and the poled sample 82 are sandwiched together with the reflecting surface 81 placed against the sample surface 85 on the opposite side of the sample 10 from the surface 84. In certain such embodiments, the two surfaces 81, 85 are in contact with one another, while in other embodiments, there is empty space or a spacer material between the two surfaces 81, 85.
In certain embodiments, placing the sample 10 proximate to the supplemental sample 20 comprises sandwiching the sample 10 and the supplemental sample 20 together. In certain such embodiments, the sample 10 and the supplemental sample 20 are clamped together, while in other embodiments, the sample 10 and the supplemental sample 20 are glued together. Other methods of placing the sample 10 proximate to the supplemental sample 20 are compatible with embodiments of the present invention.
FIG. 5 illustrates exemplary Maker fringe profiles measured from a common sample 60 (shown as open circles) and from a composite sample 30 (shown as crosses). As described above, the measured fringe profile corresponds to the Fourier transform of the second-order optical nonlinearity profile. In certain embodiments, measuring the Fourier transform comprises measuring the Maker fringe profile of the composite sample 30. The composite sample 30 was formed by cutting the common sample 60 in half and placing the two halves proximate to one another with the surfaces near the poled regions in contact with one another. The fringe profile of the composite sample 30 is more intense and is shifted towards lower angles than that of the common sample 60 because the nonlinear region of the composite sample 30 is twice as thick as the nonlinear region of the common sample 60.
FIG. 6A illustrates the second-order nonlinearity profile d33(z) of the composite sample 30 obtained by calculating the inverse Fourier transform of the measured fringe profile from the composite sample 30 as illustrated in FIG. 5. The abscissa z=0 corresponds to the boundary between the mated surfaces of the sandwiched sample 10 and the supplemental sample 20. By construction, the nonlinearity profile of the composite sample 30 is the juxtaposition of the d33(z) profile of the sample 10 and its mirror image d33(−z) from the supplemental sample 20. By retaining only the z>0 portion of the nonlinearity profile of FIG. 6A, the original nonlinearity profile d33(z) of the sample 10 is directly obtained, as illustrated in FIG. 6B. The nonlinearity profile of FIG. 6B represents an unambiguously derived nonlinearity profile of the thermally poled silica sample 10. The need for the phase information has been eliminated by artificially creating a symmetric nonlinearity profile based on the nonlinearity profile of the sample 10.
FIG. 6B also includes information regarding the depth location of the nonlinearity, which would not be available using prior methods, even if the phase information was retrieved. FIG. 6B shows that d33(z) changes sign three times and that its peak value is −0.8 picometer per volt (pm/V). This result is the highest reliable value of d33(z) reported for thermally poled silica. The peak of the nonlinearity profile is located just under the surface, and the poled region extends approximately 25 μm under the surface.
The main mechanism believed to be responsible for the second-order optical susceptibility X(2) in thermally poled silica is DC rectification of the third-order optical susceptibility X(3) of silica. As described more fully by D. Faccio et al. and T. G. Alley et al. (cited above), the second-order susceptibility X(2) is proportional to X(3)E(z), where E(z) is the permanent electric field that develops inside the glass during poling.
The charge distribution within the glass which generates the permanent electric field E(z) can be determined from the nonlinearity profile d33(z). Since the poled region is essentially a thin infinite plane, the electric field E(z) is related to the charge density ρ(z) by the one-dimensional form of Maxwell's equation, δE/δz=ρ(z)/ε, where ε is the dielectric susceptibility of the medium. Since d33(z) is proportional to E(z), the charge distribution ρ(z) can be derived by differentiating the profile d33(z). Applied to the nonlinearity profile of FIG. 6B, this method yields the charge distribution of FIG. 7. The solid curve of FIG. 7 is derived directly from the Maker fringe data by taking the derivative of the d33(z) profile of FIG. 6B. The oscillatory behavior is a numerical artifact resulting from the finite angular sampling of the measured Maker fringe data of FIG. 5. The dashed curve is a best-fit polynomial function to the solid curve.
The charge distribution ρ(z) shows an accumulation of positive charge near the surface, with a negatively charged region deeper in the sample 10. The centers of the two regions are separated by approximately 15 μm. A smaller, positively charged-region is also apparent approximately 45 μm into the sample 10. The charge distribution illustrated by FIG. 7 agrees qualitatively with results obtained with alternative techniques. Examples of these alternative techniques are described by P. G. Kazansky et al. in Thermally Poled Silica Glass: Laser Induced Pressure Pulse Probe of Charge Distribution, Applied Physics Letters, Vol. 68, 1996, pages 269-271, which is incorporated in its entirety by reference herein, and by D. Faccio et al. (cited above).
The nonlinearity profile derived in accordance with embodiments of the present invention can be independently verified by comparing the measured Maker fringe profile from the original sample 10 (shown as open circles in FIG. 5 and FIG. 8) with the theoretical fringe profile (solid curve in FIG. 8) calculated from d33(z) obtained from the measured SH signal from the composite sample 30 (FIG. 6B). This comparison, illustrated in FIG. 8, shows that the two fringe profiles agree reasonably well, and that in particular, the spatial uniformity of the second-order susceptibility of the sample 10 was sufficient to infer the nonlinearity profile reliably.
Embodiments of the present invention enable the second-order nonlinearity profile of an optically nonlinear film to be inferred unambiguously from a Maker fringe profile measurement. As described above, the nonlinearity profile of an exemplary thermally poled silica sample has been determined to: (i) have a peak value of approximately −0.8 pm/V, (ii) extend approximately 25 μm below the sample surface, and (iii) take both positive and negative values. Such magnitude and spatial information of the nonlinearity profile and of the charge distribution has significant implications in the design of future devices based on thermally poled silica.
Embodiments described above can be considered to be special cases of a more general method of determining the second-order nonlinearity profile. In the embodiments described above in which the sample 10 has been cut into two pieces which form the composite sample 30, two assumptions have been made: (1) the nonlinearity profiles of both the sample 10 and the supplemental sample 20 have the same functional dependence normal to the sample surface (i.e., ƒ(z)); and (2) both the sample 10 and the supplemental sample 20 have the same nonlinear strength normal to the sample surface (i.e., d33(z)=K·ƒ(z), where K is the same constant for both the sample 10 and the supplemental sample 20). These assumptions in principle limit the application of above-described embodiments to nonuniform poled samples.
In another embodiment, the nonlinearity profile of a sample is uniquely determined without either of the two assumptions described above. In addition, the sample 10 need not be cut into two pieces to determine the nonlinearity profile.
FIG. 9 is a flow diagram of a method 150 of measuring a sample nonlinearity profile 15 of a sample 10. In an operational block 152, a sample 10 having at least one sample surface and having a sample nonlinearity profile 15 along a sample line through a predetermined point on the sample surface is provided. The sample line is oriented perpendicularly to the sample surface. In an operational block 154, a Fourier transform of the sample nonlinearity profile 15 is measured. In an operational block 156, a reference material having at least one reference surface and having a reference nonlinearity profile along a reference line through a predetermined point on the reference surface is provided. The reference line is oriented perpendicularly to the reference surface. In an operational block 158, a Fourier transform of the reference nonlinearity profile is obtained. In an operational block 160, a first composite sample having a first composite nonlinearity profile is formed. The first composite sample is formed by placing the sample 10 and the reference material proximate to one another in a first configuration with the sample line substantially collinear with the reference line. In an operational block 162, a Fourier transform of the first composite nonlinearity profile is measured. In an operational block 164, a second composite sample having a second composite nonlinearity profile inequivalent to the first composite nonlinearity profile is formed. The second composite sample is formed by placing the sample 10 and the reference material proximate to one another in a second configuration with the sample line substantially collinear with the reference line. In an operational block 166, a Fourier transform of the second composite nonlinearity profile is measured. In an operational block 168, the sample nonlinearity profile 15 is calculated using the Fourier transforms of the sample nonlinearity profile, the reference nonlinearity profile, the first composite nonlinearity profile, and the second composite nonlinearity profile.
In certain embodiments, the Fourier transform of the sample nonlinearity profile is measured in the operational block 154 by measuring the Maker fringe (MF) data of the sample in a single-pass configuration as schematically illustrated by FIG. 10A. The MF data of the sample (MF1) is thus representative of the Fourier transform of the sample nonlinearity profile along a sample line through a predetermined point (labelled “A” in FIG. 10A) on the sample surface. The sample line is oriented perpendicularly to the sample surface. Similarly, as schematically illustrated in FIG. 10B, the Fourier transform of the reference nonlinearity profile can be measured by measuring the MF data of the reference material in a single-pass configuration. The MF data of the reference material (MF2) is thus representative of the Fourier transform of the reference nonlinearity profile along a reference line through a predetermined point (labelled “B” in FIG. 10B) on the reference surface. As used below, the MF data of the sample is expressed as:

MF1(ƒ)=|DA(ƒ)|2  (2)

and the MF data of the reference sample is expressed as:

MF2(ƒ)=|DB(ƒ)|2  (3)

where the Fourier transform of the sample nonlinearity profile dA(z) is denoted by 



  
    
      
        
          d
          A
        
        ⁡
        
          (
          z
          )
        
      
      ⁢
      
        ⟶
        FT
      
      ⁢
      
        
          D
          A
        
        ⁡
        
          (
          f
          )
        
      
    
    =
    
      
        
        
          
            D
            A
          
          ⁡
          
            (
            f
            )
          
        
        
      
      ⁢
      
        e
        
          
            jφ
            A
          
          ⁡
          
            (
            f
            )
          
        
      
    
  
  ,




the Fourier transform of the reference nonlinearity profile dB(z) is denoted by 



  
    
      
        
          d
          B
        
        ⁡
        
          (
          z
          )
        
      
      ⁢
      
        ⟶
        FT
      
      ⁢
      
        
          D
          B
        
        ⁡
        
          (
          f
          )
        
      
    
    =
    
      
        
        
          
            D
            B
          
          ⁡
          
            (
            f
            )
          
        
        
      
      ⁢
      
        e
        
          
            jφ
            B
          
          ⁡
          
            (
            f
            )
          
        
      
    
  
  ,




and ƒ is the spatial frequency.

FIGS. 11A-11D schematically illustrate various configurations for forming composite samples in accordance with embodiments of the present invention. Each of the composite samples of FIGS. 11A-11D has a corresponding composite nonlinearity profile. In FIG. 11A, the composite sample is formed by placing the anodic surface of the sample and the anodic surface of the reference material proximate to one another, and is referred to herein as an anode-to-anode configuration. In FIG. 11B, the composite sample is formed by placing the cathodic surface of the sample and the cathodic surface of the reference material proximate to one another, and is referred to herein as a cathode-to-cathode configuration. In FIG. 11C, the composite sample is formed by placing the anodic surface of the sample and the cathodic surface of the reference material proximate to one another, and is referred to herein as an anode-to-cathode configuration. In FIG. 11D, the composite sample is formed by placing the cathodic surface of the sample and the anodic surface of the reference material proximate to one another, and is referred to herein as a cathode-to-anode configuration.
In certain embodiments, the Fourier transform of the first composite linearity profile is measured in the operational block 162 by measuring the MF data of the first composite sample in a double-pass configuration. In embodiments in which the first composite sample has a first configuration as schematically illustrated by one of FIGS. 11A-11D, the MF data of the first composite sample is thus representative of the Fourier transform of the first composite nonlinearity profile along the dashed line of the corresponding one of FIGS. 11A-11D.
In certain embodiments, the Fourier transform of the second composite nonlinearity profile is measured in the operational block 166 by measuring the MF data of the second composite sample in a double-pass configuration. The configuration of the second composite sample is chosen to provide a second composite nonlinearity profile inequivalent to the first composite nonlinearity profile. For example, if the first configuration of the first composite sample is that of FIG. 11A or FIG. 11B, then the second configuration of the second composite sample can be that of FIG. 11C or FIG. 11D. Similarly, if the first configuration of the first composite sample is that of FIG. 11C or FIG. 11D, then the second configuration of the second composite sample can be that of FIG. 11A or FIG. 11B. The MF data of the second composite sample is thus representative of the Fourier transform of the second composite nonlinearity profile along the dashed line of the corresponding one of FIGS. 11A-11D.
The MF data of the anode-to-anode configuration of FIG. 11A (MFS1) contains the same information as the MF data of the cathode-to-cathode configuration of FIG. 11B (MFS2). Similarly, the MF data of the anode-to-cathode configuration of FIG. 11C (MFS3) contains the same information as the MF data of the cathode-to-anode configuration of FIG. 11D (MFS4). The MF data corresponding to the four possible configurations shown in FIGS. 11A-11D can be written as:

MFS1=|DA|2+|DB|2+2|DA∥DB|cos (φA+φB)  (4)

MFS2=|DA|2+|DB|2+2|DA∥DB|cos (φA+φB)  (5)

MFS3=|DA|2+|DB|2+2|DA∥DB|cos (φA+φB)  (6)

MFS4=|DA|2+|DB|2+2|DA∥DB|cos (φA+φB)  (7)

From Equations 2, 3, 4, and 5, it can be seen that MFS1 and MFS2 are equivalent to one another (i.e., they have the same information in terms of φA and ΦB). The extra modulation term φ0 does not contribute information regarding the nonlinearity profiles since it corresponds to a modulation term due to the summed thicknesses of the sample and reference material. Similarly, MFS3 and MFS4 are equivalent to one another, but are inequivalent to MFS1, and MFS2. Thus, in certain embodiments, either MFS1 or MFS2 is used as a first independent source of information, and either MFS3 or MFS4 is used as a second independent source of information. While the description below focuses on the configurations of FIGS. 11A and 11C, in certain embodiments, one or both of these configurations can be substituted by its equivalent configuration in FIGS. 11B and 11D, respectively.

For configurations of FIGS. 11A and 11C, the effective nonlinearity profiles can respectively be written as: dS1(z)=dA(z)+dB(−z) and dS2(z)=dA(z)+dB(z+L) where S1 and S2 respectively denote the anode-to-anode and anode-to-cathode configurations of FIGS. 11A and 11C. As used herein, the z=0 point is assumed to be at the boundary between the sample and the reference material. Also, as used herein, the Fourier transforms of these functions are denoted as: 



  
    
      
        d
        S1
      
      ⁡
      
        (
        z
        )
      
    
    ⁢
    
      ⟶
      FT
    
    ⁢
    
      
        D
        S1
      
      ⁡
      
        (
        f
        )
      
    
  
  ⁢
  
     
  
  ⁢
  and
  ⁢
  
     
  
  ⁢
  
    
      
        
          d
          S2
        
        ⁡
        
          (
          z
          )
        
      
      ⁢
      
        ⟶
        FT
      
      ⁢
      
        
          D
          S2
        
        ⁡
        
          (
          f
          )
        
      
    
    .
  




The Fourier transforms of the sample nonlinearity profile, the reference nonlinearity profile, the first composite nonlinearity profile, and the second composite nonlinearity profile are used to calculate the sample nonlinearity profile in the operational block 168. The MF measurements corresponding to the two composite samples of FIGS. 11A and 11C can be expressed as:

MFS1=|DS1(ƒ)|2=|DA(ƒ)+DB(−ƒ)|2=∥DA(ƒ)|ejφA(ƒ)+|DB(ƒ)|e−jφB(ƒ)|2  (8)

MFS2=|DS2(ƒ)|2=|DA(ƒ)+DB(ƒ)·ej2πƒL|2=∥DA(ƒ)|·ejφA(ƒ)+|DB(ƒ)|·ej[φB(ƒ)+φ0(ƒ)]∥2  (9)

where φ0(ƒ)=2 πƒL. Expanding the absolute value sign in Equations 8 and 9, MFS1 and MFS2 can be expressed as:

MFS1=|DA|2+|DB|2+2|DA∥DB|cos (φA+φB)  (10)

MFS2=|DA|2+|DB|2+2|DA∥DB|cos (φA−φB−φ−φ0)  (11)

where, the frequency dependencies of all the functions have been dropped for convenience.

Once the four sets of MF data (MF1, MF2, MFS1 and MFS2, given by Equations 2, 3, 10, and 11, respectively) are either measured or otherwise obtained, they can be used to express the following quantities: 



  
    
      
        
          cos
          ⁢
          
             
          
          ⁢
          
            (
            
              
                φ
                A
              
              +
              
                φ
                B
              
            
            )
          
        
        =
        
          
            
              
                MF
                S1
              
              -
              
                MF
                1
              
              -
              
                MF
                2
              
            
            
              2
              ⁢
              
                
                  
                    MF
                    1
                  
                  ·
                  
                    MF
                    2
                  
                
              
            
          
          =
          
            α
            Δ
          
        
      
    
    
      
        (
        12
        )
      
    
  
  
    
      
        
          cos
          ⁢
          
             
          
          ⁢
          
            (
            
              
                φ
                A
              
              -
              
                φ
                B
              
              -
              
                φ
                0
              
            
            )
          
        
        =
        
          
            
              
                MF
                S2
              
              -
              
                MF
                1
              
              -
              
                MF
                2
              
            
            
              2
              ⁢
              
                
                  
                    MF
                    1
                  
                  ·
                  
                    MF
                    2
                  
                
              
            
          
          =
          
            β
            Δ
          
        
      
    
    
      
        (
        13
        )
      
    
  




Note that α, β, and Δ are functions of frequency, ƒ, and are fully determined by the MF data from the sample, reference material, and the first and second composite samples. Equations 12 and 13 can be rewritten in the following form: 



  
    
      
        
          
            φ
            A
          
          +
          
            φ
            B
          
        
        =
        
          
            2
            ⁢
            
              π
              ·
              m
            
          
          ±
          
            
            
              
                cos
                
                  -
                  1
                
              
              ⁡
              
                (
                
                  α
                  Δ
                
                )
              
            
            
          
        
      
    
    
      
        (
        14
        )
      
    
  
  
    
      
        
          
            φ
            A
          
          -
          
            φ
            B
          
        
        =
        
          
            
              2
              ⁢
              
                π
                ·
                n
              
            
            ±
            
              
              
                
                  cos
                  
                    -
                    1
                  
                
                ⁡
                
                  (
                  
                    β
                    Δ
                  
                  )
                
              
              
            
          
          +
          
            φ
            0
          
        
      
    
    
      
        (
        15
        )
      
    
  




where m and n can take any integer value (0,±1,±2, . . . ). Note also that the output of the inverse cosine function is between 0 and π. Equations 14 and 15 can be combined to express the following quantities: 



  
    
      
        
          
            φ
            A
          
          -
          
            
              φ
              0
            
            2
          
        
        =
        
          
            
              π
              ·
              k
            
            ±
            
              
                1
                2
              
              ⁢
              
                
                
                  
                    cos
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      α
                      Δ
                    
                    )
                  
                
                
              
            
          
          ±
          
            
              1
              2
            
            ⁢
            
              
              
                
                  cos
                  
                    -
                    1
                  
                
                ⁡
                
                  (
                  
                    β
                    Δ
                  
                  )
                
              
              
            
          
        
      
    
    
      
        (
        16
        )
      
    
  
  
    
      
        
          
            φ
            B
          
          +
          
            
              φ
              0
            
            2
          
        
        =
        
          
            
              π
              ·
              l
            
            ±
            
              
                1
                2
              
              ⁢
              
                
                
                  
                    cos
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      α
                      Δ
                    
                    )
                  
                
                
              
            
          
          ∓
          
            
              1
              2
            
            ⁢
            
              
              
                
                  cos
                  
                    -
                    1
                  
                
                ⁡
                
                  (
                  
                    β
                    Δ
                  
                  )
                
              
              
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




where k and l are any integers. By taking the cosine of both sides and taking their absolute values, Equations 16 and 17 can be rewritten in the following form: 



  
    
      
        
          
          
            cos
            ⁡
            
              (
              
                
                  φ
                  A
                
                -
                
                  
                    φ
                    0
                  
                  2
                
              
              )
            
          
          
        
        =
        
          
          
            cos
            ⁡
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                    
                      
                        cos
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          α
                          Δ
                        
                        )
                      
                    
                    
                  
                
                ±
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                    
                      
                        cos
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          β
                          Δ
                        
                        )
                      
                    
                    
                  
                
              
              )
            
          
          
        
      
    
    
      
        (
        18
        )
      
    
  
  
    
      
        
          
          
            cos
            ⁡
            
              (
              
                
                  φ
                  B
                
                +
                
                  
                    φ
                    0
                  
                  2
                
              
              )
            
          
          
        
        =
        
          
          
            cos
            ⁡
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                    
                      
                        cos
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          α
                          Δ
                        
                        )
                      
                    
                    
                  
                
                ∓
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                    
                      
                        cos
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          β
                          Δ
                        
                        )
                      
                    
                    
                  
                
              
              )
            
          
          
        
      
    
    
      
        (
        19
        )
      
    
  




Equations 18 and 19 provide useful information towards the calculation of the sample nonlinearity profile. Note that the right-hand side of both Equations 18 and 19 have the same two possible values, which are given by: 



  
    
      
        
          P
          1
        
        =
        
          
          
            cos
            ⁡
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                    
                      
                        cos
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          α
                          Δ
                        
                        )
                      
                    
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                    
                      
                        cos
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          β
                          Δ
                        
                        )
                      
                    
                    
                  
                
              
              )
            
          
          
        
      
    
    
      
        (
        20
        )
      
    
  
  
    
      
        
          P
          2
        
        =
        
          
          
            cos
            ⁡
            
              (
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                    
                      
                        cos
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          α
                          Δ
                        
                        )
                      
                    
                    
                  
                
                -
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                    
                      
                        cos
                        
                          -
                          1
                        
                      
                      (
                      
                        β
                        Δ
                      
                      )
                    
                    
                  
                
              
              )
            
          
          
        
      
    
    
      
        (
        21
        )
      
    
  




where P1 and P2 denote these two possible values. Notice also that if 



  
    
      
      
        cos
        ⁡
        
          (
          
            
              φ
              A
            
            -
            
              
                φ
                0
              
              2
            
          
          )
        
      
      
    
    =
    
      P
      1
    
  
  ,
  
    
      then
      ⁢
      
        
           
        
        ⁢
        
           
        
      
      ⁢
      
        
        
          cos
          ⁡
          
            (
            
              
                φ
                B
              
              +
              
                
                  φ
                  0
                
                2
              
            
            )
          
        
        
      
    
    =
    
      
        P
        2
      
      .
    
  




The reverse is also true, i.e., if 



  
    
      
      
        cos
        ⁡
        
          (
          
            
              φ
              A
            
            -
            
              
                φ
                0
              
              2
            
          
          )
        
      
      
    
    =
    
      P
      2
    
  
  ,
  
    
      then
      ⁢
      
         
      
      ⁢
      
        
        
          cos
          ⁡
          
            (
            
              
                φ
                B
              
              +
              
                
                  φ
                  0
                
                2
              
            
            )
          
        
        
      
    
    =
    
      
        P
        1
      
      .
    
  




Using the Fourier transforms of the sample and reference material nonlinearity profiles, the following quantities can be expressed: 



  
    
      
        
          
            d
            A
          
          ⁡
          
            (
            
              z
              -
              
                L
                2
              
            
            )
          
        
        +
        
          
            
              
                d
                A
              
              ⁡
              
                (
                
                  
                    -
                    z
                  
                  -
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              ⟶
              FT
            
            ⁢
            2
          
          ⁢
          
            
            
              D
              A
            
            
          
          ⁢
          
            cos
            ⁡
            
              (
              
                
                  φ
                  A
                
                -
                
                  
                    φ
                    0
                  
                  2
                
              
              )
            
          
        
      
    
    
      
        (
        22
        )
      
    
  
  
    
      
        
          
            d
            B
          
          ⁡
          
            (
            
              z
              +
              
                L
                2
              
            
            )
          
        
        +
        
          
            
              
                d
                B
              
              ⁡
              
                (
                
                  
                    -
                    z
                  
                  +
                  
                    L
                    2
                  
                
                )
              
            
            ⁢
            
              ⟶
              FT
            
            ⁢
            2
          
          ⁢
          
            
            
              D
              B
            
            
          
          ⁢
          
            cos
            ⁡
            
              (
              
                
                  φ
                  B
                
                +
                
                  
                    φ
                    0
                  
                  2
                
              
              )
            
          
        
      
    
    
      
        (
        23
        )
      
    
  




FIGS. 12A and 12B schematically illustrate two symmetric nonlinearity profiles for two arbitrary functions dA(z) and dB(z). In FIGS. 12A and 12B, the sharp boundaries at z=±L/2 correspond to the anodic surfaces of the sample and reference material. The major difference between FIGS. 12A and 12B is that the nonzero nonlinearity profile is in the 



  
    
    z
    
  
  <
  
    L
    2
  




region

for 



  
    
      
        d
        A
      
      ⁡
      
        (
        
          z
          -
          
            L
            2
          
        
        )
      
    
    +
    
      
        d
        A
      
      ⁡
      
        (
        
          
            -
            z
          
          -
          
            L
            2
          
        
        )
      
    
  
  ,




whereas for 



  
    
      
        d
        B
      
      ⁡
      
        (
        
          z
          +
          
            L
            2
          
        
        )
      
    
    +
    
      
        d
        B
      
      ⁡
      
        (
        
          
            -
            z
          
          +
          
            L
            2
          
        
        )
      
    
  
  ,




the nonzero nonlinearity

profile is in the 



  
    
    z
    
  
  >
  
    L
    2
  




region.

Equations 22 and 23 are connected to Equations 18 and 19 because of the 



  
    
    
      cos
      ⁡
      
        (
        
          
            φ
            A
          
          -
          
            
              φ
              0
            
            2
          
        
        )
      
    
    
  
  ⁢
  
     
  
  ⁢
  and
  ⁢
  
     
  
  ⁢
  the
  ⁢
  
     
  
  ⁢
  
    
    
      cos
      ⁡
      
        (
        
          
            φ
            B
          
          +
          
            
              φ
              0
            
            2
          
        
        )
      
    
    
  




terms. This connection can be used to remove the final ambiguity of whether 



  
    
    
      cos
      ⁡
      
        (
        
          
            φ
            A
          
          -
          
            
              φ
              0
            
            2
          
        
        )
      
    
    
  
  =
  
    
      
        P
        1
      
      ⁢
      
         
      
      ⁢
      or
      ⁢
      
         
      
      ⁢
      
        
        
          cos
          ⁡
          
            (
            
              
                φ
                A
              
              -
              
                
                  φ
                  0
                
                2
              
            
            )
          
        
        
      
    
    =
    
      
        P
        2
      
      .
    
  




Since 



  
    
      d
      A
    
    ⁡
    
      (
      
        z
        -
        
          L
          2
        
      
      )
    
  
  +
  
    
      d
      A
    
    ⁡
    
      (
      
        
          -
          z
        
        -
        
          L
          2
        
      
      )
    
  




is a symmetric and real function, its Fourier transform magnitude is sufficient to uniquely recover it. Note that for a real and symmetric function, the Hartley transform is the same as the Fourier transform. The same is also true for 



  
    
      
        d
        B
      
      ⁡
      
        (
        
          z
          +
          
            L
            2
          
        
        )
      
    
    +
    
      
        d
        B
      
      ⁡
      
        (
        
          
            -
            z
          
          +
          
            L
            2
          
        
        )
      
    
  
  ,




which is also a symmetric and real function. The final ambiguity can then be removed.

In certain embodiments, the ambiguity is removed in the following manner. If 



  
    
      
        
        
          cos
          ⁡
          
            (
            
              
                φ
                A
              
              -
              
                
                  φ
                  0
                
                2
              
            
            )
          
        
        
      
      =
      
        P
        1
      
    
    ,
  
  ⁢
  
     
  




then the function 2|DA|P1 is going to be the Fourier transform magnitude of 



  
    
      d
      A
    
    ⁡
    
      (
      
        z
        -
        
          L
          2
        
      
      )
    
  
  +
  
    
      d
      A
    
    ⁡
    
      (
      
        
          -
          z
        
        -
        
          L
          2
        
      
      )
    
  




(see Equation 22). If the inverse Fourier transform gives the poled region in 



  
    
      
      z
      
    
    <
    
      L
      2
    
  
  ,
  
    
      then
      ⁢
      
         
      
      ⁢
      
        
        
          cos
          ⁡
          
            (
            
              
                φ
                A
              
              -
              
                
                  φ
                  0
                
                2
              
            
            )
          
        
        
      
    
    =
    
      
        P
        1
      
      .
    
  




Otherwise, 



  
    
    
      cos
      ⁡
      
        (
        
          
            φ
            A
          
          -
          
            
              φ
              0
            
            2
          
        
        )
      
    
    
  
  =
  
    
      P
      2
    
    .
  




This result can be double checked by computing the inverse Fourier transform of the Fourier transform magnitude 2|DA|P2 and confirming that the poled region is given by 



  
    
    z
    
  
  <
  
    
      L
      2
    
    .
  




Note finally that the inverse Fourier transforms of both Fourier transform magnitudes 2|DA|P1 and 2|DA|P2 do not at the same time give a poled region in 



  
    
      
      z
      
    
    <
    
      L
      2
    
  
  ,




because either 2|DA|P1 or 2|DA|P2 has to yield a poled region in 



  
    
    z
    
  
  >
  
    L
    2
  




for the function 



  
    
      d
      B
    
    ⁡
    
      (
      
        z
        +
        
          L
          2
        
      
      )
    
  
  +
  
    
      d
      B
    
    ⁡
    
      (
      
        
          -
          z
        
        +
        
          L
          2
        
      
      )
    
  




(see Equation 23).

In certain other embodiments, the ambiguity is removed by computing the inverse Fourier transform of both 2|DA|P1 and 2|DA|P2. The results should yield two symmetric functions as shown in FIGS. 12A and 12B. Taking only the z>0 portion of the resulting profiles yields two alternative profiles for dA(z). The same procedure can be applied for dB(z). By computing the theoretical MF curve of these two possible dA(z) profiles and comparing the results with the measured MF1(ƒ)=|DA(ƒ)|2 data, it is straightforward to choose the correct possibility.
Thus, it is possible to uniquely recover the two arbitrary nonlinearity profiles (dA(z) and dB(z)) using four sets of MF data. This result is significant since (1) the technique is applicable to even nonuniform poled samples; and (2) cutting the sample into two halves is not necessary. The technique utilizes a reference material with known MF data. In certain embodiments, obtaining the MF data of the reference material comprises measuring the MF data. In other embodiments, the MF data of the reference material is previously measured and stored in memory, and obtaining the MF data comprises reading the MF data from memory. Embodiments in which the MF data is previously stored are preferable, since the number of MF measurements will be reduced from 4 to 3. Note that it is not necessary to know the reference nonlinearity profile for this technique to work, since only its corresponding MF data is used. In addition, one can use a common reference sample to characterize a plurality of samples.
In an exemplary embodiment, the sample and reference material are cut from the same poled wafer and the MF data from the sample and reference material (given by Equations 2 and 3, respectively) are related by a constant factor (i.e., MF1=C·MF2). Since points A and B are from the same poled surface, it can be assumed that:

dA(z)=√{square root over (C)}·dB(z)  (24)

It follows from Equation (24) that |DA|=√{square root over (C)}·|DB| and φA=φB. In this exemplary embodiment, the MF data of the second composite sample (MFS2, given by Equation 11) is not required to determine the sample nonlinearity profile. This result can be seen by inserting φA=φB into Equation 11 and observing that all the phase information related to the sample nonlinearity profile is removed from Equation 11. Such embodiments are preferable because the mathematics of the solution is significantly more simple and is obtained with an experimental simplicity. In this preferred embodiment, the mathematical derivation stops at Equation 12, thereby avoiding the subsequent equations.

Using the fact that |DA|=√{square root over (C)}·|DB| and φA=φB, Equation 12 can be rewritten as: 



  
    
      
        
          2
          ·
          
            
            
              D
              A
            
            
          
          ·
          
            
            
              cos
              ⁡
              
                (
                
                  φ
                  A
                
                )
              
            
            
          
        
        =
        
          
            
              
                MF
                S1
              
              -
              
                
                  MF
                  1
                
                ·
                
                  
                    (
                    
                      1
                      -
                      
                        C
                      
                    
                    )
                  
                  2
                
              
            
            
              C
            
          
        
      
    
    
      
        (
        25
        )
      
    
  




The left-hand-side of Equation 25 is the Fourier transform magnitude of dA(z)+dA(−z) and for such real and symmetric functions, the Fourier transform magnitude is sufficient to uniquely recover the original profile. The unique recovery of the sample nonlinearity profile dA(z)=√{square root over (C)}·dB(z) is thus achieved using Equation 25. Note that embodiments in which C=1 yield the same solution as do embodiments in accordance with the method of FIG. 2. This result can be verified by putting C=1 in Equation 25.
Certain embodiments described above yield the sample nonlinearity profile as well as the reference nonlinearitly profile. In such embodiments, the reference material can comprise a second sample with a second sample nonlinearity profile to be measured. Thus, the nonlinearity profiles of two samples can be measured concurrently.
In certain embodiments, the same reference material can be used for measuring the nonlinearity profiles of a plurality of samples. If the same reference material is used with different samples, the measured reference nonlinearity profile should be substantially the same from each of the measurements. Comparison of the measured reference nonlinearity profiles accompanying each of the plurality of measured sample nonlinearity profiles then provides an indication of the consistency of the measurements across the plurality of samples.
Embodiments described herein can also be used to measure other physical functions. FIG. 13 is a flowchart of a method 200 in accordance with embodiments of the present invention. In an operational block 210, a symmetric composite function is formed by superimposing the physical function with a substantially identical physical function. In an operational block 220, a Fourier transform of the symmetric composite function is obtained. In an operational block 230, an inverse Fourier transform of the obtained Fourier transform is calculated. The calculated inverse Fourier transform provides information regarding the physical function.
For example, instead of forming a symmetric composite function in the spatial domain as described above in relation to the nonlinearity profile of poled silica, other embodiments form a symmetric intensity profile in the time domain by the use of time reversal. In such embodiments, the symmetric composite function can have utility where phase information is needed but unavailable (e.g., ultra-short pulse diagnosis using auto-correlators). An example of time reversal is described by D. A. B. Miller in Time Reversal of Optical Pulses by Four-Wave Mixing, Optics Letters Vol. 5, 1980, pages 300-302, which is incorporated in its entirety by reference herein.
FIG. 14 is a flow diagram of a method 400 of determining the temporal waveform 312 of a laser pulse 310 in accordance with embodiments of the present invention. FIG. 15 schematically illustrates one configuration for utilizing four-wave mixing (FWM) to provide the time-reversed pulse 320 in accordance with embodiments of the present invention. Other configurations are also compatible with the method of FIG. 14. Referring to the elements illustrated in FIG. 15, in an operational block 410, a laser pulse 310 is provided. The laser pulse 310 has a temporal waveform 312. In an operational block 420, a time-reversed pulse 320 is provided. The time-reversed pulse 320 has a time-reversed temporal waveform 322 which corresponds to the temporal waveform 312 after being time-reversed. In an operational block 430, the temporal waveform 312 of the laser pulse 310 and the time-reversed temporal waveform 322 of the time-reversed pulse 320 form a symmetric composite waveform. In an operational block 440, a Fourier transform of the symmetric composite waveform is obtained. In an operational block 450, an inverse Fourier transform of the obtained Fourier transform is calculated. The calculated inverse Fourier transform provides information regarding the temporal waveform 312 of the laser pulse 310.
FWM has been a subject of interest in applications such as aberration compensation, spatial information processing, frequency filtering, pulse envelope shaping, and dispersion compensation. As illustrated in FIG. 15, a nonlinear medium 330 of length L is pumped by two pulsed pump waves 340, 350. An input laser pulse 310 (with temporal waveform 312 given by Ep(x,t)) launched into the nonlinear medium 330 generates a phase conjugate pulse 320 (with time-reversed temporal waveform 322 given by Ec(x,t)), which is the time-reversed version of the temporal waveform 312 of the input pulse 310. In the embodiment described below, the input pulse 310, two pump waves 340, 350, and the nonlinear medium are at the same place for the duration of the input pulse 310. In addition, the input pulse 310, and the two pump waves 340, 350 overlap in the frequency domain.
Illustratively, the temporal waveform 312 of the input pulse 310 can be written in the following form: 



  
    
      
        
          
            E
            p
          
          ⁡
          
            (
            
              x
              ,
              t
            
            )
          
        
        =
        
          
            
              1
              2
            
            ⁢
            
              
                u
                p
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              e
              
                j
                ⁡
                
                  (
                  
                    
                      
                        ω
                        p
                      
                      ⁢
                      t
                    
                    -
                    kx
                  
                  )
                
              
            
          
          +
          
            complex
            ⁢
            
               
            
            ⁢
            
              conjugate
              .
            
          
        
      
    
    
      
        (
        26
        )
      
    
  




where up(t) is the modulation of the carrier ej(ωpt−kx). The Fourier transform of up(t) has the following form:

ŪP(ω)=∫up(t)e−jωtdt.  (27)

The temporal waveform of the resultant conjugate pulse 320 has the following form: 



  
    
      
        
          
            E
            c
          
          ⁡
          
            (
            
              x
              ,
              t
            
            )
          
        
        =
        
          
            
              1
              2
            
            ⁢
            
              
                u
                c
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              e
              
                j
                ⁡
                
                  (
                  
                    
                      
                        ω
                        c
                      
                      ⁢
                      t
                    
                    +
                    kx
                  
                  )
                
              
            
          
          +
          
            complex
            ⁢
            
               
            
            ⁢
            conjugate
          
        
      
    
    
      
        (
        28
        )
      
    
  




where “c” stands for “conjugate.” Note that the k-vector of the conjugate pulse Ec(x,t) has the reverse sign as expected. The Fourier transform of the envelope function uc(t) is defined the same way:

Ūc(ω)=∫uc(t)e−jωtdt.  (29)

The relationship between the carrier frequencies ωc, ωp, as defined above, and the center frequencies ωpump,1 and ωpump,2 of the two pumps 340, 350 is:

ωpump,1+ωpump,2−ωp=ωc  (30)

With these definitions, the envelope function uc(t) can be expressed as:

uc(t)=∫h(ω)Ūp*(−ω)ejωtdω  (31)

where h(ω) is the response function of the nonlinear material 330. For broadband conjugators (with respect to the spectrum of up(t)), h(ω) can be taken as a constant (K), giving uc(t) the following form:

uc(t)=K∫Ūp*(−ω)ejωtdω.  (32)

The foregoing forms of the envelope function uc(t) were obtained using the teachings of R. A. Fischer et al. in Transient Analysis of Kerr-Like Phase Conjugators Using Frequency-Domain Techniques, Physical Review A, Vol. 23, 1981, pages 3071-3083, which is incorporated in its entirety by reference herein.

The above equations can be used to show that for continuous-wave (CW) pumping, FWM can not time-reverse pulsed fields. This property of FWM in CW pumping has been extensively studied for dispersion compensation in fiber links. Examples of such work include A. Yariv et al. in Compensation for Channel Dispersion by Nonlinear Optical Phase Conjugation, Optics Letters Vol. 4, 1979, pages 52-54, and S. Watanabe et al. in Compensation of Chromatic Dispersion in a Single-Mode Fiber by Optical Phase Conjugation, IEEE Photonics Technical Letters, Vol. 5, 1993, pages 92-95, both of which are incorporated in their entirety by reference herein.
However, pulsed pumping can be used to achieve time reversal of amplitude pulses. Following the derivation of D. A. B. Miller (cited above), uc(t) can be expressed as:

uc(t)=K′up*(−t+τ0)  (33)

where K′ and τ0 are constants. The −t term in Equation 33 indicates the time reversal operation. Note that Equation 26 and Equation 30 are still valid for this case. The assumptions made in the derivation of Equation 33 are that:

    
    
        (1) the nonlinear medium 330 has a length L equal to or longer than the spatial length of the input pulse 310 (i.e., large L assumption) so that the input pulse 310 is completely within the nonlinear medium 330 at some time during the interaction;
        (2) the pump pulses 340, 350 are perpendicular to the nonlinear medium as shown in FIG. 15;
        (3) pump pulses 340, 350 are short compared with the input pulse 310 (i.e., the spectra of both pump pulses 340, 350 are broad enough so that all of the frequency components of the input pulse 310 see a substantially uniform pump spectral power density for both pumps);
        (4) as a consequence of (3), the thickness of the nonlinear medium 330 is preferably equal to or slightly greater than the spatial length of the pump pulses 340, 350); and
        (5) the timing of the pulses is such that when pulse 310 spatially fills the nonlinear medium 330 (i.e. the input pulse 310 is fully within the medium), the pump pulses 340 and 350 are timed to be overlapping in space with the input pulse 310 across the nonlinear medium filled by the input pulse.

Some of the details of these assumptions can be found in D. A. B. Miller's work (cited above). As described below, the apparatus schematically illustrated by FIG. 15 can serve as a phase conjugate mirror which generates the time-reversed waveform corresponding to an input waveform.

    
    


In other embodiments, the time-reversed pulse 320 can be provided using holographic techniques in accordance with embodiments of the present invention. Femtosecond spectral holography can be considered as a temporal analog of the classical spatial domain holography. In classical holography, a spatial information carrying beam (signal) and a uniform reference beam interfere in a recording medium, thereby recording a hologram comprising a set of fringes. Illumination of the hologram with a uniform test beam reconstructs either a real or conjugate image of the signal beam, depending on the geometry.
Femtosecond spectral holography for the time reversal process comprises a recording phase and a read-out phase, as schematically illustrated respectively by FIGS. 16A and 16B. In the recording phase (FIG. 16A), the reference pulse 360 is short with a relatively flat and broad spectrum, and the input pulse 310 has a temporal waveform 312 that has a shape indicative of information carried by the input pulse 310. During the recording of the hologram, a grating 370 is used to disperse both the reference pulse 360 and the input pulse 310 into their spectral components, which then propagate through a collimating lens system 372. The interference pattern formed by the complex amplitudes of each spectral component is then recorded in the holographic recording plate 375.
In the read-out phase (FIG. 16B), a short test pulse 380 is dispersed by the grating 370 and then recollimated by the lens 372. The illumination of the holographic plate 375 with this recollimated dispersed test beam 392 produces the beam 396. Using the lens 387 for recollimation and the second grating 385, a time-reversed replica 390 of the original input pulse 310 is produced. Also as a by-product, the transmitted test beam 392 appears at the output. The details of this technique are described more fully by A. M. Weiner et al. in Femtosecond Spectral Holography, IEEE Journal of Quantum Electronics, Vol. 28, 1992, pages 2251-2261, and A. M. Weiner et al. in Femtosecond Pulse Shaping for Synthesis, Processing and Time-to-Space Conversion of Ultrafast Optical Waveforms, IEEE Journal of Selected Topics in Quantum Electronics, Vol. 4, 1998, pages 317-331, both of which are incorporated in their entireties by reference herein.
The envelope of the output pulse 390 can be expressed as:

uout(t)≈ut(t)*ur(−t)*us(t)ej{overscore (K)}1{overscore (r)}+ut(t)*ur(t)*us(−t)ej{overscore (K)}2{overscore (r)}  (34)

where uout(t), ut(t), ur(t), and us(t) are the complex envelope functions of the electric fields of the output pulse 390, test pulse 380, reference pulse 360, and input pulse 310, respectively. The sign ‘*’ denotes the convolution function, and {overscore (K)}1={overscore (k)}t−{overscore (k)}r+{overscore (k)}s and {overscore (K)}2={overscore (k)}t+{overscore (k)}r−{overscore (k)}s.

When the test pulse 380 and the reference pulse 360 are considerably shorter than the input pulse 310, us(t), ut(t), and ur(t) will act as delta functions with respect to us(t), modifying the envelope of the output pulse 390 to be:

uout(t)≈us(t)ej{overscore (K)}1{overscore (r)}+us(−t)ej{overscore (K)}2{overscore (r)}.  (35)

Therefore, as a result of the illumination of the holographic plate with the test pulse 380, the output pulse 390 serves as the time-reversed signal pulse 320 in the {overscore (K)}2 direction. As described below, the apparatus schematically illustrated by FIGS. 16A and 16B can serve as a phase conjugate mirror which generates the time-reversed waveform corresponding to an input waveform.

Note that embodiments of both the pulse-pumped FWM and the spectral holography techniques use shorter pulses than the input pulse 310 to time-reverse the input pulse 310. For pulse-pumped FWM, shorter pump pulses 340, 350 are used, and for holography, shorter test pulses 380 and reference pulses 360 are used.
In certain embodiments for estimating the temporal waveform of a laser pulse, a classical correlator configuration is used with a second-harmonic generating (SHG) crystal 510. FIG. 17 schematically illustrates one embodiment of a classical intensity correlator utilizing a Michelson interferometer 500. The input pulse 310 is split into a first pulse 520 and a second pulse 522 by the beamsplitter 530. The first pulse 520 is reflected by a fixed mirror 550 back through the beamsplitter 530, thereby transmitting a third pulse 524. The second pulse 522 is reflected by a movable mirror 552 back through the beamsplitter 530, thereby transmitting a fourth pulse 526 towards the focusing lens 540. The movable mirror 552 adds a variable delay τ to the fourth pulse 526 with respect to the third pulse 524. The focusing lens 540 focuses the third pulse 524 and the fourth pulse 526 onto the SHG crystal 510. The SHG crystal 510 generates a second-harmonic (SH) field 538 which is a cross-correlation product of the third pulse 524 and the fourth pulse 526. Other SH fields 534, 536 generated by the SHG crystal 510 do not carry information regarding the third and fourth pulses 524, 526 simultaneously, and these other SH fields are blocked by the pinhole 560 placed after the SHG crystal 510. After passing through a filter 570 which blocks light at the fundamental wavelength and which passes the SH light, the cross-correlation product 538 is detected by a photomultiplier tube 580.
The third pulse 524 emerging from one arm of the Michelson interferometer 500 has a waveform given by:

E1(t)=u1(t)ejωt  (36)

where u1(t) is the complex envelope function of the input pulse 310. The fourth pulse 526 emerging from the other arm of the Michelson interferometer 500 has a waveform given by:

E2(t)=u2(t)ejωt=u1(t−τ)ejω(t−τ)  (37)

where τ is the relative time delay between the third pulse 524 and the fourth pulse 526 imposed by the delay τ between the two arms. Because of the focusing lens 540, the third pulse 524 and the fourth pulse 526 have different k-vectors at the surface of the SHG crystal 510. Consequently, the SH field 538 generated by the SHG crystal 510 that is not blocked by the pinhole 560 includes only terms due to the interaction of the third pulse 524 and the fourth pulse 526, and has a waveform given by:

E2ω(t)=u2ω(t)ej2ωt=ηu1(t)u2(t)ej2ωt=ηu1(t)u1(t−τ)e−jωτej2ωt  (38)

where η is a conversion efficiency factor corresponding to the SHG crystal and the system geometry. The signal detected by the photomultiplier 580 (i.e., the autocorrelation function) is given by:

APMT(τ)=∫E2ω(t)E2ω*(t)dt=∫|η|2|u1(t)|2|u1(t−τ)|2dt=|η|2∫I(t)I(t−τ)dt  (39)

where I(t)=|u1(t)|2 is the input signal intensity. In general, η is a function of frequency and the functions u1(t) and u2(t) can be expanded as plane waves with Fourier transform amplitudes (i.e., ui(t)=∫Ūi(ω)ejωtdω). However, it is assumed here that η is constant over the frequency range of the combined pulses, which is equivalent to assuming that X(2) is independent of frequency. This assumption holds when the effective crystal length is shorter than the coherence length of the harmonic generation over the pulse bandwidth.

By taking the Fourier transform of both sides of Equation 39 (the signal detected by the photomultiplier 580 for a single pulse) provides the Fourier transform of the autocorrelation function and is given by:

ĀPMT—single(ƒ)=|η|2|Ī(ƒ)|2  (40)

where ĀPMT—single(ƒ) and Ī(ƒ) are the Fourier transforms of APMT(τ) and I (τ), respectively. Thus, the signal from the Michelson interferometer 500 of FIG. 12 provides the magnitude of the Fourier transform of the input pulse intensity I(t).

However, the magnitude of the Fourier transform is not sufficient information to retrieve uniquely the input pulse intensity I(t). To do so would also require the knowledge of the phase of the Fourier transform, which this classical autocorrelation method does not provide. This difficulty in uniquely determining the input pulse intensity I(t) is analogous to that of retrieving the second-order susceptibility spatial profile of a nonlinear material, as described above. In practice, a number of methods can be used to avoid this difficulty. For example, a certain shape for the pulse intensity (e.g., a Gaussian) can be assumed, but there is no way to independently verify that the assumed shape is the true temporal waveform. This method therefore provides only an estimate of the pulse shape.
By using a time-reversal technique compatible with embodiments of the present invention (e.g., pulse-pumped FWM or femtosecond spectral holography), a time-reversed pulse with a time-reversed temporal waveform I(−t) of the temporal waveform I(t) of an arbitrary pulse can be produced. A symmetric composite waveform can then be formed by delaying the time-reversed pulse with respect to the original pulse and combining the temporal waveform with the time-reversed temporal waveform, for example with a beamsplitter.
FIG. 18 schematically illustrates a general configuration 600 for converting a periodically repeated sequence of pulses into a periodically repeated sequence of symmetric pulses using a movable phase conjugative mirror 610. The input pulse sequence 620 comprises a series of original pulses 622. The original pulses 622 can be symmetric or asymmetric. Each of these original pulses 622 is split by the beamsplitter 630 into two portions 640, 642, with one portion 640 being reflected by a phase conjugate mirror 610. In certain embodiments, the phase conjugate mirror 610 comprises an apparatus as schematically illustrated by FIG. 15 in which the input pulse 310 is used to generate the output pulse 320. In certain other embodiments, the phase conjugate mirror 610 comprises an apparatus as schematically illustrated by FIGS. 16A and 16B in which the input pulse 310 is used to generate the output pulse 390. These portions 640, 642 are then combined to form a symmetric composite waveform 650 comprising the temporal waveforms of the original pulses 622 and of the corresponding phase conjugate (time-reversed) pulses 652. The amount of delay Δt between an original pulse 622 and the corresponding time-reversed pulse 652 in the symmetric pulse sequence 650 is adjustable by moving the phase conjugative mirror 610.
The symmetric composite waveform 650 has the following form: 



  
    
      
        
          
            I
            symmetric
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            I
            ⁡
            
              (
              
                
                  -
                  t
                
                -
                
                  
                    Δ
                    ⁢
                    
                       
                    
                    ⁢
                    t
                  
                  2
                
              
              )
            
          
          +
          
            I
            ⁡
            
              (
              
                t
                -
                
                  
                    Δ
                    ⁢
                    
                       
                    
                    ⁢
                    t
                  
                  2
                
              
              )
            
          
        
      
    
    
      
        (
        41
        )
      
    
  




where Δt is the variable time delay between the time-reversed pulse 652 and the original pulse 622. The Fourier transform of Equation 41 is given by: 



  
    
      
        
          
            
              I
              _
            
            symmetric
          
          ⁡
          
            (
            f
            )
          
        
        =
        
          
            
              
                I
                _
              
              ⁡
              
                (
                
                  -
                  f
                
                )
              
            
            ⁢
            
              e
              
                j
                ⁢
                
                   
                
                ⁢
                f
                ⁢
                
                  
                    Δ
                    ⁢
                    
                       
                    
                    ⁢
                    t
                  
                  2
                
              
            
          
          +
          
            
              
                I
                _
              
              ⁡
              
                (
                f
                )
              
            
            ⁢
            
              
                e
                
                  
                    -
                    j
                  
                  ⁢
                  
                     
                  
                  ⁢
                  f
                  ⁢
                  
                    
                      Δ
                      ⁢
                      
                         
                      
                      ⁢
                      t
                    
                    2
                  
                
              
              .
            
          
        
      
    
    
      
        (
        42
        )
      
    
  




Since I(t) is real, |Ī(ƒ)|=|Ī(−ƒ)| and Φ(ƒ)=−Φ(ƒ), where Ī(ƒ) has been defined as: Ī(ƒ)=|Ī(ƒ)|ejφ(ƒ). Using these identities, together with Equations 40 and 42, the Fourier transform of the autocorrelation function corresponding to the symmetric composite waveform 650 has the following form:

ĀPMT—double(ƒ)=2|η|2|Ī(ƒ)|2|1+cos (2φ(ƒ)−ƒΔt)|.  (43)

In embodiments in which ĀPMT—double is real and greater than zero for all frequencies ω (i.e., there are no zero crossings of ĀPMT—double from the cosine term), then the inverse Fourier transform of Equation 43 provides the intensity of the symmetric pulse sequence 650 without any ambiguity. Once the intensity of the symmetric pulse sequence 650 is calculated in this way, the intensity of the original pulse can be found by separating the pulse 650 in the middle.
FIG. 19 illustrates an exemplary temporal waveform of an input pulse 622 compatible with embodiments of the present invention. The autocorrelation technique commonly used to estimate the temporal profile of an optical pulse is applied to this input pulse as follows. By using the configuration schematically illustrated in FIG. 17, the autocorrelation function (Equation 39) corresponding to the temporal waveform is measured. The magnitude of the Fourier transform of the signal APMT—single(τ) recorded by the photomultiplier tube 580 is schematically illustrated in FIG. 16. The frequency ƒ shown in FIG. 20 is 1/τ, and is not to be confused with the optical frequency.
In the following, an embodiment is illustrated analytically by showing the result of numerical simulations using the input pulse of FIG. 19. First, the time-reversal of the input pulse is obtained using one of the phase conjugate mirror schemes described above. FIG. 21 illustrates the symmetric composite waveform 650 defined by Equation 41, which is obtained by combining the input pulse and the time-reversed pulse. The symmetric composite waveform 650 comprises the temporal waveform of the original pulse 622 of FIG. 18 plus the time-reversed temporal waveform of the corresponding time-reversed pulse 652, separated by some time delay.
In the second step, the autocorrelator of FIG. 17 is used to generate the autocorrelation function corresponding to the symmetric composite waveform 650. FIG. 22 illustrates the magnitude of ĀPMT—double(ƒ) (expressed by Equation 43), which is the Fourier transform of the measured APMT—double(τ). The envelope of the Fourier transform magnitude of the measured APMT—double(τ) is a scaled version of the Fourier transform magnitude (as shown in FIG. 20) of the signal APMT—single(τ) measured by the photomultiplier tube 580. Therefore, |Ī(ƒ)|2 or |Ī(ƒ)| can be recovered from only the envelope of the Fourier transform magnitude of the measured APMT—double(τ) (i.e., from only the measurement of the autocorrelation function of the symmetric composite waveform 650). This result implies that the measurement of the autocorrelation function for the single pulse as schematically illustrated in FIG. 19 is redundant. However, since it does not add complexity to the measurement, certain embodiments also comprise obtaining |Ī(ƒ)|2 data from a separate source for error checking purposes.
In the third step, the function g(ƒ)=1+cos(2φ(ƒ)−ƒΔt) is determined by dividing both sides of Equation 43 by |Ī(ƒ)|2. From the knowledge of g(ƒ)−1, the function 



  
  
    cos
    ⁡
    
      (
      
        
          φ
          ⁡
          
            (
            f
            )
          
        
        -
        
          f
          ⁢
          
            
              Δ
              ⁢
              
                 
              
              ⁢
              t
            
            2
          
        
      
      )
    
  
  




can be determined. Using Equation 42, the Fourier transform of the symmetric temporal waveform can be expressed as: 



  
    
      
        
          
            
              I
              _
            
            symmetric
          
          ⁡
          
            (
            f
            )
          
        
        =
        
          2
          ⁢
          
            
            
              
                I
                _
              
              ⁡
              
                (
                f
                )
              
            
            
          
          ⁢
          
            
              cos
              ⁡
              
                (
                
                  
                    φ
                    ⁢
                    
                      (
                      f
                      )
                    
                  
                  -
                  
                    f
                    ⁢
                    
                      
                        Δ
                        ⁢
                        
                           
                        
                        ⁢
                        t
                      
                      2
                    
                  
                
                )
              
            
            .
          
        
      
    
    
      
        (
        44
        )
      
    
  




Therefore, since |Ī(ƒ)| is known from FIG. 20 or from the envelope of FIG. 22, the only information needed to determine Isymmetric(t) directly by taking the inverse Fourier transform of Equation 44 is the knowledge of 



  
    cos
    ⁡
    
      (
      
        
          φ
          ⁡
          
            (
            f
            )
          
        
        -
        
          f
          ⁢
          
            
              Δ
              ⁢
              
                 
              
              ⁢
              t
            
            2
          
        
      
      )
    
  
  .




To determine the function 



  cos
  ⁡
  
    (
    
      
        φ
        ⁡
        
          (
          f
          )
        
      
      -
      
        f
        ⁢
        
          
            Δ
            ⁢
            
               
            
            ⁢
            t
          
          2
        
      
    
    )
  




from the function 



  
    
    
      cos
      ⁡
      
        (
        
          
            φ
            ⁡
            
              (
              f
              )
            
          
          -
          
            f
            ⁢
            
              
                Δ
                ⁢
                
                   
                
                ⁢
                t
              
              2
            
          
        
        )
      
    
    
  
  ,




two possible cases can be analyzed. In the first case, if there are no zero crossings of the term 



  
    
    
      cos
      ⁡
      
        (
        
          
            φ
            ⁡
            
              (
              f
              )
            
          
          -
          
            f
            ⁢
            
              
                Δ
                ⁢
                
                   
                
                ⁢
                t
              
              2
            
          
        
        )
      
    
    
  
  ,




then there is no ambiguity due to the absolute value sign since intensity has to be non-negative (−Isymmetric(t) is not a possible solution). In the second case, if there are some zero crossings of the term 



  
    
    
      cos
      ⁡
      
        (
        
          
            φ
            ⁡
            
              (
              f
              )
            
          
          -
          
            f
            ⁢
            
              
                Δ
                ⁢
                
                   
                
                ⁢
                t
              
              2
            
          
        
        )
      
    
    
  
  ,




the sign ambiguities of the cosine term between the zero crossings can be removed by using a property of Fourier transforms. For a real and symmetric function such as Isymmetric(t), the Fourier transform Īsymmetric(ƒ) is equivalent to the Hartley transform 



  
    
      I
      symmetric
      Ha
    
    ⁡
    
      (
      f
      )
    
  
  .




Therefore, the magnitude of the Hartley transform of Isymmetric(t) (i.e., 



  (
  
    
      i
      .
      e
      .
    
    ,
    
      
        
        
          
            I
            symmetric
            Ha
          
          ⁡
          
            (
            f
            )
          
        
        
      
      =
      
        
          
          
            
              
                I
                _
              
              symmetric
            
            ⁡
            
              (
              f
              )
            
          
          
        
        =
        
          2
          ⁢
          
            
            
              
                I
                _
              
              ⁡
              
                (
                f
                )
              
            
            
          
          ⁢
          
            
            
              cos
              ⁡
              
                (
                
                  
                    φ
                    ⁡
                    
                      (
                      f
                      )
                    
                  
                  -
                  
                    f
                    ⁢
                    
                      
                        Δ
                        ⁢
                        
                           
                        
                        ⁢
                        t
                      
                      2
                    
                  
                
                )
              
            
            
          
        
      
    
  
  )




can be determined from the knowledge of 



  
  
    cos
    ⁡
    
      (
      
        
          φ
          ⁡
          
            (
            f
            )
          
        
        -
        
          f
          ⁢
          
            
              Δ
              ⁢
              
                 
              
              ⁢
              t
            
            2
          
        
      
      )
    
  
  




and |Ī(ƒ)|. For a real and compact support function (i.e., one that equals zero outside a finite region), such as Isymmetric(t), the intensity of the Hartley transform is enough to uniquely recover the original function. See, e.g., N. Nakajima in Reconstruction of a real function from its Hartley-transform intensity, Journal of the Optical Society of America A, Vol. 5, 1988, pages 858-863, and R. P. Millane in Analytic Properties of the Hartley Transform and their Implications, Proceedings of the IEEE, Vol. 82, 1994, pages 413-428, both of which are incorporated in their entirety by reference herein.

FIG. 23A illustrates the recovered symmetric temporal waveform and FIG. 23B illustrates the difference between the recovered symmetric temporal waveform and the temporal waveform of the original pulse (as shown in FIG. 21). As illustrated by FIG. 23B, the two waveforms are in excellent agreement (within approximately 0.15) with one another. The difference shown in FIG. 23B is in fact just a numerical calculation artifact, which can be improved with increased accuracy. Thus, the configuration schematically illustrated in FIG. 17 is capable of recovering ultra-short temporal waveforms unambiguously when used with a time-reversal scheme, such as that illustrated by FIG. 18.
FIG. 24 schematically illustrates a system 700 for another embodiment for determining the temporal waveform of a laser pulse. The input pulse 710 impinges onto a grating 720 which disperses the input pulse 710 into its spectral components. A lens 730 recollimates the spectral components and images them onto different elements of a CCD imaging device 740.
An arbitrary input pulse 710 has the following form:

us(t)=ũs(t)ejωct  (45)

where ũs(t) is the complex envelope function and ωc is the carrier frequency. Equation 45 can be rewritten as:

us(t)=∫Ũs(ω−ωc)ejωtdω  (46)

where Ũs(ω) denotes the Fourier transform of ũs(t). The input pulse 710 is decomposed by the grating 720 into several monochromatic plane waves with amplitudes Ũs(ω−ωc). By finding the response of the system 700 to each individual harmonic component (i.e. Ũs(ω−ωc)ejωt), the overall response of the system 700 can be determined using the integral given in Equation 46.

The field of a single harmonic Ũs(ω−ωc)ejωt at the plane 722 immediately after being dispersed by the grating 720 can be written in the following form: 



  
    
      
        
          
            
              U
              ~
            
            
              1
              ⁢
              s
            
          
          ⁡
          
            (
            
              
                x
                ;
                ω
              
              ,
              t
            
            )
          
        
        =
        
          
            
              
                U
                ~
              
              s
            
            ⁡
            
              (
              
                ω
                -
                
                  ω
                  c
                
              
              )
            
          
          ⁢
          
            e
            
              j
              ⁢
              
                 
              
              ⁢
              ω
              ⁢
              
                 
              
              ⁢
              t
            
          
          ⁢
          
            w
            ⁡
            
              (
              x
              )
            
          
          ⁢
          
            e
            
              
                -
                j
              
              ⁢
              
                 
              
              ⁢
              
                x
                ⁡
                
                  (
                  
                    
                      
                        ω
                        -
                        
                          ω
                          c
                        
                      
                      c
                    
                    ⁢
                    sin
                    ⁢
                    
                       
                    
                    ⁢
                    θ
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        47
        )
      
    
  




where w(x) is the pupil function of the optical field on the grating 720, c is the speed of light, x is the coordinate along the plane 722, and θ is the incident angle of the input pulse 710 to the grating 720. This form of the single harmonic field is described by P. C. Sun et al. in Femtosecond Pulse Imaging: Ultrafast Optical Oscilloscope, Journal of the Optical Society of America, Vol. 14, 1997, pages 1159-1170, which is incorporated in its entirety by reference herein.

The last exponential term of Equation 47 accounts for the diffraction experienced by the spectral components of the input pulse 710 due to the grating 720, assuming only first-order diffraction. The grating 720 is arranged such that the first diffraction order for the spectral component at ω=ωc propagates along the direction of the optical axis of the system 700. The lens 730 transforms the image at the plane 722 into an image at plane 742. The fields at the two planes 722, 742 are related by a spatial Fourier transform. By taking the spatial Fourier transform of Equation 23, the field at the plane 742 can be written as: 



  
    
      
        
          
            
              U
              ~
            
            
              2
              ⁢
              s
            
          
          ⁡
          
            (
            
              
                
                  f
                  
                    x
                    ′
                  
                
                ;
                ω
              
              ,
              t
            
            )
          
        
        =
        
          
            
              
                U
                ~
              
              s
            
            ⁡
            
              (
              
                ω
                -
                
                  ω
                  c
                
              
              )
            
          
          ⁢
          
            e
            
              j
              ⁢
              
                 
              
              ⁢
              ω
              ⁢
              
                 
              
              ⁢
              t
            
          
          ⁢
          
            W
            ⁡
            
              (
              
                
                  f
                  
                    x
                    ′
                  
                
                +
                
                  
                    
                      ω
                      -
                      
                        ω
                        c
                      
                    
                    
                      2
                      ⁢
                      π
                      ⁢
                      
                         
                      
                      ⁢
                      c
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                     
                  
                  ⁢
                  θ
                
              
              )
            
          
        
      
    
    
      
        (
        48
        )
      
    
  




where W(ƒx′) is the spatial Fourier transform of w(x). x′ is the coordinate along the plane 742, and ƒx′ is the spatial frequency which can be written as: 



  
    
      
        
          f
          
            x
            ′
          
        
        =
        
          
            ω
            ⁢
            
               
            
            ⁢
            
              x
              ′
            
          
          
            2
            ⁢
            π
            ⁢
            
               
            
            ⁢
            cF
          
        
      
    
    
      
        (
        49
        )
      
    
  




where F is the focal length of the lens.

The total response of the system 700 is the spectral decomposition of the field us(t) of the input pulse 710 and can be found by integrating Equation 48 over the frequency range, i.e.,:

u2s(x1;t)=∫Ũ2s(x′;ω,t)dω  (50)

By using the paraxial approximation and by assuming a large illumination window w(x), the total response given by Equation 50 can be simplified to the following form: 



  
    
      
        
          
            u
            
              2
              ⁢
              s
            
          
          ⁡
          
            (
            
              
                x
                ′
              
              ;
              t
            
            )
          
        
        ≈
        
          
            
              
                U
                ~
              
              s
            
            ⁡
            
              (
              
                -
                
                  
                    
                      x
                      ′
                    
                    ⁢
                    
                      ω
                      c
                    
                  
                  
                    F
                    ⁢
                    
                       
                    
                    ⁢
                    sin
                    ⁢
                    
                       
                    
                    ⁢
                    θ
                  
                
              
              )
            
          
          ⁢
          
            w
            ⁡
            
              (
              
                
                  [
                  
                    1
                    -
                    
                      
                        x
                        ′
                      
                      
                        F
                        ⁢
                        
                           
                        
                        ⁢
                        sin
                        ⁢
                        
                           
                        
                        ⁢
                        θ
                      
                    
                  
                  ]
                
                ⁢
                
                  ct
                  
                    sin
                    ⁢
                    
                       
                    
                    ⁢
                    θ
                  
                
              
              )
            
          
          ⁢
          
            
              e
              
                j
                ⁢
                
                   
                
                ⁢
                
                  
                    ω
                    c
                  
                  ⁡
                  
                    (
                    
                      
                        [
                        
                          1
                          -
                          
                            
                              x
                              ′
                            
                            
                              F
                              ⁢
                              
                                 
                              
                              ⁢
                              sin
                              ⁢
                              
                                 
                              
                              ⁢
                              θ
                            
                          
                        
                        ]
                      
                      ⁢
                      t
                    
                    )
                  
                
              
            
            .
          
        
      
    
    
      
        (
        51
        )
      
    
  




By generating a phase conjugate pulse as schematically illustrated in FIG. 18, the total output waveform can be expressed as: utotal(t)=us(t)+uPC(t−Δt), where PC denotes “phase conjugate.” The function uPC(t) is dependent on the type of phase conjugate mirror used to generate the phase conjugate pulses. For phase conjugate pulses formed using pulse-pumped FWM, the total field utotal(t) can be expressed, using Equation 34, in the following form:

utotal(t)=(ũs(t)+ũs*(−t−Δt))ejωct  (52)

where Δt is the time delay between the original pulse and its corresponding phase conjugate pulse. Feeding this total field into the system 700 illustrated in FIG. 24, the resultant image field at the plane 742 of the CCD imaging device 740, using Equation 51, can be expressed as: 



  
    
      
        
          
            
              
                
                  
                    u
                    
                      total
                      ,
                      2
                    
                  
                  ⁡
                  
                    (
                    
                      
                        x
                        ′
                      
                      ;
                      t
                    
                    )
                  
                
                ≈
                
                  
                    
                      
                        
                          U
                          ~
                        
                        s
                      
                      ⁡
                      
                        (
                        
                          -
                          
                            
                              
                                x
                                ′
                              
                              ⁢
                              
                                ω
                                c
                              
                            
                            
                              F
                              ⁢
                              
                                 
                              
                              ⁢
                              sin
                              ⁢
                              
                                 
                              
                              ⁢
                              θ
                            
                          
                        
                        )
                      
                    
                    ⁢
                    
                      w
                      ⁡
                      
                        (
                        
                          
                            [
                            
                              1
                              -
                              
                                
                                  x
                                  ′
                                
                                
                                  F
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  sin
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  θ
                                
                              
                            
                            ]
                          
                          ⁢
                          
                            ct
                            
                              sin
                              ⁢
                              
                                 
                              
                              ⁢
                              θ
                            
                          
                        
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        j
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            ω
                            c
                          
                          ⁡
                          
                            (
                            
                              
                                [
                                
                                  1
                                  -
                                  
                                    
                                      x
                                      ′
                                    
                                    
                                      F
                                      ⁢
                                      
                                         
                                      
                                      ⁢
                                      sin
                                      ⁢
                                      
                                         
                                      
                                      ⁢
                                      θ
                                    
                                  
                                
                                ]
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                    
                  
                  +
                
              
              ⁢
              
                 
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    
                      U
                      ~
                    
                    s
                    *
                  
                  ⁡
                  
                    (
                    
                      -
                      
                        
                          
                            x
                            ′
                          
                          ⁢
                          
                            ω
                            c
                          
                        
                        
                          F
                          ⁢
                          
                             
                          
                          ⁢
                          sin
                          ⁢
                          
                             
                          
                          ⁢
                          θ
                        
                      
                    
                    )
                  
                
                ⁢
                
                  w
                  ⁡
                  
                    (
                    
                      
                        [
                        
                          1
                          -
                          
                            
                              x
                              ′
                            
                            
                              F
                              ⁢
                              
                                 
                              
                              ⁢
                              sin
                              ⁢
                              
                                 
                              
                              ⁢
                              θ
                            
                          
                        
                        ]
                      
                      ⁢
                      
                        ct
                        
                          sin
                          ⁢
                          
                             
                          
                          ⁢
                          θ
                        
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    j
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        ω
                        c
                      
                      ⁡
                      
                        (
                        
                          
                            [
                            
                              1
                              -
                              
                                
                                  x
                                  ′
                                
                                
                                  F
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  sin
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  θ
                                
                              
                            
                            ]
                          
                          ⁢
                          t
                        
                        )
                      
                    
                  
                
                ⁢
                
                  
                    e
                    
                      j
                      ⁢
                      
                         
                      
                      ⁢
                      Δ
                      ⁢
                      
                         
                      
                      ⁢
                      
                        t
                        ⁡
                        
                          (
                          
                            -
                            
                              
                                
                                  x
                                  ′
                                
                                ⁢
                                
                                  ω
                                  c
                                
                              
                              
                                F
                                ⁢
                                
                                   
                                
                                ⁢
                                sin
                                ⁢
                                
                                   
                                
                                ⁢
                                θ
                              
                            
                          
                          )
                        
                      
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        53
        )
      
    
  




The CCD imaging device 740 at the plane 742 is responsive to intensity, which can be written as:

I(x′)=∫|utotal,2(x′;t)|2dt.  (54)

Defining Ũs(ω)=|Ũs(ω)|ejΦ(ω) and A=−F sin θ, Equation 54 can be rewritten as: 



  
    
      
        
          
            I
            double
          
          ⁡
          
            (
            
              x
              ′
            
            )
          
        
        =
        
          2
          ⁢
          
            (
            
              ∫
              
                
                  
                    
                    
                      w
                      ⁡
                      
                        (
                        
                          
                            [
                            
                              1
                              +
                              
                                
                                  x
                                  ′
                                
                                A
                              
                            
                            ]
                          
                          ⁢
                          
                            ct
                            
                              sin
                              ⁢
                              
                                 
                              
                              ⁢
                              θ
                            
                          
                        
                        )
                      
                    
                    
                  
                  2
                
                ⁢
                
                   
                
                ⁢
                
                  d
                  t
                
              
            
            )
          
          ⁢
          
             
          
          ⁢
          
            
              
              
                
                  
                    U
                    ~
                  
                  s
                
                ⁡
                
                  (
                  
                    
                      
                        x
                        ′
                      
                      ⁢
                      
                        ω
                        c
                      
                    
                    A
                  
                  )
                
              
              
            
            2
          
          ⁢
          
             
          
          ⁢
          
            
              
              
                1
                +
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        2
                        ⁢
                        
                          Φ
                          ⁡
                          
                            (
                            
                              
                                
                                  x
                                  ′
                                
                                ⁢
                                
                                  ω
                                  c
                                
                              
                              A
                            
                            )
                          
                        
                      
                      -
                      
                        Δ
                        ⁢
                        
                           
                        
                        ⁢
                        t
                        ⁢
                        
                          
                            
                              x
                              ′
                            
                            ⁢
                            
                              ω
                              c
                            
                          
                          A
                        
                      
                    
                    )
                  
                
              
              
            
            .
          
        
      
    
    
      
        (
        55
        )
      
    
  




Furthermore, Equation 55 can be rewritten as: 



  
    
      
        
          
            I
            double
          
          ⁡
          
            (
            
              x
              ′
            
            )
          
        
        =
        
          2
          ⁢
          
            G
            ⁡
            
              (
              
                x
                ′
              
              )
            
          
          ⁢
          
            
              
              
                
                  
                    U
                    ~
                  
                  s
                
                ⁢
                
                  (
                  
                    
                      
                        x
                        ′
                      
                      ⁢
                      
                        ω
                        c
                      
                    
                    A
                  
                  )
                
              
              
            
            2
          
          ⁢
          
            
            
              1
              +
              
                cos
                ⁢
                
                  (
                  
                    
                      2
                      ⁢
                      
                        Φ
                        ⁡
                        
                          (
                          
                            
                              
                                x
                                ′
                              
                              ⁢
                              
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        (
        56
        )
      
    
  




with 



  
    G
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  =
  
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Equation 56 is very similar to Equation 43, which was obtained for embodiments utilizing the intensity correlator of FIG. 17. Furthermore, the intensity profile on the CCD imaging device 740 for a single pulse us(t) can be expressed as: 



  
    
      
        
          
            I
            single
          
          ⁡
          
            (
            
              x
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            )
          
        
        =
        
          
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        (
        57
        )
      
    
  




Equation 57 is also very similar to Equation 40, which was derived above in relation to the intensity correlator embodiment as shown in FIG. 17.

To recover the complex envelope function ũs(t), as defined in Equation 45, the same algorithm described above for recovering I(t) can be applied. This process is illustrated in the FIGS. 25 through 30 using computer-generated simulations. FIGS. 25A and 25B illustrate the magnitude and phase of an arbitrary asymmetric complex envelope function ũs(t) to be characterized, respectively. (FIG. 25A is the same input pulse as shown in FIG. 19). FIG. 26 illustrates the intensity profile for this complex field, including the carrier frequency oscillations. Note that FIG. 19 described above did not include the carrier frequency oscillations and was just the envelope of the intensity. In a first step, the general configuration 600 of FIG. 18 is used to generate the symmetric temporal waveform whose complex envelope function is defined by Equation 52. FIG. 27 illustrates the symmetric temporal waveform, including the carrier frequency, generated by this first step. The symmetric temporal waveform comprises the temporal waveform of the original pulse plus the time-delayed temporal waveform of the time-reversed pulse.
In a second step, the system of FIG. 24 is used to measure the Fourier transform of the symmetric pulse with the CCD imaging device. FIG. 28A illustrates the detected intensity I(x′) on the CCD imaging device at the plane 742 for the symmetric pulse of FIG. 27. The intensity I(x′) was calculated using Equation 56. For comparison, FIG. 28B shows the detected intensity I(x′) on the CCD imaging device at the plane 742 using the original pulse of FIG. 26, which is a prior art measurement technique. The intensity I(x′) was calculated using Equation 57.
In a third step, and as discussed above in relation to determining I(t) using the intensity correlation configuration of FIG. 17, the CCD image (FIG. 28A) is used to calculate numerically both amplitude and the phase of the Fourier transform of the complex envelope function ũs(t). One can calculate the complex envelope function ũs(t) using inverse Fourier transform once both the amplitude and the phase functions of the Fourier transform are recovered. FIGS. 28A and 28B illustrate the recovered symmetric temporal waveform and the original temporal waveform, respectively, including the carrier frequencies. FIG. 29 illustrates these waveforms overlaid with one another in an expanded time frame for comparison purposes. This comparison establishes that the prediction of the pulse shape made available by this invention is excellent. The discrepancy between the two curves of FIG. 30 is in fact a numerical artifact that can be removed by increasing the computation accuracy.
Therefore, using the same process as described above in relation to Equation 43, the system 700 of FIG. 24 can be used to recover the complex envelope function ũs(t) of any given input pulse using only the Fourier transform amplitude of the symmetrized composite waveform. In addition, both the envelope of the intensity profile I(t) and the underlying optical oscillations are recoverable, as illustrated by FIG. 30. Recovering both the envelope and the underlying oscillations is an improvement with respect to the prior art intensity correlation embodiment described above, which only recovers the envelope of I(t).
As described above, the waveform of an ultra-short optical pulse is determined in certain embodiments by using time-reversal techniques to yield a time-reversed replica of the pulse and to obtain a symmetric optical pulse. The Fourier transform magnitude of the symmetric optical pulse is used to uniquely recover the original time-domain optical pulse waveform. Such time-reversal techniques utilize shorter optical pulses with respect to the input optical pulse waveform being measured. For example, to determine the temporal pulse waveform of a femtosecond-scale optical pulse, sub-femtosecond-scale pulses are used to form the symmetric optical pulse waveform by time reversal.
This experimental limitation of using pulses shorter than the input pulse waveform can present a challenge which can hinder the application of the method in certain circumstances. However, in certain embodiments, a method determines the temporal waveform of an optical pulse using the Fourier transforms of four pulse waveforms and avoiding the limitation of short pulses. FIG. 31 is a flow diagram of a method 800 in accordance with embodiments of the present invention. In an operational block 810, a sample optical pulse having a sample temporal waveform is provided. In an operational block 820, a Fourier transform of the sample temporal waveform is measured. In an operational block 830, a reference optical pulse having a reference temporal waveform is provided. In an operational block 840, a Fourier transform of the reference temporal waveform is obtained. In an operational block 850, a first composite optical pulse having a first composite temporal waveform is formed. The first composite optical pulse comprises the sample optical pulse followed by the reference optical pulse. In an operational block 860, a Fourier transform of the first composite temporal waveform is measured. In an operational block 870, a time-reversed optical pulse having a time-reversed temporal waveform is provided. The time-reversed temporal waveform corresponds to the reference temporal waveform after being time-reversed. In an operational block 880, a second composite optical pulse having a second composite temporal waveform is formed. The second composite optical pulse comprises the sample optical pulse followed by the time-reversed optical pulse. In an operational block 890, a Fourier transform of the second composite temporal waveform is measured. In an operational block 900, the sample temporal waveform is calculated using the Fourier transform of the sample temporal waveform, the Fourier transform of the reference temporal waveform, the Fourier transform of the first composite temporal waveform, and the Fourier transform of the second composite temporal waveform.
The sample optical pulse of certain embodiments can have an ultra-short temporal waveform (e.g., on the order of femtoseconds or sub-femtoseconds). The reference optical pulse of certain embodiments can have a temporal waveform which is significantly more broad than the temporal waveform of the sample optical pulse. For example, for a femtosecond sample temporal waveform, the width of the reference temporal waveform can be on the order of nanoseconds. By allowing the use of reference optical pulses with broader temporal waveforms, embodiments of the present invention allow for easier time-reversal processes, since the optical pulses used to time-reverse the reference optical pulse can be correspondingly broader. In certain such embodiments, the sample optical pulse itself can be used to time-reverse the reference optical pulse.
In certain embodiments, the first composite optical pulse includes a time delay between the sample optical pulse and the reference optical pulse. In certain such embodiments, the time delay can be adjusted to make the measurement of the Fourier transform easier. The time delay used can be dependent upon the particular technique used to measure the Fourier transform. Similarly, in certain embodiments, the second composite optical pulse includes a time delay between the sample optical pulse -and the time-reversed optical pulse, and the time delay can be adjusted to ease the measurement of the Fourier transform depending on the particular measurement technique used.
In certain embodiments, the Fourier transforms of the sample temporal waveform, the reference temporal waveform, the first composite temporal waveform, and the second composite temporal waveform are measured using an auto-correlator as described above, or using other holographic techniques. In certain embodiments, obtaining the Fourier transform of the reference temporal waveform comprises measuring the Fourier transform. In other embodiments, the Fourier transform of the reference temporal waveform is previously measured and stored in memory, and obtaining the Fourier transform comprises reading the Fourier transform from memory.
Using the four Fourier transforms in the same manner as described above for measuring the sample nonlinearity profile, the sample temporal waveform can be measured. Note that the reference temporal waveform can also be calculated by embodiments of the present invention concurrently with the calculation of the sample temporal waveform. However, because the reference optical pulse is temporally broad, its temporal waveform is of less interest than that of the sample optical pulse. In embodiments in which the same reference optical pulse is used to measure the temporal waveforms of a series of sample optical pulses, the calculated series of reference temporal waveforms can be compared to one another, thereby providing a check of the validity of the measurements across the series of calculations.
This invention may be embodied in other specific forms without departing from the essential characteristics as described herein. The embodiments described above are to be considered in all respects as illustrative only and not restrictive in any manner. The, scope of the invention is indicated by the following claims rather than by the foregoing description. Any and all changes which come within the meaning and range of equivalency of the claims are to be considered within their scope.

## Claims

1. A method of measuring a nonlinearity profile of a sample, the method comprising:
providing a sample having at least one sample surface and having a sample nonlinearity profile along a sample line through a predetermined point on the sample surface, the sample line oriented perpendicularly to the sample surface; 
measuring a Fourier transform of the sample nonlinearity profile; 
providing a reference material having at least one reference surface and having a reference nonlinearity profile along a reference line through a predetermined point on the reference surface, the reference line oriented perpendicularly to the reference surface; 
obtaining a Fourier transform of the reference nonlinearity profile; 
forming a first composite sample having a first composite nonlinearity profile by placing the sample and the reference material proximate to one another in a first configuration with the sample line substantially collinear with the reference line; 
measuring a Fourier transform of the first composite nonlinearity profile; 
forming a second composite sample having a second composite nonlinearity profile which is inequivalent to the first composite nonlinearity profile by placing the sample and the reference material proximate to one another in a second configuration with the sample line substantially collinear with the reference line; 
measuring a Fourier transform of the second composite nonlinearity profile; and 
calculating the sample nonlinearity profile using the Fourier transforms of the sample nonlinearity profile, the reference nonlinearity profile, the first composite nonlinearity profile, and the second composite nonlinearity profile.

2. The method of claim 1, wherein:
the sample has a first sample surface and has a second sample surface substantially parallel to the first sample surface; 
the first configuration has the first sample surface proximate to the reference surface; and 
the second configuration has the second sample surface proximate to the reference surface.

3. The method of claim 1, wherein:
the reference material has a first reference surface and has a second reference surface substantially parallel to the first reference surface; 
the first configuration has the sample surface proximate to the first reference surface; and 
the second configuration has the sample surface proximate to the second reference surface.

4. The method of claim 1, wherein obtaining the Fourier transform of the reference nonlinearity profile comprises measuring the Fourier transform of the reference nonlinearity profile.

5. The method of claim 1, wherein obtaining the Fourier transform of the reference nonlinearity profile comprises retrieving the Fourier transform of the reference nonlinearity profile from a memory module.

6. The method of claim 1, wherein the sample comprises poled silica.

7. The method of claim 1, wherein the sample comprises a nonlinear organic material.

8. The method of claim 1, wherein the sample comprises a nonlinear inorganic material.

9. The method of claim 8, wherein the sample comprises an amorphous material.

10. The method of claim 1, wherein the first configuration comprises the sample and the reference material sandwiched together in an anode-to-anode configuration in which an anodic surface of the sample is proximate to an anodic surface of the reference material.

11. The method of claim 10, wherein the second configuration comprises the sample and the reference material sandwiched together in an anode-to-cathode configuration in which an anodic surface of the sample is proximate to a cathodic surface of the reference material.

12. The method of claim 10, wherein the second configuration comprises the sample and the reference material sandwiched together in an cathode-to-anode configuration in which a cathodic surface of the sample is proximate to an anodic surface of the reference material.

13. The method of claim 1, wherein the first configuration comprises the sample and the reference material sandwiched together in a cathode-to-cathode configuration in which a cathodic surface of the sample is proximate to a cathodic surface of the reference material.

14. The method of claim 13, wherein the second configuration comprises the sample and the reference material sandwiched together in an anode-to-cathode configuration in which an anodic surface of the sample is proximate to a cathodic surface of the reference material.

15. The method of claim 13, wherein the second configuration comprises the sample and the reference material sandwiched together in an cathode-to-anode configuration in which a cathodic surface of the sample is proximate to an anodic surface of the reference material.

16. The method of claim 1, wherein the first configuration comprises a spacer material placed between the sample and the reference material.

17. The method of claim 1, wherein the second configuration comprises a spacer material placed between the sample and the reference material.

18. The method of claim 1, wherein measuring the Fourier transform of the sample nonlinearity profile comprises measuring the Maker fringe profile of the sample.

19. The method of claim 1, wherein measuring the Fourier transform of the sample nonlinearity profile comprises focusing a pulsed laser beam onto the sample at an incident angle, generating a second-harmonic signal, and measuring the generated second-harmonic signal as a function of the incident angle.

20. The method of claim 1, wherein measuring the Fourier transform of the first composite nonlinearity profile comprises measuring the Maker fringe profile of the first composite sample.

21. The method of claim 1, wherein measuring the Fourier transform of the first composite nonlinearity profile comprises focusing a pulsed laser beam onto the first composite sample at an incident angle, generating a second-harmonic signal, and measuring the generated second-harmonic signal as a function of the incident angle.

22. The method of claim 1, wherein measuring the Fourier transform of the second composite nonlinearity profile comprises measuring the Maker fringe profile of the second composite sample.

23. The method of claim 1, wherein measuring the Fourier transform of the second composite nonlinearity profile comprises focusing a pulsed laser beam onto the second composite sample at an incident angle, generating a second-harmonic signal, and measuring the generated second-harmonic signal as a function of the incident angle.

24. The method of claim 1, wherein the reference material comprises a second sample and the reference nonlinearity profile comprises a second sample nonlinearity profile, the method further comprising calculating the second sample nonlinearity profile using the Fourier transforms of the sample nonlinearity profile, the reference nonlinearity profile, the first composite nonlinearity profile, and the second composite nonlinearity profile.

25. The method of claim 1, further comprising calculating the reference nonlinearity profile using the Fourier transforms of the sample nonlinearity profile, the reference nonlinearity profile, the first composite nonlinearity profile, and the second composite nonlinearity profile.

26. A method of measuring nonlinearity profiles of a plurality of samples, the method comprising:
(a) measuring a first nonlinearity profile of a first sample using the method of claim 25; 
(b) measuring a second nonlinearity profile of a second sample using the method of claim 25, wherein the same reference material is used to measure the first and second nonlinearity profiles; and 
comparing the calculated reference nonlinearity profiles from (a) and (b) to provide an indication of the consistency of the measurements of the first and second nonlinearity profiles.

27. A method of measuring a nonlinearity profile of a sample, the method comprising:
providing a sample having at least one sample surface and having a sample nonlinearity profile along a sample line through a predetermined point on the sample surface, the sample line oriented perpendicularly to the sample surface; 
measuring a Fourier transform of the sample nonlinearity profile; 
providing a reference material having at least one reference surface and having a reference nonlinearity profile along a reference line through a predetermined point on the reference surface, the reference line oriented perpendicularly to the reference surface; 
obtaining a Fourier transform of the reference nonlinearity profile; 
forming a first composite sample having a first composite nonlinearity profile by placing the sample and the reference material proximate to one another in a first configuration with the sample line substantially collinear with the reference line; 
measuring a Fourier transform of the first composite nonlinearity profile; 
forming a second composite sample having a second composite nonlinearity profile which is inequivalent to the first composite nonlinearity profile by placing the sample and the reference material proximate to one another in a second configuration with the sample line substantially collinear with the reference line; 
measuring a Fourier transform of the second composite nonlinearity profile; and 
using the Fourier transforms of the sample nonlinearity profile, the reference nonlinearity profile, the first composite nonlinearity profile, and the second composite nonlinearity profile to calculate the sample nonlinearity profile.

28. A method of measuring a sample temporal waveform of a sample optical pulse, the method comprising:
providing a sample optical pulse having a sample temporal waveform; 
measuring a Fourier transform of the sample temporal waveform; 
providing a reference optical pulse having a reference temporal waveform; 
obtaining a Fourier transform of the reference temporal waveform; 
forming a first composite optical pulse comprising the sample optical pulse followed by the reference optical pulse, the first composite optical pulse having a first composite temporal waveform; 
measuring the Fourier transform of the first composite temporal waveform; 
providing a time-reversed pulse having a time-reversed temporal waveform corresponding to the reference temporal waveform after being time-reversed; 
forming a second composite optical pulse comprising the sample optical pulse followed by the time-reversed optical pulse; 
measuring the Fourier transform of the second composite temporal waveform; and 
calculating the sample temporal waveform using the Fourier transform of the sample temporal waveform, the Fourier transform of the reference temporal waveform, the Fourier transform of the first composite temporal waveform, and the Fourier transform of the second composite temporal waveform.

29. The method of claim 28, wherein the time-reversed optical pulse is provided using four-wave mixing.

30. The method of claim 28, wherein the time-reversed optical pulse is provided using spectral holography.

31. The method of claim 28, wherein the reference optical pulse is broader than the sample optical pulse.

32. The method of claim 31, wherein the time-reversed optical pulse is provided using the sample optical pulse to time-reverse the reference optical pulse.

33. The method of claim 31, wherein the sample optical pulse has a temporal width on the order of femtoseconds and the reference optical pulse has a temporal width on the order of nanoseconds.

34. The method of claim 28, wherein the first composite optical pulse includes a time delay between the sample optical pulse and the reference optical pulse.

35. The method of claim 28, wherein the second composite optical pulse includes a time delay between the sample optical pulse and the time-reversed pulse.

36. The method of claim 28, further comprising calculating the reference temporal waveform using the Fourier transforms of the sample temporal waveform, the reference temporal waveform, the first composite temporal waveform, and the second composite temporal waveform.

37. A method of measuring temporal waveforms of a plurality of optical pulses, the method comprising:
(a) measuring a first temporal waveform of a first optical pulse using the method of claim 28; 
(b) measuring a second temporal waveform of a second optical pulse using the method of claim 28, the same reference optical pulse is used to measure the first and second temporal profiles; and 

comparing the calculated reference temporal waveforms from (a) and (b) to provide an indication of the consistency of the measurements of the first and second temporal profiles.