Instrumentation, Techniques, and Measurement

Alternative phase-shifting technique for measuring full-field refractive index

[+] Author Affiliations
Kun-Huang Chen

Feng Chia University, Department of Electrical Engineering, 100 Wenhwa Road, Seatwen, Taichung 40724, Taiwan

Jing-Heng Chen

Feng Chia University, Department of Photonics, 100 Wenhwa Road, Seatwen, Taichung 40724, Taiwan

Jiun-You Lin

National Changua University of Education, Department of Mechatronics Engineering, No. 2, Shi-Da Road, Changhua City 50074, Taiwan

Yen-Chang Chu

Feng Chia University, PhD Program of Electrical and Communications Engineering, 100 Wenhwa Road, Seatwen, Taichung 40724, Taiwan

Opt. Eng. 54(9), 094101 (Sep 10, 2015). doi:10.1117/1.OE.54.9.094101
History: Received May 5, 2015; Accepted August 3, 2015
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Abstract.  This study proposes an alternative and simple method for measuring full-field refractive index. This method is based on the phase-shifting technique with a modulated electro-optical (EO) modulator and the phenomenon of total internal reflection. To this purpose, a linear polarized light is expanded and incident on the interface between the prism and the tested specimen, and the reflected light passes through an analyzer for interference. The phase difference between the s- and p-polarized light is sensitive to the refractive index of the tested specimen when the total internal reflection appears on this interface. Based on this effect, the resulting phase differences make it possible to analyze the refractive index of the tested specimen through a phase-shifting technique with a modulated EO modulator. The feasibility of this method was verified by experiment, and the measurement resolution can reach a value of refractive index unit of at least 3.552×104. This method has advantages of simple installation, ease of operation, and fast measurement.

Figures in this Article

The measurement of full-field refractive index exhibits its significance and has many important applications in many specific fields, including the inspection of optical material and thin-film characteristic as well as research in biology, biochemistry, and medicine. Currently, some methods have been proposed to measure full-field refractive index, such as the interferometric methods,16 digital holographic microscopy,7 and ellipsometric methods.8 In the interferometric methods, the phase difference is relative to the refractive index. To obtain the phase difference distribution to measure full-field refractive index, the phase-shifting technique is an important and effective method that features simple structures, easy operation, and rapid measurement. Generally, the phase-shifting methods introduce successive phase steps into interferometric signals with various phase-shifting strategies, such as piezoelectric transducer,1,2 holographic grating,3 and tunable wavelength.4 Aforementioned methods could suffer from mechanical perturbations, and their costs are high. In addition, to reveal the phase distribution, techniques of three-step and four-step phase-shifting methods are commonly applied. However, the introduced values of phase-shifting must be specified. Therefore, this study proposes an alternative and simple method based on the Carré phase-shifting technique with modulated electro-optical (EO) modulator for measuring full-field refractive index. A linear polarized light is expanded and incident on the interface between the prism and the tested specimen. When the total internal reflection appears on this interface, a significant phase difference between the s- and p-polarized light is introduced. This phase difference is relative to the refractive index of the tested specimen, and can be precisely measured with phase-shifting technique with modulated EO modulator. Accordingly, the full-field refractive index of the tested specimen can be estimated. To prove the feasibility of the proposed method, the tested specimen combining with pure water and castor oil is measured. The experimental data correspond well with the theoretical values. Due to the introduction of EO modulation and Carré phase-shifting techniques, the proposed measurement method can avoid mechanical perturbation and has merits of simple installation, ease of operation, and fast measurement.

Figure 1 shows the optical configuration of the proposed method. For convenience, the z-axis is set in the direction of light propagation, and the x-axis is set perpendicular to the plane of the paper. Linear polarized laser light with the polarization state at 45 deg with respect to the x-axis passes through an EO modulator with the optic axis at 0 deg with respect to the x-axis. Using a DC power supply (DC) and a linear-voltage amplifier (LVA) to drive the EO modulator, the Jones vector of the electric field of the resulted laser light Ein can be written as Display Formula

Ein=12(eiΓ2eiΓ2)eiω0t,(1)
where ω0 is the light frequency, and Γ denotes the phase retardation between the s- and p-polarizations, which can be written as9Display Formula
Γ=πVzVπ,(2)
where Vπ is the half-wave voltage of the EO modulator, and Vz is an external voltage applied to the EO modulator. Then, the light beam is expanded and collimated by a spatial filter to form a plane wave, and incident on the undersurface of a prism at an angle θ. The refractive indices of the prism and the tested specimen are n1 and n2, respectively. When the incident angle is larger than the critical angle θc=sin1(n2/n1), the total internal reflection appears. The reflected light passes through an analyzer with the transmission axis at 45 deg with respect to the x-axis, and finally reaches a CMOS camera (CCD) by a telecentric imaging lens (TIL). The Jones vector of the light at the CMOS camera can be written as Display Formula
E(x,y)=122[|rp(x,y)|eiΓ2+iφp(x,y)+|rs(x,y)|eiΓ2+iφs(x,y)]·(11),(3)
where rp(x,y), rs(x,y), φp(x,y), and φs(x,y) denote the reflection coefficients and phase differences of p- and s-polarizations, respectively. These parameters can be written as10Display Formula
rp(x,y)=n2(x,y)cosθisin2θn2(x,y)n2(x,y)cosθ+isin2θn2(x,y)=|rp(x,y)|eiφp(x,y),(4)
Display Formula
rs(x,y)=cosθisin2θn2(x,y)cosθ+isin2θn2(x,y)=|rs(x,y)|eiφs(x,y),(5)
where n=n2/n1 is the relative refractive index. Therefore, the intensity measured at the CMOS camera can be expressed as Display Formula
I(x,y)=|E(x,y)|2=14{|rp(x,y)|2+|rs(x,y)|2+2|rp(x,y)||rs(x,y)|cos[Γ+φ(x,y)]},(6)
where φ(x,y) is the phase difference between the p- and s-polarized light from the reflection at the boundary surface under the conditions of total internal reflection, and it can be expressed as Display Formula
φ(x,y)=φp(x,y)φs(x,y)=arg[rp(x,y)]arg[rs(x,y)].(7)

Substituting Eqs. (4) and (5) into Eq. (7), the relationship of the phase difference φ(x,y) and the relative refractive index n(x,y) can be expressed as Display Formula

φ(x,y)=2tan1[sin2θn2(x,y)tanθsinθ],(8)

Eq. (8) can also be rewritten as Display Formula

n2(x,y)=n1sinθ{1tan2[φ(x,y)2]·tan2θ}1/2.(9)

Graphic Jump Location
Fig. 1
F1 :

Schematic diagram for measuring the full-field refractive index, P: polarizer; EO: electro-optic modulator; LVA: linear-voltage amplifier; DC: DC power supply; MO: microscopic objective; PH: pinhole; L: collimating lens; AN: analyzer; TIL: telecentric imaging lens; CCD: CMOS camera; PC: personal computer.

According to Eq. (9), the refractive index n2(x,y) of the tested specimen is the function of the phase difference φ(x,y). Hence, the refractive index n2(x,y) of the tested specimen can be obtained by an accurate measurement of the phase difference φ(x,y).

To measure the phase difference φ(x,y), the phase-shifting technique with modulated EO modulator is introduced. From Eq. (6), the different voltages of (3/2)V1, (1/2)V1, (1/2)V1, and (3/2)V1 are sequentially entered into the EO modulator to obtain different phase shifting. Then, the four sets of interferometric signals can be obtained and can be respectively expressed as Display Formula

I1(x,y)=14{|rp(x,y)|2+|rs(x,y)|2+2|rp(x,y)||rs(x,y)|cos[φ(x,y)3πV12Vπ]},(10)
Display Formula
I2(x,y)=14{|rp(x,y)|2+|rs(x,y)|2+2|rp(x,y)||rs(x,y)|cos[φ(x,y)πV12Vπ]},(11)
Display Formula
I3(x,y)=14{|rp(x,y)|2+|rs(x,y)|2+2|rp(x,y)||rs(x,y)|cos[φ(x,y)+πV12Vπ]},(12)
Display Formula
I4(x,y)=14{|rp(x,y)|2+|rs(x,y)|2+2|rp(x,y)||rs(x,y)|cos[φ(x,y)+3πV12Vπ]}.(13)

By using the Carré algorithm,11,12 the phase difference φ(x,y) can be calculated as Display Formula

φ(x,y)=tan1{[I1(x,y)I4(x,y)]+[I2(x,y)I3(x,y)]}{3[I2(x,y)I3(x,y)][I1(x,y)I4(x,y)]}[I2(x,y)+I3(x,y)][I1(x,y)+I4(x,y)].(14)

Therefore, the phase difference φ(x,y) can be determined with accurately measured intensities I1(x,y)I4(x,y). According to Eq. (9), the full-field refractive index n2(x,y) of the tested specimen then can be obtained.

To prove the feasibility of the proposed method, the tested specimen, a mixture of pure water (nw=1.331 at 632.8 nm) and castor oil (no=1.480 at 632.8 nm) with a refractive index distribution n2(x,y), was measured at room temperature of 25°C. A He–Ne laser with a wavelength of 632.8 nm that was expanded with a diameter of 4.5 mm was used as the test light source. The EO modulator (Mode 4002, New Focus) is driven by a DC power supply (GPD3303, GWINSTEK) and an LVA (Mode 3211, New Focus). The different phase retardations of EO modulator were modulated by sequentially supplying different voltage values of 120V, 40V, 40 V, and 120 V. A high-resolution motorized rotation stage (Model SGSP-60-WPQ, Sigma Koki, Inc.) with an angular resolution of 0.005 deg was used to mount and rotate the tested apparatus. The tested apparatus consisted of an SF11 right-angle prism (n1=1.778) with the tested specimen on its base. To achieve high sensitivity, the incident angle θ of light at the base of the prism was set to 57 deg. A telecentric lens (Silver Series Telecentric Lens, Edmund Optics Inc.) with a primary magnification of 0.25× and a working distance of 160 mm was mounted on an 8-bit gray level CMOS camera (XCD-U100CR, Sony Electronics Inc.) for imaging the interferometric signals. The pixels of the CMOS camera were 1600×1200 with a cell size of 4.4μm×4.4μm (sensor format 1/1.8-type). A personal computer and the MATLAB® software were used to analyze the captured images. The experimental results are shown in Figs. 2345. Figure 2 shows the four interferometric images of various phase distributions. Figures 3 and 4 show the measured phase difference φ(x,y) distribution and refractive index n2(x,y) distribution, respectively. Figure 5 shows the distribution of refractive index along the x-axis at y=3.5mm, in which the refractive indices of 1.336 and 1.482 are corresponding to pure water and castor oil, respectively. The measured results are in good agreement with the reference values, demonstrating the capability of the proposed method.

Graphic Jump Location
Fig. 2
F2 :

Four interferometric images of various phase distributions with EO modulation driving on (a) 120V, (b) 40V, (c) 40 V, and (d) 120 V.

Graphic Jump Location
Fig. 3
F3 :

The result of phase difference distribution.

Graphic Jump Location
Fig. 4
F4 :

The result of refractive index distribution.

Graphic Jump Location
Fig. 5
F5 :

The distribution of refractive index along the x-axis at y=3.5mm.

Considering the errors of phase and incident angle, the refractive index measurement resolution Δn2 can be estimated. According to Eq. (9), it can be expressed as Display Formula

Δn2=|n2φ|Δφerr+|n2θ|×Δθerr=|Bsec2(φ2)tanθ2A|×Δφerr+|An1cosθBtan(φ2)sec2θA|×Δθerr,(15)
where Display Formula
A=1tan2(φ2)tan2θ,(16)
and Display Formula
B=n1tan(φ2)sinθtanθ,(17)
where |Δφ| denotes the total phase error in the experiment, |Δφ| represents the error of the incident angle at the base of the right-angle prism. The Δφ can be estimated by considering the CMOS camera resolved-phase error, the polarization-mixing error, and the phase-shifting modulated error. The theoretical resolution of gray-level interferometric signals is about ΔI1/256=3.9×103 with an 8-bit CMOS camera. Therefore, the CMOS camera resolved-phase error was 0.0056 deg by substituting the resolution of interferometric signal ΔI into Eqs. (10) to (14). The extinction ratio of the polarizer (Newport Inc.) is 1×103, so the polarization-mixing error was estimated with value of 0.0072 deg. In this experimental setup, a DC power supply with the voltage resolution of 0.03% was used to drive the EO modulator, the phase-modulated error ΔΓ was about 1.4595 deg. Accordingly, the phase-shifting modulated error was 0.0042 deg by substituting the error of EO modulator ΔΓ into Eqs. (10) to (14). Consequently, the total phase error Δφ calculated in our experiment was 0.0170 deg. In addition, considering the fabricated tolerance of the right-angle prism and the resolution of the rotational stage, the value of |Δφ| was found to be 0.0195 deg. Substituting these values and the related experimental conditions into Eqs. (15) to (17), the relationship of the measurement resolution Δn2 and the refractive index n2 can be obtained, as shown in Fig. 6. Because of the simultaneous phase difference and phase errors in Carré phase-shifting algorithm, the curve appears as a nonlinear relation. The result in Fig. 6 displays that the measurement resolution has the highest value at 3.552×104 refractive index unit (RIU), when the refractive index n2 equals 1.460. Consequently, in our method, the measurement resolution can reach a value of at least 3.552×104 RIU.

Graphic Jump Location
Fig. 6
F6 :

Relationship of Δn2 versus n2.

This paper proposes a simple method for measuring full-field distribution of refractive index. It is based on the phenomenon of total internal reflection and phase-shifting technique with modulated EO modulator. Experiments confirm the feasibility of this method. The measurement resolution can reach a value of at least 3.552×104 RIU. This method offers the benefits of simple installation, ease of operation, and rapid measurement.

This work was supported by the Ministry of Science and Technology, Taiwan, under Contract MOST 103-2221-E-035-031.

Jian  Z. C.  et al., “A method for measuring two-dimensional refractive index distribution with the total internal reflection of p-polarized light and the phase-shifting interferometry,” Opt. Commun.. 268, , 23 –26 (2006). 0030-4018 CrossRef
Yassien  K. M., “Comparative study on determining the refractive index profile of polypropylene fibres using fast Fourier transform and phase-shifting interferometry,” J. Opt. A: Pure Appl. Opt.. 11, , 075701  (2009)(10pp).CrossRef
Lichtenberg  S.  et al., “Refractive-index measurement of gases with a phase-shift keyed interferometer,” Appl. Opt.. 44, , 4659 –4665 (2005). 0003-6935 CrossRef
Okada  K.  et al., “Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable-source phase shifting interferometry,” Appl. Opt.. 29, , 3280 –3285 (1990). 0003-6935 CrossRef
Zhao  W.  et al., “Investigation of the refractive index distribution in precision compression glass molding by use of 3D tomography,” Meas. Sci. Technol.. 20, , 055109  (2009). 0957-0233 CrossRef
Angelsky  O. V., , Hanson  S. G., and Maksimyak  P. P., “Use of optical correlation techniques for characterizing scattering objects and media,” Vol. PM71, ,  SPIE Press ,  Bellingham, Washington  (1999).
Lin  Y. C., and Cheng  C. J., “Determining the refractive index profile of micro-optical elements using transflective digital holographic microscopy,” J. Opt.. , 12, , 115402  (2010).CrossRef
Liu  Z.  et al., “Near-field ellipsometry for thin film characterization,” Opt. Express. 18, , 3298 –3310 (2010). 1094-4087 CrossRef
Lawrence  W. E., “Electron-electron scattering in the low-temperature resistivity of the noble metals,” Phys. Rev. B. 13, , 5316 –5539 (1976).CrossRef
Born  M., and Wolf  E., Principles of Optics. , 7th ed., pp. 40 ,  The Press Syndicate of the University of Cambridge ,  Cambridge, United Kingdom.  (1999).
Gasvik  K. J., Optical Metrology. , 3rd ed., pp. 269 –296,  John Wiley & Sons, Ltd. ,  Chichester, United Kingdom  (2002).
Hariharan  P., , Oreb  B. F., and Eiju  T., “Digital phase shifting interferometry: a simple error compensating phase calculation algorithm,” Appl. Opt.. 26, , 2504 –2505 (1987). 0003-6935 CrossRef

Kun-Huang Chen received his BS degree from the Physics Department of Chung Yuan Christian University, Taiwan, in 2000 and his PhD from the Institute of Electro-Optical Engineering of National Chiao Tung University, Taiwan, in 2004. In 2004, he joined the faculty of Feng Chia University, where he is currently a professor with the Department of Electrical Engineering. His current research interests include optical metrology and optical sensors.

Jing-Heng Chen received his BS degree from the Physics Department of Tunghai University, Taiwan, in 1997 and his MS and PhD degrees from the Institute of Electro-Optical Engineering, National Chiao Tung University, Taiwan, in 1999 and 2004, respectively. In 2004 he joined the faculty of Feng Chia University, where he is currently a professor with the Department of Photonics. His current research interests include optical testing and holography.

Jiun-You Lin received his MS degree from the Institute of Electro-Optical Engineering of National Chiao Tung University, Taiwan, in 2000 and his PhD from the Institute of Electro-Optical Engineering of National Chiao Tung University, Taiwan, in 2004. He joined the faculty of National Changhua University of Education in 2005, where he is currently an associate professor with the Department of Mechatronics Engineering. His current research interests include optical metrology and measurement of optical constants of a chiral medium.

Yen-Chang Chu received his BS and MS degrees from the Department of Electrical Engineering of Feng Chia University, Taiwan, in 2008 and 2011, respectively. He is now working toward his PhD in white-light-emitting diodes in the PhD program of Electrical and Communications Engineering, Feng Chia University. His current research interests include optical testing and white-light-emitting diodes.

© The Authors. Published by SPIE under a Creative CommonsAttribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requiresfull attribution of the original publication, including its DOI.

Citation

Kun-Huang Chen ; Jing-Heng Chen ; Jiun-You Lin and Yen-Chang Chu
"Alternative phase-shifting technique for measuring full-field refractive index", Opt. Eng. 54(9), 094101 (Sep 10, 2015). ; http://dx.doi.org/10.1117/1.OE.54.9.094101


Figures

Graphic Jump Location
Fig. 1
F1 :

Schematic diagram for measuring the full-field refractive index, P: polarizer; EO: electro-optic modulator; LVA: linear-voltage amplifier; DC: DC power supply; MO: microscopic objective; PH: pinhole; L: collimating lens; AN: analyzer; TIL: telecentric imaging lens; CCD: CMOS camera; PC: personal computer.

Graphic Jump Location
Fig. 2
F2 :

Four interferometric images of various phase distributions with EO modulation driving on (a) 120V, (b) 40V, (c) 40 V, and (d) 120 V.

Graphic Jump Location
Fig. 3
F3 :

The result of phase difference distribution.

Graphic Jump Location
Fig. 4
F4 :

The result of refractive index distribution.

Graphic Jump Location
Fig. 5
F5 :

The distribution of refractive index along the x-axis at y=3.5mm.

Graphic Jump Location
Fig. 6
F6 :

Relationship of Δn2 versus n2.

Tables

References

Jian  Z. C.  et al., “A method for measuring two-dimensional refractive index distribution with the total internal reflection of p-polarized light and the phase-shifting interferometry,” Opt. Commun.. 268, , 23 –26 (2006). 0030-4018 CrossRef
Yassien  K. M., “Comparative study on determining the refractive index profile of polypropylene fibres using fast Fourier transform and phase-shifting interferometry,” J. Opt. A: Pure Appl. Opt.. 11, , 075701  (2009)(10pp).CrossRef
Lichtenberg  S.  et al., “Refractive-index measurement of gases with a phase-shift keyed interferometer,” Appl. Opt.. 44, , 4659 –4665 (2005). 0003-6935 CrossRef
Okada  K.  et al., “Separate measurements of surface shapes and refractive index inhomogeneity of an optical element using tunable-source phase shifting interferometry,” Appl. Opt.. 29, , 3280 –3285 (1990). 0003-6935 CrossRef
Zhao  W.  et al., “Investigation of the refractive index distribution in precision compression glass molding by use of 3D tomography,” Meas. Sci. Technol.. 20, , 055109  (2009). 0957-0233 CrossRef
Angelsky  O. V., , Hanson  S. G., and Maksimyak  P. P., “Use of optical correlation techniques for characterizing scattering objects and media,” Vol. PM71, ,  SPIE Press ,  Bellingham, Washington  (1999).
Lin  Y. C., and Cheng  C. J., “Determining the refractive index profile of micro-optical elements using transflective digital holographic microscopy,” J. Opt.. , 12, , 115402  (2010).CrossRef
Liu  Z.  et al., “Near-field ellipsometry for thin film characterization,” Opt. Express. 18, , 3298 –3310 (2010). 1094-4087 CrossRef
Lawrence  W. E., “Electron-electron scattering in the low-temperature resistivity of the noble metals,” Phys. Rev. B. 13, , 5316 –5539 (1976).CrossRef
Born  M., and Wolf  E., Principles of Optics. , 7th ed., pp. 40 ,  The Press Syndicate of the University of Cambridge ,  Cambridge, United Kingdom.  (1999).
Gasvik  K. J., Optical Metrology. , 3rd ed., pp. 269 –296,  John Wiley & Sons, Ltd. ,  Chichester, United Kingdom  (2002).
Hariharan  P., , Oreb  B. F., and Eiju  T., “Digital phase shifting interferometry: a simple error compensating phase calculation algorithm,” Appl. Opt.. 26, , 2504 –2505 (1987). 0003-6935 CrossRef

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