Accurate chirped synthetic wavelength interferometer

Abstract

A system is provided for measuring distance or displacement, comprising: first and second laser sources configured to provide first and second laser outputs; a beam combiner configured to receive and combine at least part of the first and second laser outputs into a combined laser output; a signal calibrator configured to receive at least part of the first laser output, the second laser output, or the combined laser output, and output a calibration signal; a plurality of optical paths, including a first optical path, a second optical path, the plurality of optical paths being configured to direct at least part of the combined beam onto an optical detector to produce an interference signal; and a signal processor configured to receive the interference signal and determine a pathlength difference between the first and second optical paths.

Claims

1. A system comprising: a first chirped laser source configured to provide a first laser output having a first optical frequency varying at a first chirp rate; a calibration unit comprising at least one gas cell having at least one absorption line, the calibration unit configured to receive a first portion of the first laser output and to output a calibration signal; a first plurality of optical paths configured to direct a second portion of the first laser output onto an optical detector to produce a first interference signal; and a signal processor configured to receive the first interference signal and the calibration signal, the signal processor further configured to: determine a first time based on when the first optical frequency matches a feature of the absorption line of the gas cell during a chirp; and determine an interferometer phase based on the first time.

2. The system of claim 1 additionally comprising: a second chirped laser source configured to provide a second laser output having a second optical frequency varying at a second chirp rate; and a second plurality of optical paths configured to direct a portion of the second laser output onto the optical detector, or a different optical detector, to produce a second interference signal, wherein the signal processor is additionally configured to determine a second time based on when the second optical frequency matches a feature of an absorption line of the gas cell, or a different gas cell, during a chirp.

3. The system of claim 2, wherein the first plurality of optical paths and the second plurality of optical paths share at least one common optical path.

4. The system of claim 1, wherein the signal processor is further configured to determine a distance based, at least in part, on the determined interferometer phase.

5. The system of claim 1, wherein the determined interferometer phase is a standard interferometer phase.

6. The system of claim 1, wherein the determined interferometer phase is synthetic interferometer phase.

7. The system of claim 2, wherein the signal processor is further configured to calculate a fast Fourier transform or related transform of the first interference signal, the second interference signal, or both.

8. The system of claim 2, wherein the signal processor is further configured to calculate a Hilbert or related transform of the first interference signal, the second interference signal, or both.

9. The system of claim 2, wherein the signal processor is further configured to apply an electronic or digital filter to the first interference signal, the second interference signal, or both.

10. The system of claim 2, wherein the first chirp rate is different from the second chirp rate.

11. The system of claim 2, wherein the first chirp rate is the same as the second chirp rate.

12. The system of claim 7, wherein the signal processor is further configured to calculate a range profile based on the fast Fourier transform or related transform.

13. A method comprising: chirping a first laser output having a first optical frequency by varying the first optical frequency at a first chirp rate; directing a first portion of the first laser output to a calibration unit comprising at least one gas cell having at least one absorption line, wherein the calibration unit is configured to output a calibration signal; directing a second portion of the first laser output through a first plurality of optical paths onto a detector to produce a first interference signal; and determining a distance based at least in part on the first interference signal and the calibration signal, wherein determining the distance comprises: determining a first time when the first optical frequency matches a feature of the absorption line of the gas cell during the chirping; and determining an interferometer phase based, at least in part, on the first time.

14. The method of claim 13, additionally comprising: chirping a second laser output having a second optical frequency by varying the second optical frequency at a second chirp rate; and directing a portion of the second laser output through a second plurality of optical paths onto the optical detector, or a different optical detector, to produce a second interference signal, wherein determining the distance is also based on the second interference signal, and wherein determining the distance further comprises determining a second time when the second optical frequency matches a feature of an absorption line of the gas cell, or a different gas cell, during a chirp.

15. The method of claim 14, wherein the first plurality of optical paths and the second plurality of optical paths share at least one common optical path.

16. The method of claim 14, wherein the first chirp rate is different from the second chirp rate.

17. The method of claim 14, wherein the first chirp rate is the same as the second chirp rate.

18. The method of claim 14 wherein determining the distance additionally comprises: applying a fitting operation to the first interference signal, the second interference signal, or both, based on fitting parameters, wherein the fitting parameters comprise at least an RF frequency of the first interference signal or the second interference signal.

19. The method of claim 14, wherein determining the distance further comprises applying a fast Fourier transform or related transform to the first interference signal, the second interference signal, or both.

20. The method of claim 19, further comprising producing a range profile based on fast Fourier transform, wherein the range profile comprises a first range peak based on the first interference signal and a second range peak based on the second interference signal, and wherein the first range peak and the second range peak are sub-resolved.

21. The method of claim 14, wherein determining the distance further comprises applying a Hilbert or related transform to the interference signal.

22. The method of claim 14, wherein the determining the distance further comprises applying an electronic or digital filter to the interference signal.

23. The method of claim 18, wherein the fitting parameters additionally comprise an RF phase of the first interference signal or the second interference signal.

Description

DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a diagram of a chirped synthetic wavelength interferometer according to disclosed embodiments;

(2) FIG. 2 is a flowchart showing the operation of the chirped synthetic wavelength interferometer according to disclosed embodiments;

(3) FIG. 3 is a diagram showing the frequency bandwidths and separation for linearized frequency-chirped lasers, as well as the absorption manifold of an example reference gas, according to disclosed embodiments;

(4) FIG. 4A is a plot of model results showing frequency-modulated continuous-wave (FMCW) beat notes in the frequency domain corresponding to one surface, but two lasers with different chirp rates, according to disclosed embodiments;

(5) FIG. 4B is a graph of model results showing the resulting FMCW (1.274 m standard deviation) and synthetic wavelength (35 nm standard deviation) precisions, according to disclosed embodiments;

(6) FIG. 5 is a diagram showing components used in a single-laser chirped synthetic wavelength interferometry (SWI), according to disclosed embodiments.

(7) FIG. 6 is a graph of experimental data showing two pairs of intensity modulation sidebands corresponding to two surfaces, according to disclosed embodiments;

(8) FIG. 7 is a graph of experimental data showing that the disclosed technique is insensitive to sample motion, while the standard FMCW technique exhibits speckle and Doppler errors;

(9) FIG. 8 is a flowchart showing possible steps to isolate, filter, and measure the range to or separation between multiple surfaces for a chirped synthetic wavelength interferometer according to disclosed embodiment;

(10) FIG. 9 is a graph of simulated range profile data, fit guess, and fit, according to disclosed embodiments;

(11) FIG. 10 is a flowchart showing possible steps to determine the distance to or separation between multiple sub-resolved surfaces according to disclosed embodiments; and

(12) FIG. 11 is a graph of data showing that the positions of two surfaces can be determined far better than the range resolution by fitting even when the surfaces are closely spaced and sub-resolved, according to disclosed embodiments.

DETAILED DESCRIPTION

(13) Chirped Synthetic Wavelength Interferometer

(14) FIG. 1 is a diagram of a chirped synthetic wavelength interferometer 100 according to disclosed embodiments. As shown in FIG. 1, the chirped synthetic wavelength interferometer 100 includes a first frequency-chirped laser 110, a second frequency-chirped laser 115, a beam combiner/splitter 120, calibration unit 125 that may contain a length or frequency reference, an optical circulator 130, a reference surface 135, a first sample surface 140, a second sample surface 145, and a detection and processing unit 150.

(15) The first frequency-chirped laser 110 and the second frequency-chirped laser 115 each output light of an optical frequency that changes substantially linearly (chirps) in time over a given chirp duration.

(16) The beam combiner/splitter 120 is configured to receive and combine at least part of the first and second laser outputs into a combined laser output. In some embodiments, a single laser may produce an output with both frequency-chirped components, in which case the beam combination occurs internal to the laser.

(17) The combined laser output from the beam combiner/splitter 120 is then split into a first portion and a second portion. The first portion is directed to the calibration unit 125. The second portion may be directed through the circulator 130 and a plurality of optical paths configured to direct at least part of the combined beam onto an optical detector to produce an interference signal.

(18) In FIG. 1, an optical path may include a transmitted portion denoted Tx. An optical path may include reflection from the reference surface 135, the reflected portion from the reference surface 135 being denoted LO. An optical path may include reflection from a first sample surface 140, the reflected portion from the first sample surface 140 being denoted Rx1. An optical path may include reflection from a second sample surface 145, the reflected portion from the second sample surface 145 being denoted Rx2. At least two of the optical paths are configured to direct light via the circulator 130 to the detection and processing unit 150 to produce an interference signal. It is understood that a sample surface 140, 145 may also serve as the reference surface. Moreover, it is understood that an optical path may not include a surface. For instance, as is common for coherent detection, a reference optical path may be an optical fiber that bypasses the optical circulator 130 and is combined with the Rx after the circulator 130 and before the detection and processing unit 125. It is understood that any optical path is included in this disclosure. The detected interference signal is processed with information from the calibration unit 125 to determine the absolute distance separation between any of the surfaces.

(19) A mathematical description of the interference signal produced by combining the LO and Rx1 resulting from one of the frequency-chirped laser outputs may follow from the formalism provided in Reference [Z. W. Barber, et al., Accuracy of active chirp linearization for broadband frequency modulated continuous wave ladar, Appl. Opt., 49, 213 (2010)]. However, it is understood that alternate formalism may be applied to other combinations of surfaces and frequency-chirped laser outputs. The time-varying electric field for the LO may be represented in the form:
E.sub.LO(t,z=0)=E.sub.0e.sup.i(.sup.0.sup.t+1/2t.sup.2.sup.)(4)

(20) where .sub.0 is the angular optical frequency at the beginning of the chirp, and =2 is the angular chirp rate, where is the frequency chirp rate. The Rx1 field may be modeled by propagating the LO field to the sample surface and back. To accomplish this, one may first Fourier transform the LO field to the frequency domain to yield

(21) $\begin{array}{cc}{E}_{\mathrm{LO}}\left(,z=0\right)=\frac{E_0}{\sqrt{2}}{}_{-}^{}{e}^{-i\left({}_{0}t+\frac{1}{2}{t}^2\right)}{e}^{it}\mathrm{dt}=E_0\frac{1-i}{\sqrt{2}}{e}^{\frac{{i\left(-{}_0\right)}^2}{2}}& \left(5\right)\end{array}$

(22) To model the return field, one may then propagate the LO field to a sample surface and back to the reference surface by applying a Taylor expanded form of the propagator e.sup.iz to yield
E.sub.Rx(,Z=2R)=E(,z=0)e.sup.i2.sup.0.sup.Re.sup.i2.sup.1.sup.(-.sup.0.sup.)Re.sup.i2.sup.2.sup.(-.sup.0.sup.).sup.2.sup.R.(6)

(23) Here R is the range or separation between the reference surface 135 and a sample surfaces, 140, 145, and

(24) ${{{}_0=\frac{{}_{0}n}{c},{}_1=\frac{}{}.Math.}_{={}_0}=\frac{1}{v_g},\mathrm{and}{}_2=\frac{{}^2}{{}^2}.Math.}_{={}_0}.$
Also, n is the refractive index of the medium between the reference and a sample surface, and v.sub.g is the group velocity in the medium. The time-domain description of the field reflected from a sample surface 140, 145, back to the reference surface 135, may be given by:

(25) $\begin{array}{cc}{E}_{\mathrm{Rx}}\left(t,z=2R\right)=E_0\sqrt{\frac{{}^{}}{}}{e}^{-i\left({}_0\left(t-2\frac{n}{c}R\right)+\frac{1}{2}{{}^{}\left(t-2{}_{1}R\right)}^2\right)}& \left(7\right)\end{array}$

(26) where:

(27) $\begin{array}{cc}{}^{}=\frac{}{1+2R{}_2}& \left(8\right)\end{array}$

(28) The interference between the fields E.sub.LO and E.sub.Rx may generate a signal of the form

(29) $\begin{array}{cc}S\left(t\right)E_{\mathrm{LO}}\left(t,z=0\right)E_{\mathrm{Rx}}\left(t,z=2R\right)e^{-i\left(2{}_0\frac{n}{c}R-\frac{1}{2}{{}^{}\left(2{}_{1}R\right)}^2+2R{}^{}{}_{1}t+\frac{1}{2}\left(-{}^{}\right)t^2\right)}& \left(9\right)\end{array}$

(30) For cases where dispersion is small, the terms involving .sub.2 and .sub.1.sup.2 may be neglected, and the signal may be adequately approximated by:

(31) $\begin{array}{cc}S\left(t\right)e^{-i\left(2R{}_{1}t+2{}_0\frac{n}{c}R\right)}& \left(10\right)\end{array}$

(32) To allow for two simultaneous frequency-chirped laser outputs and an arbitrary number of sample surfaces, Equation (10) may be generalized to the form:

(33) $\begin{array}{cc}S\left(t\right)\underset{j=1}{\overset{2}{.Math.}}\underset{k}{.Math.}{e}^{-1\left(2f_{\mathrm{beat},j,k}t+{}_{j,k}^{}\right)}& \left(11\right)\end{array}$

(34) where the subscript j is used to identify the laser chirp (1=first or 2=second) and the subscript k is used to identify the particular pair of surfaces (typically the reference and one sample surface) that are interfering.

(35) The variable f.sub.beat,j,k is the measurable FMCW beat frequency and is given by:
f.sub.beat,j,k=2R.sub.k.sub.j.sub.1,j.(12)

(36) With f.sub.beat,j,k measured, this equation may be used to determine the range between surfaces through:

(37) $\begin{array}{cc}{R}_k=\frac{f_{\mathrm{beat},j,k}}{2{}_j{}_{1,j}}& \left(13\right)\end{array}$

(38) which may be determined with a resolution and precision given by Equation (1) and Equation (3), respectively, when chirp nonlinearities and other noise sources can be neglected. The phase term in Equation (11) is defined by:

(39) $\begin{array}{cc}{}_{j,k}^{}=2{}_j\frac{2n_j{R}_k}{c}& \left(14\right)\end{array}$

(40) where .sub.j=.sub.j/2 is the optical frequency of the j-th laser at the start of the chirp. .sub.j,k represents the traditional interferometer phase, which can only be measured modulo 2. The physically measurable phase is given by:

(41) 0$\begin{array}{cc}{}_{j,k}={}_{j,k}^{}-2m_{j,k}=2\left({}_j\frac{2n_j{R}_k}{c}-m_{j,k}\right)& \left(15\right)\end{array}$

(42) where m.sub.j,k is an integer representing the number of standard interferometer fringes between the two surfaces for the laser with starting frequency .sub.j. Equation (15) can be rearranged to determine the range between the surfaces through

(43) $\begin{array}{cc}{R}_k=\frac{{}_j}{2n_j}\left(\frac{{}_{j,k}}{2}+m_{j,k}\right),& \left(16\right)\end{array}$

(44) where .sub.j is the vacuum wavelength of the j.sup.th laser at the start of the chirp. While this equation may be used to determine the range with very high precision given by Equation (3) with Equation (2) in the ideal case, it also highlights the standard interferometer fringe ambiguity problem because m.sub.j,k is typically large and unknown. However, if the range can be determined via a separate measurement to considerably better than a fringe (i.e. break the fringe ambiguity), then one may take full advantage of Equation (16). The fringe number may be obtained from a separate measurement by solving the equation

(45) $\begin{array}{cc}{m}_{j,k}=\mathrm{ROUND}\left[\frac{2n_j{R}_k}{{}_j}\right]& \left(17\right)\end{array}$

(46) Another useful quantity that can be extracted from the measurement described by Equation (11) is the synthetic interferometer phase, which is given by

(47) $\begin{array}{cc}{}_{2,k}-{}_{1,k}={}_{2,k}^{}-{}_{1,k}^{}-2l_k=2\left[\left(n_2{}_2-n_1{}_1\right)\frac{2R_k}{c}-l_k\right],& \left(18\right)\end{array}$

(48) where l.sub.k is an integer representing the number of synthetic interferometer fringes between the two surfaces. The synthetic fringe number may be determined by a separate measurement of R.sub.k and may be calculated using

(49) $\begin{array}{cc}{l}_k=\mathrm{ROUND}\left[\frac{2R_k}{{}_{\mathrm{synth}}}\right]& \left(19\right)\end{array}$

(50) where the well-known synthetic wavelength is given by

(51) $\begin{array}{cc}{}_{\mathrm{synth}}=\frac{1}{.Math.\frac{n_2{}_2-n_1{}_1}{c}.Math.}=\frac{{}_1{}_2}{.Math.n_2{}_1-n_1{}_2.Math.}& \left(20\right)\end{array}$

(52) The synthetic fringe number l.sub.k may be smaller than m.sub.j,k and may therefore be measured more easily from a practical standpoint. With known l.sub.k, Equation (18) can be rearranged to determine the range through

(53) $\begin{array}{cc}{R}_k=\frac{{}_{\mathrm{synth}}}{2}\left(\frac{{}_{2,k}-{}_{1,k}}{2}+l_k\right),& \left(21\right)\end{array}$

(54) Similarly, Equation (21) can be used to determine the absolute distance or separation between the surfaces with very high precision given by Equation (3) with Equation (2) in the ideal case.

(55) All three measurable quantities (the FMCW heterodyne beat frequency of Equation (12), the synthetic interferometer phase of Equation (18), and the standard interferometer phase of Equation (15) may be obtained from one measurement, represented by Equation (11). It is possible to use these measurable quantities, or a subset of them, to determine the range from the reference surface 135 to a sample surface 140, 145 with very high precision and accuracy.

(56) Method of Operation

(57) FIG. 2 is a flowchart 200 showing the operation of the chirped synthetic waveform interferometer 100 according to disclosed embodiments.

(58) As shown in FIG. 2, a sequence of steps to determine the range from the reference surface 135 to a sample surface 140, 145 with very high precision and accuracy may include:

(59) Measuring the frequency-modulated continuous-wave (FMCW) beat f.sub.beat,j,k using at least one chirped laser to determine the range R.sub.k coarsely using Equation (13). (210)

(60) Using the coarse range measurement to determine the synthetic fringe number, l.sub.k using Equation (19). (220)

(61) Using the synthetic fringe number l.sub.k and a measured synthetic interferometer phase to determine an intermediate range through Equation (21). (230)

(62) Using the intermediate range measurement to determine the interferometer fringe number, m.sub.j,k for at least one laser using Equation (17). (240)

(63) Using the interferometer fringe number m.sub.j,k and the measured standard interferometer phase to determine the fine range through Equation (16). (250)

(64) Two Linearized Frequency-Chirped Laser Embodiment

(65) In order to determine the fringe numbers l.sub.k and m.sub.j,k without error, the coarse and intermediate ranges R.sub.k should be measured with uncertainty much better than the synthetic and standard interferometer wavelengths, respectively. However, the prior art identifies nonlinearities in the frequency chirp or phase as factors that degrade the resolution, precision, and accuracy of the results or that necessitate complex components and processing to mitigate. To improve upon the prior art, the disclosed design uses linearized frequency-chirped lasers, which enable range measurements to closely approach the resolution and precision shown in Equation (1) and Equation (3), respectively.

(66) While optical sideband chirps by means of external modulation (e.g. an electro-optic modulator) can be straightforward to achieve up to about 10 GHz and exhibit very low excursions from linearity, this technique can become complex and expensive much beyond 10 GHz. On the other hand, >10 GHz chirps of an optical carrier can be routinely achieved by changing the current, temperature, or mechanical cavity length of, for instance, semiconductor distributed feedback (DFB) lasers, vertical cavity surface emitting lasers (VCSELs), and external cavity diode lasers (ECDLs). However, not until recently have passive and active techniques been developed to linearize optical chirps over large bandwidths (>>10 GHz).

(67) One example embodiment uses two actively linearized frequency-chirped lasers 110, 115 in the configuration shown in FIG. 1. While active linearization may be preferred over passive linearization due to the improved laser coherence that active linearization affords, passively linearized lasers may also be used. Moreover, it is also understood that the one may also adjust the measurement clock to produce uniform K-space sampling as performed in Reference [Opt Express. 2010 Apr. 26; 18(9): 9511-9517]. To achieve critical aspects of this invention, the frequency excursions from perfect linearity are preferably less than about 10 MHz during the chirp. Each linearized frequency-chirped laser for this example embodiment may exhibit a chirp over about 100 GHz of optical bandwidth as shown in FIG. 3.

(68) FIG. 3 is a diagram 300 showing the frequency bandwidths and separation for two linearized frequency-chirped lasers, as well as the absorption manifold of an example reference gas that may be used in the calibration unit 125, according to disclosed embodiments.

(69) The lasers may be chirped simultaneously in opposite directions, as disclosed in U.S. Pat. No. 5,371,587. Using at least one of these lasers, the coarse range, R.sub.k, may be determined using Equation (13). With linearized frequency-chirped lasers this measurement may be made with a resolution of about 1.5 mm and a precision of less than about 5 m from Equation (1) and Equation (3), respectively, assuming 50 dB SNR for a 1 ms integration time.

(70) For this embodiment the centers of the two laser chirp bandwidths may be spaced by about 4 THz, as shown in FIG. 3. In this case, the coarse range measurement precision of <5 m is much smaller than the synthetic wavelength, which is about 40 m. Therefore, the coarse range measurement may be used to accurately determine the synthetic fringe number using Equation (19) with or without averaging multiple measurements when linearized frequency-chirped lasers are used.

(71) With knowledge of the correct synthetic fringe number, measurement of the synthetic interferometer phase may be used to determine the intermediate range through Equation (21). By using linearized frequency-chirped lasers, and therefore linear phase, the intermediate range may be determined with a precision of about 60 nm using Equation (2) and Equation (3), and again assuming a 50 dB SNR. Using lasers with a wavelength near 1.5 m, 60 nm precision is sufficient to enable accurate determination of the standard interferometer fringe number, m.sub.j,k, for either or both lasers, through Equation (17).

(72) FIG. 4A is a plot 400 of model results showing frequency-modulated continuous-wave (FMCW) beat notes in the frequency domain corresponding to one surface, but two lasers, each with a different chirp rate, according to disclosed embodiments; and FIG. 4B is a graph of the range measurement results showing the resulting FMCW (1.393 m standard deviation) and synthetic wavelength (33 nm standard deviation) precisions, according to disclosed embodiments.

(73) In particular, FIG. 4A shows the simulated range profile for one sample surface and two lasers (one peak for each laser). The lasers are separated in optical frequency by 4 THz. FIG. 4B shows the resulting simulated precision that can be obtained by fitting the range peaks to find their centers. The squares 310 represent coarse range measurements (precision=1.393 m), while the circles 320 represent the intermediate range measurements (precision=33 nm).

(74) With knowledge of the correct standard fringe number, measurement of the standard interferometer phase may be used to determine the fine range through Equation (16). By using linearized frequency-chirped lasers, and therefore linear phase, the fine range may be determined with a precision of about 2 nm using Equation (2) and Equation (3), and again assuming a 50 dB SNR. In determining a range for which the laser light enters a surface from low to high index of refraction (e.g. from air to glass), the well-known 180 phase flip of the reflected light from that surface should be included in the calculation of the range. In that case, .sub.j,k may be replaced by .sub.j,k in Equation (16). It is also understood that refractive index and dispersion in the material between surfaces may need to be accounted for to accurately determine range. This example embodiment shows that the use of linearized frequency-chirped lasers enable a range measurement with 2 nm Cramer-Rao-limited precision in a 1 ms measurement time.

(75) In determining the absolute distance, accuracy may be equally as important as precision and again the use of linearized frequency-chirped lasers may be important to achieve fine accuracy. Therefore, one of the output paths of the beam splitter/combiner 120 in FIG. 1 (a first portion) may be received by the calibration unit 125 of the system to accurately determine the chirp rate .sub.j for each laser for the FMCW measurement, the difference in the optical frequencies chirps (.sub.2.sub.1), and the optical frequency for at least one laser chirp .sub.j.

(76) In this embodiment, a spectroscopic gas cell may be used in the calibration unit 125 to determine these needed parameters. Fundamental atomic or molecular absorption lines provide wavelength references that are very stable under changing environmental conditions, such as temperature and pressure variations or the presence of electromagnetic fields. (See, e.g., S. L. Gilbert, W. C. Swann, and Chih-Ming Wang, Hydrogen Cyanide H13C14N Absorption Reference for 1530 nm to 1565 nm Wavelength CalibrationSRM 2519a, NIST Special Publication 260-137, 2005 Edition.)

(77) In this embodiment, a NIST-traceable H13C14N gas cell may be used with typical absorption line spacings of about 50-105 GHz. During each laser's chirp the optical frequency may be swept over at least two absorption lines. With well-known frequency separation of the lines, linearized frequency chirps, and accurate measurement timing, it is therefore readily possible to calculate the chirp rate through .sub.j=/t where is the frequency separation between two absorption lines and t is the time for the laser frequency to be swept between the same two lines. It is understood that other absorption characteristics such as the width of a single line could also be used in addition to or instead of the separation between absorption lines.

(78) While the absolute frequencies of the absorption lines are known to about 5-25 MHz, the most accurately known lines can be chosen and in practice the frequency spacing between adjacent lines can be determined better than this. With an absorption line separation known to better than 1 MHz, and a frequency separation of 100 GHz between absorption lines, the chirp rate .sub.j, and thus the FMCW distance measurement, may therefore be calibrated by this method to better than about one part in 10. However, this determination of .sub.j also requires that the deviations of the laser frequency from a perfectly linear chirp remain significantly under about 10 MHz. This condition may be achieved using either passive or active chirp linearization techniques, or by adjusting the measurement clock to produce uniform K-space sampling.

(79) The quantity (.sub.2.sub.1), which determines the synthetic wavelength, may be calibrated in a similar manner, but the separation between the applicable absorption lines is about 4 THz for the present embodiment, as shown in FIG. 3. In this case, the absolute optical frequency at the start of the chirp for each laser may be determined by chirping across a suitable H13C14N absorption line. One may measure the time difference, t, between the start of the chirp and the time that the laser frequency matches the center frequency of an absorption line, .sub.ref. By using a linearized frequency-chirped laser with frequency deviations from linearity of <10 MHz and calibrated chirp rate, .sub.j, one may then determine the optical frequency of the start of the chirp using .sub.j=.sub.ref.sub.jt. By using a gas cell for which the separation (.sub.2.sub.1) can also be determined to better than 10 MHz, this may provide calibration of the distance based on the synthetic interferometer phase to an accuracy of better than about three parts in 10.sup.6. The synthetic wavelength distance measurement may therefore break the standard interferometer fringe ambiguity for the present embodiment for distances and thickness of up to about 30 cm.

(80) Similarly, by chirping across an absorption line, at least one of the laser start frequencies .sub.j can be determined to about 5 parts in 10.sup.8 (10 MHz accuracy out of 200 THz optical frequency). This indicates that absolute distance may be calibrated to the nanometer and sub-nanometer scale for distances and thickness up to about 30 cm. Improved wavelength references and chirp linearization may be used to increase this distance. The disclosed technique of sweeping over absorption features and utilizing the frequency and phase linearity of linearized frequency chirped lasers to calibrate the chirp rate and the optical frequencies .sub.1 and .sub.2 in this patent is simpler and potentially faster than the previously disclosed technique of stabilizing the lasers to absorption features. It is understood that the refractive index, group refractive index and dispersion may need to be known to the same or better accuracy than the optical frequencies and chirp rates in order to achieve the desired performance.

(81) Single Laser Embodiment

(82) It is also possible to perform a form of chirped synthetic wavelength measurements using a single frequency-chirped laser instead of two or more. The disclosed technique is related to previous incoherent length metrology techniques that measure the RF phase of optical intensity modulation. (See, e.g., I. Fujimay, S. Iwasaki and K. Seta, High-resolution distance meter using optical intensity modulation at 28 GHz, Meas. Sci. Technol. 9 (1998) 1049-1052, and A. Barker, Performance enhancement of intensity-modulated laser rangefinders on natural surfaces, SPIE Vol. 5606 (2004).) However, the previous methods may be degraded by reflections from surfaces other than the sample surface or by low signal levels, and may require high receiver bandwidth.

(83) FIG. 5 is a diagram showing components used in a single-laser synthetic wavelength interferometer (SWI) 500, according to disclosed embodiments. As shown in FIG. 5, the single-laser synthetic wavelength interferometer 500 includes a frequency-chirped laser 510, beam splitter 520, a first modulator 525, a circulator 530, a measurement surface 535, a second measurement surface 537, a second modulator 540, a signal generator 545, a beam combiner 550, and a detection and processing unit 555.

(84) As shown in FIG. 5, the output from a frequency-chirped laser may be split into two output portions: a first frequency-chirped laser output portion (Tx) and a second frequency-chirped laser output portion (LO).

(85) The first frequency-chirped laser output portion (Tx) may be transmitted through the first modulator 525 and modulated at frequency f.sub.mod. This portion may be transmitted to the measurement surface 535, 537. A portion of the reflected or scattered light (Rx1, Rx2) may be directed through the second modulator 540 and demodulated at a slightly different modulation frequency (f.sub.mod+f), where f may be made small to accommodate low detector bandwidths if desired.

(86) Light from the second modulator 540 may be recombined with the second frequency-chirped laser output portion (LO) and the combined light may be directed onto an optical detector in the detection and processing unit 555 to produce an interference signal. The range to target may be obtained by measuring the RF phase shift for f.sub.mod accumulated over the transit from the first modulator 525 to the second modulator 540, which is also present at the difference frequency f. This phase shift can be determined by measuring the relative phase difference between two of the RF modulation sidebands corresponding to the coherent carrier.

(87) Mathematically, the detected interference signal may be given by Equation (10) for one laser, one pair of surfaces, where the LO path in FIG. 5 may replace the optical path to the reference surface, and with the inclusion of intensity modulation:
S(t)cos(2f.sub.mod t+.sub.1)cos[2(f.sub.mod+f)t+.sub.2]e.sup.i(2f.sup.beat.sup.t)(22)

(88) where f.sub.beat is the standard FMCW heterodyne beat note, .sub.1 is the RF phase of modulator 1, .sub.2 is the RF phase of modulator 2, the standard interferometer phase term has been neglected, and the DC bias phase of each intensity modulators has been set to zero to simplify the mathematics. Retaining only terms that oscillate at frequencies near f.sub.beat yields
S(t)e.sup.i(2(f.sup.beat.sup.f)t)+e.sup.i(2(f.sup.beat.sup.+f)t+)(23)

(89) where =.sub.2.sub.1 is the RF phase difference between the LO path and the Rx path, which contains the physical path length difference. This equation shows that the effect of modulation and demodulation of the chirped laser is to add intensity modulation sidebands (offset from f.sub.beat by f) to the standard FMCW carrier beat note. When the DC bias phase of the intensity modulators is set to zero, as has been assumed here, the carrier beat note is suppressed, leaving only the sidebands separated by 2f.

(90) FIG. 6 is a plot of experimental data showing two pairs of intensity modulation sidebands corresponding to two surfaces, 535, 537, according to disclosed embodiments. In particular, FIG. 6 shows experimental data of two surfaces, each with a pair of intensity modulated sidebands separated by 2f.

(91) The important distance information is contained in the RF phase that may be obtained by subtracting the phase evolution of the upper sideband from that of the lower sideband corresponding to a single surface, which yields
.sub.usb(t).sub.lsb(t)=2(2ft+)(24)

(92) The phase evolution of each sideband may be obtained by applying a band-pass filter to the desired sideband. An example digital filter function is shown as the black line 610 in FIG. 6. Once this sideband is isolated, one may determine the phase evolution by techniques such as Hilbert transform. One may then demodulate electronically at the offset frequency f. Alternatively, one may measure the offset frequency using one of the measurement surfaces 535 as a reference surface in the combined Tx/Rx path after the circulator. This case results in multiple pairs of peaks, as shown in FIG. 6, each pair corresponding to a surface in the beam path. In this case, one may subtract the phase evolution obtained from the one surface 537 as in Eqn (24), from the analogous equation obtained from the reference surface 535. Since f is common to both equations, only the RF phase difference between the sample surface and the reference surface remains. This phase difference will yield the distance separation between the reference surface 535 and the sample surface 537. The range between the reference surface 535 and sample surface 537 is then given by

(93) $\begin{array}{cc}R=\frac{1}{2}\left(\frac{}{2}+m_{\mathrm{RF}}\right)\left(\frac{c}{2f_{\mathrm{mod}}}\right),& \left(25\right)\end{array}$

(94) where m.sub.RF is an integer that may be determined by a different measurement to remove the RF fringe ambiguity. In particular, the FMCW range measurement can be used to remove the RF fringe measurement.

(95) For this embodiment, frequency and phase noise due to speckle, Doppler, nonlinear frequency chirps, or other sources may be common mode for both of the RF sidebands. Therefore, the subtraction of the phase between the sidebands may suppress such common-mode noise.

(96) FIG. 7 is a graph 700 of experimental data showing that the disclosed technique is insensitive to sample motion, while the standard FMCW technique exhibits speckle and Doppler errors. In particular, FIG. 7 illustrates a benefit of the phase and frequency insensitivity. For the data in the figure, a sample surface is put into motion starting near measurement number 260. The data shows that range measurements using this disclosed embodiment (circles 720) is insensitive to the motion as compared to the standard FMCW technique (squares 710), which shows significant errors due to Doppler or speckle for the points where motion is present (i.e., points greater than 260 in this embodiment). This insensitivity to errors due to motion can be an advantage over purely FMCW measurements. On the other hand, important advantages over a purely incoherent measurement are that the signal amplitude is amplified by the LO, as it is for the coherent FMCW measurement, and that multiple surfaces may be range-resolved or distinguished. A final advantage is that this measurement technique can be calibrated accurately with accurate knowledge of the RF modulation frequency.

(97) While this embodiment (FIG. 5) shows a frequency-chirped laser for the FMCW technique and certain modulators in certain positions for illustrative purposes, many variants are possible. For instance, other forms of FMCW ladar, such as that using sideband chirps, and other coherent detection techniques may replace the role of a frequency-chirped laser for the FMCW technique to provide discrimination between different surfaces or to capitalize on amplification by the LO. Also, intensity modulators were used for this embodiment, but the use of other modulators (such as frequency or phase) may also be possible.

(98) The first and second modulators 525, 540 may also be placed at different locations in the setup and still enable extraction of the RF phase. For instance, the second modulators 540 may be placed in the LO path. Or, one modulator 525, 540 may be used with both modulation and demodulation frequencies in the combined Tx/Rx path after the circulator. Also, the modulation could be performed using direct modulation of the laser source. If higher RF bandwidth is acceptable, only one modulator or direct modulation is possible and demodulation is not necessary.

(99) Measurement of Multiple Surfaces

(100) A significant drawback of traditional interferometry is the fact that light from multiple surfaces can give distance errors because the multiple surfaces may contribute to one indistinguishable interferometric signal. However, as shown in FIG. 6, the contributions from multiple resolved surfaces can be distinguished by their distinct signal frequencies using the disclosed invention for either the Two Linearized Frequency-Chirped Laser Embodiment or the Single-Laser Embodiment. Such measurements of multiple surfaces may be categorized into one of two cases.

(101) The first multi-surface case is when the surfaces of interest are well resolved by the FMCW measurement, such as those shown in FIG. 6. In this case, the peaks do not interfere or otherwise affect one another. In this case, one may apply the steps shown in FIG. 8 to determine the range to, or separation between, multiple resolved surfaces. The steps are only shown as an exemplary case. Other alternate steps and methods are also possible to separate the signals corresponding to different surfaces, filter to isolate signals corresponding to different surfaces, or process the signals corresponding to different surfaces to determine the ranges, and optionally determine the separation between multiple resolved surfaces. Moreover, some steps may be omitted. For the exemplary steps shown in FIG. 8, one may first detect and digitize the interference signal resulting from multiple sample surfaces, and possibly multiple lasers. Next, one may perform a fast Fourier Transform to separate the resolved peaks in the RF frequency domain and effectively create a range profile. One may next apply bandpass filters, such as the digital bandpass filters shown by the black trace 610 in FIG. 6, to the signal in the frequency domain to isolate peaks corresponding to different surfaces. This isolation of the peaks may allow one to process the contributions from each surface independently without interference from the other surfaces. For instance, one may next perform a fast Inverse Fourier Transform on the filtered signal to transform back to the time domain. Next, one may perform a Hilbert or other transform to produce the phase evolution of the filtered signal in time (B. Boashash, Estimating and interpreting the instantaneous frequency of a signal. II. Algorithms and applications, Proceedings of the IEE, vol. 80, no. 4, pp. 540-568, 1992). With this phase evolution signal, one may calculate the parameters f.sub.beat,j,k and .sub.j,k that can be used to determine range to the sample surface. f.sub.beat,j,k may be obtained from the slope of the phase evolution signal and .sub.j,k may be obtained by the offset of the phase evolution signal. The process may be performed for multiple lasers when the chirp rates of the lasers are different or the peaks from the different lasers are well resolved by some other means. One may optionally subtract the range determinations for multiple sample surfaces to determine the separation between sample surfaces. This technique may be used, for instance, to precisely measure the position of sample surfaces, and therefore determine separation between two sample surfaces by subtracting their ranges.

(102) The second case for measurement of multiple surfaces is when the surfaces are not well resolved by the FMCW measurement so that the peaks may interfere or otherwise affect one another. In this case, the disclosed embodiments teach how it is still possible to accurately determine the range of each peak, even if the peaks are closely spaced. In this case, it may not be possible to separate and filter the peaks associated with sample surfaces, but the disclosed embodiments teach how it still may be possible to accurately or precisely determine the range to sub-resolved sample surfaces, and their separations, even if the peaks are closely spaced. This determination is complicated by the fact that the measurement is coherent, so the optical phase of the reflection from each surface can dramatically alter the peak shape and corrupt the range determination. FIG. 9 shows simulated range profile data (gray dots) with noise for one laser and two sub-resolved surfaces on a linear vertical scale. Although two surfaces are present, they are not resolved, so only one peak is visually observed in the range profile. Solid vertical lines indicate the true simulated surface locations in the range profile.

(103) One may apply the steps shown in FIG. 10 to determine the range to, or separation between, multiple sub-resolved surfaces. The steps are only shown as an exemplary case. It is understood that other alternate steps and methods are also possible to determine the range and possibly the separation between multiple sub-resolved surfaces, and possibly multiple lasers. Moreover, steps may also be omitted. For the exemplary steps shown in FIG. 10, the interference signal resulting from multiple sample surfaces, and possible multiple lasers, may first be detected and digitized (1005). Next, one may perform a fast Fourier Transform (1010) to produce a range profile such as that shown in FIG. 4A or FIG. 9. One may separately start with a mathematical model of the interference signal (1030) such as that described by Eqn (11), which may include multiple sample surfaces and multiple lasers. Fixed parameters (1040) and initial guesses (1020) for fitting parameters may be used. Approximations to the mathematical form may also be used. The fixed parameter inputs (1040) to the mathematical model may include laser wavelengths (1042), chirp rates (1046), and number of sample surfaces (1044), which may be known or determined from other measurements. However, it is understood that the fixed input parameters shown may not be used and other inputs are possible. Also, some fixed parameters shown may also or instead be fitting parameters, such as number of surfaces. Then one may perform a fast Fourier Transform of the modeled interference signal (1050) to produce a model range profile. One may then perform mathematical fitting (1015), such as least squares fitting, between the data and model to determine the range to, or separation between (1070), multiple surfaces. The general mathematical techniques of fitting are well known (P. G. Guest, Numerical Methods of Curve Fitting, Cambridge University Press; Reprint edition (Dec. 13, 2012), ISBN: 9781107646957.) and it is understood that many different fitting techniques may be used. Fitting parameters may include fringe numbers (1068), range (1061, 1062), amplitude (1063, 1064), and phase (1065, 1067) corresponding to different surfaces and lasers. It is understood that not all of these fitting parameters must be used and other fitting parameters may also be used, such as number of surfaces. It is also understood that other configurations and steps that use fitting to determine the position of the sub-resolved peaks are possible. For instance, instead of starting with a mathematical model in the time domain (1030), one may develop a mathematical expression for the interference signal directly in the frequency domain, or an approximation thereof, and apply fitting (1015) without taking the Fast Fourier Transform 1050.

(104) Using fitting of the known functional form of the surfaces given by Eqn (11), or approximations to that functional form, with amplitude (1063, 1064), range (1061, 1062), and phase (1065, 1067) of the measured signals as the fit parameters, the locations of the two surfaces may be determined far better than the range resolution. The dashed curves in FIG. 9 show the range profile resulting from initial parameter guesses for fitting (vertical dashed lines show the initial range guess values). The solid curves show the resulting least squares fit signal for the combined surfaces. The fit range locations are visually indistinguishable from the true locations on the plot. The process of fitting can also be performed iteratively to improve the measurement. For instance, one may perform the fitting process using the interference signal from just one laser to determine a coarse range and thus the synthetic fringe number, l.sub.j,k. The fitting process can then be performed again with l.sub.j,k as an input parameter and with two lasers to determine the intermediate range measurement or the standard interferometer fringe number, m.sub.j,k. Similarly, the fitting process may then be performed again with m.sub.j,k as an input parameter and with one or more lasers to determine the fine range measurement.

(105) FIG. 11 shows the true (black lines) and calculated by fit position (gray circles and squares) positions of the two surfaces for different surface separations. For the case shown, the FMCW range resolution is about 1.5 mm (given by Eqn (1)), but the surface positions can be measured and determined to better than about 10 m even when the surface spacing is reduced to 10 m. Moreover, when two lasers are used and the starting optical frequency of each laser is known, Eqn (16) and Eqn (21) can additionally be used to provide more constraints with l.sub.k and/or m.sub.j,k as additional fit parameters to further improve the measurement.