Abstract:
Techniques for measuring spatial locations. An apparatus for measuring spatial locations includes a first object at a first location, a second object at a second location, and a waveguide-based interferometer coupled between the first object and the second object. The waveguide-based interferometer includes a waveguide material. The waveguide material is adapted to carry at least a reference beam and a measurement beam. The reference beam traverses a first path defined within the waveguide material. The measurement beam traverses a second path defined within the waveguide material and a third path defined outside of the waveguide material. The third path is related to at least one of the first location and the second location.

Description:
CROSS-REFERENCES TO RELATED APPLICATIONS 
     This application is a continuation-in-part application to U.S. application Ser. No. 10/051,122 filed Jan. 22, 2002, now U.S. Pat. No. 6,947,621, which is incorporated by reference herein. 
     STATEMENT AS TO RIGHTS TO INVENTIONS MADE UNDER FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     Not Applicable 
     REFERENCE TO A “SEQUENCE LISTING,” A TABLE, OR A COMPUTER PROGRAM LISTING APPENDIX SUBMITTED ON A COMPACT DISK. 
     Not Applicable 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates generally to interferometer techniques. More particularly, the invention provides a method and system for using waveguide in an interferometer. Merely by way of example, the invention has been applied to measuring spatial locations, but it would be recognized that the invention has a much broader range of applicability. 
     Interferometers have been widely used for spatial measurements. The interferometers include homodyne interferometers, such as conventional Michelson interferometer, and heterodyne interferometers. Homodyne interferometers use laser beams at the same frequency, while heterodyne interferometers use laser beams at different frequencies. 
     Interferometers measure the phase of a continuous signal for as long as the signal remains stable over all portions of a reference path and a measurement path. The major practical problem with high resolution interferometry is that changes of optical path-lengths in the system, including ones that are not intended, are measured. After optical beams are split in the interferometer, movement of optical components, especially mirrors, add or subtract optical path-length from one of the two beams separately, resulting in a signal that is unrelated to the measurement. Therefore, stability of optical components in the separate legs of the interferometer is usually important. See J. D. Trolinger,  Ultra High Resolution Interferometry,  Proc. SPIE Vol. 2816, pp. 114–123 (1996). 
     Hence, it is desirable to improve interferometer techniques. 
     BRIEF SUMMARY OF THE INVENTION 
     The present invention relates generally to interferometer techniques. More particularly, the invention provides a method and system for using waveguide in interferometer. Merely by way of example, the invention has been applied to measuring spatial locations, but it would be recognized that the invention has a much broader range of applicability. 
     According to a specific embodiment of the present invention, an apparatus for measuring spatial locations includes a first object at a first location, a second object at a second location, and a waveguide-based interferometer coupled between the first object and the second object. The waveguide-based interferometer includes a waveguide material. The waveguide material is adapted to carry at least a reference beam and a measurement beam. The reference beam traverses a first path defined within the waveguide material. The measurement beam traverses a second path defined within the waveguide material and a third path defined outside of the waveguide material. The third path is related to at least one of the first location and the second location. 
     According to another embodiment of the present invention, an apparatus for measuring spatial locations includes a first object at a first location, a second object at a second location, and a waveguide-based interferometer coupled between the first object and the second object. The waveguide-based interferometer includes a waveguide material. The waveguide material is adapted to carry at least a reference beam at a reference frequency and a corresponding reference wavelength and a measurement beam at a measurement frequency and a corresponding measurement wavelength. The reference frequency is different from the measurement frequency. The reference beam traverses a first path defined within the waveguide material. The measurement beam traverses a second path defined within the waveguide material and a third path defined outside of the waveguide material. The third path related to at least one of the first location and the second location. 
     According to yet another embodiment of the present invention, an apparatus for measuring spatial locations includes a first object at a first location, a second object at a second location, and a waveguide-based interferometer coupled between the first object and the second object. The waveguide-based interferometer includes a waveguide material. The waveguide material is adapted to carry at least a first reference beam at a first frequency and a corresponding first wavelength, a second reference beam at a second frequency and a corresponding second wavelength, a first measurement beam at the first frequency and the corresponding first wavelength, and a second measurement beam at the second frequency and the corresponding wavelength. The first frequency is different from the second frequency. The first reference beam traverses a first reference path defined within the waveguide material. The second reference beam traverses a second reference path defined within the waveguide material. The first measurement beam traverses a first measurement path defined within the waveguide material and an external path defined outside of the waveguide material. The external path is related to at least one of the first location and the second location. The second measurement beam traverses a second measurement path defined within the waveguide material. The first measurement path and the first reference path have a first common end. The second measurement path and the second reference path have a second common end. 
     According to yet another embodiment, an apparatus for measuring spatial locations includes a first object at a first location, a second object at a second location, and a waveguide-based interferometer coupled between the first object and the second object. The waveguide-based interferometer includes a waveguide material. The waveguide material is adapted to carry at least a first reference beam at a first frequency and a corresponding first wavelength, a second reference beam at a second frequency and a corresponding second wavelength, a first measurement beam at the first frequency and the corresponding first wavelength, and a second measurement beam at the second frequency and the corresponding wavelength. The first frequency is different from the second frequency. The first reference beam traverses a first reference path defined within the waveguide material. The second reference beam traverses a second reference path defined within the waveguide material. The first measurement beam traverses a first measurement path defined within the waveguide material and a first external path defined outside of the waveguide material. The first external path is related to the first location. The second measurement beam traverses a second measurement path defined within the waveguide material and a second external path defined outside of the waveguide material. The second external path is related to the second location. The first measurement path and the first reference path have a first common end. The second measurement path and the second reference path have a second common end. 
     According to yet another embodiment of the present invention, an apparatus for measuring spatial locations includes a first object at a first location and a waveguide-based interferometer coupled to the first object. The waveguide-based interferometer includes a waveguide material. The waveguide material is adapted to carry at least a first beam and a second beam. The first beam traverses a first path defined within the waveguide material. The second beam traverses a second path defined within the waveguide material and a third path defined outside of the waveguide material. The third path is related to the first location. 
     Many benefits are achieved by way of the present invention over conventional techniques. For example, certain embodiments of the present invention replaces many of the optical components, their mountings and mechanically variable paths in a conventional interferometer with photolithographically defined components in an inherently stable single mode planar optical waveguide circuit. The waveguide circuit is a planar lightwave circuit (PLC) fabricated in silica on silicon, for example, as described in M. Kawachi,  Silica waveguides on silicon and their application to integrated - optic components,  Optical and Quantum Electronics, Vol. 22, pp391–416 (1990). Other material systems may be used but PLC in silica on silicon usually has mechanical robustness, thermal stability, low coefficient of thermal expansion and commercial fabrication process maturity. Some embodiments of the present invention use a PLC containing at least two circulating optical waveguide circuits. The two circuits are non-interfering until combined at an output coupler. Circuit or waveguide paths on the PLC are photolithographically defined so that thermally induced path-length differences between the two non-interfering circuits are compensated. 
     Various additional objects, features and advantages of the present invention can be more fully appreciated with reference to the detailed description and accompanying drawings that follow. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  represents an embodiment of a heterodyne interferometer according to the present invention; 
         FIG. 1A  is a simplified diagram for mirror systems; 
         FIG. 1B  is a simplified diagram for collimating system; 
         FIG. 1C  is a simplified diagram for alignment of single mode fiber and waveguide; 
         FIG. 1D  is a simplified diagram for silicon V-groove blocks according to one embodiment of the present invention; 
         FIG. 1E  is a simplified diagram for positioning single mode fiber in silicon V-groove blocks according to one embodiment of the present invention. 
         FIG. 2  represents a schematic drawing of an embodiment of a planar lightwave circuit included in the embodiment of a heterodyne interferometer represented in  FIG. 1 ; 
         FIG. 2A  is a simplified diagram for a homodyne interferometer according to one embodiment of the present invention; 
         FIG. 3  represents another embodiment of a heterodyne interferometer according to the present invention; 
         FIG. 3A  is a simplified diagram for mirror systems; 
         FIG. 4  represents a schematic drawing of an embodiment of a planar lightwave circuit included in the embodiment of a heterodyne interferometer represented in  FIG. 3 ; 
         FIG. 5  represents a schematic drawing of another embodiment of a planar lightwave circuit according to the present invention; 
         FIG. 6  represents a schematic drawing of yet another embodiment of a planar lightwave circuit according to the present invention; 
         FIG. 7  represents a perspective view and a cross-section of a planar lightwave circuit according to an embodiment of the present invention; 
         FIG. 8  represents a schematic drawing of an embodiment of a balanced detector; 
         FIG. 9  is simplified diagram illustrating relationship between reflected signal and surface angle; 
         FIG. 10  represents a TM mode stripper according to an embodiment of the present invention; and 
         FIG. 11  is a simplified diagram illustrating the effectiveness of the TM mode stripper as shown in  FIG. 10 . 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The present invention relates generally to interferometer techniques. More particularly, the invention provides a method and system for using waveguide in an interferometer. Merely by way of example, the invention has been applied to measuring spatial locations, but it would be recognized that the invention has a much broader range of applicability. 
     Certain embodiments of the present invention use a Planar Lightwave Circuit (PLC). For example, the embodiment shown in  FIG. 1  uses a PLC. The circuit is also commonly referred to as a Photonic Integrated Circuit (PIC), a Lightwave Integrated Optic (LIO) circuit or an Integrated Optic (IO) circuit among other common nomenclatures in the literature. PLC fabrication technology has been developed in a number of material systems including glasses, polymers, lithium niobates and III-V semiconductors. Among the glasses are silica (SiO 2 ) glasses with or without additional network forming oxides such as GeO 2 , TiO 2 , P 2 O 5  and/or B 2 O 3 . These glasses and the circuit components therein may be deposited onto various substrates such as silicon (Si) or quartz (SiO 2 ). 
     Some embodiments of the present invention will be described in terms of the silica on silicon technology due to the mature state of commercial development of this material system along with the desirable mechanical, optical and thermal properties of this material system. However, the present invention can be realized in any of the material systems available for PLC development. The silica on silicon PLC may be fabricated by any of the processes available for this material system including but not limited to flame hydrolysis deposition (FHD), vacuum deposition (VD) and various chemical vapor deposition processes (CVD) including low pressure CVD (LPCVD), atmospheric pressure CVD (APCVD) and plasma enhanced CVD (PECVD). 
     A PLC according to an embodiment of the present invention can include single mode optical waveguides with one or more of waveguide bends, waveguide crossings, waveguide couplers and waveguide splitters. The waveguide bends usually are constrained to radii of curvature sufficiently large to yield acceptably low radiation loss from the bends. This radiation loss will be determined by the refractive index difference between the waveguide core and cladding materials (the “index contrast”) and will be fixed by the particular waveguide properties chosen. Waveguide crossing angles may be determined by a minimum crossing angle that provides an acceptably low level of crosstalk between the two crossing waveguides, which is a function of the index contrast. Minimum bend radii and minimum crossing angles may determine the minimum physical size of the PLC. 
     Waveguide couplers and/or splitters are identified in the embodiments of  FIGS. 1–6  by the letters A, B, E, F, G, H and I. Splitters A and B in  FIGS. 1 and 2  along with splitters G and H in  FIGS. 5 and 6  may be Y-branch splitters, directional couplers or multimode interference (MMI) devices. All output couplers, E, H and I, may be directional couplers or MMI devices to provide the indicated balanced output. Directional couplers will be described in the various embodiments of the present invention although Y-branch or MMI devices represent examples of applicable alternatives. 
       FIG. 1  is a simplified diagram for a heterodyne interferometer according to one embodiment of the present invention. This diagram is merely an example, which should not unduly limit the scope of the present invention. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. Heterodyne interferometer  100  includes mirror system  110  also called M R , mirror system  112  also called M B , collimating system  120 , collimating system  122 , laser system  130 , frequency shifter system  140 , single mode fibers  150 , silicon V-groove blocks  160 , single mode fibers  170 , reference balanced detector  180 , and measurement balanced detector  182 , and PLC system  190 . Although the above has been shown using systems  110 ,  112 ,  120 ,  122 ,  130 ,  140 ,  150 ,  160 ,  170 ,  180 ,  182 , and  190 , there can be many alternatives, modifications, and variations. For example, some of the systems may be expanded and/or combined. For example, mirror system  110  or  112  may be combined with PLC system  190 . Other systems may be inserted to those noted above. Depending upon the embodiment, the specific systems may be replaced. Further details of these systems are found throughout the present specification and more particularly below. 
     Mirror system  110  receives a laser beam from PLC system  190  and sends the laser beam to mirror system  112 . Mirror system  112  receives the laser beam and sends the laser beam back to PLC system  190 . Between mirror systems  110  and  112 , the laser beam travels outside PLC system  190 . 
       FIG. 1A  is a simplified diagram for mirror systems  110  and  112 . The diagram is merely an example, which should not unduly limit the scope of the present invention. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. Mirror system  110  includes mirrors  114  and  115 , and mirror system  112  includes mirrors  116  and  117 . Mirror  114  forms a right angle with mirror  115 , and mirror  116  forms a right angle with mirror  117 . As shown in  FIG. 1A , the laser beam leaves PLC system  190  at C and travels towards mirror system  110  substantially in parallel to direction  118 . At mirror system  110 , the laser beam is reflected off of mirrors  114  and  115  and then travels towards mirror system  112  substantially in parallel to direction  119 . At mirror system  112 , the laser beam is reflected off mirrors  116  and  117  and subsequently returns to PLC system  190  at D. As discussed above and further emphasized here,  FIG. 1A  is merely an illustration. For example, mirror systems  110  and  112  may each comprise more or less than two mirrors and the mirrors may take various types of configurations. In one embodiment of the present invention, mirror systems  110  and  112  each comprise a cube comer reflector. 
     Collimating system  120  collimates the laser beam exiting from PLC system  190  at C, and collimating system  122  focuses the laser beam returning to PLC system  190  at D.  FIG. 1B  is a simplified diagram for collimating system  120 . The diagram is merely an illustration, which do not unduly limit the scope of the claims. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. As shown in  FIG. 1B , PLC system  190  is formed within cladding layer  123  on top of substrate layer  124 . Substrate layer  125  and collimating system  120  are attached to support layer  125 . Collimating system  120  receives the laser beam exiting out of PLC system  190  at C. Exit C is located at a focal point of collimating system  120  and serves substantially as a point laser source. After the laser beam passes through collimating system  120 , the laser beam becomes substantially collimated. Additionally, collimating system  120  is designed so that its thermal expansion in optical length cancels out the thermal expansion of region  127  in support layer  125 . This design can be accomplished by various conventional methods, such as CTE technique. Additionally, collimating system  120  may include one lens as illustrated in  FIG. 1B  or include several lenses. Additionally, collimating system  122  may be constructed with a structure similar to the one shown in  FIG. 1B  and consistent with the above discussions. 
     Laser system  130  provide a laser beam that travels through PLC system  190  and is eventually detected by reference detector  180  and measurement detector  182 . For example, the laser beam has a wavelength ranging from a near ultraviolet wavelength to a far infrared wavelength. The near ultraviolet wavelength may equal 250 nm, and the far infrared wavelength may equal 50 μm. To improve accuracy of spatial measurement, laser system  130  should have a good phase stability. Laser beam  132  emitted by laser source  130  usually have a coherent length that exceeds or equals the distance over which the laser beam travels outside PLC system  190  between C, M R , M B , and D. For example, the coherent length is several meters. Laser system  130  may use a He—Ne laser source or other type of laser source. 
     Frequency shifter system  140  changes the frequency of input laser beam by a predetermined amount. Frequency shifter system  140  may include an acousto-optic frequency shifter, an electro-optic frequency shifter, or other type of frequency shifter. Under certain circumstances, the acousto-optic frequency shifter is preferred. The electro-optic frequency shifter may generate a large number of harmonics and hence contaminate the frequency purity of the laser beam. Additionally, frequency shifter system  140  may include an offset frequency signal generator driving the acousto-optic frequency shifter or the electro-optic frequency shifter. 
     Single mode fibers  150  transmit the laser beam to PLC system  190  while maintaining polarization of the laser beam. The laser beam emitted from laser system  130  is polarized, and single mode fibers  150  is designed to maintain such polarization. But the purity of the polarization usually degrades with traveling distance in single mode fibers  150 . Additionally, the polarization angle with respect to the planar waveguide circuit in PLC system  190  can be difficult to control. Similarly, single mode fibers  170  transmit the laser beam to reference balanced detector  180  and measurement balanced detector  182  while maintaining polarization of the laser beam. 
     V-groove blocks  160  align center of each of single mode fibers  150  and  170  with their respective entrance to or exit from PLC system  190 .  FIG. 1C  is a simplified diagram for alignment of single mode fiber and waveguide. The diagram is merely an illustration, which do not unduly limit the scope of the claims. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. As shown in  FIG. 1C , cladding layer  123  surrounds waveguide core  164 . For example, waveguide core  164  is usually about 5-μm in diameter, and cladding layer  123  is usually about 15-μm thick above or below waveguide core  164 . Single mode fiber  162  may be one of single mode fibers  150  and  170 . Single mode fiber  162  has fiber core  163 . For example, single mode fiber  162  is about 125 μm in diameter and fiber core is about 9 μm in diameter. Alignment between fiber core  163  and waveguide core  123  prefers high precision. The high precision may be achieved by using silicon V-groove blocks in some embodiments of the present invention. 
       FIG. 1D  is a simplified diagram for silicon V-groove blocks according to one embodiment of the present invention. The diagram is merely an illustration, which should not unduly limit the scope of the claims. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. As shown in  FIG. 1D , a silicon V-groove block includes side surfaces  166  and  167 . Side surfaces  166  and  167  forms angle  168 . Additionally, the silicon V-grove block has opening width  169 . For example, side surfaces  166  and  167  are  110  silicon surfaces, and angle  168  equals about 55°. 
       FIG. 1E  is a simplified diagram for positioning single mode fiber in silicon V-groove blocks according to one embodiment of the present invention. The diagram is merely an illustration, which should not unduly limit the scope of the claims. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. As shown in  FIG. 1E , single mode fiber  162  with fiber core  163  are placed in a V-groove block and in contact with side surface  166  and  167 . By adjusting opening width  169 , the position of single mode fiber  162  and its fiber core  163  may be accurately controlled with respect to top surface  165  of the silicon layer. Hence the alignment between fiber core  163  and waveguide core  123  may be improved. 
     Reference balanced detector  180  and measurement balanced detector  190  each measure the phase of a heterodyne interference signal by determining the time when the intensity of the signal drops to zero. The phase of each heterodyne interference signal may also be measured by observing other characteristics of the interference signal by other techniques well know in the art. 
     PLC system  190  is within the rectangular region outlined by the broken line. PLC system  190  includes a planar lightwave circuit and TM mode stripping devices  192 . The distance to be measured is the separation between mirrors M R  and M B . The measurement beam may be considered to enter PLC system  190  at coupler B, traverse the path BC on PLC system  190 , exit PLC system  190  at C, pass though collimating system  120  and reflect from mirror M R  onto mirror M B . From mirror M B , the beam is reflected back onto the PLC at D after passing though another collimating system  122 . The measurement beam then traverses path DE on the PLC and exits the PLC at coupler E. 
     The path length on PLC system  190  of the measurement beam is BC+DE and the external path length of the measurement beam is CM R +M R M B +M B D. A reference path for the measurement beam is established by the path length BI. The phase of the measurement optical signal entering the PLC at coupler at B is uncontrolled and variable due to phase noise in laser source  130 , mechanical and thermal effects in one or more single mode fibers  150  coupling laser source  130  to PLC system  190  at B, and coupling effects between single mode fibers  150  and laser system  130  or PLC system  190 . This unknown and uncontrolled optical phase at the input to coupler B may be defined as φ R (t). 
     The heterodyne optical signal may be considered to enter PLC system  190  at coupler A. A portion of the beam exits the PLC through the measurement coupler output at E after traversing the path length AE on the PLC. In this embodiment, the path length AE is selected such that AE=BC+DE so that thermal variations of measurement signal path on chip are equal to thermal variations of heterodyne signal path. A reference path for the heterodyne optical signal may be established by the path length AI such that AI=BI so that thermal variations of these two optical paths are also matched. 
     The phase of the heterodyne optical signal entering PLC system  190  at coupler at A may be uncontrolled and variable due to phase noise in laser source  130 , phase noise in the frequency shifter system  140 , mechanical and thermal effects in one or several single mode fibers  150  coupling the laser beam to frequency shifter system  140 , one or several single mode fibers coupling frequency shifter system  140  to PLC system  190  at A, and coupling effects between single mode fibers  150  and laser source  130 , frequency shifter system  140 , or PLC system  190 . This unknown and uncontrolled optical phase at the input to coupler A may be defined as φ B (t). The embodiment of an interferometer shown  FIG. 1  and its corresponding PLC schematic shown in  FIG. 2  will be referred to herein as a “Racetrack” configuration since the measurement beam exits PLC system  190  at port C, traverses a loop around PLC system  190  and reenters PLC system  190  through port D. In certain embodiments, this configuration accommodates the required PLC path-length equalities, including AE=BC+DE and AI=BI, by appropriate location of the couplers A and B on the PLC along with an extra “bulge” in the path AE to lengthen the inside track of AE. 
     In the embodiment of  FIGS. 1 and 2 , the optical intensity interference pattern at the reference signal output coupler, I, is given by the following Equation 1:
 
 P   ref∓ =( P   R   +P   B ∓2√{square root over ( P   R   P   B )}cos[(ω R −ω B ) t +(φ R −φ B )+(θ BI −θ AI )])/4   (1)
 
     where P R  is the optical power, ω R  is the optical frequency and φ R (t) is the optical phase at the input to coupler B while P B  is the optical power, ω B  is the optical frequency and φ B (t) is the optical phase at the input to coupler A. Equation 1 does not assume that any optical loss will occur. Optical loss would only appear as a reduction in the values of P R  and P B . It would not affect the phase terms that are of interest in the heterodyne interferometer. The upper (minus) sign in the equation corresponds to the optical power in the upper waveguide exiting the coupler I while the lower (plus) sign corresponds to the optical power in the lower waveguide exiting the coupler I. 
     The phase terms, θ BI  and θ AI  correspond to the optical path lengths θ BI =ω R n eff BI/c and θ AI =ω B n eff AI/c where n eff  is the effective refractive index of the guided optical waves and c is the velocity of light in vacuum. The difference of these two phase terms in the output heterodyne interference reference signal may be defined by the following Equation 2:
 
θ BI −θ AI   =n   eff (ω R   BI−ω   B   AI )/ c=ω   R   n   eff ( BI−AI (1+Δω/ω R ))/ c   (2)
 
     where Δω is the offset frequency difference, ω B −ω R . Since ω R ≈1.2×10 15  radians per second and Δω≈10 5  to 10 8  radians per second, the term involving Δω/ω≈10 −7  to 10 −10  may be neglected. As a result, θ BI −θ AI ≈ω R n eff (BI−AI)/c≡θ 1 , where θ 1  is a small, temperature insensitive phase angle since BI≈AI by design. Hence Equation (1) may be simplified as follows:
 
 P   ref∓ =( P   R   +P   B ∓2√{square root over ( P   R   P   B )}cos[(ω R −ω B ) t+(φ   R −φ B )])/4  (2A)
 
     A similar equation for the optical intensity interference pattern at the measurement output coupler, E, may be given by the following Equation 3:
 
 P   meas     ∓   =( P   R   +P   B ∓2√{square root over ( P   R   P   B )}cos[(ω R −ω B ) t +(φ R −φ B )+θ E +2φ L ])/4  (3)
 
     where θ E ≡θ BC+DE −θ AE  is a small, temperature insensitive phase angle since AE≈BC+DE by design and 2φ L =ω R (CM R +M R M B +M B D)/c contains the external path length of the measurement beam. Hence Equation 3 may be simplified as follows:
 
 P   meas     ∓   =( P   R   +P   B ∓2√{square root over ( P   R   P   B )}cos[(ω R −ω B ) t +(φ R −φ B )+2φ L ])/4  (3A)
 
     Comparing the phases of the measurement interference pattern, P meas     ∓   , and reference interference pattern, P ref     ∓   , we see that the unknown, uncontrolled phase function, φ R −φ B  may be determined from P ref     ∓    by reference balanced detector  180 . φ R −φ B  is subtracted from the phase of P meas     ∓    determined by measurement balanced detector  190  and the subtraction results in 2φ L . Consequently, the distance between M R  and M B  is determined as follows:
 
 CM   R   +M   R   M   B   +M   B   D= 2× c×φ   L /ω R   (3B)
 
       FIG. 2A  is a simplified diagram for a homodyne interferometer according to one embodiment of the present invention. The diagram is merely an example, which should not unduly limit the scope of the present invention. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. As shown in  FIG. 2A , the homodyne interferometer includes at least PLC system  210  which in turn includes TM Mode Stripping Devices  212  and a planar lightwave circuit. The optical beam from a laser system enters PLC system  210  at coupler A, traverses the path AC on the PLC of PLC system  210 , exits the PLC at C, passes though a collimating system and reflect from mirror M R  onto mirror M B . From mirror M B , the beam is reflected back onto the PLC at D after passing though another collimating system. The beam then traverses path DE on the PLC and exits PLC system  210  at coupler E. Additionally, the optical beam enters the PLC at coupler A, traverse the path AE on the PLC, and exits PLC system  210  at coupler E. In this embodiment, the path length AE is selected such that AE=AC+DE so that thermal variations of two signal paths on the PLC are equal to each other. At coupler E, the signals from the two signal paths interfere with each other and form an interference pattern substantially independent of time. This interference pattern can be detected by a conventional detector, and consequently the distance between mirrors, i.e., CM R +M R M B +M B D, can be determined. 
       FIG. 3  is a simplified diagram for a heterodyne interferometer according to another embodiment of the present invention. This diagram is merely an example, which should not unduly limit the scope of the present invention. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. Heterodyne interferometer  300  includes mirror system  310  also called M R , mirror system  312  also called M B , collimating system  320 , collimating system  322 , laser system  330 , frequency shifter system  340 , single mode fibers  350 , silicon V-groove blocks  360 , single mode fibers  370 , balanced detector  380 , and balanced detector  382 , and PLC system  390 . Although the above has been shown using systems  310 , 312 ,  320 ,  322 , 330 ,  340 ,  350 ,  360 ,  370 ,  380 ,  382 , and  390 , there can be many alternatives, modifications, and variations. For example, some of the systems may be expanded and/or combined. For example, mirror system  310  or  312  may be combined with PLC system  390 . Other systems may be inserted to those noted above. Depending upon the embodiment, the specific systems may be replaced. Further details of these systems are found throughout the present specification and more particularly below. 
     Collimating system  322 , laser system  330 , frequency shifter system  340 , single mode fibers  350 , silicon V-groove blocks  360 , single mode fibers  370 , balanced detector  380 , and balanced detector  382  are substantially similar to collimating system  122 , laser system  130 , frequency shifter system  140 , single mode fibers  150 , silicon V-groove blocks  160 , single mode fibers  170 , balanced detector  180 , and balanced detector  182  respectively. 
       FIG. 3A  is a simplified diagram for mirror systems  310  and  312 . The diagram is merely an example, which should not unduly limit the scope of the present invention. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. Mirror system  310  includes mirror  314 , and mirror system  312  includes mirror  316 . As shown in  FIG. 3A , the laser beam leaves PLC system  390  at C and travels towards mirror system  310  substantially perpendicular to mirror  310 . At mirror system  310 , the laser beam is reflected off of mirror  314 , travels towards PLC system  390 , and then reenters PLC system  390  at C. Additionally, the laser beam leaves PLC system  390  at D and travels towards mirror system  312  substantially perpendicular to mirror  316 . At mirror system  310 , the laser beam is reflected off of mirror  316 , travels towards PLC system  390 , and then reenters PLC system  390  at D. As discussed above and further emphasized here,  FIG. 3A  is merely an illustration. For example, mirror systems  310  and  312  may each comprise more than one mirror and the mirrors may take various types of configurations. In one embodiment of the present invention, mirror systems  110  and  112  each comprise a cube corner reflector. 
     PLC system  390  is within the rectangular region outlined by the broken line. PLC system  390  includes a planar lightwave circuit and TM mode stripping devices  392 .  FIG. 4  schematically represents the embodiment of the PLC circuit incorporated in the interferometer of  FIG. 3 . The distance to be measured is the sum of the distance from the PLC at C to the mirror at L R  and the distance from the PLC at D to the mirror at L B . One optical source may be considered to enter PLC system  390  at coupler B with optical power, P R , optical frequency, ω R , and optical phase, φ R (t). In this embodiment, the phase of the measurement optical signal entering PLC system  390  at coupler B, φ R (t), is uncontrolled and variable due to phase noise in laser source  330 , mechanical and thermal effects in one or more single mode fibers  350  coupling laser source  330  to PLC system  390 , and coupling effects between one or more single mode fibers  350  and laser source  330  or PLC system.  390 . A second optical source may be considered to enter the PLC at coupler A with optical power, P B , optical frequency, ω B , and optical phase, φ B (t). In this embodiment, the phase of the heterodyne optical signal entering PLC system  390  at coupler A, φ B (t), is uncontrolled and variable due to phase noise in laser source  330 , phase noise in frequency shifter system  340 , mechanical and thermal effects in one or more single mode fibers coupling laser source  330  to frequency shifter system  340  and one or more single mode fibers coupling frequency shifter system  340  to PLC system  390 , and coupling effects between one or more single mode fibers and laser source  330 , frequency shifter system  340  or PLC system  390 . 
     One branch of the path of the measurement optical signal entering PLC system  390  at B traverses the path BC, exits PLC system  390  at C and after an external path of 2L R , reenters PLC system  390  at C and traverses the path CBF to the output coupler at F. The path length of this signal on PLC system  390  is 2BC+BF. One branch of the path of the heterodyne optical signal entering PLC system  390  at A traverses the path AF to the output coupler at F. These two path lengths are set equal, i.e. 2BC+BF=AF to compensate thermal effects on PLC system  390 . 
     Another branch of the measurement optical signal entering the PLC at B traverses the path BE to the output coupler at E. Another branch of the heterodyne optical signal entering PLC system  390  at A traverses the path AD, exits the PLC at D and after an external path of 2L B , reenters PLC system  390  at D and traverses the path DAE to the output coupler at E. The path length of this signal on PLC system is 2AD+AE. These later two path lengths are also set equal, i.e. 2AD+AE=BE to compensate thermal effects on PLC system  390 . The interferometer of  FIG. 3  and its corresponding PLC schematic shown in  FIG. 4  are referred to herein as the “Trombone.” In some embodiments, the PLC path length equalities, 2BC+BF=AF and 2AD+AE=BE are easily accommodated by appropriate adjustment of the lengths of the horizontal straight segments in  FIG. 4 . 
     In the embodiment shown in  FIGS. 3 and 4 , the optical intensity interference pattern at the optical signal output coupler, E, may be defined by the following Equation 4: 
               P     E   ±       =         P   R     4     +         P   B     8     ±             P   R     ⁢     P   B           2     3   /   2         ⁢           ⁢     sin   ⁡     [         (       ω   R     -     ω   B       )     ⁢   t     +     (       ϕ   R     -     ϕ   B       )     +     (       θ   BE     -     2   ⁢           ⁢     θ   AD       -     θ   AE       )     -     2   ⁢           ⁢     ϕ     DL   R           ]                     (   4   )             
 
     where P R  is the optical power, ω R  is the optical frequency and φ R (t) is the optical phase at the input to coupler B while P B  is the optical power, ω B  is the optical frequency and φ B (t) is the optical phase at the input to coupler A as indicated in  FIG. 4 . No optical loss has been assumed in this expression. Optical loss would only appear as a reduction in the values of P R  and P B . It would not affect the phase terms that are of interest in the heterodyne interferometer. The upper (plus) sign in the equation corresponds to the optical power in the upper waveguide exiting the coupler E while the lower (minus) sign corresponds to the optical power in the lower waveguide exiting the coupler E. 
     The phase terms, θ BE  and 2θ AD +θ AE  correspond to the optical path lengths, θ BE =ω R n eff BE/c and 2θ AD +θ AE =ω B n eff (2AD+AE)/c, where n eff  is the effective refractive index of the guided optical waves and c is the velocity of light in vacuum. The difference of these two phase terms may be defined by the following Equation: 
                       θ   BE     -     2   ⁢           ⁢     θ   AD       -     θ   AE       =         n   eff     ⁡     (         ω   R     ⁢   BE     -       ω   B     ⁡     (       2   ⁢   AD     +   AE     )         )       /   c                 =       ω   R     ⁢         n   eff     ⁡     (     BE   -       (       2   ⁢   AD     +   AE     )     ⁢     (     1   +     Δ   ⁢           ⁢     ω   /     ω   R           )         )       /   c                     (   5   )             
 
     where Δω is the offset frequency difference, ω B −ω R . Since ω R ≈1.2×10 15  radians per second and Δω≈10 5  to 10 8  radians per second, the term involving Δω/ω R ≈10 −7  to 10 −10  may be neglected. Also, θ BE −2θ AD −θ AE ≈ω R n eff (BE−(2AD+AE))/c≡θ E , where θ E  is a small, temperature insensitive phase angle since BE≈2AD+AE by design. The phase term 2θ DL     B   =2ω B L B /c contains the round trip path length from PLC system  390  at exit point D to mirror system M B  and back to point D. 
     A similar equation for the optical intensity interference pattern at the optical signal output coupler, F, is given by the following Equation 6: 
               P     F   ±       =         P   R     4     +         P   B     8     ±             P   R     ⁢     P   B           2     3   /   2         ⁢           ⁢     sin   ⁡     [         (       ω   R     ⁢     ω   B       )     ⁢   t     +     (       ϕ   R     -     ϕ   B       )     +     (       2   ⁢           ⁢     θ   BC       +           ⁢     θ   BF     -     θ   AF       )     +     2   ⁢           ⁢     ϕ     CL   R           ]                     (   6   )             
 
     where θ F ≡2θ BC +θ BF −θ AF  is a small, temperature insensitive phase angle since AF≈2BC+BF by design. The upper (plus) sign in the equation corresponds to the optical power in the upper waveguide exiting the coupler F while the lower (minus) sign corresponds to the optical power in the lower waveguide exiting the coupler F. The phase term 2φ DL     R   =2ω R L R /c contains the round trip path length from PLC system  390  at exit point C to mirror M R  and back to point C. If the phase of the signal out of coupler E is subtracted from the phase of the signal out of coupler F, the result 2(φ CL     R   +φ DL     B   )+θ F −θ E  is obtained, which contains the separation of the two mirrors plus two small, fixed, temperature insensitive terms, θ F  and θ E . θ F  and θ E  are usually negligible and Δω is also usually negligible, so the sum of L R  and L B  can be calculated as follows:
 
 L   R   +L   B   =c ×(φ DL     R   +φ DL     B   )/ω R   (6A)
 
     While the interferometer configuration shown in  FIG. 3  is such that measurement of the sum of the two distances, L R  and L B  allows elimination of the uncontrolled phase fluctuations at the input couplers, φ R −φ B , it is also possible to add a separate measurement circuit to determine this quantity as shown the schematic representations of PLC circuits shown in  FIGS. 5 and 6 . 
     In  FIG. 5 , the additional couplers G and H are arranged such that the input phase signal is coupled out on the left side of the PLC. This embodiment is referred to herein as the “Lightbulb”. The design and operation of this circuit is essentially the same as the “Trombone” except that some optical power is coupled out at I to provide the reference signal. 
     As shown in  FIG. 5 , PLC system  590  is within the rectangular region outlined by the broken line. PLC system  590  includes a planar lightwave circuit and TM mode stripping devices  592 . One branch of the measurement optical signal entering PLC system  590  at B traverses the path BC, exits the PLC at C and after an external path of 2L R , reenters the PLC at C and traverses the path CBF to the output coupler at F. One branch of the heterodyne optical signal entering PLC system  590  at A traverses the path AD, exits the PLC at D and after an external path of 2L B , reenters the PLC at D and traverses the path DAE to the output coupler at E. 
     Additionally, another branch of the measurement optical signal entering PLC system  590  at B traverses the path BGE to the output coupler at E. Another branch of the heterodyne optical signal entering the PLC at A traverses the path AHF and exits PLC system  590  at F. Moreover, yet another branch of the measurement optical signal entering PLC system  590  at B traverses the path BGI to the output coupler at I. Yet another branch of the heterodyne optical signal entering PLC system  590  at A traverses the path AHI to the output coupler at I. 
     If the optical signal inputs are as described for the Trombone circuit discussed above, i.e., P R  (ω R ,φ R ) into coupler B and P B (ω B ,φ B ) into coupler A, then the optical intensity interference pattern at the optical signal output coupler, E, is given by the following Equation 7: 
               P     E   ±       =         P   R     8     +         P   B     8     ±             P   R     ⁢     P   B         4     ⁢           ⁢     sin   ⁡     [         (       ω   R     -     ω   B       )     ⁢   t     +     (       ϕ   R     -     ϕ   B       )     +     (       θ     BG   +   GE       -     2   ⁢           ⁢     θ   AD       -     θ   AE       )     -     2   ⁢           ⁢     ϕ     DL   R           ]                     (   7   )             
 
     The upper (plus) sign in the equation corresponds to the optical power in the upper waveguide exiting the coupler E while the lower (minus) sign corresponds to the optical power in the lower waveguide exiting the coupler E. The phase angle θ BG+GE −2θ AD −θ AE ≡θ E  is made small and temperature invariant or insensitive by setting the path lengths BG+GE=2AD+AE. θ E  is usually negligible. 
     The optical intensity interference pattern at the optical signal output coupler, F, is given by the following Equation 8: 
               P     F   ±       =         P   R     8     +         P   B     8     ±             P   R     ⁢     P   B         4     ⁢           ⁢     sin   ⁡     [         (       ω   R     -     ω   B       )     ⁢   t     +     (       ϕ   R     -     ϕ   B       )     +     (       2   ⁢           ⁢     θ   BC       +     θ   BF     -     θ     AH   +   HF         )     +     2   ⁢           ⁢     ϕ     CL   R           ]                     (   8   )             
 
     The upper (plus) sign in the equation corresponds to the optical power in the upper waveguide exiting the coupler F while the lower (minus) sign corresponds to the optical power in the lower waveguide exiting the coupler F. The phase angle 2θ BC +θ BF −θ AH+HF ≡θ F  is made small and temperature invariant or insensitive by setting the path lengths 2BC+BF=AH+HF. θ F  is usually negligible. Finally, the optical intensity interference pattern at the optical signal output coupler, I, is given by the following Equation 9: 
               P     I   ±       =         P   R     8     +         P   B     8     ±             P   R     ⁢     P   B         4     ⁢           ⁢     sin   ⁡     [         (       ω   R     -     ω   B       )     ⁢   t     +     (       ϕ   R     -     ϕ   B       )     +     θ     BG   +   GI       -     θ     AH   +   HI         ]                     (   9   )             
 
     The upper (plus) sign in the equation corresponds to the optical power in the upper waveguide exiting the coupler I while the lower (minus) sign corresponds to the optical power in the lower waveguide exiting the coupler I. The phase angle θ BG+GI −θ AH+HI ≡θ I  is made small and temperature invariant or insensitive by setting the path lengths BG+GI=AH+HI. θ I  is usually negligible. As discussed above, ω B −ω R  is also usually negligible. From the phase of Equation 9, the unknown and uncontrolled phase term, φ R −φ B , may be obtained for an independent determination of L R  and L B  in Equations (7) and (8). 
       FIG. 6  illustrates an embodiment in which the additional couplers G and H are arranged such that the input phase signal is coupled out on the right side of the PLC. This embodiment is referred to herein as the “Jellyfish”. The design and operation of this circuit is similar to the Lightbulb circuit as shown in  FIG. 5 . 
     As shown in  FIG. 6 , PLC system  690  is within the rectangular region outlined by the broken line. PLC system  590  includes a planar lightwave circuit and TM mode stripping devices  692 . One branch of the path of the measurement optical signal entering PLC system  690  at B traverses the path BC, exits PLC system  690  at C and after an external path of 2L R , reenters PLC system  690  at C and traverses the path CBF to the output coupler at F. The path length of this signal on PLC system is 2BC+BF. One branch of the heterodyne optical signal entering the PLC at A traverses the path AD, exits PLC system  690  at D and after an external path of 2L B , reenters PLC system  690  at D and traverses the path DAE to the output coupler at E. The path length of this signal on PLC system  690  is 2AD+AE. 
     Another branch of the measurement optical signal entering PLC system  690  at B traverses the path BHE to the output coupler at E. Another branch of the heterodyne optical signal entering PLC system  690  at A traverses the path AGF, exits PLC system  690  at F. Additionally, yet another branch of the measurement optical signal entering PLC system  690  at B traverses the path BHI to the output coupler at I. Yet another branch of the heterodyne optical signal entering the PLC at A traverses the path AGI to the output coupler at I. 
     If the optical signal inputs are as described for the Trombone circuit discussed above, i.e., P R (ω R ,θ R ) into coupler B and P B (ω B ,θ B ) into coupler A, then the optical intensity interference pattern at the optical signal output coupler, E, is given by the following Equation 9A: 
               P     E   ±       =         P   R     8     +         P   B     8     ±             P   R     ⁢     P   B         4     ⁢           ⁢     sin   ⁡     [         (       ω   R     -     ω   B       )     ⁢   t     +     (       ϕ   R     -     ϕ   B       )     +     (       θ     BH   +   HE       -     2   ⁢           ⁢     θ   AD       -     θ   AE       )     -     2   ⁢           ⁢     ϕ     DL   R           ]                     (     9   ⁢   A     )             
 
     The upper (plus) sign in the equation corresponds to the optical power in the upper waveguide exiting the coupler E while the lower (minus) sign corresponds to the optical power in the lower waveguide exiting the coupler E. The phase angle θ BH+HE −2θ AD −θ AE ≡θ E  is made small and temperature invariant or insensitive by setting the pathlengths BH+HE=2AD+AE. θ E  is usually negligible. 
     The optical intensity interference pattern at the optical signal output coupler, F, is given by the following Equation 10: 
               P     F   ±       =         P   R     8     +         P   B     8     ±             P   R     ⁢     P   B         4     ⁢           ⁢     sin   ⁡     [         (       ω   R     -     ω   B       )     ⁢   t     +     (       ϕ   R     -     ϕ   B       )     +     (       2   ⁢           ⁢     θ   BC       +     θ   BF     -     θ     AG   +   GF         )     +     2   ⁢           ⁢     ϕ     CL   R           ]                     (   10   )             
 
     The upper (plus) sign in the equation corresponds to the optical power in the upper waveguide exiting the coupler F while the lower (minus) sign corresponds to the optical power in the lower waveguide exiting the coupler F. The phase angle 2θ BC +θ BF −θ AG+GF ≡θ F  is made small and temperature invariant or insensitive by setting the path lengths 2BC+BF=AG+GF. θ F  is usually negligible. 
     The optical intensity interference pattern at the optical signal output coupler, I, is given by the following Equation 11: 
               P     I   ±       =         P   R     8     +         P   B     8     ±             P   R     ⁢     P   B         4     ⁢           ⁢     sin   ⁡     [         (       ω   R     -     ω   B       )     ⁢   t     +     (       ϕ   R     -     ϕ   B       )     +     θ     BH   +   HI       -     θ     AG   +   GI         ]                     (   11   )             
 
     The upper (plus) sign in the equation corresponds to the optical power in the upper waveguide exiting the coupler I while the lower (minus) sign corresponds to the optical power in the lower waveguide exiting the coupler I. The phase angle θ BH+HI −θ AG+GI ≡θ I , is made small and temperature invariant or insensitive by setting the path lengths BH+HI=AG+Gl. θ I  is usually negligible. As discussed above, ω B −ω R  is also usually negligible. From the phase of Equation 11, the unknown and uncontrolled phase term, φ R −φ B , may be obtained for an independent determination of L R  and L B  in Equations 9A and 10. 
     The present invention provide advantages in various applications. For example, certain embodiments of the present invention can provide a number of advantages in the fabrication process. Along these lines, some embodiments of the present invention may be employed to replace a number of discreet, precision optical components and the attendant assembly processes with a single component manufactured with standard industrial processes which have grown out of a combination of optical fiber fabrication technology and microelectronic large scale integration technology.  FIG. 7  illustrates a perspective view and a cross-section of a planar lightwave circuit described in conjunction with PLC system  390  in  FIGS. 3 and 4 . In  FIG. 7 , optical waveguides are embedded in a silica layer on top of a silicon layer. If the planar lightwave circuit according to some embodiments of the present invention employs telecom industry wavelengths, there is no change or perturbation to standard PLC process lines, even though the photolithographic mask to define the specific circuit is required as a special item. 
     The optical waveguide size and minimum bending radii may be determined by the specific index contrast utilized in a particular fabrication facility or process. Representative values are given in Table 1 from M. Kawachi,  Recent progress in silica based planar lightwave circuits on silicon,  IEE Proc.-Optoelectron., Vol. 143, pp. 257–262 (1996). 
     
       
         
               
             
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Silica on Silicon Waveguide Parameters 
               
             
          
           
               
                 Waveguide type 
                 Low Δn 
                 Middle Δn 
                 High Δn 
               
               
                   
               
             
          
           
               
                 Core/cladding index 
                 0.25 
                 0.45 
                 0.75 
               
               
                 contrast Δn(%) 
               
               
                 Core size, μm 
                 8 × 8 
                 7 × 7 
                 6 × 6 
               
               
                 Loss, dB/cm 
                 &lt;0.1 
                 &lt;0.1 
                 &lt;0.1 
               
               
                 Fiber coupling loss, dB/point 
                 &lt;0.1 
                 0.1 
                 0.5 
               
               
                 Minimum bending radius, mm 
                 25 
                 15 
                 5 
               
               
                   
               
             
          
         
       
     
     From the PLC circuits laid out as schematically indicated in  FIGS. 2 , and  4 – 6 , the minimum bending radii given in Table 1 will establish typical minimum PLC circuit dimensions equal to or greater than about 5 cm by about 5 cm for a low index contrast process to about 1 cm by about 1 cm for a high index contrast process since all circuits involve arcs of about 90° or about 180° in various orientations. 
     The total thickness of glass on the silicon substrate typically varies from about 10 to about 50 μm. This thickness is dependent on the established processes of various manufacturers and does not affect the operation of the PLC. It may, however, influence the post process fabrication of TM mode stripping structures as discussed below. 
     In the above analyses, it was assumed that critical phase relations were established at the input couplers (A and B in  FIGS. 1 ,  2 ,  3 ,  4 ,  5 , and  6 ) and persisted only through the output couplers (E, F and I in the  FIGS. 1 ,  2 ,  3 ,  5 , and  6 ). This is true since the phase of the two signals exiting coupler A are fixed with respect to one another at the input of coupler A as are the two signals exiting coupler B fixed with respect to one another at the input of coupler B. When the signals exit the couplers E, F or I, the beat note between the two signals leaving the couplers is established and no further interaction occurs. Thus, the entire heterodyne interaction occurs in the robust, monolithic, photolithographically defined structure of silica on silicon. 
     Some embodiments of the present invention uses couplers, combiners or splitters that equally distribute its output power to two terminals. The equal distribution of output power facilitates balanced detection optimization. However, there may be advantages to use other splitting ratios in certain cases.  FIG. 8  schematically represents the output of a balanced detector as being the difference in photocurrents generated in two series connected photodiodes. The input optical powers to a directional coupler are P R  (ω R , θ R ) to the upper waveguide and P B (ω B ,θ B ) to the lower waveguide. For example, the directional coupler may be coupler I or E as shown in  FIG. 1 . The coupling of energy from the lower to upper waveguide and visa versa is dependent upon the coupling constant k between the waveguides and the length L of the interacting region. There is a sinusoidal variation of the coupling with length and coupling constant as indicated in  FIG. 8 . The upper waveguide existing the PLC has intensity P up  as shown below.
 
 P   up   =P   R  cos 2    kL+P   B  sin 2    kL− 2√{square root over ( P   R   P   B )}sin[(ω R −ω B ) t+θ   R −θ B ]sin  kL  cos kL   (12)
 
Additionally, the lower waveguide existing the PLC has intensity P down  as shown below.
 
 P   down   =P   R  sin 2   kL+P   B  cos 2   kL+ 2√{square root over ( P   R   P   B )}sin[(ω R −ω B ) t+θ   R −θ B ]sin  kL  cos  kL   (13)
 
     The balanced detector subtracts the response current corresponding to P down  from the response current corresponding to P up  and obtain the signal current i sig , as shown below.
 
 i   sig   =A   res ( P   R   −P   B )(cos 2    kL −sin 2    kL )−4 A   res √{square root over ( P   R   P   B  )}sin[(ω R −ω B ) t+θ   R −θ B ]sin  kL  cos  kL   (14)
 
     where A res  is a constant measuring sensitivity of the detector. If the coupling constant and interaction length are such that kL=π/4, then the coupling ratio is 50:50 and the signal intensity, i sig , in the balanced detector contains only the beat frequency or heterodyne term. Also the signal current i sig  reaches the following maximum when kL=π/4,
 
 i   sig =−2 A   res √{square root over ( P   R   P   B  )}sin[(ω R −ω B ) t+θ   R −θ B ]
 
     Furthermore, if the total optical power into the coupler, P opt =P R +P B  is fixed, then the signal current is maximized for P R =P B =P opt /2 . 
     If the total input optical power to the PLC is fixed by laser system  130  or  330  as indicated in  FIG. 1  or  3  but the optical loss in one channel is excessive (typically in the measurement path to external mirrors M R  and M B ), then further circuit optimization may be possible by adjusting the coupling ratios of the input couplers, A and B. In the case of the Racetrack circuit,  FIGS. 1 and 2 , the input coupler splitting ratio may be designed to provide excess power into the measurement path. In the case of the Lightbulb and Jellyfish circuits shown in  FIGS. 5 and 6 , the reference splitters G and H may be designed to optimize the division of power between the reference output port at I and the measurement ports at E and F. 
     Further advantages of certain embodiments of the present invention relate to temperature insensitivity of the devices. In the description of the various embodiments of the PLC circuit, certain path lengths were set equal to minimize thermal effects. These are summarized in Table 2. The small phase offset terms maybe written as
 
(ω R   n   eff   /c )[Δ L−LΔω/ω   R ] or (2 πn   eff /λ oR )[Δ L−LΔω/ω   R ]
 
where λ oR  is the free space wavelength of the optical signal at frequency ω R .
 
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Phase imbalance or off-set terms 
               
             
          
           
               
                 Circuit 
                 Matching Paths 
                 Phase Off-set 
               
               
                   
               
               
                 Racetrack 
                 AE = BC + DE 
                 θ E  = ω R n eff [BC + DE − 
               
               
                 FIGS. 1–2 
                   
                 AE(1 + Δω/ω R )]/c 
               
               
                   
                 AI = BI 
                 θ I  = ω R n eff [BI − AI(1 + 
               
               
                   
                   
                 Δω/ω R )]/c 
               
               
                 Trombone 
                 BE = 2AD + AE 
                 θ E  = ω R n eff [BE − (2AD + 
               
               
                 FIGS. 3 and 4 
                   
                 AE)(1 + Δω/ω R )]/c 
               
               
                   
                 AF = 2BC + BF 
                 θ F  = ω R n eff [2BC + BF − 
               
               
                   
                   
                 AF(1 + Δω/ω R )]/c 
               
               
                 Lightbulb 
                 BG + GE = 
                 θ E  = ω R n eff [BG + GE − 
               
               
                 FIG. 5 
                 2AD + AE 
                 (2AD + AE)(1 + Δω/ω R )]/c 
               
               
                   
                 AH + HF = 
                 θ F  = ω R n eff [2BC + BF − 
               
               
                   
                 2BC + BF 
                 (AH + HF)(1 + Δω/ω R )]/c 
               
               
                   
                 AH + HI = 
                 θ I  = ω R n eff [BG + GI − 
               
               
                   
                 BG + GI 
                 (AH + HI)(1 + Δω/ω R )]/c 
               
               
                 Jellyfish 
                 BH + HE = 
                 θ E  = ω R n eff [BH + HE − 
               
               
                 FIG. 6 
                 2AD + AE 
                 (2AD + AE)(1 + Δω/ω R )]/c 
               
               
                   
                 AG + GF = 
                 θ F  = ω R n eff [2BC + BF − 
               
               
                   
                 2BC + BF 
                 (AG + GF)(1 + Δω/ω R )]/c 
               
               
                   
                 AG + GI = 
                 θ I  = ω R n eff [BH + HI − 
               
               
                   
                 BH + HI 
                 (AG + GI)(1 + Δω/ω R )]/c 
               
               
                   
               
             
          
         
       
     
     For example, ΔL equals BC+DE−AE for Racetrack PLC and may easily be maintained at equal to or less than about 5 micrometers by the photolithographic design/fabrication process, including the location of the PLC edges at C and D in  FIGS. 1–6 . The edges of the PLC may be identified by a series of fiducial lines included on the photolithographic mask and precision edge polishing following PLC fabrication. The properties of silica on silicon PLCs is such that the thermal coefficient of optical path length changes in a silica based waveguide is 
           1     Δ   ⁢           ⁢   L       ⁢     ⅆ     ⅆ   T       ⁢     (       n   eff     ⁢           ⁢   Δ   ⁢           ⁢   L     )       =     1   ×       10     -   5       ⁢           [     1   ⁢     /     ⁢   °   ⁢           ⁢     C   .       ]           
 
so that if ΔL≦about 5 μm, δΔL≦about 5×10 −5 δT μm. A change in temperature of approximately one degree Celsius will result in only about 50 picometers of PLC unbalance change.
 
     From the embodiments of interferometer PLCs indicated in  FIGS. 2 ,  2 A,  4 ,  5  and  6 , the PLC optical path lengths of all paths from A or B to E, F, or I will be L˜2πR min , where R min  is the minimum radius of curvature from Table 1. Since 5≦R min ≦25 mm and 10 −10 ≦Δω/ω R ≦10 −7 , the maximum thermal contribution of the n eff LΔω/ω R  term to the optical path-length change will be δL≦1.6×10 −8  δT micrometers. A temperature change of about one degree Celsius will result in only about 0.016 picometers of PLC unbalance change, which is completely negligible compared to the thermal effects associated with ΔL changes. 
     Still additionally advantages of some embodiments of the present invention relate to immunity of devices from spurious signals. Spurious signals at the output couplers at the two frequencies ω R  and ω B , which have traversed paths other than the intended path will introduce measurement errors. These spurious signals may arise from reflections (especially at PLC interfaces with the external world), cross coupling (at waveguide crossings) and polarization mode dispersion. 
     The use of “angle lapping” is well known in the fiber optic industry to reduce reflections at fiber-fiber interfaces. This technique is also used for fiber PLC interfaces. An additional interface refection reduction at fiber PLC interfaces is accomplished by the use of index matching bonding agents at the silicon V-block fiber assembly structures indicated in  FIGS. 1 and 3 . For the interferometer, reflection reduction at the PLC free space interfaces, ports C and D in the embodiments shown in  FIGS. 1–7 , by angle lapping will result in reflection reduction as indicated in  FIG. 9 . The angle may vary from zero degree to about 7.5 degrees. Further reduction may be achieved by anti reflection coating of the PLC free space interface. 
     Single mode optical fibers and single mode waveguides in PLCs may sustain two orthogonal polarizations that may propagate at slightly different velocities. It may therefore be necessary to preferentially excite only one polarization mode at each input port, such as ports A and B in the embodiments shown in  FIGS. 1–7 , and to remove any optical power scattered from the desired mode into the orthogonal mode in the interferometer. This “mode stripping” operation may be accomplished at the output ports, such as ports E, F and I in the embodiments shown in  FIGS. 1–7 , as well as at portions of PLCs close to output interface C or/and D. 
     Preferential excitation of the desired mode may be accomplished by transporting the polarized laser beam to the PLC by polarization maintaining fibers as indicated for example in  FIGS. 1 and 3 . A mode stripping structure may be fabricated in PLCs as indicated in  FIGS. 10 and 11 . This structure can operate on the differing optical currents associated with the TE and TM modes that are induced in a metallic conductor. The TM optical mode is characterized by a large transverse optical frequency magnetic field oriented parallel to the plane of the PLC while the TE optical mode is characterized by a large optical frequency magnetic filed oriented perpendicular to the plane of the PLC. The tangential magnetic field induces optical currents in adjacent metallic conductors.  FIG. 11  illustrates optical attenuation in the TM and TE modes of a high-index-contrast single mode waveguide due to an Al metal film. The index contrast is represented by Δn. The thick top waveguide cladding layer may be thinned to the desired depth by various processes such as reactive ion etching, ion beam milling and/or wet chemical etching. As shown in  FIG. 11 , upper clad thickness  1110  may range from 2 to 4 μm. The chosen etch procedure may be adjusted to provide a smooth surface for the deposition of the metallic film. The chosen example metal, Al, was selected due to the large magnitude of both the real, n, and imaginary, k, components of the optical index for Al at a wavelength of about 1.56 μm. These mode stripping patches may be located outside of the PLC paths before the input couplers A and B, after the output couplers E, F, and I in  FIGS. 1–7 , or close to PLC interface C or D. 
     Certain embodiments of the present invention provide a solution to problems existing with known devices. Along these lines, these embodiments provide a small, lightweight and robust optical gauge. A planar lightwave circuit (PLC) according to the present invention replaces large bulk components typically utilized in known devices. Use of the PLC according to some embodiments of the present invention results in smaller devices that are less sensitive or insensitive to thermal changes. The top surface of a PLC system may be less than 25 cm 2 , as shown by PLC system  190  in  FIGS. 1 and 2 , by PLC system  210  in  FIG. 2A , by PLC system  390  in  FIGS. 3 and 4 , and  7 , by PLC system  590  in  FIG. 5 , and by PLC system  690  in  FIG. 6 . The PLC system may weigh less than 20 grams. Additionally, a device according to certain embodiments of the present invention can be fiber coupled to inputs and outputs, such as sensors and light sources, and contains few components. Features of some embodiments of the present invention can permit devices to measure distance changes, for example, that are on the order of about one millionth of the wavelength of light utilized in an application of the device. 
     Some embodiments of the present invention can provide a device that can be utilized in the control of large structures by providing position, velocity, and acceleration data for control loops, characterization of turbulent flow by measuring velocity spectra, characterization of sound by measurement of motion of surfaces, among other variables. Due to its size, construction, and operating characteristic, certain embodiments of the present invention are particularly suited to use for control of large, remotely controlled optical systems. Devices according to the present invention could also be utilized on assembly lines, where process controls require the non-contact control of positions and velocity. Devices according to the present invention are particularly useful where a rugged design is required. 
     The foregoing description of the invention illustrates and describes the present invention. Additionally, the disclosure shows and describes only the preferred embodiments of the invention, but as aforementioned, it is to be understood that the invention is capable of use in various other combinations, modifications, and environments and is capable of changes or modifications within the scope of the inventive concept as expressed herein, commensurate with the above teachings, and/or the skill or knowledge of the relevant art. For example, the PLC circuits in  FIGS. 4 ,  5 , and  6  may be modified to form homodyne interferometers. As another example, the waveguide circuits used in various embodiments of the present invention are not planar. The embodiments described hereinabove are further intended to explain best modes known of practicing the invention and to enable others skilled in the art to utilize the invention in such, or other, embodiments and with the various modifications required by the particular applications or uses of the invention. Accordingly, the description is not intended to limit the invention to the form disclosed herein. Also, it is intended that the appended claims be construed to include alternative embodiments.