Alternative phase-shifting techniquefor measuring full-field refractiveindex

Kun-Huang ChenJing-Heng ChenJiun-You LinYen-Chang Chu

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Alternative phase-shifting technique for measuringfull-field refractive index

Kun-Huang Chen,a,* Jing-Heng Chen,b Jiun-You Lin,c and Yen-Chang Chud

aFeng Chia University, Department of Electrical Engineering, 100 Wenhwa Road, Seatwen, Taichung 40724, TaiwanbFeng Chia University, Department of Photonics, 100 Wenhwa Road, Seatwen, Taichung 40724, TaiwancNational Changua University of Education, Department of Mechatronics Engineering, No. 2, Shi-Da Road, Changhua City 50074, TaiwandFeng Chia University, PhD Program of Electrical and Communications Engineering, 100 Wenhwa Road, Seatwen, Taichung 40724, Taiwan

Abstract. This study proposes an alternative and simple method for measuring full-field refractive index. Thismethod is based on the phase-shifting technique with a modulated electro-optical (EO) modulator and the phe-nomenon of total internal reflection. To this purpose, a linear polarized light is expanded and incident on theinterface between the prism and the tested specimen, and the reflected light passes through an analyzer forinterference. The phase difference between the s- and p-polarized light is sensitive to the refractive index ofthe tested specimen when the total internal reflection appears on this interface. Based on this effect, the resultingphase differences make it possible to analyze the refractive index of the tested specimen through a phase-shift-ing technique with a modulated EO modulator. The feasibility of this method was verified by experiment, and themeasurement resolution can reach a value of refractive index unit of at least 3.552 × 10−4. This method hasadvantages of simple installation, ease of operation, and fast measurement. © The Authors. Published by SPIE undera Creative CommonsAttribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requiresfull attribution of the originalpublication, including its DOI. [DOI: 10.1117/1.OE.54.9.094101]

Keywords: refractive index; total internal reflection; electro-optical modulator; phase-shifting interferometry.

Paper 150575 received May 5, 2015; accepted for publication Aug. 3, 2015; published online Sep. 10, 2015.

1 IntroductionThe measurement of full-field refractive index exhibits itssignificance and has many important applications in manyspecific fields, including the inspection of optical materialand thin-film characteristic as well as research in biology,biochemistry, and medicine. Currently, some methods havebeen proposed to measure full-field refractive index, such asthe interferometric methods,1–6 digital holographic micros-copy,7 and ellipsometric methods.8 In the interferometricmethods, the phase difference is relative to the refractiveindex. To obtain the phase difference distribution to measurefull-field refractive index, the phase-shifting technique isan important and effective method that features simple struc-tures, easy operation, and rapid measurement. Generally,the phase-shifting methods introduce successive phasesteps into interferometric signals with various phase-shiftingstrategies, such as piezoelectric transducer,1,2 holographicgrating,3 and tunable wavelength.4 Aforementioned methodscould suffer from mechanical perturbations, and their costsare high. In addition, to reveal the phase distribution, tech-niques of three-step and four-step phase-shifting methodsare commonly applied. However, the introduced values ofphase-shifting must be specified. Therefore, this study pro-poses an alternative and simple method based on the Carréphase-shifting technique with modulated electro-optical(EO) modulator for measuring full-field refractive index.A linear polarized light is expanded and incident on the inter-face between the prism and the tested specimen. When thetotal internal reflection appears on this interface, a significantphase difference between the s- and p-polarized light is

introduced. This phase difference is relative to the refractiveindex of the tested specimen, and can be precisely measuredwith phase-shifting technique with modulated EO modulator.Accordingly, the full-field refractive index of the testedspecimen can be estimated. To prove the feasibility of theproposed method, the tested specimen combining withpure water and castor oil is measured. The experimental datacorrespond well with the theoretical values. Due to theintroduction of EO modulation and Carré phase-shiftingtechniques, the proposed measurement method can avoidmechanical perturbation and has merits of simple installa-tion, ease of operation, and fast measurement.

2 PrinciplesFigure 1 shows the optical configuration of the proposedmethod. For convenience, the z-axis is set in the directionof light propagation, and the x-axis is set perpendicularto the plane of the paper. Linear polarized laser light withthe polarization state at 45 deg with respect to the x-axispasses through an EO modulator with the optic axis at 0 degwith respect to the x-axis. Using a DC power supply (DC)and a linear-voltage amplifier (LVA) to drive the EO modu-lator, the Jones vector of the electric field of the resulted laserlight Ein can be written as

EQ-TARGET;temp:intralink-;e001;326;149Ein ¼1ffiffiffi2

p�

eiΓ2

e−iΓ2

�eiω0t; (1)

where ω0 is the light frequency, and Γ denotes the phaseretardation between the s- and p-polarizations, which canbe written as9*Address all correspondence to: Kun-Huang Chen, E-mail: chenkh@fcu.edu.tw

Optical Engineering 094101-1 September 2015 • Vol. 54(9)

Optical Engineering 54(9), 094101 (September 2015)

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EQ-TARGET;temp:intralink-;e013;63;501Γ ¼ πVz

Vπ; (2)

where Vπ is the half-wave voltage of the EO modulator, andVz is an external voltage applied to the EO modulator. Then,the light beam is expanded and collimated by a spatial filterto form a plane wave, and incident on the undersurfaceof a prism at an angle θ. The refractive indices of theprism and the tested specimen are n1 and n2, respectively.When the incident angle is larger than the critical angleθc ¼ sin−1ðn2∕n1Þ, the total internal reflection appears.The reflected light passes through an analyzer with the trans-mission axis at 45 deg with respect to the x-axis, and finallyreaches a CMOS camera (CCD) by a telecentric imaging lens(TIL). The Jones vector of the light at the CMOS camera canbe written asEQ-TARGET;temp:intralink-;e003;63;328

Eðx;yÞ¼ 1

2ffiffiffi2

phjrpðx;yÞjeiΓ

2þiϕpðx;yÞ þ jrsðx;yÞje−iΓ

2þiϕsðx;yÞ

i

·

�1

1

�; (3)

where rpðx; yÞ, rsðx; yÞ, ϕpðx; yÞ, and ϕsðx; yÞ denotethe reflection coefficients and phase differences of p- ands-polarizations, respectively. These parameters can bewritten as10

EQ-TARGET;temp:intralink-;e004;63;209

rpðx; yÞ ¼n2ðx; yÞ cos θ − i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin2θ − n2ðx; yÞ

pn2ðx; yÞ cos θ þ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin2θ − n2ðx; yÞ

p¼ jrpðx; yÞjeiϕpðx;yÞ; (4)

EQ-TARGET;temp:intralink-;e005;63;139rsðx; yÞ ¼cos θ − i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin2θ − n2ðx; yÞ

pcos θ þ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin2θ − n2ðx; yÞ

p ¼ jrsðx; yÞjeiϕsðx;yÞ;

(5)

where n ¼ n2∕n1 is the relative refractive index. Therefore,the intensity measured at the CMOS camera can beexpressed as

EQ-TARGET;temp:intralink-;e006;326;734

Iðx; yÞ ¼ jEðx; yÞj2

¼ 1

4fjrpðx; yÞj2 þ jrsðx; yÞj2

þ 2jrpðx; yÞjjrsðx; yÞj cos½Γþ ϕðx; yÞ�g; (6)

where ϕðx; yÞ is the phase difference between the p- ands-polarized light from the reflection at the boundary surfaceunder the conditions of total internal reflection, and it can beexpressed asEQ-TARGET;temp:intralink-;e007;326;629

ϕðx; yÞ ¼ ϕpðx; yÞ − ϕsðx; yÞ¼ arg½rpðx; yÞ� − arg½rsðx; yÞ�: (7)

Substituting Eqs. (4) and (5) into Eq. (7), the relationshipof the phase difference ϕðx; yÞ and the relative refractiveindex nðx; yÞ can be expressed as

EQ-TARGET;temp:intralink-;e008;326;546ϕðx; yÞ ¼ −2tan−1� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sin2 θ − n2ðx; yÞp

tan θ sin θ

�; (8)

Eq. (8) can also be rewritten as

EQ-TARGET;temp:intralink-;e009;326;490n2ðx; yÞ ¼ n1 sin θ

�1 − tan2

�ϕðx; yÞ

2

�· tan2θ

�1∕2

: (9)

According to Eq. (9), the refractive index n2ðx; yÞ of thetested specimen is the function of the phase differenceϕðx; yÞ. Hence, the refractive index n2ðx; yÞ of the testedspecimen can be obtained by an accurate measurement ofthe phase difference ϕðx; yÞ.

To measure the phase difference ϕðx; yÞ, the phase-shift-ing technique with modulated EO modulator is introduced.From Eq. (6), the different voltages of −ð3∕2ÞV1, −ð1∕2ÞV1,ð1∕2ÞV1, and ð3∕2ÞV1 are sequentially entered into the EOmodulator to obtain different phase shifting. Then, the foursets of interferometric signals can be obtained and can berespectively expressed asEQ-TARGET;temp:intralink-;e010;326;315

I1ðx;yÞ¼1

4

�jrpðx;yÞj2þjrsðx;yÞj2

þ2jrpðx;yÞjjrsðx;yÞjcos�ϕðx;yÞ−3πV1

2Vπ

��; (10)

EQ-TARGET;temp:intralink-;e011;326;254

I2ðx;yÞ¼1

4

�jrpðx;yÞj2þjrsðx;yÞj2

þ2jrpðx;yÞjjrsðx;yÞjcos�ϕðx;yÞ−πV1

2Vπ

��; (11)

EQ-TARGET;temp:intralink-;e012;326;197

I3ðx; yÞ ¼1

4

�jrpðx; yÞj2 þ jrsðx; yÞj2

þ 2jrpðx; yÞjjrsðx; yÞj cos�ϕðx; yÞ þ πV1

2Vπ

��; (12)

EQ-TARGET;temp:intralink-;e013;326;140I4ðx; yÞ ¼1

4

�jrpðx; yÞj2 þ jrsðx; yÞj2

þ 2jrpðx; yÞjjrsðx; yÞj cos�ϕðx; yÞ þ 3πV1

2Vπ

��.

(13)

By using the Carré algorithm,11,12 the phase differenceϕðx; yÞ can be calculated as

Rotation stage

θ

MOPH L

SF11 Prism (n1)

AN(45 deg)

PC

Tested specimen n2(x, y):

a mixture of pure water and castor oil

TIL

CCD

Spatial filterP(45 deg)

EO

LVA

DC z

y

Laser

Fig. 1 Schematic diagram for measuring the full-field refractive index,P: polarizer; EO: electro-optic modulator; LVA: linear-voltage ampli-fier; DC: DC power supply; MO: microscopic objective; PH: pinhole;L: collimating lens; AN: analyzer; TIL: telecentric imaging lens; CCD:CMOS camera; PC: personal computer.

Optical Engineering 094101-2 September 2015 • Vol. 54(9)

Chen et al.: Alternative phase-shifting technique for measuring full-field refractive index

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EQ-TARGET;temp:intralink-;e014;63;722ϕðx; yÞ ¼ tan−1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif½I1ðx; yÞ − I4ðx; yÞ� þ ½I2ðx; yÞ − I3ðx; yÞ�gf3½I2ðx; yÞ − I3ðx; yÞ� − ½I1ðx; yÞ − I4ðx; yÞ�gp

½I2ðx; yÞ þ I3ðx; yÞ� − ½I1ðx; yÞ þ I4ðx; yÞ�: (14)

Therefore, the phase difference ϕðx; yÞ can be determinedwith accurately measured intensities I1ðx; yÞ ∼ I4ðx; yÞ.According to Eq. (9), the full-field refractive indexn2ðx; yÞ of the tested specimen then can be obtained.

3 Experimental Results and DiscussionsTo prove the feasibility of the proposed method, the testedspecimen, a mixture of pure water (nw ¼ 1.331 at 632.8 nm)and castor oil (no ¼ 1.480 at 632.8 nm) with a refractiveindex distribution n2ðx; yÞ, was measured at room tempera-ture of 25°C. A He–Ne laser with a wavelength of 632.8 nmthat was expanded with a diameter of 4.5 mm was usedas 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). Thedifferent phase retardations of EO modulator were modu-lated by sequentially supplying different voltage values of−120 V, −40 V, 40 V, and 120 V. A high-resolution motor-ized rotation stage (Model SGSP-60-WPQ, Sigma Koki,Inc.) with an angular resolution of 0.005 deg was usedto mount and rotate the tested apparatus. The tested appa-ratus consisted of an SF11 right-angle prism (n1 ¼ 1.778)with the tested specimen on its base. To achieve highsensitivity, the incident angle θ of light at the base of theprism was set to 57 deg. A telecentric lens (Silver SeriesTelecentric Lens, Edmund Optics Inc.) with a primarymagnification of 0.25× and a working distance of 160 mmwas mounted on an 8-bit gray level CMOS camera(XCD-U100CR, Sony Electronics Inc.) for imaging theinterferometric signals. The pixels of the CMOS camerawere 1600 × 1200 with a cell size of 4.4 μm × 4.4 μm(sensor format 1/1.8-type). A personal computer and theMATLAB® software were used to analyze the capturedimages. The experimental results are shown in Figs. 2–5.Figure 2 shows the four interferometric images of variousphase distributions. Figures 3 and 4 show the measuredphase difference ϕðx; yÞ distribution and refractive indexn2ðx; yÞ distribution, respectively. Figure 5 shows the dis-tribution of refractive index along the x-axis at y ¼ 3.5 mm,in which the refractive indices of 1.336 and 1.482 arecorresponding to pure water and castor oil, respectively.The measured results are in good agreement with the refer-ence values, demonstrating the capability of the proposedmethod.

Considering the errors of phase and incident angle, therefractive index measurement resolution Δn2 can be esti-mated. According to Eq. (9), it can be expressed asEQ-TARGET;temp:intralink-;e015;63;158

Δn2 ¼ ∂n2∂ϕ

Δϕerrþ ∂n2∂θ

× Δθerr

¼−B sec2

ϕ2

�tan θ

2A

× Δϕerr

þAn1 cos θ −

B tanϕ2

�sec2 θ

A

× Δθerr; (15)

where

EQ-TARGET;temp:intralink-;e016;326;668A ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − tan2

�ϕ

2

�tan2 θ

s; (16)

and

EQ-TARGET;temp:intralink-;e017;326;610B ¼ n1 tan

�ϕ

2

�sin θ tan θ; (17)

where jΔϕj denotes the total phase error in the experiment,jΔϕj represents the error of the incident angle at the baseof the right-angle prism. The Δϕ can be estimated byconsidering the CMOS camera resolved-phase error, thepolarization-mixing error, and the phase-shifting modulatederror. The theoretical resolution of gray-level interferometricsignals is about ΔI ≅ 1∕256 ¼ 3.9 × 10−3 with an 8-bitCMOS camera. Therefore, the CMOS camera resolved-phase error was 0.0056 deg by substituting the resolutionof interferometric signal ΔI into Eqs. (10) to (14). Theextinction ratio of the polarizer (Newport Inc.) is 1 × 10–3,so the polarization-mixing error was estimated with valueof 0.0072 deg. In this experimental setup, a DC power supplywith the voltage resolution of 0.03% was used to drive theEO modulator, the phase-modulated error ΔΓ was about

(a) (b)

(c) (d)

I1 I2

I3 I4

Fig. 2 Four interferometric images of various phase distributionswith EO modulation driving on (a) −120 V, (b) −40 V, (c) 40 V,and (d) 120 V.

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1.4595 deg. Accordingly, the phase-shifting modulated errorwas 0.0042 deg by substituting the error of EO modulatorΔΓ into Eqs. (10) to (14). Consequently, the total phaseerror Δϕ calculated in our experiment was 0.0170 deg. Inaddition, considering the fabricated tolerance of the right-angle prism and the resolution of the rotational stage, thevalue of jΔϕj was found to be 0.0195 deg. Substitutingthese values and the related experimental conditions intoEqs. (15) to (17), the relationship of the measurement reso-lution Δn2 and the refractive index n2 can be obtained, asshown in Fig. 6. Because of the simultaneous phase differ-ence and phase errors in Carré phase-shifting algorithm, thecurve appears as a nonlinear relation. The result in Fig. 6displays that the measurement resolution has the highestvalue at 3.552 × 10−4 refractive index unit (RIU), whenthe refractive index n2 equals 1.460. Consequently, in ourmethod, the measurement resolution can reach a value ofat least 3.552 × 10–4 RIU.

4 ConclusionThis paper proposes a simple method for measuring full-fielddistribution of refractive index. It is based on the phenome-non of total internal reflection and phase-shifting techniquewith modulated EO modulator. Experiments confirm thefeasibility of this method. The measurement resolution canreach a value of at least 3.552 × 10−4 RIU. This methodoffers the benefits of simple installation, ease of operation,and rapid measurement.

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

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Fig. 4 The result of refractive index distribution.

Fig. 6 Relationship of Δn2 versus n2.Fig. 3 The result of phase difference distribution.

Fig. 5 The distribution of refractive index along the x -axis aty ¼ 3.5 mm.

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Chen et al.: Alternative phase-shifting technique for measuring full-field refractive index

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Kun-Huang Chen received his BS degree from the PhysicsDepartment of Chung Yuan Christian University, Taiwan, in 2000and his PhD from the Institute of Electro-Optical Engineering of

National Chiao Tung University, Taiwan, in 2004. In 2004, he joinedthe faculty of Feng Chia University, where he is currently a professorwith the Department of Electrical Engineering. His current researchinterests include optical metrology and optical sensors.

Jing-Heng Chen received his BS degree from the Physics Depart-ment of Tunghai University, Taiwan, in 1997 and his MS and PhDdegrees from the Institute of Electro-Optical Engineering, NationalChiao Tung University, Taiwan, in 1999 and 2004, respectively. In2004 he joined the faculty of Feng Chia University, where he iscurrently a professor with the Department of Photonics. His currentresearch 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, in2000 and his PhD from the Institute of Electro-Optical Engineeringof National Chiao Tung University, Taiwan, in 2004. He joined the fac-ulty of National Changhua University of Education in 2005, where he iscurrently an associate professor with the Department of MechatronicsEngineering. His current research interests include optical metrologyand measurement of optical constants of a chiral medium.

Yen-Chang Chu received his BS and MS degrees from the Depart-ment of Electrical Engineering of Feng Chia University, Taiwan, in2008 and 2011, respectively. He is now working toward his PhD inwhite-light-emitting diodes in the PhD program of Electrical and Com-munications Engineering, Feng Chia University. His current researchinterests include optical testing and white-light-emitting diodes.

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