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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 11 — May. 26, 2008
  • pp: 7778–7788
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High speed interferometric ellipsometer

Chien-Chung Tsai, Hsiang-Chun Wei, Sheng-Lung Huang, Chu-En Lin, Chih-Jen Yu, and Chien Chou  »View Author Affiliations


Optics Express, Vol. 16, Issue 11, pp. 7778-7788 (2008)
http://dx.doi.org/10.1364/OE.16.007778


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Abstract

A novel high speed interferometric ellipsometer (HSIE) is proposed and demonstrated. It is based on a novel differential-phase decoder which is able to convert the phase modulation into amplitude modulation in a polarized heterodyne interferometer. Not only high detection sensitivity but also fast response ability on ellipsometric parameters (EP) measurements based on amplitude-sensitive method is constructed whereas different amplitudes with respect to P and S polarized heterodyne signals in this phase to amplitude modulation conversion is discussed. The ability of HSIE was verified by testing a quarter wave plate while a real time differential-phase detection of a liquid crystal device versus applied voltage by using HSIE was demonstrated too. These results confirm that HSIE is able to characterize the optical property of specimen in terms of EP at high speed and high detection sensitivity experimentally.

© 2008 Optical Society of America

1. Introduction

2. Methodology and experimental setup

IP(δωt)=AP12+AP22+2AP1AP2cos[δωt+δϕP],
(1)
IS(δωt)=AS12+AS22+2AS1AS2cos[δωt+δϕS],
(2)

where δϕP=ϕ P1-ϕ P2 and δϕS=ϕ S1-ϕ S2 are the phase retardations between P1 and P2 waves and between S1 and S2 waves respectively. The beat frequency of the heterodyne signal is δω=ω 1-ω 2 where ω 1 and ω 2 are the driving frequencies of acousto-optic modulators in the reference and the signal arms. If only AC terms are considered, Eqs. (1) and (2) become

IP(δωt)=2AP1AP2cos[δωt+δϕP],
(3)
IS(δωt)=2AS1AS2cos[δωt+δϕS],
(4)

where no specimen is inserted into the signal channel. IP(δωt) and IS(δωt) is under the conditions of Δ≡δϕS-δϕP≅0° and ψ≡tan-1(A S1 A S2/A P1 A P2)=45° because ϕ S1=ϕ P1, ϕ S2=ϕ P2, A S1=A P1, and A S2=A P2 are satisfied at the same time. In contrast, when the specimen is inserted into signal channel for testing, both ellipsometric parameters, the phase difference Δ and the arctangent of amplitude ratio ψ of P2 and S2 polarized waves, can be obtained by use of a lock-in amplifier (LIA) properly. However, a slow response of the measurement is resulted because the phase lock-in technique is involved. In order to avoid from using LIA and also able to enhance the time response of the EP measurements, we set α=δωt+δϕS+δϕP2 , β=δϕSδϕP2 , κp=2A p1 A p2, and κs=2A s1 A s2 in Eqs. (3) and (4), then

IP(δωt)κPcos[αβ],
(5)
IS(δωt)κScos[α+β],
(6)

whereas, A P1=A S1 and ϕ P1=ϕ S1 are satisfied in the reference channel by calibration. Let (κS-κP)cos β=cos γ and (κS+κP)sin β=sin γ, then the output signal IDiff(δωt), from DA in Fig. 1 can be expressed by

IDiff(δωt)=IS(δωt)IP(δωt)
=(κSκP)cosβcosα(κS+κP)sinβsinα
=cosγcosαsinγsinα=κS2+κP22κSκPcos(δϕSδϕP)cos(α+γ),
=κDiffcos(α+γ)
(7)

Δ=cos1[κP2+κS2κDiff22κPκS],
(8)
ψ=tan1(κSκP).
(9)

The EP of specimen are then able to be measured precisely in terms of κS, κP, and κDiff in real time.

Fig. 1. The optical setup: He-Ne: laser source, BS1, BS2: beam splitters, AOM1, AOM2: acousto-optic modulators, M1, M2: mirrors, Po1, Po2: polarizers, C: compensator, S: sample, PBS: polarization beam splitter, Dp, Ds: detectors, BPF: band-pass filter, DA: differential amplifier, FG: function generator, DAU: data acquisition unit, PC: personal computer

3. Experimental results

Fig. 2. Experimental results of (a) the amplitudes of P polarized, S polarized heterodyne signals, and their difference, (b) the amplitude ratio of P and S heterodyne signals, (c) the parameter ψ, (d) the absolute value of phase retardation versus rotation angle of 180° dynamic range, (e) the recovered phase retardation with DC phase bias, (f) the phase retardation of zero DC phase bias is consistent with the experimental curve measured by LIA (solid line).

Meanwhile, in order to check the ability of real time measurement of HSIE, a homogeneous parallel aligned liquid crystal device (PALCD) was tested. It can be measured by measuring time-dependent phase retardation of PALCD which is controlled by an external applied voltage dynamically. Figure 3(a) shows the structure of LC molecule which is equivalent to an index ellipsoid as shown in Fig. 3(b) [15

15. C. C. Tsai, H. C. Wei, C. H. Hsieh, L. P. Yu, C. R. Yu, H. S. Huang, and C. Chou, “Characterization of a nematic PALC at large oblique incidence angle,” Opt. Express 15, 10381–10389 (2007). [CrossRef] [PubMed]

]. The spatial orientation of LC molecule is further defined in Fig. 3(c) in which the incident light is along z-axis. The azimuth angle θ, is the angle between the projected direction of LC molecule onto x-y plane and x-axis. In the mean time, the pretilt angle θpre, is the angle between LC molecules and the parallel plane of glass substrate at off-state as shown in Fig. 3(c). When AC voltage is applied, the LC molecules between two Indium-Tin-Oxide (ITO) layers are symmetrically tilted as shown in Fig. 3(d). The relation of effectively tilted angle θet, can be described by ne(θet)=(∫d z=0 ne(θ(z))dz)/d where θ is function of z and d is the LC cell gap [6

6. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, New York, 1999), Chap. 5.

].

Fig. 3. The molecular alignment of homogeneous LC at (a) parallel-aligned condition at offstate with initial orientation θpre, (b) the effective index ellipsoid model of homogeneous LC, (c) the side view of PALCD in order to find the relation between PALCD and the incident laser beam, (d) symmetrically tilted condition of molecules to the central surface (z=d/2) between two ITO films with applied voltage at on-state where the effective tilt angle is θet.

In order to ensure that the mean direction of off-state homogeneous LC molecules are in y-z plane under the condition of zero voltage applied, PALCD is rotated until the phase retardation is maximized by using LIA in this experiment. And then, the AC voltage is applied by increments until 12 Volts. However, the change of phase retardation as shown in Fig. 4 was out of the dynamic range (0°≤Δ≤180°) measured by LIA. Thus, a suitably adjustment on the azimuth angle θ (see Fig. 3(c)) is adjusted until Δ of PALCD is within 0°≤Δ≤180°. Figure 4 shows the response of phase retardation versus applied AC voltage. There is a plateau happened at small applied voltage and then drop rapidly until a smooth response is reached. From the measurement, we can clearly see the non-uniform response of LC molecules to the applied voltage which is similar to the transmission response of TN-LCD [6

6. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, New York, 1999), Chap. 5.

]. This also can be applied to in-plane-switching (IPS) mode on PALCD [6

6. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, New York, 1999), Chap. 5.

].

Fig. 4. Experimental results of phase retardation versus AC square-wave voltage (1 KHz) on homogeneous PALCD by slowly increasing the voltage from 0 V to 12 V.

The time response of Δ of PALCD versus applied square wave voltage at 1 KHz is shown in Fig. 5. It is obviously seen that the rise time and decay time are different and can be measured very precisely and dynamically. In this experiment, τrise=186 ms and τdecay=213 ms were measured. This result proves that the ability of HSIE on high speed Δ measurement is applicable. Experimentally, the frequency response of Δ is limited by the frequency response of DVM in DAU shown in Fig. 1. The sampling rate was 500 (point/sec) in this experiment. Besides, the repeatability on phase retardation measurement at high speed was tested too.

Fig. 5. The rise time τrise and decay time τdecay of phase retardation of homogeneous PALC under the condition of applying 10 VAC square-wave voltage (1 KHz). In this experiment, τrise=186 ms and τdecay=213 ms were measured.

The excellent consistence is shown in Fig. 6. This is because of a common-path configuration of HSIE which immunes the environmental disturbance efficiently. In addition, the novel balanced detector [11

11. C. Chou, C. W. Lyu, and L. C. Peng, “Polarized differential phase laser scanning microscope,” Appl. Opt. 40, 95–99 (2001). [CrossRef]

] is setup that the excess noise of the laser intensity fluctuation is reduced as well and the shot-noise-limited detection is applicable theoretically [12

12. C. Chou, H. K. Teng, C. C. Tsai, and L. P. Yu, “Balanced detector interferometric ellipsometer,” J. Opt. Soc. Am. A 23, 2871–2879 (2006). [CrossRef]

, 13

13. Y. Q. Li, D. Guzun, and M. Xiao, “Sub-shot-noise-limited optical heterodyne detection using an amplitude-squeezed local oscillator,” Phys. Rev. Lett 82, 5225–5228 (1999). [CrossRef]

]. During the measurement, the phase stability of ±0.01° was achieved in this experiment.

Fig. 6. The repeatable phase retardation measurements of test1 and test2 at different time.

4. Error analysis

In order to evaluate the accuracy of this experimental system, the numerical error analyses of the phase retardation Δ and the parameter ψ corresponding to Eqs. (8) and (9) are derived in Eqs. (10) and (11), respectively.

δΔ=[(ΔkP)2(δkP)2+(ΔkS)2(δkS)2+(ΔkDiff)2(δkDiff)2]12
=[(κSκPcosΔ)2(δκPκP)2+(κPκScosΔ)2(δκSκS)2+2(κSκP+κPκScosΔ)2(δκDiffκDiff)22(1cos2Δ)]12.
=[(σcosΔ)2(δκPκP)2+(1σcosΔ)2(δκSκS)2+2(σ+1σcosΔ)2(δκDiffκDiff)22(1cos2Δ)]12
(10)
δψ=[(ψkP)2(δkP)2+(ψkS)2(δkS)2]12
=[(1kPkS+kSkP)2((δkPkP)2+(δkSkS)2)]12
=[(11tan1ψ+tan1ψ)2((δkPkP)2+(δkSkS)2)]12.
(11)

In Eq. (10), σ=(κS/κP) is defined and ((δκP/κP), (δκS/κS), (δκDiff/κDiff))=(0.01, 0.01,0.0025) is assumed. The (δκP/κP), (δκS/κS), and (δκDiff/κDiff are defined as SNR-1 of κP, κS, and κDiff respectively which are able to be calculated from Fig. 2(a). Then the phase error δΔ can be simulated near 0° or 180° for the case of cosΔ~±1.0 as shown in Figs. 7(a) and 7(b). In Figs. 7(a) and 7(b), δΔ becomes large when Δ is close to 0° or 180° where σ=1, σ=0.5, σ=0.1 and σ=0.01 are considered. In the experiment of testing QWP in Fig. 2, the error on Δ (~90°) measurement about 0.02° is seen clearly in Fig. 7(c) in which σ ~0.5 is assumed from Fig. 2(a). Similarly, to calculate δψ from Eq. (11), when the SNR-1 is also set at (δκP/κP)≅(δκS/κS)=0.01, then δψ is calculated and shown in Fig. 7(d) in which δψ is symmetric to ψ=45° in the full range of 0°≤ψ≤90°.

Fig. 7. The error of Δ under the conditions of (δκP/κP)=0.01, (δκS/κS)=0.01, and (δκDiff/κDiff)=0.0025 when (a) the Δ is from 0° to 2°, (b) the Δ is from 178° to 180°, (c) the Δ is from 45° to 135°. The error of ψ from 0° to 90° is shown in (d).

5. Conclusions and discussion

Acknowledgment

This research was supported partially by National Science Council of Taiwan through the project NSC-96-2221-E-010-002-MY2.

+C. C. Tsai and H. C. Wei are now with the Graduate Institute of Photonics and Optoelectronics, National Taiwan University, Taipei, Taiwan 106.

References and links

1.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

2.

C. H. Lin, C. Chou, and K. S. Chang, “Real time interferometric ellipsometry with optical heterodyne and phase lock-in techniques,” Appl. Opt. 29, 5159–5162 (1990). [CrossRef] [PubMed]

3.

R. E. Ziemer and W. H. Tranter, Principles of Communications (Wiley, New York, 1995), Chap. 6.

4.

Y. Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985). [CrossRef] [PubMed]

5.

M. Sato, K. Seino, K. Onodera, and N. Tanno, “Phase-drift suppression using harmonics in heterodyne detection and its application to optical coherence tomography,” Opt. Commun. 184, 95–104 (2000). [CrossRef]

6.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, New York, 1999), Chap. 5.

7.

C. Yang, A. Wax, R. R. Dasari, and M. S. Feld , “2π ambiguity-free optical distance measurement with subnanometer precision with a novel phase-crossing low-coherence interferometer,” Opt. Lett. 27, 77–79 (2002). [CrossRef]

8.

T. E. Jenkins, “Multiple-angle-of-incidence ellipsometry,” J. Phys. D. Appl. Phys. 32, R45–R56 (1999). [CrossRef]

9.

K. Riedling, Ellipsometry for Industrial Applications (Springer-Verlag, New York, 1988). [CrossRef]

10.

L. R. Watkins and M. D. Hoogerland, “Interferometric ellipsometer with wavelength-modulated laser diode source,” Appl. Opt. 43, 4362–4366 (2004). [CrossRef] [PubMed]

11.

C. Chou, C. W. Lyu, and L. C. Peng, “Polarized differential phase laser scanning microscope,” Appl. Opt. 40, 95–99 (2001). [CrossRef]

12.

C. Chou, H. K. Teng, C. C. Tsai, and L. P. Yu, “Balanced detector interferometric ellipsometer,” J. Opt. Soc. Am. A 23, 2871–2879 (2006). [CrossRef]

13.

Y. Q. Li, D. Guzun, and M. Xiao, “Sub-shot-noise-limited optical heterodyne detection using an amplitude-squeezed local oscillator,” Phys. Rev. Lett 82, 5225–5228 (1999). [CrossRef]

14.

C. Chou, Y. C. Huang, and M. Chang, “Effect of elliptical birefringence on the measurement of the phase retardation of a quartz wave plate by an optical heterodyne polarimeter,” J. Opt. Soc. Am. A 14, 1367–1372 (1997). [CrossRef]

15.

C. C. Tsai, H. C. Wei, C. H. Hsieh, L. P. Yu, C. R. Yu, H. S. Huang, and C. Chou, “Characterization of a nematic PALC at large oblique incidence angle,” Opt. Express 15, 10381–10389 (2007). [CrossRef] [PubMed]

16.

L. C. Peng, C. Chou, C. W. Lyu, and J. C. Hsieh, “Zeeman laser-scanning confocal microscopy in turbid media,” Opt. Lett. 26, 349–351 (2001). [CrossRef]

OCIS Codes
(040.2840) Detectors : Heterodyne
(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.5050) Instrumentation, measurement, and metrology : Phase measurement

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: April 3, 2008
Revised Manuscript: May 9, 2008
Manuscript Accepted: May 9, 2008
Published: May 14, 2008

Citation
Chien-Chung Tsai, Hsiang-Chun Wei, Sheng-Lung Huang, Chu-En Lin, Chih-Jen Yu, and Chien Chou, "High speed interferometric ellipsometer," Opt. Express 16, 7778-7788 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-11-7778


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References

  1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  2. C. H. Lin, C. Chou, and K. S. Chang, "Real time interferometric ellipsometry with optical heterodyne and phase lock-in techniques," Appl. Opt. 29, 5159-5162 (1990). [CrossRef] [PubMed]
  3. [1] . R. E. Ziemer and W. H. Tranter, Principles of Communications (Wiley, New York, 1995), Chap. 6.
  4. Y. Y. Cheng and J. C. Wyant, "Phase shifter calibration in phase-shifting interferometry," Appl. Opt. 24, 3049-3052 (1985). [CrossRef] [PubMed]
  5. M. Sato, K. Seino, K. Onodera and N. Tanno, "Phase-drift suppression using harmonics in heterodyne detection and its application to optical coherence tomography," Opt. Commun.  184, 95-104 (2000). [CrossRef]
  6. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, New York, 1999), Chap. 5.
  7. C. Yang, A. Wax, R. R. Dasari, and M. S. Feld, "2�? ambiguity-free optical distance measurement with subnanometer precision with a novel phase-crossing low-coherence interferometer," Opt. Lett. 27, 77-79 (2002). [CrossRef]
  8. T. E. Jenkins, "Multiple-angle-of-incidence ellipsometry," J. Phys. D. Appl. Phys. 32, R45-R56 (1999). [CrossRef]
  9. K. Riedling, Ellipsometry for Industrial Applications (Springer-Verlag, New York, 1988). [CrossRef]
  10. L. R. Watkins and M. D. Hoogerland, "Interferometric ellipsometer with wavelength-modulated laser diode source," Appl. Opt. 43, 4362-4366 (2004). [CrossRef] [PubMed]
  11. C. Chou, C. W. Lyu, and L. C. Peng, "Polarized differential phase laser scanning microscope," Appl. Opt. 40,95-99 (2001). [CrossRef]
  12. C. Chou, H. K. Teng, C. C. Tsai, and L. P. Yu, "Balanced detector interferometric ellipsometer," J. Opt. Soc. Am. A 23, 2871-2879 (2006). [CrossRef]
  13. Y. Q. Li, D. Guzun, and M. Xiao, "Sub-shot-noise-limited optical heterodyne detection using an amplitude-squeezed local oscillator," Phys. Rev. Lett 82, 5225-5228 (1999). [CrossRef]
  14. C. Chou, Y. C. Huang, and M. Chang, "Effect of elliptical birefringence on the measurement of the phase retardation of a quartz wave plate by an optical heterodyne polarimeter," J. Opt. Soc. Am. A 14, 1367-1372 (1997). [CrossRef]
  15. C. C. Tsai, H. C. Wei, C. H. Hsieh, L. P. Yu, C. R. Yu, H. S. Huang, and C. Chou, "Characterization of a nematic PALC at large oblique incidence angle," Opt. Express 15, 10381-10389 (2007). [CrossRef] [PubMed]
  16. L. C. Peng, C. Chou, C. W. Lyu, and J. C. Hsieh, "Zeeman laser-scanning confocal microscopy in turbid media," Opt. Lett. 26, 349-351 (2001). [CrossRef]

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