Abstract

A time-domain technique for quantitatively measuring linear retardance with an unprecedented degree of accuracy is described both theoretically and experimentally. This novel approach builds upon the unique capabilities afforded by the pulsed ring-down paradigm, as augmented by the insertion of polarization-selective components into the light injection stage and signal detection train of a stable, high-finesse cavity that contains the optical component under investigation. Application to a quarter-waveplate of the compound zero-order design highlights the robust and versatile nature of the proposed scheme while simultaneously demonstrating the ability to resolve retardation imperfections with (one standard deviation) uncertainties of better than λ106.

© 2007 Optical Society of America

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    [CrossRef]
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  61. At λ=355nm, the empty cavity lifetime, as extracted from the sum of I⊥ and I‖ signals [cf., Eq. ], was typically 3.790±0.035μs, corresponding to an effective mirror reflectivity of R=99.85%. The intracavity losses incurred by insertion of the targeted quarter-waveplate decreased these values to 1.206±0.012μs and R=99.53%.
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    [CrossRef]

2005 (4)

S. Shen, J. She, and T. Tao, "Optimal design of achromatic true zero-order waveplates using twisted nematic liquid crystal," J. Opt. Soc. Am. A 22, 961-965 (2005).
[CrossRef]

W.-C. Kuo, K.-Y. Liao, G.-J. Jan, H.-K. Teng, and C. Chou, "Simultaneous measurement of phase retardation and fast-axis angle of phase retardation plate," Jpn. J. Appl. Phys., Part 1 44, 1095-1100 (2005).
[CrossRef]

S. M. Wilson, K. B. Wiberg, J. R. Cheeseman, M. J. Frisch, and P. H. Vaccaro, "Nonresonant optical activity of isolated organic molecules," J. Phys. Chem. A 109, 11752-11764 (2005).
[CrossRef] [PubMed]

C. Vallance, "Innovations in cavity ring-down spectroscopy," New J. Chem. 29, 867-874 (2005).
[CrossRef]

2004 (3)

S. Y. Lee, J. F. Lin, and Y. L. Lo, "A compact circular heterodyne interferometer for simultaneous measurements of variation in the magnitude of phase retardation and principal axis angle," Meas. Sci. Technol. 15, 978-982 (2004).
[CrossRef]

K.-C. Lang, H.-K. Teng, H.-F. Chang, C.-Y. Han, and C. Chou, "Two-dimensional linear birefringence vector measurement by polarization-stepping interferometer," Jpn. J. Appl. Phys., Part 1 43, 1633-1637 (2004).
[CrossRef]

A. V. Samoylov, V. S. Samoylov, A. P. Vidmachenko, and A. V. Perekhod, "Achromatic and super-achromatic zero-order waveplates," J. Quant. Spectrosc. Radiat. Transf. 88, 319-325 (2004).
[CrossRef]

2003 (3)

2002 (5)

F. Brandi, E. Polacco, and G. Ruoso, "Stress-optic modulator: A novel device for high sensitivity linear birefringence measurements," Meas. Sci. Technol. 12, 1503-1508 (2002).
[CrossRef]

S. Cattaneo, O. Zehnder, P. Gunter, and M. Kauranen, "Nonlinear optical technique for precise retardation measurement," Phys. Rev. Lett. 88, 243901 (2002).
[CrossRef] [PubMed]

T. Müller, K. B. Wiberg, and P. H. Vaccaro, "An optical mounting system for cavity ring-down polarimetry," Rev. Sci. Instrum. 73, 1340-1342 (2002).
[CrossRef]

T. Müller, K. B. Wiberg, P. H. Vaccaro, J. R. Cheeseman, and M. J. Frisch, "Cavity ring-down polarimetry (CRDP): Theoretical and experimental characterization," J. Opt. Soc. Am. B 19, 125-141 (2002).
[CrossRef]

H.-K. Teng, C. Chou, C.-N. Chang, C.-W. Lyu, and Y.-C. Huang, "Linear birefringence measurement with a differential-phase optical polarimeter," Jpn. J. Appl. Phys., Part 1 41, 3140-3144 (2002).
[CrossRef]

2001 (2)

Y. Zhang, S. Zhang, Y. Han, Y. Li, and X. Xu, "Method for the measurement of retardation of waveplates based on laser frequency-splitting technology," Opt. Eng. (Bellingham) 40, 1071-1075 (2001).
[CrossRef]

B. Wang and W. Hellman, "Accuracy assessment of linear birefringence measurement system using a Soleil-Babinet compensator," Rev. Sci. Instrum. 72, 4066-4070 (2001).
[CrossRef]

2000 (5)

T. Müller, K. B. Wiberg, and P. H. Vaccaro, "Cavity ring-down polarimetry (CRDP): A new scheme for probing circular birefringence and circular dichroism in the gas phase," J. Phys. Chem. A 104, 5959-5968 (2000).
[CrossRef]

G. Berden, R. Peeters, and G. Meijer, "Cavity ring-down spectroscopy: Experimental schemes and applications," Int. Rev. Phys. Chem. 19, 565-607 (2000).
[CrossRef]

J. Schirmer and T. Schmidt-Kaler, "Liquid crystal phase retarder with broad spectral range," Opt. Commun. 176, 313-317 (2000).
[CrossRef]

P. Hariharan and P. E. Ciddor, "Superachromatic circular polarizer," Meas. Sci. Technol. 11, N117-N118 (2000).
[CrossRef]

W.-Q. Zhang, "New phase shift formulas and stability of waveplate in oblique incident beam," Opt. Commun. 176, 9-15 (2000).
[CrossRef]

1999 (2)

J. P. Lesso, A. J. Duncan, W. Sibbett, and M. J. Padgett, "A technique for modeling the performance of birefringent waveplates," Opt. Quantum Electron. 31, 645-653 (1999).
[CrossRef]

B. Wang and T. C. Oakberg, "A new instrument for measuring both the magnitude and angle of low level linear birefringence," Rev. Sci. Instrum. 70, 3847-3854 (1999).
[CrossRef]

1998 (2)

M. D. Wheeler, S. M. Newman, A. J. Orr-Ewing, and M. N. R. Ashfold, "Cavity ring-down spectroscopy," J. Chem. Soc., Faraday Trans. 94, 337-351 (1998).
[CrossRef]

P. Hariharan, "Broad-band superachromatic retarders," Meas. Sci. Technol. 9, 1678-1681 (1998).
[CrossRef]

1997 (4)

1996 (3)

C. Chou, Y.-C. Huang, and M. Chang, "Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz waveplate," Jpn. J. Appl. Phys., Part 1 35, 5526-5529 (1996).
[CrossRef]

M. H. Chiu, C. D. Chen, and D.-C. Su, "Method for determining the fast axis and phase retardation of a waveplate," J. Opt. Soc. Am. A 13, 1924-1929 (1996).
[CrossRef]

P. Hariharan, "Achromatic and apochromatic halfwave and quarterwave retarders," Opt. Eng. (Bellingham) 35, 3335-3337 (1996).
[CrossRef]

1995 (3)

E. A. West and M. H. Smith, "Polarization errors associated with birefringent waveplates," Opt. Eng. (Bellingham) 34, 1574-1580 (1995).
[CrossRef]

E. A. West and M. H. Smith, "Polarization errors associated with birefringent waveplates," Opt. Eng. (Bellingham) 34, 1574-1580 (1995).
[CrossRef]

K. Pietraszkiewicz, W. A. Wozniak, and P. Kurzynowski, "Effect of multiple reflections in retardation plates with elliptical birefringence," J. Opt. Soc. Am. A 12, 420-424 (1995).
[CrossRef]

1994 (1)

1993 (4)

1992 (1)

1991 (1)

1990 (1)

1988 (3)

1971 (1)

1966 (1)

1964 (2)

1960 (1)

1948 (1)

Appl. Opt. (14)

P. D. Hale and G. W. Day, "Stability of birefringent linear retarders (waveplates)," Appl. Opt. 27, 5146-5153 (1988)
[CrossRef] [PubMed]

J. M. Beckers, "Achromatic linear retarders," Appl. Opt. 10, 973-975 (1971).
[CrossRef] [PubMed]

B. R. Grunstra and H. B. Perkins, "A method for the measurement of optical retardation angles near 90 degrees," Appl. Opt. 5, 585-588 (1966).
[CrossRef] [PubMed]

S. D. Chidester, J. W. Harvey, and R. P. Hubbard, "Measurement of crystal retarders," Appl. Opt. 30, 12-14 (1991).
[CrossRef] [PubMed]

D. H. Goldstein, "Mueller matrix dual-rotating retarder polarimeter," Appl. Opt. 31, 6676-6683 (1992).
[CrossRef] [PubMed]

D. B. Chenault and R. A. Chipman, "Measurements of linear diattenuation and linear retardance spectra with a rotating sample spectropolarimeter," Appl. Opt. 32, 3513-3519 (1993).
[CrossRef] [PubMed]

G. C. Nechev, "Analytical phase-measuring technique for retardation measurements," Appl. Opt. 33, 6621-6625 (1994).
[CrossRef] [PubMed]

P. A. Williams, A. H. Rose, and C. M. Wang, "Rotating-polarizer polarimeter for accurate retardation measurement," Appl. Opt. 36, 6466-6472 (1997).
[CrossRef]

S. Nakadate, "High-precision retardation measurement using phase detection of Young's fringes," Appl. Opt. 29, 242-246 (1990).
[CrossRef] [PubMed]

K. B. Rochford and C. M. Wang, "Accurate interferometric retardance measurements." Appl. Opt. 36, 6473-6479 (1997).
[CrossRef]

H.-K. Teng, C. Chou, C.-N. Chang, and H.-T. Wu, "Application of phase-to-amplitude conversion technique to linear birefringence measurements," Appl. Opt. 42, 1798-1804 (2003).
[CrossRef] [PubMed]

H. Takasaki, M. Isobe, T. Masaki, A. Konda, T. Agastuma, and Y. Watanabe, "An automatic retardation meter for automatic polarimetry by means of an ADP polarization modulator," Appl. Opt. 3, 345-350 (1964).
[CrossRef]

L.-H. Shyu, C.-L. Chen, and D.-C. Su, "Method for measuring the retardation of a waveplate," Appl. Opt. 32, 4228-4230 (1993).
[CrossRef] [PubMed]

J. E. Hayden and S. D. Jacobs, "Automated spatially scanning ellipsometer for retardation measurements of transparent materials," Appl. Opt. 32, 6256-6263 (1993).
[CrossRef] [PubMed]

Chem. Rev. (Washington, D.C.) (1)

J. J. Scherer, J. B. Paul, A. O'Keefe, and R. J. Saykally, "Cavity ringdown laser absorption spectroscopy: History, development, and application to pulsed molecular beams," Chem. Rev. (Washington, D.C.) 97, 25-51 (1997).
[CrossRef]

Int. Rev. Phys. Chem. (1)

G. Berden, R. Peeters, and G. Meijer, "Cavity ring-down spectroscopy: Experimental schemes and applications," Int. Rev. Phys. Chem. 19, 565-607 (2000).
[CrossRef]

J. Chem. Soc., Faraday Trans. (1)

M. D. Wheeler, S. M. Newman, A. J. Orr-Ewing, and M. N. R. Ashfold, "Cavity ring-down spectroscopy," J. Chem. Soc., Faraday Trans. 94, 337-351 (1998).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (3)

J. Phys. Chem. A (2)

S. M. Wilson, K. B. Wiberg, J. R. Cheeseman, M. J. Frisch, and P. H. Vaccaro, "Nonresonant optical activity of isolated organic molecules," J. Phys. Chem. A 109, 11752-11764 (2005).
[CrossRef] [PubMed]

T. Müller, K. B. Wiberg, and P. H. Vaccaro, "Cavity ring-down polarimetry (CRDP): A new scheme for probing circular birefringence and circular dichroism in the gas phase," J. Phys. Chem. A 104, 5959-5968 (2000).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (1)

A. V. Samoylov, V. S. Samoylov, A. P. Vidmachenko, and A. V. Perekhod, "Achromatic and super-achromatic zero-order waveplates," J. Quant. Spectrosc. Radiat. Transf. 88, 319-325 (2004).
[CrossRef]

Jpn. J. Appl. Phys., Part 1 (5)

H.-K. Teng, C. Chou, C.-N. Chang, C.-W. Lyu, and Y.-C. Huang, "Linear birefringence measurement with a differential-phase optical polarimeter," Jpn. J. Appl. Phys., Part 1 41, 3140-3144 (2002).
[CrossRef]

C. Chou, Y.-C. Huang, and M. Chang, "Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz waveplate," Jpn. J. Appl. Phys., Part 1 35, 5526-5529 (1996).
[CrossRef]

K.-C. Lang, C. Chou, and H.-K. Teng, "Polarized ring interferometer for linear birefringence measurement," Jpn. J. Appl. Phys., Part 1 42, 5826-5830 (2003).
[CrossRef]

K.-C. Lang, H.-K. Teng, H.-F. Chang, C.-Y. Han, and C. Chou, "Two-dimensional linear birefringence vector measurement by polarization-stepping interferometer," Jpn. J. Appl. Phys., Part 1 43, 1633-1637 (2004).
[CrossRef]

W.-C. Kuo, K.-Y. Liao, G.-J. Jan, H.-K. Teng, and C. Chou, "Simultaneous measurement of phase retardation and fast-axis angle of phase retardation plate," Jpn. J. Appl. Phys., Part 1 44, 1095-1100 (2005).
[CrossRef]

Meas. Sci. Technol. (4)

S. Y. Lee, J. F. Lin, and Y. L. Lo, "A compact circular heterodyne interferometer for simultaneous measurements of variation in the magnitude of phase retardation and principal axis angle," Meas. Sci. Technol. 15, 978-982 (2004).
[CrossRef]

F. Brandi, E. Polacco, and G. Ruoso, "Stress-optic modulator: A novel device for high sensitivity linear birefringence measurements," Meas. Sci. Technol. 12, 1503-1508 (2002).
[CrossRef]

P. Hariharan and P. E. Ciddor, "Superachromatic circular polarizer," Meas. Sci. Technol. 11, N117-N118 (2000).
[CrossRef]

P. Hariharan, "Broad-band superachromatic retarders," Meas. Sci. Technol. 9, 1678-1681 (1998).
[CrossRef]

New J. Chem. (1)

C. Vallance, "Innovations in cavity ring-down spectroscopy," New J. Chem. 29, 867-874 (2005).
[CrossRef]

Opt. Commun. (2)

J. Schirmer and T. Schmidt-Kaler, "Liquid crystal phase retarder with broad spectral range," Opt. Commun. 176, 313-317 (2000).
[CrossRef]

W.-Q. Zhang, "New phase shift formulas and stability of waveplate in oblique incident beam," Opt. Commun. 176, 9-15 (2000).
[CrossRef]

Opt. Eng. (Bellingham) (4)

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Other (13)

At λ=355nm, the empty cavity lifetime, as extracted from the sum of I⊥ and I‖ signals [cf., Eq. ], was typically 3.790±0.035μs, corresponding to an effective mirror reflectivity of R=99.85%. The intracavity losses incurred by insertion of the targeted quarter-waveplate decreased these values to 1.206±0.012μs and R=99.53%.

In practice, the waveplate under investigation was tilted away from normal incidence by a minute amount (≤0.1mrad deviation) to avoid slight, yet perceptible, interference effects from the creation of secondary cavities (i.e., between mirror and waveplate) in the resonator assembly.

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of polarimetric instrumentation. A spatially filtered and mode-matched laser pulse ( < 10 ns duration) traverses a linear polarizer before entering a stable (near-confocal), high-finesse cavity of L = 1.706 m length through the planar rear surface of the input mirror. The retardation component under investigation (a λ 4 -plate) is placed within this resonator assembly, where it is aligned to be at a near-normal angle of incidence and to have its principal axes set nominally at 45° relative to the direction of input polarization. Light emerging from the output mirror is polarization analyzed and imaged onto identical detectors that monitor the intensities for two linear-polarization components aligned parallel ( I ) and perpendicular ( I ) to that of the initially injected optical field.

Fig. 2
Fig. 2

Effect of retardance error on polarimetric ratio signal. The predicted polarimetric ratio signal for an imperfect quarter-wave retardation plate [cf., Eq (11)] is plotted as a function of cavity ring-down time for relative retardance errors [ ξ ( 2 π 4 ) ] of (a) 0.004, (b) 0.008, (c) 0.02, and (d) 0.04. In each case, it has been assumed that α = 0 and t r t = 10 ns , with the incident optical pulse having a Gaussian profile of 8 ns duration (FWHM).

Fig. 3
Fig. 3

Effect of instrumental response function on polarimetric ratio signal. The predicted polarimetric ratio signal for an imperfect quarter-wave retardation plate [cf., Eq. (11)] is plotted as a function of cavity ring-down time for a Gaussian instrumental response function (or incident laser pulseshape) of (a) 1 ns , (b) 5 ns , (c) 10 ns , and (d) 15 ns duration (FWHM). In each case, it has been assumed that α = 0 and t r t = 10 ns , with the relative retardance error of the waveplate being held constant at 0.02 (or ξ = 0.01 π ).

Fig. 4
Fig. 4

Polarimetric analysis of a quarter-waveplate. A compound zero-order λ 4 -plate was inserted into the resonator assembly illustrated in Fig. 1 and probed at the design wavelength of λ 0 = 355 nm . The top panel highlights the polarimetric ratio signal derived from measured ring-down traces while the bottom panel depicts the results of a nonlinear least-squares regression based upon the model suggested by Eqs. (7, 10, 11). The slight decrease in the amplitude of I r a t i o ( t ) observed with increasing time stems from imperfect background subtraction of the experimental data, as modeled by incorporating adjustable constants into the predicted forms for I ( t ) and I ( t ) . The slowly varying envelope that modulates the comb-like pattern of alternating-polarity pulses provides clear evidence for a substantial waveplate imperfection, with detailed analysis finding the effective retardance error to be ξ = 1.71322 ( 42 ) × 10 2 rad . This approach also allowed the cavity round-trip time and duration of the Gaussian response function to be extracted, yielding t r t = 11.391290 ( 52 ) ns and τ F W H M = 10.73 ( 10 ) ns , respectively.

Fig. 5
Fig. 5

Wavelength dependence of retardance error. The dependence of observed retardance error (ξ in radians) upon incident wavelength is plotted for a λ 4 -plate of the compound zero-order design. The circular symbols denote the results of polarimetric measurements performed at discrete wavelengths, with the two overlapping points at 355 nm affording an indication for experimental precision. In particular, the latter were acquired independently at the beginning and the end of the data run (i.e., corresponding to a time interval of 3 h ) without any attempt being made to actively stabilize the temperature of the apparatus. The superimposed curve follows from a linear least-squares regression that gave an overall slope of 5.37888 ( 68 ) × 10 3 rad nm . Although fabricated for use at λ 0 = 355 nm , the waveplate under investigation appears to have more ideal performance characteristics (for normal angle of incidence) at 358.42 nm .

Equations (13)

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Δ φ ( λ ) = 2 π λ Δ n ( λ ) d ,
I ( N ) = R 2 N 2 { 1 + sin 2 ( 2 α ) cos 2 ( 2 α ) sin [ N π + 2 ( N + 1 2 ) ξ ] } ,
I ( N ) = R 2 N 2 cos 2 ( 2 α ) { 1 + sin [ N π + 2 ( N + 1 2 ) ξ ] } ,
sin [ N π + 2 ( N + 1 2 ) ξ ] ξ = 0 sin [ N π ] = 0 .
I d i f f ( N ) = I ( N ) I ( N ) = R 2 N { cos 2 ( 2 α ) sin [ N π + 2 ( N + 1 2 ) ξ ] sin 2 ( 2 α ) } = R 2 N { cos 2 ( 2 α ) cos [ 2 ( N + 1 2 ) ( ξ + π 2 ) ] + sin 2 ( 2 α ) } ,
I s u m ( N ) = I ( N ) + I ( N ) = R 2 N ,
I r a t i o ( N ) = I d i f f ( N ) I s u m ( N ) = cos 2 ( 2 α ) cos [ 2 ( N + 1 2 ) ( ξ + π 2 ) ] sin 2 ( 2 α ) ξ = 0 cos 2 ( 2 α ) cos [ ( N + 1 2 ) π ] sin 2 ( 2 α ) = sin 2 ( 2 α ) .
I r a t i o ( N ) cos [ 2 ( N + 1 2 ) ( ξ + π 2 ) ] ξ = 0 cos [ ( N + 1 2 ) π ] = 0 ,
I r a t i o ( N ) cos [ 2 ( N + 1 2 ) ( ξ + π 2 ) ] = cos [ 2 ( N + 1 2 ) ξ ] cos [ ( N + 1 2 ) π ] + sin [ 2 ( N + 1 2 ) ξ ] sin [ ( N + 1 2 ) π ] = ( 1 ) N sin [ 2 ( N + 1 2 ) ξ ] .
I r a t i o ( t ) = N = 0 I r a t i o ( N ) δ ( t N t r t ) ,
I r a t i o ( t ) = + r ( t t ) I r a t i o ( t ) d t , = N = 0 I r a t i o ( N ) [ + r ( t t ) δ ( t N t r t ) d t ] = N = 0 I r a t i o ( N ) r ( t N t r t ) .
2 ( N n + 1 2 ) ξ = n π n = 0 , 1 , 2 , .
I r a t i o ( N ) = I d i f f ( N ) I s u m ( N ) = cos 2 ( 2 α ) cos [ 2 ( N + 1 2 ) Δ φ ] sin 2 ( 2 α ) ,

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