Phase change measurement of birefringent optical devices with white light interferometry Khos-Ochir Tsogvoo,1 Purevdorj Munkhbaatar,2 Byung Kwan Yang,3 Jin Seung Kim,1,2,4 and Kim Myung-Whun1,2,4,* 1

Department of Nanoscience and Technology, Chonbuk National University, Jeonju 561-756, South Korea 2

Department of Physics, Chonbuk National University, Jeonju 561-756, South Korea 3

Jiny Photonics Co. Ltd., Wan-ju 565-854, South Korea

4

Institute of Photonics and Information Technology, Chonbuk National University, Jeonju 561-756, South Korea *Corresponding author: [email protected]

Received 2 October 2013; revised 26 November 2013; accepted 27 November 2013; posted 3 December 2013 (Doc. ID 198718); published 24 December 2013

This paper describes a method to determine the phase retardation of birefringent optical components by combining spectral interferometry and the Fourier transform method. The retardation of each orthogonal polarization component was resolved by using two rotatable linear polarizers in the interferometer. The phase retardation measured by using suggested method was compared to that measured using the conventional polarimetric method. The results of independent methods were well matched, which confirms the validity of the proposed method. © 2013 Optical Society of America OCIS codes: (120.3180) Interferometry; (230.6120) Spatial light modulators; (120.5410) Polarimetry; (160.3710) Liquid crystals. http://dx.doi.org/10.1364/AO.53.000141

1. Introduction

The phase of a light beam changes as the beam passes through optical devices. The wavelength dependence of the phase change is crucial information for the precise control and measurement of the broadband light, in particular of ultrashort laser pulses [1,2]. Group-delay dispersion of optical devices is a critical parameter for the generation, measurement, and control of ultrashort laser pulses. White-light interferometry has been widely utilized as an effective method to measure the dispersion of optical devices such as lenses, dielectric mirrors, and prism pairs [3–8]. In some experiments, the light is passed through birefringent optical devices such as waveplates for 1559-128X/14/010141-06$15.00/0 © 2014 Optical Society of America

manipulation of the polarization state of the light beam. Conventional white-light interferometric methods cannot afford complete information about the dispersion of the birefringent optical devices. The index of refraction in such devices depends not only on the wavelength but also on the polarization state of the light beam. As a result, the phase difference between the two orthogonal polarization components is accumulated in the beam path of an interferometer, and the difference makes the interferometric analysis of some birefringent optical devices quite complicated [9]. Sophisticated fitting methods or time-of-flight methods utilizing using ultrashort laser pulse can be employed to overcome the problem [10–12]. However, it would be a considerable advantage if the phase retardation and the high order phase change of the birefringent materials could be measured directly in a single, and relatively simple interferometric setup with a conventional 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS

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white light source such as a tungsten halogen lamp (HL) and without complicated fitting processes. In this report, a polarimetric-method-combined white light interferometric method was suggested to measure the phase change of wavelength components in birefringent optical devices. We demonstrated the method’s accuracy by measuring the high order phase change and the phase retardation of an achromatic quarter waveplate (QWP) and by comparing the experimental result with the theoretical result. The method also was applied to analyze the phase change of white light beam due to a twisted nematic (TN) liquid crystal spatial light modulator (SLM). 2. Experiment

Two setups were used: one for polarimetry and the other one for interferometry. Figure 1(a) shows the polarimetric setup. The light source was a white light tungsten HL. The transmitted light through the optical fiber cable (C) was collimated by a lense (L). The light was sent to the linear polarizer (P). The polarized light was transmitted through a rotating QWP and analyzed by using a second linear polarizer [analyzer (A)]. The transmitted light was measured using a spectrometer (SM). Figure 1(b) shows the interferometric setup. The light source was the same as the previous setup. The polarized light was divided into two beams (one for reference and the other for signal) by using a beam splitter (BS1 ). The reference beam was reflected from the movable mirrors (MMs) and was

Fig. 2. Spectral interferogram of a BK7 glass plate (thin solid line) showed sharp interference. The spectral intensity of reference beam (thick solid line) and signal beam (dashed line) are presented.

transmitted through the second beam splitter (BS2 ). The signal beam was reflected from the fixed mirrors (M1 and M2 ) and transmitted through the sample (S). The reference beam and the signal beam were combined into a single beam using the second beam splitter. The combined beam was reflected from a fixed mirror (M3 ) to be directed to the second linear polarizer (analyzer). Finally, the spectrometer measured the combined beam to produce a spectral interferogram. A typical spectral interferogram is shown in Fig. 2. A BK7 glass plate, an achromatic QWP, and an SLM were used as samples. 3. Analysis Methods

A light beam contains amplitude and phase information. The amplitude can be easily determined from irradiance measurement. However, obtaining the phase information requires some sophisticated techniques. When a light beam passes a transparent material, its phase change relative to a beam passing the air is Δφω 

ω · d · nω − 1 : c

(1)

Here, ω is the angular frequency of light, c is the velocity of light in vacuum, d and nω are the thickness and refractive index of the material, respectively. Δφω can be approximated by Taylor expansion with respect to the central frequency ω0 as follows: 1 Δφω  Δφ0  Δφ1 ω − ω0   Δφ2 ω − ω0 2  … 2 (2)

Fig. 1. (a) Experimental setup for polarimetry. (b) Experimental setup for spectral interferometry: HL, halogen lamp; C, multimode optical fiber cable; L, collimating lense; P, polarizer; QWP, quarter waveplate; A, analyzer; SM, spectrometer; BS, beam splitters; S, sample; M, mirror; MM, movable mirror. 142

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Here, Δφ0 is a constant; Δφ1 is the linear term determining the phase delay (time delay); Δφ2 is the second-order derivative representing the quadratic variation of the phase and thus affects the phase delay dispersion. Δφ2 and higher-order terms are important for broadband light. To obtain Δφω, the phase of the light beam transmitted through a sample should be estimated from

the interferogram as illustrated in Fig. 1(b). A Fourier transformation spectral interferometry (FTSI) was used to extract Δφω from the interferogram. Basic principles of the method can be briefly reviewed as follows [13]. The spectrum is obtained in the wavelength domain in most experiments. However, it is convenient to handle the spectrum in the frequency domain for the mathematical transformation. The electric field of a light beam can be described in the frequency domain by using the following expression: Eω 

p Sω · exp−iφω:

(3)

Here, Sω is the spectral irradiance and φω is the spectral phase. As shown in the setup of Fig. 1(b), the reference beam precedes the signal beam by the time delay caused by the path difference. The interferogram made by the reference beam and the signal beam were measured by the spectrometer. The spectral irradiances of the reference and signal beams were independently measured by using the shutters on the two arms. The interference term, Sω  Ref2f ω expiωτg;

4. Results and Discussion

The proposed method was checked by measuring the phase change due to a BK7 glass plate. The retardation effect for BK7 is negligible and only the high order phase change is relevant. The second- and third-order coefficients of Eq. (2) were found by fitting the polynomial function with the experimental data. The second-order coefficient was 113.50 fs2 ∕rad and the third-order was 26.95 fs3 ∕rad2. Theoretical coefficients were estimated by Sellmeier function based on the parameters provided by the glass manufacturer. The theoretical second- and third-order coefficients were 111.78 fs2 ∕rad and 26.66 fs3 ∕rad2 , respectively, consistent with the experimental results. The higher-order phase change per unit thickness was calculated and was shown in Fig. 3(a). The solid circles represented the experimental data and the solid line illustrated the theoretical fitting function. The higher-order phase changes of an achromatic QWP and an SLM were experimentally determined as shown in Fig. 3(a). These samples were birefringent. In birefringent materials, the phase delays for different polarization components result in the travel distance difference. The phase difference due to the travel

(4)

was extracted by subtracting the two spectral irradiances from the interferogram. Here, f ω  ER ωES ω. ER ω and ES ω are the electric fields of the reference beam and signal beam, respectively. By inverse Fourier transforming the experimental spectrum was obtained: F −1 Sω  f t − τ  f −t − τ:

(5)

Here f t − τ and f −t − τ are the time reserved form of each other and thus they have the same properties. Only f t − τ was singled out by multiplying a Heaviside function Θt to Eq. (5). Finally, the relative phase of the signal beam with respect to the reference beam can be obtained by the inverse Fourier transform and by taking the argument of the complex electric field: Δϕω  ArgEω:

(6)

The phase change Δφ; can be found from the difference of two relative phases obtained with and without the sample (i.e., Δφ  Δϕsample − Δϕempty ). Sample and other dispersive optical devices in the setup may cause the unbalance of the optical paths in the two interferometer arms for all wavelengths. As a result, the white-light correlogram can be distorted and the phase and intensity of the spectral interferogram also can be modified [14,15]. To minimize the errors, the moving mirror position should be carefully checked and changed after the insertion of the sample or after the realignment of the optical elements.

Fig. 3. (a) Solid circles are the interferometrically measured higher-order phase change due to a BK7 glass plate. The overlaid solid line is fitted by using Eq. (2). The solid squares and solid triangles are the higher-order phase change due to an achromatic QWP and TN-SLM, respectively. The solid line is the fitting function. (b) The difference between higher-order phase changes of two orthogonal polarizations for the QWP (solid circles) and the polynomial fitting curve (dashed line). The same data for a TN SLM (open circles) and the polynomial fitting curve (dotted–dashed line). The polarization directions were 45° and −45° against the horizontal direction (see text). 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS

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distance difference was called the phase retardation in this study. The polarization dependence of the phase change was checked for two orthogonal polarizations. For BK7 glass, the difference between the phase changes for two orthogonal polarizations was approximately 0.01 rad∕mm and, as expected, there was no frequency dependence. Therefore, 0.01 rad∕mm can be considered the measurement error. Figure 3(b) shows the difference in higher-order phase change Δφ between the fast axis (Δφfast ) and the slow axis (Δφslow ), i.e., Δφfast − Δφslow due to an achromatic QWP (solid circles) and an SLM (open circles). The result showed that the polarization dependence of the higher-order phase change was larger (∼0.1 rad∕mm) than the experimental error (0.01 rad∕mm) but smaller than the wavelength variation of the high order phase change (>1 rad∕mm) shown in Fig. 3(a). The retardation effect on Δφ of the achromatic QWP was estimated using an interferometric method. First, the fast axis of the QWP was aligned to one of the two orthogonal polarization directions (i.e., horizontal and vertical polarization). Here, the horizontal plane was defined as the interferometer arm plane. Then Δφ for the two orthogonal polarization directions was obtained using the interferometric method (Δφh and Δφv ). The phase retardation of the QWP was measured by subtracting Δφh of the horizontal polarization from Δφv of the vertical polarization. To obtain the horizontal polarization, linear polarizers (P and A) were put in front of the first beam splitter (BS1 ) and in front of the spectrometer (S) as shown in Fig. 1(b). The retardation of the QWP is shown in Fig. 4 (dashed line). The difference in phase change between the fast and slow axes was a quarter wave (0.25 wave) consistent to the expectation.

The rotating sample method is accurate as demonstrated; however, the method is not appropriate for some samples that cannot be rotated. A more advanced method was developed to overcome this difficulty. The signal beam was elliptically polarized when it passed through a sample. The electric field E of any elliptically polarized light can be decomposed into two orthogonal components (e.g., Ex and Ey ). The signal beam electric field can be written by Jones vector as follows: E⃗ sig  e−iφx



Ex

Ey eiφx −φy 

 :

(7)

The reference beam was linearly polarized in the horizontal plane. The horizontal plane direction was defined as 0°. For the two orthogonal directions of analyzer, 45° and −45° were chosen to minimize the asymmetry of the spectrometer response. The reference beam electric field also can be decomposed into two orthogonal components as follows:   1 E⃗ ref  EL e−iφL : 1

(8)

The transmitted total electric field through the analyzer can be determined by Jones calculation: E⃗  J A E⃗ sig  E⃗ ref    cos2 θ − 45° sinθ − 45°cosθ − 45°  sinθ − 45°cosθ − 45° sin2 θ − 45°   Ex e−iφx  EL e−iφL · : (9) Ey e−iφy  EL e−iφL Here J A is Jones matrix of the analyzer and θ is the analyzer angle with respect to the horizontal direction. When the analyzer is aligned along x axis (θ  45°) the measured spectral intensity can be estimated as: p 2 2 Stot x  jEx j  jEL j  2 jEx jjEL j cosφx − φL ; (10) and φx − φL  can be determined from the result by employing the FTSI method. Similarly, when analyzer angle was aligned along y axis (θ  −45°) the measured spectral intensity can be estimated as: q 2 2 Stot y  jEy j  jEL j  2 jEy jjEL j cosφy − φL ; (11)

Fig. 4. Phase retardation due to an achromatic QWP interferometrically measured while rotating the sample (dashed line) and rotating the analyzer (solid line). While rotating the analyzer, the transmission axis of the polarizer (P) was fixed at 0° and the fast axis of the QWP at 45° against the horizontal direction. The analyzer was aligned to 45° and −45° against the horizontal direction. The polarimetric data (dotted–dashed line) confirms the validity of the interferometric methods. 144

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and φy − φL  also can be determined using the same method. The phase difference between two orthogonal components (φx − φy ) can be obtained by subtracting those two values. Here θ  45° for x axis and θ  −45° for y axis. The phase difference (φ45° λ − φ−45° λ) obtained at two analyzer angles was presented in Fig. 4 (thick solid line).

The validity of the proposed method was confirmed by comparing the result with those independently measured using a polarimetric method. The polarimeter was shown in Fig. 1(a). Here, the optical system consisted of a light source, a linear polarizer, the achromatic QWP, and an analyzer. When the polarizer and analyzer were fixed, their transmission axes were aligned to the horizontal direction 0°. The achromatic QWP was rotated and its fast axis was measured at the angle θ with respect to the horizontal direction 0°. To determine the transmittance of the optical system, Jones matrix method was used [16–18]:

Eout  J A · J 4λ · J P · Ein    1 0 cos2 θ  eiΔϕx −Δϕy  sin2 θ iΔϕx  · ·e 0 0 1 − eiΔϕx −Δϕy   · cos θ · sin θ

Here J λ∕4 and J P are the Jones matrix for the achromatic QWP, and the polarizer. Ein and Eout are the electric fields of the input and the output beams. Δϕx and Δϕy are the phase changes of the horizontally and vertically polarized light beams, respectively. Δϕ Δϕx − Δϕy  is the phase retardation of the achromatic QWP. The transmittance of the optical system depends on the rotation angle θ of the achromatic QWP and wavelength λ [18]: jEout j2 jEin j2 1  f3  cos 4θ  1 − cos 4θ cosΔϕλg: 4 (13)

Tθ; λ 

The intensity of the transmitted beam was measured and recorded while rotating the achromatic QWP from 0° to 180° at 5° increments. The measured intensity was normalized and then fitted using Eq. (8) as shown in Fig. 4 (dotted–dashed line). The polarimetric data were matched quite well to the interferometric data. Therefore, the interferometric method can be considered consistent with the polarimetric method. The rotating analyzer interferometric method was used to measure the phase retardation of a TN-SLM. The phase retardation of a TN-SLM can be controlled by the external voltage applied to the device. The degree of phase retardation of the liquid crystal molecules can be scaled by the gray levels from 0 to 255. In our TN-SLM, the 0 level corresponded to the fully twisted state of liquid crystal molecules and the 255 level corresponded to the parallelly

aligned state. The measured phase retardation [φ45° λ − φ−45° λ] was shown in Fig. 5 for some gray levels. At around 800 nm, the phase retardation of the TN-SLM could be adjusted from 0.3 wave to 2.7 waves. The residual birefringence of the beam splitters affected the accuracy of the rotating analyzer interferometric method. This effect was investigated for three types of beam splitters: a pellicle, a plate, and a polarization-insensitive cube beam splitter. Figure 6 shows the phase retardation of the achromatic QWP. A pellicle beamsplitter showed difference of approximately 0.02 wave between the data obtained

1 − eiΔϕx −Δϕy   · cos θ · sin θ sin2 θ  eiΔϕx −Δϕy  cos2 θ

  1 · 0

0 0

 · Ein :

(12)

with the two methods: interferometric and polarimetric. This not-so-small discrepancy was probably due to the multiple reflections due to the multilayer structure of the pellicle beam splitter. The plate beam splitter showed relatively large discrepancy of approximately 0.05 wave. This was probably due to the difference in transmittance coefficient between pand s-polarization components. The result obtained by a polarization-insensitive beam splitter cube was fairly consistent with the polarimetric data. The wavy structure was the artifact produced in the Fourier transformation. If the polarization sensitivity of beamsplitters can be reduced further, the accuracy of the proposed method could be improved.

Fig. 5. Phase retardation due to SLM for numerous gray levels measured by using the interferometric method when the analyzer was aligned to 45° and −45° against the horizontal direction. The inset shows the gray level dependence of the phase retardation at 800 nm marked by the vertical dotted line. 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS

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Fig. 6. Phase retardation of an achromatic QWP interferometrically measured with various beam splitters while rotating the analyzer. The dotted–dashed line is polarimetric data for comparison. The solid circles, the solid squares, and the solid triangles represent the data measured with the pellicle, the cube, and the plate beam splitter, respectively.

5. Summary

We developed a method to determine the phase change of a broadband light by using rotating analyzer Fourier transform spectral interferometry. The validity of obtaining higher-order phase change was confirmed by comparing experimental data with theoretical estimation for a BK7 glass plate. The method was also applied to the measurements of phase change of light beams passing through a QWP and a TN SLM. The good agreement with the polarimetric method in retardation measurements confirmed the validity of the proposed method. This research was supported by the National Research Foundation of Korea (Grants Nos. 2012R1A1A4A01010025 and 220-2011-1-C00016). References 1. P. F. Curley, Ch. Spielmann, T. Brabec, F. Krausz, E. Wintner, and A. J. Schmidt, “Operation of a femtosecond Ti:sapphire solitary laser in the vicinity of zero group-delay dispersion,” Opt. Lett. 18, 54–56 (1993). 2. V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference 4: characterization and compensation of the spectral phase of ultrashort laser pulses,” Opt. Lett. 29, 775–777 (2004).

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3. V. Nirmal Kumar and D. Narayana Rao, “Using interference in the frequency domain for precise determination of thickness and refractive indices of normal dispersive materials,” J. Opt. Soc. Am. B 12, 1559–1563 (1995). 4. T. Imran, K.-H. Hong, T. J. Yu, and C. H. Nam, “Measurement of the group-delay dispersion of femtosecond optics using white-light interferometry,” Rev. Sci. Instrum. 75, 2266– 2270 (2004). 5. D. X. Hammer, A. J. Welch, G. D. Noojin, R. J. Thomas, D. J. Stolarski, and B. A. Rockwell, “Spectrally resolved white-light interferometry for measurement of ocular dispersion,” J. Opt. Soc. Am. A 16, 2092–2102 (1999). 6. S. Diddams and J.-C. Diels, “Dispersion measurements with white-light interferometry,” J. Opt. Soc. Am. B 13, 1120–1129 (1996). 7. K. Naganuma, K. Mogi, and H. Yamada, “Group-delay measurement using the Fourier transform of an interferometric cross correlation generated by white light,” Opt. Lett. 15, 393–395 (1990). 8. P. Zhang, Y. Tan, W. Liu, and W. Chen, “Methods for optical phase retardation measurement: a review,” Sci. China Tech. Sci. 56, 1155–1163 (2013). 9. J. M. DeFreitas and M. A. Player, “Polarization effects in heterodyne interferometry,” J. Mod. Opt. 42, 1875–1899 (1995). 10. H. Delbarre, C. Przygodzki, M. Tassou, and D. Boucher, “High-precision index measurement in anisotropic crystals using white-light spectral interferometry,” Appl. Phys. B 70, 45–51 (2000). 11. H. Delbarre, C. Przygodzki, and D. Boucher, “Determination of dispersion characteristics in anisotropic materials using ultrashort pulses,” Appl. Phys. B 66, 169–173 (1998). 12. H. Delbarre, C. Przygodzki, and D. Boucher, “A temporal method for dispersion measurements with ultrashort pulses in the visible and near-infrared region,” Int. J. Infrared Milli. 19, 441 (1998). 13. L. Lepetit, G. Chériaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12, 2467–2474 (1995). 14. P. Pavlíček and J. Soubusta, “Measurement of the influence of dispersion on white-light interferometry,” Appl. Opt. 43, 766–770 (2004). 15. P. Pavlíček and O. Hybl, “White-light interferometry on rough surfaces measurement uncertainty caused by surface roughness,” Appl. Opt. 47, 2941–2949 (2008). 16. K. Ohkubo and J. Ohtsubo, “Evaluation of LCTV as a spatial light modulator,” Opt. Commun. 102, 116–124 (1993). 17. K. Lu and B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical phase modulator,” Opt. Eng. 29, 240–246 (1990). 18. T. Khos-Ochir, P. Munkhbaatar, B. K. Yang, H. W. Kim, J. S. Kim, and K. Myung-Whun, “Polarimetric measurement of Jones matrix of a twisted nematic liquid crystal spatial light modulator,” J. Opt. Soc. Korea 16, 443–448 (2012).

Phase change measurement of birefringent optical devices with white light interferometry.

This paper describes a method to determine the phase retardation of birefringent optical components by combining spectral interferometry and the Fouri...
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