- Email: [email protected]
Fast digital polarimeter for stokes parameter measurement
Volume 58, number 6
OPTICS COMMUNICATIONS
15 July 1986
FAST DIGITAL POLARIMETER FOR STOKES PARAMETER MEASUREMENT Kazuyoshi I T O H 1, Masanori S U G A W A R A and Yoshihiro O H T S U K A Department of Engineering Science, Faculty of Engineering Hokkaido University, Sapporo 060, Japan Received 24 February 1986
A polarimeter for analyzing rapid variations of the four Stokes parameters of a partially polarized light beam is described. A LiNbO 3 crystal is used as a fast polarization modulator for obtaining photometric signals from which the parameters are evaluated. The modulator crystal is controlled by a microprocessor, which also monitors and decodes the photometric signals.
1. Introduction The state of polarization of a quasi-monochromatic light beam is completely described by the coherency matrix or equivalently by the Stokes parameters [1]. Here, we present a method for evaluating rapidly all the four Stokes parameters and a demonstration experiment. The method can be applied to the statistical analyses of the polarization variations in an optical fiber under various stimulations. The techniques of automated polarimetry or ellipsometry [ 2 - 9 ] can be used or modified for the rapid measurement of the Stokes parameters. The fastest group characterized by the configuration of multiple detectors operating in parallel is best suited for the polarization detector in real-time feedback systems [ 3 - 5 ] , but appears lacking in accuracy and stability. The single-harmonic-polarization-modulation techniques [ 6 - 8 ] are quite acceptable in speed and precision but need more than one lock-in amplifier. In this regard, the technique based on the linear-retardationmodulation (LRM) [9] and the rotating analyzer systems [6] are more advantageous, because the photometric signal consists of a single harmonic component and only one demodulator circuit is needed. It is even possible to replace the demodulator circuit by a digital computer [9]. The present method employs the LRM principle. 1 Present address: Department of Applied Physics, Osaka University, Yamadaoka 2-1, Suita, Osaka 565, Japan. 0 030-4018/86/$03.50 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)
However, the retardation modulation is under the control of a microprocessor and therefore is n,:,t continuous but numerical. The number of data for the parameter estimation is significantly reduced by its numerical modulation scheme. The reduction in data size lessens the computational load and the demand for storage capacity. The microprocessor also enhances the flexibility of the polarimeter system.
2. Principle The polarimeter system is shown in fig. 1. A cartesian coordinate system is taken so that the z-axis coin-
D2
P LN
~
x
" 3
I
xX v
,
I I
Fig. 1. Optical system for fast digital polarimetry. The state of polarization (SOP) of incident light is discretely modified in a rapid manner by an electro-optic crystal (LNC). A tilted polarizing-beam-splitter (PBS) separates the light beam of the modified SOP into those of orthogonally polarized light. The SOP of the incident light is estimated from the intensity variations of the separated light beams. 375
Volume 58, number 6
OPTICS COMMUNICATIONS
cides with the optical path and the y-axis with the optic axis of a uniaxial 'LiNbO3 crystal (LNC) [4,5,10]. The state of polarization (SOP) of the incident light is modulated by the electro-optic effect of LNC. The important feature of the present system is that the SOP is numerically modulated by a microprocessor (MP) via interface circuits (I/O). After modulation, the light beam is split by a tilted polarizing-beamsplitter (PBS), and each of the separated beams is detected by a photodiode (D1 or D2). Let the relative phase retardation of LNC created by the external electric field be denoted by ~ and the slant angle of PBS by 0. We also take a tilted coordinate system x ' - y ' - z ' associated with the tilted PBS. Then the optical electric field vector with respect to the tilted system is given by
E'xl ( c o s O Eyl=~-sinO
0
sin0)(1
)1
Ex
\
cos0 \ 0 exp(i~b) ~Eyexp(i~))' (1)
where (Ex, Ey exp (i6)) T is the field vector of the incident light in the original coordinate system, E x and Ey are the real amplitudes, and 6 is the relative phase r~tardation between the orthogonal components. Since PBS separates the orthogonal components of eq. (1), the detector outputs are proportional to
DI(0) Dl(n/2) D ~
1
Dl(n )
D1(3~/2)
cos 20
sin 20
cos 20
0
cos 20 -sin 20 cos 20
0
- c o s 20 -sin 20
D2(0)
0 -sin 20 0
S.
sin 20 0
(4)
Since eq. (4) is overdetermined, a least-squares (LS) solution for the Stokes vector can be found if we assume appropriate noise processes for the two detectors. The LS solution may involve almost all the data for estimating each parameter and require complex calculations. We adopt here a heuristic solution. Because the thermal-drifting and noise characteristics of the two detectors and their joint characteristics are not known exactly, it is believed that outputs from a single detector should be used as exclusively as possible for the stable estimation. It is also important that the calculations are simple. The solution is
~=
1
0
0
0
0
0
cos-120
0
sin-120
0
-sin-120
0
0
D 1(~b)= ½(1, cos 20, sin 20 cos ~, - s i n 20 sin ~b)S,
15 July 1986
-~m-~2o
0
~-'2o
1
-cos-a2 0
0
0p | D, /
.J (5)
(2) D 2 (q~)= 1(1, - c o s 20, -sin 20 cos ¢, sin 20 sin q~)S, where S is the Stokes vector defined by [1]
/
Sl
Sa
s2
s3
= 2(ExEy
cos 6"
'1 i 2 (ExEy sin 6)J
(3)
Here, si (i = 0, 1,2, 3) are called the Stokes parameters and ( ) denotes the time or ensemble average. We have to estimate the four Stokes parameters from an appropriate data set of these detector outputs. We tentatively selected the data set as (DI(0), D 1(rr/2), D 1(70, D 1(3rr/2), D 2 (0)). In the following, we see that all the four Stokes parameters can be evaluated from these outputs from the two detectors. The detection process can now be expressed as
376
where ^ means the estimate. The solution can be verified by substituting eq. (4) into (5). The estimate for the Stokes vestor is thus obtained by inelaborate manip ulation of the photometric data. Note that the slant an. gle (0) of PBS must be set other than 7r/4 rad. For the symmetry in noise characteristics, we fixed the angle tc be rr/8 rad. The present method of retardation modulation is similar to the technique of LRM suggested for the fast ellipsometry [9], since the retardation modulation (¢) for the first detector output (DI(~)) is linearly increased with a constant increment (rr/2 rad). In the present method, however, the discrete modulation scheme is employed and all the four Stokes parameters are evaluated from much fewer data. It is also noted that the LRM technique, including the present version, has a close analogy with the linear-phase-modu lation techniques in modern interferometry [11-14]. It is possible to obtain a non-redundant solution in
Volume 58, number 6
OPTICS COMMUNICATIONS
which the four Stokes parameters are estimated or determined from four measurements. The expression for the first three parameters are the same as in eq. (5) but the estimate for s 3 is replaced by ~ = (sin -1 20, - 2 sin -1 20, sin -1 20,0, 0) D .
(6)
Evidently the datum on Dl(3n/2 ) can be omitted. If the ultimate speed of measurement is sought, this solution may be used. But if uncorrelated random noise such as shot noise is noticeable in the photometric data, s3 of S is superior to ~ .
3. Experiment The speed and flexibility of the present polarimeter system is demonstrated by an experiment. He-Ne
~ "
- " - '"- - -"_ " . ~ ' - J t z - • ...._..
$ 2
-...
8
-.-
#
1 ~ 8 ~ i m e < m ~ >
Fig. 2. Records of transient variations of the four Stokes parameters (ordinates are in an arbitrary unit). The variations in Stokes parameters of the output beam from a single-mode fiber were analyzed before and after the moment the fiber was given a flick (shown by a triangle).
15 July 1986
laser light was coupled into a single mode fiber with length approximately 1.5 m. A portion of the fiber with a nylon jacket was bent with a radius of curvature of approximately 4 cm and attached to an edge of an optical bench with adhesive tape. The curved portion was then vibrated with a finger. The vibration creates rapid variations of linear birefringence in the fiber. The SOP of the fiber output was analyzed by the polarimeter. The thickness and length of the modulator crystal were 1 and 40 mm, respectively, and the half wavelength voltage was 72 V. The result is shown in fig. 2. The microprocessor has been programmed that the SOP analysis of the transient phenomenon is facilitated. The processor overwrites the latest datum on the oldest one in the buffer area of memory, while checking a sudden change in a certain Stokes parameter. When a change is detected, the processor is triggered but continues to store the data stream for a moment, resulting in a complete record of the SOP before and after the trigger. This mode of triggering, widely used by modern transient recorders, can only be realized by an intelligent system. It took 1.5 ms to measure one set of the four Stokes parameters and 80 sets were recorded within 120 ms. The trigger point is indicated by a triangle on the time axis. The speed of the present system is primarily limited by that of the driving circuit for the modulator crystal and may be improved. A system which includes much faster versions of driver, analog-to-digital (A/D) converter and processor, will measure the four parameters within a few tens of microseconds. The statistical variations of the measured parameters were investigated. Linearly polarized light of various polarization angle or circularly polarized light was used as an input. The standard deviation was evaluated and normalized by its mean value. The normalized standard deviation in so parameter was less than 0.7% for any test beam. For the other parameters the normalized error was evaluated when the polarization of the test beam matched the parameter and the mean value was maximum. The normalized errors were 0.65% for Sl, 0.80% for s2 and 1.7% for s 3. These errors may be ascribed to the instabilities in the control voltage of LNC and/or the intensity fluctuations of the light source. The noise levels are, however, allowable for the present system because, roughly speaking, they are compatible with the resolution of the 8-bit A/D con377
Volume 58, number 6
OPTICS COMMUNICATIONS
verter placed at the front end of the signal processor. Finally it is noted that the present system is provided with a program for the thermal-drifting correction and the necessary hardware. The natural birefringence in LNC tends to change with temperature and affects the SOP of incident light. This thermal drifting of natural birefringence is probed by an auxiliary light beam and detector. If the auxiliary beam is sufficiently near and parallel with the signal beam, the corrected SOP is obtained by subtracting the thermal variation of the monitored retardation angle from the signal retardation angle. In an experiment of heat cycle ranging more than ten degrees, successful correction was confirmed.
4. Concluding remarks A modified version of the linear-retardation-modulation technique for the fast and complete recording of polarization variations has been described. The simple optical system, moderate data size, and small amount of calculations are the important featuresl It has also been demonstrated that the technique is particularly suitable for a computer(processor)-based system. The suggested procedure for the parameter estimation was obtained only by intuition and is redundant. Less redundant or more optimized procedures may be found.
378
15 July 1986
Acknowledgements The authors gratefully acknowledge K. Ono for providing the LiNbO 3 crystal, and K. Tanaka for the valuable suggestions on the manuscript.
References [1] M. Born and E. Wolf, Principles of optics, 4th Ed. (Pergamon, London, 1970) Sec. 10.8. [2] E. CoUett, U.S. patent 4158506, (CI. 365/356), 19, June 1979. [3] R. Ulrich, Appl. Phys. Lett. 35 (1979) 840. [4] M. Kubota, T. Oohara, K. Furuya and Y. Suematsu, Electron. Lett. 16 (1980) 573. [5] Y. Kidoh, Y. Suematsu and K. Furuya, IEEE J. Quant. Electron. QE-17 (1981) 991. [6] R.M.A. Azzam, Ellipsometry and polarized light (NorthHolland, Amsterdam, 1977) Sec. 3.10 and Sec. 5.7. [7] G.R. Boyer, B.F. Lamouroux and B.S. Prade, Appl. Optics 18 (1979) 1217. [8] S.E. Segre, J. Opt. Soc. Am. 72 (1982) 167. [9] A. Moritani, Y. Okuda, H. Kubo and J. Nakai, Appl. Optics 22 (1983) 2429. [10] E.G. Spencer, P.V. Lenzo and A.A. Ballman, Proc. IEEE 55 (1967) 2074. [11] J.F. Walkup and J.W. Goodman, J. Opt. Soc. Am. 63 (1973) 399. [12] J.H. Bruning, D.R. Heriott, J.E. Gallagher, D.P.R. Rosenfeld, A.D. White and D.J. Brangaccio, Appl. OptiCs 13 (1974) 2693. [13] J.C. Wyant, Appl. Optics 14 (1975) 2622. [14] K. Itoh and Y. Ohtsuka, Optics Comm. 36 (1981) 250.
OPTICS COMMUNICATIONS
15 July 1986
FAST DIGITAL POLARIMETER FOR STOKES PARAMETER MEASUREMENT Kazuyoshi I T O H 1, Masanori S U G A W A R A and Yoshihiro O H T S U K A Department of Engineering Science, Faculty of Engineering Hokkaido University, Sapporo 060, Japan Received 24 February 1986
A polarimeter for analyzing rapid variations of the four Stokes parameters of a partially polarized light beam is described. A LiNbO 3 crystal is used as a fast polarization modulator for obtaining photometric signals from which the parameters are evaluated. The modulator crystal is controlled by a microprocessor, which also monitors and decodes the photometric signals.
1. Introduction The state of polarization of a quasi-monochromatic light beam is completely described by the coherency matrix or equivalently by the Stokes parameters [1]. Here, we present a method for evaluating rapidly all the four Stokes parameters and a demonstration experiment. The method can be applied to the statistical analyses of the polarization variations in an optical fiber under various stimulations. The techniques of automated polarimetry or ellipsometry [ 2 - 9 ] can be used or modified for the rapid measurement of the Stokes parameters. The fastest group characterized by the configuration of multiple detectors operating in parallel is best suited for the polarization detector in real-time feedback systems [ 3 - 5 ] , but appears lacking in accuracy and stability. The single-harmonic-polarization-modulation techniques [ 6 - 8 ] are quite acceptable in speed and precision but need more than one lock-in amplifier. In this regard, the technique based on the linear-retardationmodulation (LRM) [9] and the rotating analyzer systems [6] are more advantageous, because the photometric signal consists of a single harmonic component and only one demodulator circuit is needed. It is even possible to replace the demodulator circuit by a digital computer [9]. The present method employs the LRM principle. 1 Present address: Department of Applied Physics, Osaka University, Yamadaoka 2-1, Suita, Osaka 565, Japan. 0 030-4018/86/$03.50 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)
However, the retardation modulation is under the control of a microprocessor and therefore is n,:,t continuous but numerical. The number of data for the parameter estimation is significantly reduced by its numerical modulation scheme. The reduction in data size lessens the computational load and the demand for storage capacity. The microprocessor also enhances the flexibility of the polarimeter system.
2. Principle The polarimeter system is shown in fig. 1. A cartesian coordinate system is taken so that the z-axis coin-
D2
P LN
~
x
" 3
I
xX v
,
I I
Fig. 1. Optical system for fast digital polarimetry. The state of polarization (SOP) of incident light is discretely modified in a rapid manner by an electro-optic crystal (LNC). A tilted polarizing-beam-splitter (PBS) separates the light beam of the modified SOP into those of orthogonally polarized light. The SOP of the incident light is estimated from the intensity variations of the separated light beams. 375
Volume 58, number 6
OPTICS COMMUNICATIONS
cides with the optical path and the y-axis with the optic axis of a uniaxial 'LiNbO3 crystal (LNC) [4,5,10]. The state of polarization (SOP) of the incident light is modulated by the electro-optic effect of LNC. The important feature of the present system is that the SOP is numerically modulated by a microprocessor (MP) via interface circuits (I/O). After modulation, the light beam is split by a tilted polarizing-beamsplitter (PBS), and each of the separated beams is detected by a photodiode (D1 or D2). Let the relative phase retardation of LNC created by the external electric field be denoted by ~ and the slant angle of PBS by 0. We also take a tilted coordinate system x ' - y ' - z ' associated with the tilted PBS. Then the optical electric field vector with respect to the tilted system is given by
E'xl ( c o s O Eyl=~-sinO
0
sin0)(1
)1
Ex
\
cos0 \ 0 exp(i~b) ~Eyexp(i~))' (1)
where (Ex, Ey exp (i6)) T is the field vector of the incident light in the original coordinate system, E x and Ey are the real amplitudes, and 6 is the relative phase r~tardation between the orthogonal components. Since PBS separates the orthogonal components of eq. (1), the detector outputs are proportional to
DI(0) Dl(n/2) D ~
1
Dl(n )
D1(3~/2)
cos 20
sin 20
cos 20
0
cos 20 -sin 20 cos 20
0
- c o s 20 -sin 20
D2(0)
0 -sin 20 0
S.
sin 20 0
(4)
Since eq. (4) is overdetermined, a least-squares (LS) solution for the Stokes vector can be found if we assume appropriate noise processes for the two detectors. The LS solution may involve almost all the data for estimating each parameter and require complex calculations. We adopt here a heuristic solution. Because the thermal-drifting and noise characteristics of the two detectors and their joint characteristics are not known exactly, it is believed that outputs from a single detector should be used as exclusively as possible for the stable estimation. It is also important that the calculations are simple. The solution is
~=
1
0
0
0
0
0
cos-120
0
sin-120
0
-sin-120
0
0
D 1(~b)= ½(1, cos 20, sin 20 cos ~, - s i n 20 sin ~b)S,
15 July 1986
-~m-~2o
0
~-'2o
1
-cos-a2 0
0
0p | D, /
.J (5)
(2) D 2 (q~)= 1(1, - c o s 20, -sin 20 cos ¢, sin 20 sin q~)S, where S is the Stokes vector defined by [1]
/
Sl
Sa
s2
s3
= 2(ExEy
cos 6"
'1 i 2 (ExEy sin 6)J
(3)
Here, si (i = 0, 1,2, 3) are called the Stokes parameters and ( ) denotes the time or ensemble average. We have to estimate the four Stokes parameters from an appropriate data set of these detector outputs. We tentatively selected the data set as (DI(0), D 1(rr/2), D 1(70, D 1(3rr/2), D 2 (0)). In the following, we see that all the four Stokes parameters can be evaluated from these outputs from the two detectors. The detection process can now be expressed as
376
where ^ means the estimate. The solution can be verified by substituting eq. (4) into (5). The estimate for the Stokes vestor is thus obtained by inelaborate manip ulation of the photometric data. Note that the slant an. gle (0) of PBS must be set other than 7r/4 rad. For the symmetry in noise characteristics, we fixed the angle tc be rr/8 rad. The present method of retardation modulation is similar to the technique of LRM suggested for the fast ellipsometry [9], since the retardation modulation (¢) for the first detector output (DI(~)) is linearly increased with a constant increment (rr/2 rad). In the present method, however, the discrete modulation scheme is employed and all the four Stokes parameters are evaluated from much fewer data. It is also noted that the LRM technique, including the present version, has a close analogy with the linear-phase-modu lation techniques in modern interferometry [11-14]. It is possible to obtain a non-redundant solution in
Volume 58, number 6
OPTICS COMMUNICATIONS
which the four Stokes parameters are estimated or determined from four measurements. The expression for the first three parameters are the same as in eq. (5) but the estimate for s 3 is replaced by ~ = (sin -1 20, - 2 sin -1 20, sin -1 20,0, 0) D .
(6)
Evidently the datum on Dl(3n/2 ) can be omitted. If the ultimate speed of measurement is sought, this solution may be used. But if uncorrelated random noise such as shot noise is noticeable in the photometric data, s3 of S is superior to ~ .
3. Experiment The speed and flexibility of the present polarimeter system is demonstrated by an experiment. He-Ne
~ "
- " - '"- - -"_ " . ~ ' - J t z - • ...._..
$ 2
-...
8
-.-
#
1 ~ 8 ~ i m e < m ~ >
Fig. 2. Records of transient variations of the four Stokes parameters (ordinates are in an arbitrary unit). The variations in Stokes parameters of the output beam from a single-mode fiber were analyzed before and after the moment the fiber was given a flick (shown by a triangle).
15 July 1986
laser light was coupled into a single mode fiber with length approximately 1.5 m. A portion of the fiber with a nylon jacket was bent with a radius of curvature of approximately 4 cm and attached to an edge of an optical bench with adhesive tape. The curved portion was then vibrated with a finger. The vibration creates rapid variations of linear birefringence in the fiber. The SOP of the fiber output was analyzed by the polarimeter. The thickness and length of the modulator crystal were 1 and 40 mm, respectively, and the half wavelength voltage was 72 V. The result is shown in fig. 2. The microprocessor has been programmed that the SOP analysis of the transient phenomenon is facilitated. The processor overwrites the latest datum on the oldest one in the buffer area of memory, while checking a sudden change in a certain Stokes parameter. When a change is detected, the processor is triggered but continues to store the data stream for a moment, resulting in a complete record of the SOP before and after the trigger. This mode of triggering, widely used by modern transient recorders, can only be realized by an intelligent system. It took 1.5 ms to measure one set of the four Stokes parameters and 80 sets were recorded within 120 ms. The trigger point is indicated by a triangle on the time axis. The speed of the present system is primarily limited by that of the driving circuit for the modulator crystal and may be improved. A system which includes much faster versions of driver, analog-to-digital (A/D) converter and processor, will measure the four parameters within a few tens of microseconds. The statistical variations of the measured parameters were investigated. Linearly polarized light of various polarization angle or circularly polarized light was used as an input. The standard deviation was evaluated and normalized by its mean value. The normalized standard deviation in so parameter was less than 0.7% for any test beam. For the other parameters the normalized error was evaluated when the polarization of the test beam matched the parameter and the mean value was maximum. The normalized errors were 0.65% for Sl, 0.80% for s2 and 1.7% for s 3. These errors may be ascribed to the instabilities in the control voltage of LNC and/or the intensity fluctuations of the light source. The noise levels are, however, allowable for the present system because, roughly speaking, they are compatible with the resolution of the 8-bit A/D con377
Volume 58, number 6
OPTICS COMMUNICATIONS
verter placed at the front end of the signal processor. Finally it is noted that the present system is provided with a program for the thermal-drifting correction and the necessary hardware. The natural birefringence in LNC tends to change with temperature and affects the SOP of incident light. This thermal drifting of natural birefringence is probed by an auxiliary light beam and detector. If the auxiliary beam is sufficiently near and parallel with the signal beam, the corrected SOP is obtained by subtracting the thermal variation of the monitored retardation angle from the signal retardation angle. In an experiment of heat cycle ranging more than ten degrees, successful correction was confirmed.
4. Concluding remarks A modified version of the linear-retardation-modulation technique for the fast and complete recording of polarization variations has been described. The simple optical system, moderate data size, and small amount of calculations are the important featuresl It has also been demonstrated that the technique is particularly suitable for a computer(processor)-based system. The suggested procedure for the parameter estimation was obtained only by intuition and is redundant. Less redundant or more optimized procedures may be found.
378
15 July 1986
Acknowledgements The authors gratefully acknowledge K. Ono for providing the LiNbO 3 crystal, and K. Tanaka for the valuable suggestions on the manuscript.
References [1] M. Born and E. Wolf, Principles of optics, 4th Ed. (Pergamon, London, 1970) Sec. 10.8. [2] E. CoUett, U.S. patent 4158506, (CI. 365/356), 19, June 1979. [3] R. Ulrich, Appl. Phys. Lett. 35 (1979) 840. [4] M. Kubota, T. Oohara, K. Furuya and Y. Suematsu, Electron. Lett. 16 (1980) 573. [5] Y. Kidoh, Y. Suematsu and K. Furuya, IEEE J. Quant. Electron. QE-17 (1981) 991. [6] R.M.A. Azzam, Ellipsometry and polarized light (NorthHolland, Amsterdam, 1977) Sec. 3.10 and Sec. 5.7. [7] G.R. Boyer, B.F. Lamouroux and B.S. Prade, Appl. Optics 18 (1979) 1217. [8] S.E. Segre, J. Opt. Soc. Am. 72 (1982) 167. [9] A. Moritani, Y. Okuda, H. Kubo and J. Nakai, Appl. Optics 22 (1983) 2429. [10] E.G. Spencer, P.V. Lenzo and A.A. Ballman, Proc. IEEE 55 (1967) 2074. [11] J.F. Walkup and J.W. Goodman, J. Opt. Soc. Am. 63 (1973) 399. [12] J.H. Bruning, D.R. Heriott, J.E. Gallagher, D.P.R. Rosenfeld, A.D. White and D.J. Brangaccio, Appl. OptiCs 13 (1974) 2693. [13] J.C. Wyant, Appl. Optics 14 (1975) 2622. [14] K. Itoh and Y. Ohtsuka, Optics Comm. 36 (1981) 250.