Polarization properties imaging systems

Abstract

This disclosure is generally directed to systems for imaging polarization properties of optical-material samples. As one aspect, there is provided a system for precise, simultaneous imaging of both the in-plane and out-of-plane birefringence properties of sample material over a wide range of incidence angles. The spatially resolved imaging approach described here is amenable to determination of a wide range of polarimetric properties, in addition to the in-plane and out-of-plane birefringence measure discussed as a preferred embodiment.

Claims

1. A method of simultaneous imaging in-plane and out-of-plane birefringence properties of a sample over a wide range of incidence angles, comprising: directing a plurality of polarization-modulated light rays to a sample location such that each ray has a specific angle of incidence such that the plurality of rays defines a plurality of incidence angles relative to the sample; redirecting each of the light rays that pass through the sample to separate pixels of an imaging device so that the intensity characteristics of each ray can be detected; gating the imaging device using as a trigger signal a frequency-synthesized waveform corresponding to the polarization modulation that is applied to the light rays; and simultaneously detecting birefringence properties of the sample associated with each of the redirected light rays.

2. The method of claim 1, including modulating the polarization of the light rays with a photoelastic modulator (PEM) operating at a PEM frequency; and wherein the gating includes using a trigger signal based on the frequency synthesis of a half duty square wave and the PEM frequency.

3. The method of claim 2, wherein the modulating the polarization of the light rays includes the use of two or more PEMs and wherein the gating includes using a trigger signal based on the frequency synthesis of a half duty square wave and the frequency of the two or more PEMs.

4. A method of gating a detector in an imaging system that includes an optical setup having a polarization modulator between a light source and a detector, wherein the detector is positioned to receive light from the source and produce a detected signal in response to the received light, and wherein the polarization modulator modulates the intensity of the light directed through the polarization modulator to provide a modulator output waveform representative of that modulation, the method comprising: providing a gating mechanism that can be triggered for controlling exposure time periods during which the detector receives the light from the source; selecting a first waveform; synthesizing the modulator output waveform and the first waveform to produce a trigger signal; and applying the trigger signal to the gating mechanism for triggering the gating mechanism and thereby control exposure time periods during which the detector receives the light from the source.

5. The method of claim 4, including demodulating the detected signal.

6. The method of claim 1, further comprising calibrating a pixel intensity offset for one or more of the pixels of the imaging device, wherein the offset is associated with a polarization effect introduced by optical components providing directing and redirecting that varies with the respective incidence angles.

7. The method of claim 1, further comprising modulating the polarization of the light rays with a plurality of photoelastic modulators (PEMs) operating at separate respective PEM frequencies, wherein the gating includes using a trigger signal based on the frequency synthesis of a half duty square wave and one or more harmonics associated with a combination of the PEM frequencies.

8. The method of claim 7, wherein the PEM frequencies are faster than an operational frequency of the imaging device and the one or more harmonics associated with combination of the PEM frequencies is slower than the operational frequency.

9. The method of claim 7, further comprising driving the plurality of PEMs at a driving amplitude selected so as to allow a separation and measurement of Bessel waveforms associated with each PEM from a total waveform detected by the imaging device.

10. The method of claim 9, wherein the driving amplitude corresponds to an equivalence between absolute values of the measured Bessel waveforms at the one or more harmonics.

11. The method of claim 1, wherein the frequency-synthesized waveform has a zero or n/2 phase relation to one or more Bessel waveforms associated with the polarization-modulated light rays.

12. The method of claim 7, wherein the frequency-synthesized waveform has a zero or n/2 phase relation to one or more Bessel waveforms associated with the PEMs.

13. The method of claim 7, wherein the frequency of the frequency-synthesized waveform is selected based on an averaging of multiple PEM cycles at the PEM frequencies.

14. An apparatus comprising: an optic situated to direct a plurality of polarization-modulated light rays to a sample location such that each ray has a specific angle of incidence so as to define a plurality of respective incidence angles relative to the sample; an optic situated to redirect each of the light rays that pass through the sample; an imaging device situated to receive the redirected light rays at separate pixels so that the intensity characteristics of each ray can be detected and so as to simultaneously detect in-plane and out-of-plane birefringence properties of the sample associated with each of the redirected light rays; and a controller coupled to the imaging device and operable to gate the imaging device using as a trigger signal a frequency-synthesized waveform corresponding to the polarization modulation that is applied to the light rays.

15. The apparatus of claim 14, further comprising a plurality of photoelastic modulators (PEMs) situated to modulate the polarization of the light rays with separate respective PEM frequencies, wherein the gating includes using a trigger signal based on the frequency synthesis of a half duty square wave and one or more harmonics associated with a combination of the PEM frequencies.

16. The apparatus of claim 15, wherein the PEM frequencies are faster than an operational frequency of the imaging device and the one or more harmonics associated with combination of the PEM frequencies is slower than the operational frequency.

17. The apparatus of claim 15, wherein the frequency-synthesized waveform has a zero or n/2 phase relation to one or more Bessel waveforms associated with the PEMs.

18. An apparatus, comprising: light source; a detector situated to receive light from the source and produce a detected signal in response to the received light; at least one polarization modulator situated between light source and detector so as to modulate the intensity of the light directed through the polarization modulator to provide a modulator output waveform representative of that modulation; a gating mechanism coupled to the detector so as to control based on a trigger signal exposure time periods during which the detector receives the light from the source; and a controller coupled to the gating mechanism so as to apply the trigger signal, wherein the trigger signal is synthesized from the modulator output waveform and a first selected waveform.

19. The apparatus of claim 18, wherein the at least one polarization modulator comprises a plurality of photoelastic modulators (PEMs) operating at separate respective PEM frequencies, wherein the trigger signal is based on the frequency synthesis of a half duty square wave and one or more harmonics associated with a combination of the PEM frequencies.

20. The apparatus of claim 18, wherein the frequency-synthesized waveform has a zero or n/2 phase relation to the modulator output waveform.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a diagram showing one preferred arrangement of the optical components of a system that may be used for imaging polarization properties, such as birefringence, across an area of a sample.

(2) FIGS. 2 and 3 are diagrams illustrating one preferred method of triggering a light source at particular instances in relation to the operation of the PEM components of the optical setup.

(3) FIG. 4 is a diagram showing another preferred arrangement of the optical components of a system that may be used for imaging polarization properties, such as Stokes parameters, across an area of a sample.

(4) FIG. 5 is a diagram showing another preferred arrangement of the optical components of a system that may be used for imaging polarization properties.

(5) FIG. 6 is a diagram showing another preferred arrangement of the optical components of a system that may be used for imaging polarization properties such as differential extinction.

(6) FIG. 7 is a diagram of a preferred arrangement of the optical components of a system for simultaneously imaging both in-plane and out-of-plane birefringence properties of a sample.

(7) FIG. 8 is a diagram like FIG. 7, but illustrating the particulars of the light source and detecting optical components of the system.

(8) FIG. 9 is a diagram showing the relationship between the Bessel waveform generated by a PEM and a half duty square wave generated as part of a gating trigger signal for the imaging device of the present system.

(9) FIG. 10 is a diagram illustrating the steps employed for generating a frequency synthesized trigger signal for the gated imaging device of the present system.

(10) FIG. 11 is a diagram illustrating the harmonic relationship of a sine/cosine demodulating signal as compared to a measured square-wave demodulating signal for a single PEM system.

(11) FIGS. 12 and 13 are diagrams illustrating how the Bessel waveforms of two or more PEMs are combined for arriving at a correct amplitude for driving the PEMS for well-conditioned demodulation of the output signal from the imaging device.

DETAILED DESCRIPTION

(12) FIG. 1 depicts one preferred arrangement or setup of optical components of a system 15 that may be used for imaging polarization properties, such as birefringence, across an area of a sample. That system includes a light source 17 that directs light (upwardly in the setup of FIG. 1) through a sample 30 to an imaging device 29 such as a conventional CMOS or CCD-type camera. The light source is, preferably, a LED having individually controlled diodes emitting red (R, having, for example, a wavelength of 630 nanometers, nm), green (G, having, for example, a wavelength of 530, nm) and blue (B, having, for example, a wavelength of 470 nm) that are, preferably, sequentially turned ON and OFF, or triggered. (For measurements requiring lower precision, the system can employ an RGB Bayer filter camera and record red, green, and blue retardance values simultaneously.)

(13) The light from the source 17 passes through a diffuser 19 and lens 21 from which propagates a relatively wide, collimated beam that passes through a polarizer 23 oriented at 0 relative to the baseline axis of the setup.

(14) The polarized light emanating from the polarizer 23 is incident upon the optical element of a first photoelastic modulator (PEM) 20. In a preferred embodiment, the PEM is one manufactured by Hinds Instruments, Inc., of Hillsboro, Oreg. The use of PEMs for modulating the polarization of the source light is preferred because of its fast modulation, insensitivity to angle of propagation, large aperture, high precision, and large spectral range. It is contemplated, however, that other mechanisms could be used for modulating the polarization of the source light, such as electro-optic modulators, liquid crystal retarders, and mechanically modulated waveplates.

(15) The PEM 20 has its birefringent axis oriented at 45 and is controlled by an associated controller that imparts an oscillating birefringence to the optical element of the PEM, preferably at a nominal frequency of 42 kHz. In this regard, the controller drives one or more quartz transducers that are adhered to the optical element of the PEM 20.

(16) The oscillating birefringence of the PEM 20 introduces a time-varying phase difference between the orthogonal components of the polarized light that propagates through the PEM. At any instant in time, the phase difference represents the retardation introduced by the PEM. As noted earlier, the retardation is measurable in units of length, such as nanometers. The PEM 20 is adjustable to allow variation of the amplitude of the retardation introduced by the PEM.

(17) With continued reference to FIG. 1, the light emanating from the sample 30 is directed through a lens 25in this embodiment a microscope lensto a second PEM 22 that is controlled by an associated controller to operate at a modulation frequency (for example, 47 kHz) that is different from the modulation frequency of the first PEM 20. The second PEM 22 is oriented at 0. A second polarizer 27, oriented at 45, is between the second PEM 22 and the CCD 29.

(18) It is noteworthy here that the system configuration illustrated in FIG. 1 includes microscope lensing and, as such, is useful for imaging small or microscopic fields of, for example 200 m.sup.2 to 1 cm.sup.2 in area, with 2 to 10 magnification and 1-8 megapixel (MP) resolution. It will be appreciated that the imaging lensing components of the systems described herein can be varied as needed to accommodate much larger fields for imaging corresponding large samples. Both Khler and telecentric (including a diffuser) sample illumination methods may be employed, among others.

(19) With continued reference to FIG. 1, the CCD 29 includes individual pixels, each of which receives light passing through a particular point of the sample 30 so that the light intensity captured in each pixel corresponds to the birefringence characteristics of that particular point. In particular, for each point in the sample the quantity of interest is retardance, , and angle of the fast axis, . Expressed as a 44 Mueller matrix below.

(20) $M_{\mathrm{bir}}\left({}_S,{}_S\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& M_{11}& M_{12}& -\sin \left(2{}_S\right)\sin \left({}_S\right)\\ 0& -M_{12}& M_{22}& \cos \left(2{}_S\right)\sin \left({}_S\right)\\ 0& \sin \left(2{}_S\right)\sin \left({}_S\right)& -\cos \left(2{}_S\right)\sin \left({}_S\right)& \cos \left({}_S\right)\end{array}\right]$

(21) In summary, the optical path for the dual-PEM birefringence imaging microscope system of FIG. 1 comprises an LED light source, a linear polarizer, a PEM, the sample, a second PEM, a second linear polarizer, and a CCD camera. For each pixel of the CCD, the sample's linear retardance, .sub.S, and angle of the fast axis, .sub.S, is measured.

(22) The mathematics for the optical train are best expressed with Mueller matrices. The Mueller matrix for a linear polarizer is shown below.

(23) $M_{\mathrm{pol}}\left(\right)=\left[\begin{array}{cccc}1& C_{}& S_{}& 0\\ C_{}& C_{}^2& C_{}{S}_{}& 0\\ S_{}& C_{}{S}_{}& S_{}^2& 0\\ 0& 0& 0& 0\end{array}\right]$

(24) In the equation above, C.sub.=cos , and S.sub.=sin where is the half angle of rotation of the linear polarizer.

(25) The Mueller matrix for the PEMs is given below.

(26) $M_{\mathrm{PEM}}\left(,\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& C_{}{S}_{}^2+C_{}^2& C_{}{S}_{}\left(1-C_{}\right)& -S_{}{S}_{}\\ 0& C_{}{S}_{}\left(1-C_{}\right)& C_{}{C}_{}^2+S_{}^2& S_{}{C}_{}\\ 0& S_{}{S}_{}& -S_{}{S}_{}& C_{}\end{array}\right]$
where C.sub.=cos , and S.sub.67=sin where (t)=A cos(t+), A is the modulation amplitude, is the angular velocity of the photoelastic modulator, t is time, and is the phase delay of the photoelastic modulator.

(27) The sample of interest is assumed to have a Mueller matrix that takes the form of a linear birefringent material, such as the PEM above, but here, the Mueller matrix of the sample is expressed using a generic lower-right-hand Mueller matrix as shown below.

(28) $M_S=\left[\begin{array}{cccc}{M}_{00}& 0& 0& 0\\ 0& M_{11}& M_{12}& M_{13}\\ 0& M_{12}& M_{22}& M_{23}\\ 0& -M_{13}& -M_{23}& M_{33}\end{array}\right]$

(29) The elements of the sample Mueller matrix M.sub.mn are identical to M.sub.PEM (, ) with =.sub.S and =.sub.S where .sub.S is the angle of the fast axis, and .sub.S is the retardance of the sample. These two quantities will be isolated later, but first the entire optical pathway is expressed in Mueller matrices for isolating the terms of interest.

(30) $M_{\mathrm{OT}}=M_{\mathrm{pol}}\left(0\right)M_{\mathrm{PEM}}\left(\frac{}{4},{}_2\right)M_S{M}_{\mathrm{PEM}}\left(0,{}_1\right)M_{\mathrm{pol}}\left(\frac{}{4}\right)$

(31) The time dependent intensity at the CCD is given by the 00 element of the Mueller matrix for the entire optical train, M.sub.OT, and is shown below.
I(.sub.1,.sub.2)=M.sub.00+C.sub.1C.sub.2M.sub.12+C.sub.1C.sub.2M.sub.23+S.sub.1C.sub.2M.sub.13+S.sub.1S.sub.2M.sub.33

(32) The intensity expression is then vectorized to isolate the terms of interest, which are the elements of M.sub.S.
I(.sub.1,.sub.2)=[1C.sub.1C.sub.2C.sub.1S.sub.2S.sub.1C.sub.2S.sub.1S.sub.2][M.sub.00M.sub.12M.sub.23M.sub.13 M.sub.33].sup.T

(33) In the above expression, T is the transpose operation. The light source is controlled to pulse only when both .sub.1 and .sub.2 are at a known value, as will be explained below. A number of images can be taken, and the vectors for each can be assembled into a complete matrix as shown below.

(34) $I=\mathrm{AM}$$I=\left[\begin{array}{c}I\left({}_{1A},{}_{2A}\right)\\ I\left({}_{1B},{}_{2B}\right)\\ \mathrm{.Math.}\\ I\left({}_{1N},{}_{2N}\right)\end{array}\right]$$A=\left[\begin{array}{ccccc}1& C_{1A}{C}_{2A}& C_{1A}{S}_{2A}& S_{1A}{C}_{2A}& S_{1A}{S}_{2A}\\ 1& C_{1B}{C}_{2B}& C_{1B}{S}_{2B}& S_{1B}{C}_{2B}& S_{1B}{S}_{2B}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ 1& C_{1N}{C}_{2N}& C_{1N}{S}_{2N}& S_{1N}{C}_{2N}& S_{1N}{S}_{2N}\end{array}\right]$$M={\left[M_{00}{M}_{12}{M}_{23}{M}_{13}{M}_{33}\right]}^T$

(35) In the equation above, I is a vector of intensity values at different PEM retardance values, .sub.1 and .sub.2. The analyzer matrix, A, is comprised of the cosine and sine values at the known retardance values. The analyzer matrix's condition number is then optimized given the fewest feasible images. The equation is then inverted to solve for the quantities of M.
M=A.sup.1I

(36) If more than 5 images are collected, a pseudo-inverse is used. From here, the linear retardance, .sub.S, and angle of the fast axis, .sub.S, are solved using the following equations.

(37) ${}_S=\frac{1}{2}\tan 2^{-1}\left(M_{13},M_{23}\right)$${}_S={\tan }^{-1}\left(\sqrt{{\left(\frac{M_{13}}{M_{33}}\right)}^2+{\left(\frac{M_{23}}{M_{33}}\right)}^2}\right)$

(38) Above, tan 2.sup.1 is the four quadrant arc tangent.

(39) The challenge is to detect the intensity only at given values of the PEMs. To accomplish this, the LED light source can be triggered for a short period during each cycle of the PEM as shown in FIG. 2 and discussed next.

(40) FIG. 2 shows that as the retardance of the modulator changes, the sine and cosine of the retardance alternates between 1 and 1. Therefore, to measure the amplitude of the cosine component, cos(), two images can be collected by flashing the source light LED 17 at the times indicated by the dotted regions, I.sub.A, and dotted, waved regions, I.sub.A+. The difference in intensity for these two images, I.sub.A, is proportional to the component coupled with cos().

(41) $I_A=\frac{I_{A+}-I_{A-}}{2}$

(42) In the optical train shown in FIG. 1, all the components of interest have two modulator birefringence terms. To collect these quantities, one of two methods can be used. The first method is identical to the graph and example above: collect the intensity whenever one of the two modulators is at a given phase. This will result in the cosine term of the other modulator being time-averaged as shown below in the Jacobi Anger expansion.

(43) $J_0\left[.Math.A.Math.\right]={}_0^T\frac{\cos \left[\cos \left[t\right]\right]}{T}\mathrm{dt}\underset{t=0}{\overset{T}{.Math.}}\frac{\cos \left[\cos \left[t\right]\right]}{T}$$0={}_0^T\frac{\sin \left[\cos \left[t\right]\right]}{T}\mathrm{dt}\underset{t=0}{\overset{T}{.Math.}}\frac{\sin \left[\cos \left[t\right]\right]}{T}$

(44) For example, the term C.sub.1 C.sub.2 would become cos(.sub.1)J.sub.0[|A.sub.2|] if the triggering was set to blink or pulse the LED in phase with the first modulator, and ignore the second. The problem with this scheme is that all the sine terms go to zero, and cannot be measured.

(45) To measure a sine term, the triggering must take into account the operation of both modulators. This can be accomplished by a dedicated piece of hardware that watches both square waves generated by the PEMs and creates in-phase triggering pulses as above, while accounting for multiple modulators, and runs the resulting triggers through a logical AND gate to find a common result. This is illustrated in FIG. 3.

(46) The problem with this approach (FIG. 3) is that while each modulator trigger is active for about 1% of the modulator cycle, the two triggers only fire in unison about 0.01% (1% of 1%) of the time. This infrequent occurrence of unison firing to generate a triggering signal or pulse means that exposure times for the detector/camera must be very large.

(47) One way to address this above mentioned problem requiring long exposure times in instances where two or more PEMs are included in the optical setup is to use (have turned on) only one PEM at a time, which will be discussed next with reference to FIG. 4.

(48) FIG. 4 depicts an optical setup that is particularly useful for imaging Stokes parameters of a sample. That setup includes a light source 417 that directs light (upwardly in FIG. 4) through a sample 430 to an imaging device 429 such as a conventional CMOS or CCD-type camera. The light source is, preferably, an LED. The light from the source 417 passes through a polarizer 423 oriented at 45 relative to the baseline axis of the setup and a 0 quarter-waveplate 419 before passing through the sample 430 and an imaging lens 425.

(49) The polarized light emanating from the imaging lens is incident upon the optical element of a first photoelastic modulator (PEM) 420. The PEM 420 has its birefringent axis oriented at 0 and is controlled by an associated controller that imparts an oscillating birefringence to the optical element of the PEM, preferably at a nominal frequency of 42 kHz.

(50) With continued reference to FIG. 4, the light emanating from the first PEM 420 is directed through a second PEM 422 that is controlled by an associated controller to operate at a modulation frequency (for example, 47 kHz when turned ON). The second PEM 422 is oriented at 45. A second polarizer 427, oriented at 22.5, and lens group 421 are between the second PEM 422 and the CCD 429.

(51) The trigger pulses in this embodiment are selected at desired phases during a PEM modulation cycle. This is summarized in the following Table 1, Operation Mode 1, with the second PEM 422 turned OFF, and where pulses are triggered at four phases during a PEM cycle.

(52) TABLE-US-00001 TABLE 1 Operation Mode 1: Second PEM 422 Turned Off. PEM (1.sup.st, 42 kHz, ) /6 (30) 7/6 (210), Phase (.sub.1t) 0, 5/6 (150) /2 11/6 (330) .sub.42 = sin (.sub.1t) 0 /2 /2 sin (.sub.42) 0 1 0 1 cos (.sub.42) 1 0 1 0

(53) The camera intensity (detector) signal at each PEM phase becomes:

(54) a. .sub.1t=0, or :

(55) 0$I_{42\mathrm{cp}}=I_o\left[1+\frac{\sqrt{2}}{2}\sin \left(2{}_s\right)\sin \left({}_s\right)-\frac{\sqrt{2}}{2}\cos \left(2{}_S\right)\sin \left({}_s\right)\right]$
b. .sub.1t=/6 (30), or 5 /6 (150):

(56) $I_{42\mathrm{cp}}=I_o\left[1+\frac{\sqrt{2}}{2}\sin \left(2{}_s\right)\sin \left({}_s\right)-\frac{\sqrt{2}}{2}\cos \left({}_S\right)\right]$
c. .sub.1t=/2:

(57) $I_{42\mathrm{cn}}=I_o\left[1+\frac{\sqrt{2}}{2}\sin \left(2{}_s\right)\sin \left({}_s\right)+\frac{\sqrt{2}}{2}\cos \left(2{}_S\right)\sin \left({}_s\right)\right]$
d. .sub.1t=7 /6 (210), or 11 /6 (330):

(58) $I_{42\mathrm{cn}}=I_o\left[1+\frac{\sqrt{2}}{2}\sin \left(2{}_s\right)\sin \left({}_s\right)+\frac{\sqrt{2}}{2}\cos \left({}_S\right)\right]$

(59) A measurement taken without a sample in the setup will provide I.sub.0. Combining terms (a)-(d) and normalizing to I.sub.0 provide measurements of sin(2 .sub.S)sin(.sub.S), cos(2 .sub.S)sin(.sub.S), and cos(), which are Mueller matrix elements M.sub.13, M.sub.23, and M.sub.33. The linear retardance, .sub.S, and angle of the fast axis, .sub.S, are solved using the following equations.

(60) ${}_S=\frac{1}{2}\tan 2^{-1}\left(M_{13},M_{23}\right)$${}_S={\tan }^{-1}\left(\sqrt{{\left(\frac{M_{13}}{M_{33}}\right)}^2+{\left(\frac{M_{23}}{M_{33}}\right)}^2}\right)$

(61) Above, tan 2.sup.1 is the four quadrant arc tangent.

(62) As seen above, linear birefringence can be measured using a single PEM in this optical configuration. However, one of the three terms, M.sub.13, is measured with a DC offset, which is more susceptible to measurement errors. This term can be more accurately measured at the second PEM's modulation frequency as summarized in the following Table 2, Operation Mode 2, with the first PEM 420 turned OFF, and where pulses are triggered at four phases during a PEM cycle.

(63) TABLE-US-00002 TABLE 2 Operation Mode 2: First PEM 420 Turned Off. PEM (2nd, 47 kHz, ) /6 (30) 7/6 (210), Phase (.sub.2t) 0, 5/6 (150) /2 11/6 (330) .sub.47 = sin (.sub.2t) 0 /2 /2 sin (.sub.47) 0 1 0 1 cos (.sub.47) 1 0 1 0

(64) The camera intensity (detector) signal at each PEM phase becomes:

(65) a. .sub.2t=0, or :

(66) $I_{47\mathrm{cp}}=I_o\left[1+\frac{\sqrt{2}}{2}\sin \left(2{}_s\right)\sin \left({}_s\right)-\frac{\sqrt{2}}{2}\cos \left(2{}_s\right)\sin \left({}_s\right)\right]$
b. .sub.2t=/6 (30), or 5 /6 (150):

(67) $I_{47\mathrm{cp}}=I_o\left[1+\frac{\sqrt{2}}{2}\sin \left(2{}_s\right)\sin \left({}_s\right)-\frac{\sqrt{2}}{2}\cos \left({}_s\right)\right]$
c. .sub.2t=/2:

(68) $I_{47\mathrm{cn}}=I_o\left[1+\frac{\sqrt{2}}{2}\sin \left(2{}_s\right)\sin \left({}_s\right)+\frac{\sqrt{2}}{2}\cos \left(2{}_s\right)\sin \left({}_s\right)\right]$
d. .sub.2t=7 /6 (210), or 11 /6 (330):

(69) $I_{47\mathrm{sn}}=I_o\left[1+\frac{\sqrt{2}}{2}\sin \left(2{}_s\right)\sin \left({}_s\right)+\frac{\sqrt{2}}{2}\cos \left({}_s\right)\right]$

(70) Combining terms (a) and (c) can provide measurements of sin(2 .sub.S)sin(.sub.S), which is Mueller matrix elements M.sub.13, at the second PEM's modulation. The four pieces of data from both Operation Modes 1 and 2 are collected to improve signal-to-noise by averaging.

(71) In view of the foregoing, the optical setup illustrated in FIG. 4 is amenable to three operation configurations: (1) Using only one PEM (a relatively lower cost/complexity option); (2) Using both PEMs but operated sequentially (Operation Modes 1 and 2 above); or (3) Operating both PEMs simultaneously. Selection of the appropriate configuration depends upon the time constraints (exposure time), power and wavelengths of source light used, and the field sizes and magnifications employed in a particular application.

(72) FIG. 5 shows another embodiment of an optical setup that is also useful for imaging Stokes parameters and where an electro-optic modulator 500 provides rapid gating (sequentially permitting and occluding light transmission through the modulator) of the source light reaching the CCD. (Another option is to use a CCD intensifier in lieu of an electro-optic modulator.) Such an approach is an alternative to pulsing or strobing the source light as described above.

(73) The instrument formed in accordance with the optical setup of FIG. 5 has two primary variations. In one configuration, the first polarizer 523 is aligned at 0; in the other configuration, the polarizer is aligned at 22.5. The intensity equation for the 0 configuration is given below.
I(.sub.1,.sub.2)=S.sub.0+C.sub.2S.sub.1+S.sub.1S.sub.2S.sub.3C.sub.1S.sub.2S.sub.3

(74) Where the Stokes vector is given by the following equation:

(75) $S=\left[\begin{array}{c}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right]$

(76) The prior intensity equation can also be written in vector form.

(77) 0$I\left({}_1,{}_2\right)=\left[\begin{array}{cccc}1& C_2& S_1{S}_2& -C_1{S}_2\end{array}\right]\left[\begin{array}{c}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right]$

(78) The vector formed from the state of the modulators can be given the name of the analyzer vector as above. For the configuration with the polarizer at 22.5, the analyzer vector is given below.

(79) $A\left({}_1,{}_2\right)=\left[\begin{array}{cccc}1& \frac{\sqrt{2}}{2}{C}_1+\frac{\sqrt{2}}{2}{S}_1{S}_2& \frac{\sqrt{2}}{2}{C}_2& \frac{\sqrt{2}}{2}{C}_1{S}_2-\frac{\sqrt{2}}{2}{S}_1\end{array}\right]$

(80) This configuration may be operated with each modulator alternating between ON and OFF because all the quantities of interest can be expressed as the harmonics of only one PEM. If the first modulator is turned off, the equation for the analyzer vector is given below.

(81) $A\left(0,{}_2\right)=\left[\begin{array}{cccc}1& \frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}{C}_1& \frac{\sqrt{2}}{2}{S}_2\end{array}\right]$

(82) And likewise, the analyzer vector with the second modulator turned off is given below.

(83) $A\left({}_1,0\right)=\left[\begin{array}{cccc}1& \frac{\sqrt{2}}{2}{C}_1& \frac{\sqrt{2}}{2}& -\frac{\sqrt{2}}{2}{S}_1\end{array}\right]$

(84) The drive electronics for such a system are considerably simpler as the pulse scheme described above only depends upon the currently active modulator.

(85) By choosing to use the LED illumination as discussed above, this system can also be used to measure the linear birefringence of a sample as well as the linear extinction. The particulars of such an instrument are discussed next with reference to FIG. 6.

(86) In the embodiment of FIG. 6, the retarder 625 is, ideally, a perfect achromatic quarter-waveplate and, in combination with the 0 polarizer 623, illuminates the sample with perfectly circularly polarized light. This Stokes polarimeter can measure the Stokes vector after the sample 630 as discussed above. To measure the birefringence or linear extinction, the Mueller matrix of the first half of the instrument is given below.

(87) $M_{\mathrm{SOT}}=M_S{M}_{\mathrm{ret}}\left(\frac{}{4},\frac{}{2}\right)M_{\mathrm{pol}}\left(0\right)$

(88) By isolating the first column of the resulting Mueller matrix, the Stokes vector contains all the relevant information to solve for the retardance and angle of the fast axis as above.

(89) $S=\left[\begin{array}{c}1\\ -S_{}{S}_{}\\ S_{}{C}_{}\\ C_{}\end{array}\right]=\left[\begin{array}{c}{M}_{00}\\ M_{13}\\ M_{23}\\ M_{33}\end{array}\right]$

(90) The trouble with this setup (FIG. 6) is that the retarder producing circularly polarized light is rarely purely achromatic. This results in the following Stokes vector after the sample.

(91) $S=\left[\begin{array}{c}{M}_{00}\\ C_{\mathrm{ret}}{M}_{11}+S_{\mathrm{ret}}{M}_{13}\\ C_{\mathrm{ret}}{M}_{12}+S_{\mathrm{ret}}{M}_{23}\\ -C_{\mathrm{ret}}{M}_{13}+S_{\mathrm{ret}}{M}_{33}\end{array}\right]$

(92) In the equation above, C.sub.ret is the cosine of the retardance of the waveplate and S.sub.ret is the sine of the retardance of the waveplate 625. This can be somewhat corrected by measuring the Stokes vector of the retarder alone. Assuming perfect alignment, the Stokes vector of the retarder alone is given below.

(93) $S=\left[\begin{array}{c}1\\ C_{\mathrm{ret}}\\ 0\\ S_{\mathrm{ret}}\end{array}\right]$

(94) If the light source 617 is unpolarized, and the sample 630 exhibits differential extinction, then a Stokes polarimeter setup can also measure these properties. The polarizer 623 and retarder 625 are removed from the optical train, and the measured Stokes vector is given below.

(95) $S=\left[\begin{array}{c}T\\ -\mathrm{LE}\\ -{\mathrm{LE}}^{}\\ \mathrm{CE}\end{array}\right]=\left[\begin{array}{c}{M}_{00}\\ M_{10}\\ M_{20}\\ M_{30}\end{array}\right]$

(96) Where T is the transmission, LE is the linear extinction along 0/90, LE is the linear extinction along 45/135, and CE is the circular extinction.

(97) It is noteworthy here that the amplitude of modulation and phase lag between the detected square wave from the photoelastic modulators 620, 622 and the actual retardance both need to be calibrated. The complete equation for the amplitude of the birefringence at any time is given below.
(t,A,)=A cos(2ft+)

(98) In the equation above, is the phase lag, f is the frequency of modulation, t is the time, A is the amplitude of modulation. The intensity measured at the CCD 629 is proportional to the sine and/or cosine of the retardance of each modulator, (t, A, ).
C.sub.=cos((t,A,))=cos(A cos(2ft+))
S.sub.=sin((t,A,))=sin(A cos(2ft+))

(99) The light source 614 for the optical train is only on during certain times, and the intensity at the CCD 629 is thereby taken as the average of the sine or cosine of the birefringence during this time.

(100) $C_{}\frac{{}_{t=t_1}^{t_2}\cos \left(A\cos \left(2t+\right)\right)}{t_2-t_1}$

(101) More practically, the mean of the sine or cosine can be taken as the mean over only a few discrete points during the ON time.

(102) While the pulsing, triggering and gating techniques described above are conceptually straightforward, the poor exposure time to intensity ratio can induce considerable dark current and light pollution noise. In order to alleviate this, frequency synthesis can be utilized to generate a simple waveform at the higher harmonics of interest. This approach is described in detail following discussion of a system for simultaneous imaging of both in-plane and out-of-plane birefringence properties of a sample over a wide range of incidence angles.

(103) FIG. 7 is a diagram showing a preferred arrangement of the optical-train components of a system for simultaneous imaging of both in-plane and out-of-plane birefringence properties of a sample over a wide range of incidence angles. In FIG. 7, there is shown an optical sample 30, which may be, for example, a thin polymeric film intended for use as a birefringence compensation film as discussed above. The sample is supported upon a holder 32 with an associated variable aperture 34.

(104) The system includes light source optical components 40, the particulars of which are described further below. Those components 40 produce a source beam B of polarization-modulated, collimated light. That beam B is directed through a focusing lens system 42 that has an aperture-to-focal-length ratio selected for focusing the light on the sample 30 such that the individual rays b1-b7 that reach a location on the sample 30 each arrive with a different angle of incidence. For illustration purposes, an exemplary batch of only seven such rays b1-b7 is shown in FIG. 7.

(105) The incidence angle of the central ray b4 is 0 or normal to the surface of the sample. The remaining rays b1-b3 and b5-b7 have incrementally larger incidence angles in the direction away from the normal, central ray b4 and thus cover a wide range of incidence angles. The range can be determined by the selection of the focusing lens system and dependent upon, for example, the sought-after viewing angle of the device with which the compensation film is to be employed.

(106) The variable aperture 34 (which can be located as shown or, alternatively, between the sample 30 and lens system 42) is adjusted to facilitate accurate control of the generation of the range of incidence angles.

(107) The rays b1-b7 that emanate from the sample are directed through a collimating lens system 44 that directs that collimated beam B1 through detecting optical components 46 (further described below) to a multi-pixel imaging device 48. In a preferred embodiment, that device may be a digital imaging device or camera that employs an intensified, charge-coupled device (ICCD), a standard CCD, or CMOS-type sensor. For convenience, the imaging device will be hereafter referred to as a CCD 48.

(108) With reference to FIG. 8, in one preferred embodiment, the light source optical components 40 include a light source 50, such as a light emitting diode (LED), with associated lensing for generating the source beam B. The beam B passes through a polarizer 52 that is oriented with its polarization direction at +45 relative to a baseline axis. A high-extinction polarizer, such as a Glan-Thompson calcite polarizer, is preferred.

(109) The polarized light emanating from the polarizer 52 is incident upon the optical element of a first photoelastic modulator (PEM) 54. In a preferred embodiment, the PEM is one manufactured by Hinds Instruments, Inc., of Hillsboro, Oreg. It is noteworthy here that although a PEM is preferred, other mechanisms could be used for modulating the polarization of the source light.

(110) The most accurate birefringence measurements are achieved when one minimizes the residual birefringence present in the optical components of the system. To this end, the PEM 54 is configured to eliminate residual birefringence that may be otherwise produced by the forces present in supporting the optical element of the PEM.

(111) The PEM 54 has its birefringent axis oriented at 0 and is controlled by a controller 56 that imparts an oscillating birefringence to the optical element of the PEM, preferably at a nominal frequency of 50 kHz. In this regard, the controller 56 drives one or more quartz transducers that are adhered to the optical element of the PEM.

(112) The oscillating birefringence of the PEM 54 introduces a time-varying phase difference between the orthogonal components of the polarized light that propagates through the PEM. At any instant in time, the phase difference represents the retardation introduced by the PEM. As noted earlier, the retardation is measurable in units of length, such as nanometers. The PEM is adjustable to allow variation of the amplitude of the retardation introduced by the PEM, as described more below.

(113) With continued reference to FIG. 8, the detecting optical components 46 include a second PEM 58 that is controlled by a controller 60 to operate at a modulation frequency (for example, 55 kHz) that is different from the modulation frequency of the first PEM 54. The second PEM 58 is oriented at 45. An analyzer 62, oriented at 0, is between the second PEM 58 and the CCD 48.

(114) The CCD 48 includes individual pixels, each of which receives the light from a ray b1-b7. Thus, the light intensity captured in each pixel corresponds to the birefringence characteristics of a ray having a given angle of incidence such that, for example, a central pixel in the CCD would receive light intensity information relating to a ray b4 at a normal incidence, and a pixel aligned with ray b7 would receive light intensity information relating to a ray having the greatest angle of incidence.

(115) The preferred CCD 48 includes a gain mask that is controlled by a computer or dedicated controller operated by a computer. The CCD 48 is a time-gated device having a temporal resolution or frame rate that is generally an order of magnitude slower than the frequency with which the PEMs 54, 58 are driven. As mentioned above, if operated normally, the CCD would, in each frame, average out the PEM-modulated light received in each pixel and thus render impossible the ability to calculate the birefringence properties of interest (here, the in-plane and out-of-plane birefringence).

(116) Described below is an innovative technique for triggering the gating of the CCD 48 in a manner that allows rapid collection of a useful amount of light information in instances only when the modulation state of both PEMs 54, 58 is known, thereby enabling the precise, simultaneous determination of in-plane and out-of-plane birefringence properties of the sample corresponding to each pixel of the CCD, hence over a wide range of incidence angles.

(117) Before describing the CCD gating technique just mentioned, this description proceeds with a delineation of the data analysis applied to the intensity information received in each pixel on the assumption that the triggered CCD pixel information collected is processed (demodulated) to yield only the relevant intensity information for each pixel. (It is noteworthy that each angled apart light ray b1-b7 experiences a slightly different polarization effect from the lenses and other optical components of the system. Accordingly, as a preliminary step, each pixel must be offset or calibrated to correct for the effect.)

(118) The thin film material, such as the sample 30 of interest here, typically has polymeric structure such that the normal birefringence or in-plane birefringence is significantly less than the vertical or out-of-plane birefringence. That is, n.sub.Xn.sub.Yn.sub.Z, where n.sub.Y and n.sub.X are, respectively, the indices of refraction of the sample in the orthogonal axes, X and Y, and n.sub.Z is the index of refraction of the sample that is normal to the plane defined by those X and Y axes.

(119) Data Analysis:

(120) Normal Retardation (in-Plane Retardation):

(121) The quantity .sub.N represents the magnitude of the normal retardation, in nanometers, that is measured at the normal incidence. The normal or in-plane birefringence is defined as:
n.sub.N=n.sub.Yn.sub.X

(122) If the fast axis of the in-plane birefringence is oriented at .sub.N, n.sub.Y>n.sub.X where X coincides with the fast axis orientation, the normal retardation and normal birefringence is as follows:

(123) 0$\begin{array}{cc}{}_N=n_N.Math.d.Math.1000=\left(n_Y-n_X\right)d.Math.1000\mathrm{or}{n}_N=n_Y-n_X=\frac{{}_N}{d.Math.1000}& \mathrm{eqn}.\left(1\right)\end{array}$
where d is the thickness of the thin film in micrometers.

(124) Oblique Retardation:

(125) The quantity .sub.O represents the magnitude of oblique retardation, in nanometers, that is measured at an oblique incident angle (), which is the angle (see FIG. 8) between the oblique and normal beams. If the sample has an average index of refraction of n, the incident angle inside the film sample is:

(126) $\begin{array}{cc}={\sin }^{-1}\left[\frac{\sin }{n}\right]& \mathrm{eqn}.\left(2\right)\end{array}$

(127) For the oblique incidence, the ordinary and extraordinary rays will have different indices of refraction inside the film sample, as in a uniaxial crystal. The refractive index of the ordinary ray, n.sub.O, is independent of the incidence angle, while the refractive index of the extraordinary ray is dependent on the incidence angles. The extraordinary ray in the case here is polarized in the XZ plane (E-vector vibrating in the XZ plane) and the ray has an angle of to the Z-axis (optic axis). The effective refractive index, n.sub.eff, of the extraordinary ray is:

(128) $\begin{array}{cc}{n}_{\mathrm{eff}}\left(\right)=\frac{n_o{n}_e}{\sqrt{n_e^2{\cos }^2+n_o^2{\sin }^2}}& \mathrm{eqn}.\left(3\right)\end{array}$
where n.sub.e=n.sub.Z and n.sub.O=n.sub.X, n.sub.Y (uniaxial crystal). The n.sub.eff is always between and n.sub.e and n.sub.O.

(129) Vertical (Out-of-Plane) Birefringence Calculation:

(130) When n.sub.Z<n.sub.Xn.sub.y, n.sub.eff<n.sub.y, the measured fast axis of the oblique retardation will appear to be along the X-axis. The oblique retardation in the XZ plane can be expressed as follows:

(131) $\begin{array}{cc}{}_O=\left[n_Y-\left(\frac{n_X{n}_Z}{\sqrt{n_Z^2{\cos }^2+n_X^2{\sin }^2}}\right)\right].Math.\frac{d.Math.1000}{\cos }& \mathrm{eqn}.\left(4\right)\end{array}$

(132) Rearranging equation (4) and using eqn. (1) yields:

(133) $\begin{array}{cc}\frac{{}_O\cos }{d.Math.1000}-\frac{{}_N}{d.Math.1000}=n_X-\left(\frac{n_X{n}_Z}{\sqrt{n_Z^2{\cos }^2+n_X^2{\sin }^2}}\right)& \mathrm{eqn}.\left(5\right)\end{array}$
which leads to

(134) $\begin{array}{cc}\frac{{}_O\cos }{d.Math.1000}-\frac{{}_N}{d.Math.1000}\left(n_X-n_Z\right){\sin }^2=\mathrm{Sign}\left(\mathrm{VBR}\right)\left(n_Z-n_X\right){\sin }^2& \mathrm{eqn}.\left(6\right)\\ \mathrm{or},& \\ n_V=\mathrm{Sign}\left(\mathrm{VBR}\right)\left(n_Z-n_X\right)=\frac{1}{d.Math.1000.Math.{\sin }^2}\left(\cos .Math.{}_O-{}_N\right)& \mathrm{eqn}.\left(7\right)\end{array}$
where Sign (VBR)=1 when n.sub.Z<n.sub.X.

(135) Note that the approximation used from equations. (5) to (6) is very good for small n. At n=n.sub.Xn.sub.Z=0.01, the approximation has less than 1% error.

(136) R.sub.th Calculation:

(137) When n.sub.z<n.sub.x<n.sub.y, the R.sub.th (out-of-plane retardance) is defined as follows:

(138) $\begin{array}{cc}\begin{array}{c}{R}_{\mathrm{th}}=.Math.n_Z-\frac{\left(n_X+n_Y\right)}{2}.Math.d1000\\ =\left[\frac{\left(n_X+n_Y\right)}{2}-n_Z\right]d1000\\ =\left(n_V+\frac{n_N}{2}\right)d1000.\end{array}& \mathrm{eqn}.\left(8\right)\end{array}$

(139) In terms of experimental measurements, we have:

(140) $\begin{array}{cc}{R}_{\mathrm{th}}=\frac{1}{{\sin }^2}\left(\cos .Math.{}_O-{}_N\right)+\frac{{}_N}{2}& \mathrm{eqn}.\left(9\right)\end{array}$

(141) In equation (9), the thickness of the film does not enter the calculation of R.sub.th from the measured normal and oblique retardation values because the measured retardation values have an implicit thickness.

(142) This description now turns to the above-mentioned technique for triggering the gating of the CCD 48 in a manner that allows rapid collection of a useful amount of light information and in instances only when the modulation state of both PEMs 54, 58 is known, thereby enabling the precise determination of birefringence properties of the sample that correspond to each pixel of the CCD.

(143) In particular, this technique employs frequency synthesis for generating a simple waveform at high harmonics of the PEM modulations for gating the CCD. This approach is best understood by first considering a system where the optical train includes only a single PEM (with the understanding that using only a single PEM severely limits the range of optical properties that can be measured) and then returning to consideration of the frequency synthesis approach in systems using two or more PEMs.

(144) Frequency synthesis for controlling an imaging system that uses a single PEM can employ a standard half duty square waveform. Such waveforms are simple to create, and allow for the maximum light throughput to the imaging device. Use of such a waveform, however, requires careful attention to details such as the expected intensity difference between two images (the response), and the phase of the PEM. The first issue that needs to be addressed is how to predict the expected intensity from a square-wave-gated CCD and the Bessel waveform that is generated by the PEM.

(145) The expected response for a half duty square wave combined with a Bessel waveform can be calculated by first multiplying the Fourier expansion of a square wave with the Fourier expansion for the Bessel waveform produced by the PEM. Shown below is the Fourier expansion of the Bessel waveform.

(146) $\begin{array}{cc}\cos \left[A\cos \right]=J_0\left(A\right)+2\underset{n=1}{\overset{}{.Math.}}{\left(-1\right)}^n{J}_{2n}\left(A\right)\cos \left[2n\right]& \mathrm{eqn}.\left(10\right)\\ \sin \left[A\cos \right]=-2\underset{n=1}{\overset{}{.Math.}}{\left(-1\right)}^n{J}_{2n-1}\left(A\right)\cos \left[\left(2n-1\right)\right]& \mathrm{eqn}.\left(11\right)\end{array}$

(147) Below is the Fourier expansion of the half duty square wave.

(148) $\begin{array}{cc}\frac{4}{}\underset{n=1}{\overset{}{.Math.}}\frac{\sin \left[\left(2k-1\right)\right]}{2k-1}& \mathrm{eqn}.\left(12\right)\end{array}$

(149) The expected correlation response between any two waveforms is shown below.

(150) 0$\begin{array}{cc}R=\frac{{}_{t=0}^{T}A\left(t\right)M\left(t\right)\mathrm{dt}}{{}_{t=0}^T{M\left(t\right)}^2\mathrm{dt}}& \mathrm{eqn}.\left(13\right)\end{array}$
where R is the expected scalar response of the acquired waveform, A(t), in this case, a Bessel waveform; and M(t) is the modulating waveform, a half duty square wave. If the phase between the Bessel waveform and the square wave is assumed to be zero, then the equation above can be solved as the expansions below.

(151) $\begin{array}{cc}{R}_{cos\left(\right)}=\underset{n=1}{\overset{}{.Math.}}{\left(-1\right)}^n\frac{8}{\left(2n-1\right)}{J}_{2\left(2n-1\right)}\left(A\right)& \mathrm{eqn}.\left(14\right)\\ R_{\mathrm{si}n\left(\right)}=\underset{n=1}{\overset{}{.Math.}}\frac{8}{\left(2n-1\right)}{J}_{2n-1}\left(A\right)& \mathrm{eqn}.\left(15\right)\end{array}$

(152) In the equations above, the square wave of the cosine response is double the base Bessel waveform frequency, and the sine response frequency is the same as the Bessel waveform frequency. Using the two equations (14, 15) above, the expected response of a Bessel waveform can be predicted, and utilized to solve for the optical properties of the sample. To accomplish this, two images are collected for every Bessel waveform term (the cosine or sine in this case). The difference between these two images is directly correlated to the Mueller matrix elements that describe the optical properties of the sample.

(153) As shown in the graphs of FIG. 9, the Bessel waveform 64 is assumed to be a pure cosine Bessel waveform. First, a square wave 66 with twice the frequency and no phase offset is generated. The mean intensity 68 is measured using an integrating intensity detector. Then the square wave is phase shifted 70 by half a wave and a second mean intensity measurement 72 is taken. The difference in intensity between these two measurements is given as the response in the equations (13)-(15) above.

(154) In the following portion of this description, the effects of using multiple PEMs and frequency synthesis are described primarily with reference to the block diagram of FIG. 10. The initial step is to generate a square wave at any given harmonic of the PEMs with the correct phase.

(155) When using half duty square waves with multiple PEMs in simultaneous operation, the mixed harmonics of the PEMs must be synthesized. The synthesis can most easily be accomplished with modern digital hardware. The first step in the synthesis is to measure and record in, for example, computer memory (FIG. 10, 401) the period of each PEM 54, 58 present in the optical setup, here the setup shown in FIG. 8.

(156) To measure the period of each PEM, the PEM's feedback or output square wave is measured using a high-frequency counter. The counter is selected to operate at orders of magnitude higher than the PEM in order to attain the necessary frequency accuracy. For instance, a 400 MHz counter applied to a 40 kHz PEM will have ten thousand counts per cycle for a resolution of 4 Hz. To increase this resolution, multiple PEM cycles are measured.

(157) To ensure that the synthesized frequency does not contribute noise to the PEM instrumentation, the frequency resolution needs to be lower than 0.1 Hz. To attain this resolution for the example just given, at least 40 complete PEM cycles must be measured and averaged with a 400 MHz counter clock or 80 complete cycles with a 200 MHz counter clock. For ease of implementation, measuring powers of two cycles allows for simple right-shift division. So for a 200 MHz counter clock and 40 kHz PEM, 128 PEM cycles are averaged (2.sup.7) with the counter containing a total of at least 20 bits. For a 200 MHz counter clock and a 60 kHz PEM, 256 PEM cycles (2.sup.8) are averaged with a counter containing at least 20 bits total.

(158) Once the period for each PEM is measured, the frequency can be calculated by simply inverting the period.

(159) $\begin{array}{cc}F=\frac{1}{P}& \mathrm{Eqn}.\left(16\right)\end{array}$

(160) The harmonic of interest can now be calculated by multiplying, then summing, the relevant frequencies (FIG. 10, 402). For instance, when using two PEMs operating at a frequency of 1 F.sub.1 and 1 F.sub.2, the cosine of the sample's birefringence is measured by examining the 1 F.sub.1+1 F.sub.2 harmonic.

(161) The period for the selected square wave (FIG. 10, 404) to be synthesized with the combined PEMs waveform (FIG. 10, 406) is then calculated by inverting its frequency. The resultant period is applied to another counter (FIG. 10, 408) that controls the output of gating control 410 of the CCD 48.

(162) Given the forgoing generation and application of the frequency synthesized trigger signal as just discussed, upon demodulation (FIG. 10, 412) of the CCD output signal 414 as discussed below, the data received for each pixel of the CCD will correspond to known states of the PEMs that modulate the polarization of the beams involved and thus provide data that can be used to accurately calculate the birefringence measure of interest.

(163) The phase of the CCD output signal 414 must be discerned for accurate demodulation (FIG. 10, 412) of that signal to extract the intensity information. With the use of a single PEM, the phase relationship between the PEM's Bessel waveform and a sine/cosine demodulating signal has the expected harmonic relationship, in that the angle between the sine and cosine directly allows for the calculation of the phase of the PEM. This is not the case, however, as here, for a system using square wave demodulation. With square wave demodulation, the higher harmonics of the square waves interact with the higher harmonics of the Bessel waveform, and the expected clean trigonometric relationship is polluted. For one PEM, the expected sine/cosine trigonometric relationship and the measured square wave response are graphed in FIG. 11.

(164) The differences between the square wave response and sine/cosine response with phase can be approximated as below.

(165) $\begin{array}{cc}{E}_{\mathrm{co}s}\frac{1}{4}\left[-\cos +\cos \left(3\right)\right]& \mathrm{eqn}.\left(17\right)\\ E_{\mathrm{si}n}\frac{1}{4}\left[\sin +\sin \left(3\right)\right]& \mathrm{eqn}.\left(18\right)\\ ={}_B-{}_S& \mathrm{eqn}.\left(19\right)\end{array}$

(166) In equation (19), .sub.PB is the phase of the Bessel waveform and .sub.S is the phase of demodulating square waveform. Of important note is that the square wave and sine/cosine waveform response is identical when the phase difference is zero or multiples of /2. Accordingly, if the square wave can be generated with no difference in phase relation, then the simple analytical equations herein can be used.

(167) In order to accomplish the demodulation without having to take these phase complications into account, the synthesized square wave must possess a zero phase relation to the basis Bessel waveforms. This zero phase relationship can be accomplished by starting the square waveform with an instantaneous phase offset:
.sub.O=N.sub.1.sub.1+N.sub.2.sub.2+eqn. (20)

(168) In the equation (20) above, N.sub.n is the harmonic multiple for each PEM (usually 1 or 2), and .sub.n is the instantaneous phase. In practice, the instantaneous phase for each PEM is the counter used to measure the frequency. Accordingly, the total phase offset, .sub.O, will generally be much greater than the period of the generated frequency. As such, the phase offset must be reduced an arbitrary number of periods until it is below a single period but greater than zero. Then the frequency synthesis counter (FIG. 10, 408) is started from this value. Such a routine allows for the determination of both the frequency and phase of the square wave with reference to the Bessel waveforms generated from the PEMs.

(169) Next, the amplitude of modulation must be chosen to allow the separation and measurement of the Bessel waveforms from the total waveform. When using more than one PEM, the total waveform is the linear combination of basis Bessel waveforms and their combinations as shown in FIG. 12.

(170) Mathematically, this situation can be presented as a matrix problem:

(171) $\begin{array}{cc}\left[\begin{array}{c}{I}_{2F_1}\\ I_{1F_2}\\ \mathrm{.Math.}\\ I_{2F_1+2F_2}\\ \mathrm{.Math.}\\ I_{2F_1+1F_2}\end{array}\right]=R\left[\begin{array}{c}\cos \left({}_1\right)\\ \sin \left({}_1\right)\\ \mathrm{.Math.}\\ \cos \left({}_1\right)\cos \left({}_2\right)\\ \mathrm{.Math.}\\ \cos \left({}_1\right)\sin \left({}_2\right)\end{array}\right]& \mathrm{eqn}.\left(21\right)\end{array}$

(172) In equation (21), I.sub.NF.sub.p represents the measured intensity of one measurement of the gating at the synthesized frequency, or the difference of intensity measurements with a gating frequency at the given harmonic. The goal is then to invert the matrix, R, to solve for the amplitudes of each of the Bessel waveform harmonics given a number of intensity measurements. For best results, the matrix R must be invertible and well-conditioned. The optimal condition for matrix R occurs when the absolute value of the Bessel functions at the measured harmonics are equal. In the case of measuring the first and second harmonic in the preferred embodiment, |J.sub.1(A)|=|J.sub.2(A)|, this equivalence occurs at a driving amplitude for the PEMs of about 2.63 radians.

(173) An illustrative plot of the condition number of a matrix R for a range of driving amplitudes is shown in FIG. 13 for a system that employs 4 PEMs (polarimeter) using three demodulation waveforms: Bessel, sine/cosine, and square. The best condition is found using Bessel waveform demodulation at the J.sub.0(A)=0 condition, but the high speed digital synthesis of Bessel waveforms is far from trivial. In practice, lock-in amplifiers are commonly used with sine/cosine demodulation. For both the sine and the square wave case of the present invention, the condition is optimal between about 2.4 to 2.7 radians of PEM modulation amplitude.

(174) While the present invention has been described in terms of preferred embodiments, it will be appreciated by one of ordinary skill in the art that modifications may be made without departing from the teachings and spirit of the foregoing. Moreover, it will be understood that the systems for imaging polarization properties described above, while often considering linear birefringence or retardance or Stokes parameters, can be readily adapted for many other properties, such as circular dichroism (also known as circular extinction), and circular birefringence (also known as circular retardance).