modified critical angle method for measuring the ...€¦ · the critical angle technique is...

$: Modified critical angle method for measuring the ...€¦ · The critical angle technique is modified for the accurate measurement of the refractive index of bio-optical materials$
Song et al. Vol. 12, No. 5 /May 1995/J. Opt. Soc. Am. B 797

Modified critical angle method for measuringthe refractive index of bio-optical

materials and its application to bacteriorhodopsin

Q. Wang Song, Chin-Yu Ku, and Chunping Zhang

Department of Electrical and Computer Engineering and W. M. Keck Center for Molecular Electronics,Syracuse University, Syracuse, New York 13244-1240

Richard B. Gross and Robert R. Birge

Department of Chemistry and W. M. Keck Center for Molecular Electronics,Syracuse University, Syracuse, New York 13244-4100

Richard Michalak

Rome Air Force Photonics Center, RLyOC, Griffiss Air Force Base, New York 13441-5700

Received September 14, 1994

The critical angle technique is modified for the accurate measurement of the refractive index of bio-opticalmaterials. Based on the analysis of reflection from the boundary of the material as a function of incidentangle and polarization direction, the critical illumination angle is obtained by numerical differentiation of thereflection curve. As an example, the dispersion curve of bacteriorhodopsin is given. The measurement errorand the effect of the host bovine skin gelatin on the results are analyzed.

1. INTRODUCTIONRecently, bio-optical materials such as bacteriorhodopsin(BR) have become important linear and nonlinear opticalmaterials because of their real-time holographic record-ing capabilities, information storage potential, and largeoptical nonlinearity.1–6 With their complex molecularstructures, one of the attractive features of bio-opticalmaterials is that they can be genetically or chemicallymodified on the molecular scale to suit given applications.The refractive index is one of the most important physicalproperties of an optical material. However, most of thebio-optical materials have absorption, and it is difficult tomake a good freestanding optical surface with bio-opticalmaterials. These facts make the determination of theirrefractive indices difficult.

The critical-angle method7,8 has been used for the mea-surement of the refractive index of nonabsorbing materi-als. The refractive index of an absorbing material can bededuced from ellipsometric or reflection measurements9,10

and pseudo-Brewster-angle measurements.11,12 But thetop surfaces of most bio-optical materials are not of highenough optical quality for the application of the methodsmentioned above. In this paper we describe a modifiedcritical-angle method for measuring the refractive indexof an absorbing bio-optical material. As an example, wereport the measurement of the refractive index of agelatin-based film doped with BR because it is one ofthe important bio-optical materials on account of its largeoptical nonlinearity and many application potentials inthe field of optical information processing.

0740-3224/95/050797-07$06.00

In the following sections we will first describe the the-ory of the proposed method. Then we will show how itis experimentally implemented. As an example, we pro-vide the dispersion curve of BR. The various aspects ofexperimental error are then analyzed.

2. THEORYThe principle behind the measurement is illustrated inFig. 1. In this figure medium 1 is a nonabsorbing prismof known high refractive index n1. Medium 2, which canbe either absorbing or nonabsorbing, is the test sampleto be measured. In this setup total internal reflection atthe interface between media 1 and 2 is used to determinethe refractive index n2 of medium 2. b is the vertexangle of the prism, a is the incident angle at the face of theprism, and u is the incident angle in the interface betweenthe prism and the sample. a is related to u through theequation

u b 6 arcsin

√sin a

n1

!, (1)

where the positive (negative) sign indicates that a hasa clockwise (counterclockwise) rotation from the prismsurface (normal).

The complex refractive index of medium 2 (test sample)can be written as n2 n2s1 1 ik2d, in which n2 is the realrefractive index to be measured and k2 is the attenua-tion index.13 It is convenient to use the parameters u2

and v2 to describe the reflection coefficients. These twoparameters are related to the system parameters by

1995 Optical Society of America

798 J. Opt. Soc. Am. B/Vol. 12, No. 5 /May 1995 Song et al.

Fig. 1. Diagram illustrating the principle for the index mea-surement setup.

2u22 n2

2s1 2 k22d 2 n1

2 sin2 u

1 hfn22s1 2 k2

2d 2 n12 sin2 ug2 1 4n2

4k22j1/2 ,

2v22 2 fn2

2s1 2 k22d 2 n1

2 sin2 ug

1 hfn22s1 2 k2

2d 2 n12 sin2 ug2 1 4n2

4k22j1/2 . (2)

In the case in which the electric vector is perpendicular tothe plane of incidence (TE wave) the reflection coefficientRs,' of the light beam at the interface of the prism andthe sample is given by

Rs,'

Én1 cos u 2 su2 1 iv2dn1 cos u 1 su2 1 iv2d

É2. (3)

If the electric vector is parallel to the plane of incidence(TM wave), the reflection coefficient Rs,k of the TM waveis given by

Rs,k

Éfn2

2s1 2 k22dcos u 2 n1u2g2 1 s2n2

2k2 cos u 2 n1v2d2

fn22s1 2 k2

2dcos u 1 n1u2g2 1 s2n22k2 cos u 1 n1v2d2

É2.

(4)

In the geometry shown in Fig. 1 the incident beam willundergo reflections at the air–prism, prism–sample,and prism–air interfaces. The reflection losses in theair–prism and prism–air interfaces are the same; onecan calculate them by noting that

Rp,'

ØØØØØØØØØØØØØØ

cos a 2 n1 cos

24arcsin

√sin a

n1

!35cos a 1 n1 cos

24arcsin

√sin a

n1

!35


2

(5)

for the TE wave and

Rp,k


n1

,cos

24arcsin

√sin a

n1

!35 2 1ycos a

n1

,cos

24arcsin

√sin a

n1

!35 1 1ycos a


2

(6)

for the TM wave. The overall effective reflection coeffi-cient of light from this system is given by

Rk Rs,ks1 2 Rp,kd2 (7)

for the TM wave and

R' Rs,'s1 2 Rp,'d2 (8)

for the TE wave.By combining Eqs. (1)–(8), one can calculate the effec-

tive reflections of this system for any given set of systemparameters. Figures 2 and 3 show R' and Rk, respec-tively, as a function of the incident angle a for n1 1.73,n2 1.52, and k2 0.0, 0.0005, 0.001, 0.002, 0.004, 0.006.These figures clearly show the effect of absorption inmedium 2 on the reflection. To be specific, when k2 iszero (i.e., no absorption), the total internal reflection is ob-vious. The illumination angle on the air–prism interfaceat which the total reflection occurs is defined as the criti-cal illumination angle a0

c. It is sharply indicated by theinflection point. When the absorption becomes stronger,

Fig. 2. Reflection of the TE wave as a function of the incidentangle for n1 1.73 and n2 1.52. Curves a, b, c, d, e, and f arefor k2 0.00, 0.0005, 0.001, 0.002, 0.004, and 0.006, respectively.

Fig. 3. Reflection of the TM wave as a function of the anglefor n1 1.73 and n2 1.52. Curves a, b, c, d, e, and f are fork2 0.00, 0.0005, 0.001, 0.002, 0.004, and 0.006, respectively.


(a)

(b)Fig. 4. (a) Reflection waves and (b) their derivatives forn1 1.73 and n2 1.52. The dashed curves are for the TEwave, and the solid curves are for the TM wave. The curveswith an abrupt reflection are for k2 0.0, and the smooth curvesare for k2 0.001.

the reflection curve subsequently becomes smoother, andthe inflection point of the curve is gradually lost.

Figure 4 shows the relationship between the reflec-tion curves and their derivatives. It can be seen thatthe reflection curve at the critical illumination anglechanges sharply for k2 0.0 and gradually for k2 0.001. For a nonabsorbing medium (i.e., k2 0) the an-gular positions of the maximum derivatives of the reflec-tion curves for both TE and TM waves are the sameas the critical illumination angle a0

c. For an absorb-ing medium (in this case k 0.001) the angular posi-tions of the maximum derivatives of the reflection curvesdeviate slightly from the critical illumination angle a0

c.The angular positions of the maximum derivatives forthe TE and TM waves are defined as ac,E and ac,M , re-spectively. Figures 5(a) and 5(b) show DaE ac,E 2 a0

c(TE wave) and DaM ac,M 2 a0

c (TM wave), respec-tively, as a function of attenuation k2 for the differ-ent n2. As k2 increases, both jDaE j and jDaM j firstincrease, then decrease, pass zero, and increase again.

In general, jDaE j . jDaM j. So, for a weak absorbingmedium, ac,M can be used as the approximation for a0

cto determine the refractive index of medium 2 (i.e., thetest sample).

Figure 6 shows the calculated reflection curves of theTM wave as a function of the incident angle for k2 0.004, n1 1.73, and n2 1.520, 1.525, 1.530, 1.535,1.540, 1.545. This figure shows that, for a constant k2,the reflection curve shifts as n2 changes but the shapeof the curve is unchanged. In other words, the methoddiscussed above of using the ac,M to approximate a0

c isstill valid for different values of n2.

As the value of k2 increases, the error in determin-ing a0

c from the method described above becomes larger.Two methods can be used to yield the refractive index ofa strongly absorbing medium. One is the curve-fittingmethod. After finding the measured reflection curveand knowing k2 (it can be easily measured) of the testsample, one can use Eqs. (7) and (8) and different n2 tofit the measured reflection curve and thus obtain the re-fractive index of the sample. As depicted in Fig. 6, the

(a)

(b)Fig. 5. Shift of the maximum derivatives from the critical illu-mination angle as a function of the attenuation k2: (a) TE wave,(b) TM wave. The curves from top to bottom are for n2 1.525,1.530, 1.535, and 1.540, respectively.


Fig. 6. Reflection of the TM wave as a function of the incidentangle for k2 0.004 and n 1.73. Curves a, b, c, d, e, and f arefor n2 1.520, 1.525, 1.530, 1.535, 1.540, and 1.545, respectively.

shift of the reflection curve is sensitive to different valuesof n2. So the accuracy of this curve-fitting method canbe high. The second method is the successive approxi-mation method. After obtaining the maximum deriva-tive angle ac,M1 from the measured reflection curve, wecan obtain the approximate refractive index n2,1 of thesample. Knowing k2 and n2,1 of the sample, we can ob-tain the angle shift DaM1 from the computation illustratedin Fig. 5. Then we can obtain the critical illuminationangle a

0c,1 ac,M1 1 DaM ,1. From this we arrive at the

next approximated refractive index, n2,2, of the sample.From n2,2 we repeat the above process. In this way onecan obtain a precise value of the refractive index n2 ofthe sample.

The divergence of the probing Gaussian beam also hasan effect on the bend of the reflection curve. In order todistinguish the effects of absorption and divergence of theprobing Gaussian beam on the reflection curve, we use anonabsorbing material, water, as a sample. The bound-ary that we are considering here is between the prism andthe water. Curves a, b, and c in Fig. 7 are the calculatedreflection curves of the TM-mode probe Gaussian beamwith the divergence angles 0.00, 20, and 40, respectively.Clearly, the divergence of the probe Gaussian beam alsocauses a shift and a bending of the reflection curve nearthe critical angle. By finding the location of the maximaof the derivatives, one can find that the error is muchsmaller than DaM . Therefore the effect of the divergenceof the probe Gaussian beam on the shape and the shift ofthe reflection curves can be neglected. If high accuracyis required, one can use a collimated beam to eliminatethe effect of beam divergence.

3. EXPERIMENTThe experimental setup for this measurement is shownin Fig. 1 above. A power meter is used to detect thereflection power from the interface between the prismand the sample. a is the incident angle on the incidentplane of the prism. a0

c, which is defined as the criticalillumination angle in this paper, is the incident angleat which total internal reflection occurs at the interface

between prism and sample. ac,M is used to approximatea0

c to determine the refractive index n2 of the test sample.The refractive index n2 of the sample is given by

n2 n1 sin

24b 6 arcsin

√an1

sin ac,M

!35 , (9)

where the refractive index of air is taken as 1.000 andn1, which ranges from 1.794 to 1.7193 for the wavelengthused in this experiment, is the refractive index of theprism. These values are obtained from the dispersiontable supplied by the prism vendor. The positive (nega-tive) sign indicates that ac,M is a clockwise (counterclock-wise) rotation from the prism normal.

The magnitude of the critical illumination angle de-pends on the ratio n2yn1. In general, medium 3 is air,and the total internal reflection occurs first at the inter-face of the sample and the air. This reflection makes themeasurement of the critical illumination angle difficult.Therefore the total reflection at the interface between thesample and the air should be avoided. The arrangementsshown in Fig. 8 are used for this purpose. Figures 8(a)and 8(b) are for liquid samples, and Figs. 8(c) and 8(d)are for solid samples. For example, Fig. 8(c) shows howto use a matching liquid with a refractive index higherthan that of the sample to attach another prism that hasa refractive index higher than that of the matching liq-uid. By these arrangements the total internal reflectionwill occur only at the interface between the prism andthe sample.

As an illustration, we present the measurement of therefractive index of BR film. BR is the light-harvestingprotein found in the purple membrane of the organismHalobacterium salinarium. It is considered to be one ofthe most promising bio-optical materials for holographicapplications. 1.5% (volume) bovine skin gelatin was usedto host the BR, so that it has a good optical surface whileminimizing the effect of the host on the refractive indexof pure BR film. The BR film is coated onto one of thesurfaces of a high-index equilateral prism that is mountedupon a rotating stage with a precision of 0.20. This setup

Fig. 7. Reflection of the TM wave as a function of the incidentangle. Curves a, b, and c are for divergences of the Gaussianprobe beam of 00, 20, and 40, respectively.


Fig. 8. Various schemes to eliminate the reflection on the topsurface of the test sample in order to obtain the critical illumi-nation angle.

provides an angular position measurement with a preci-sion of 0.018±. Since most of the matching liquids candamage the membrane of BR, we made the top surface ofBR film rough and then coated it with fine carbon pow-der. This procedure eliminates the reflection on the topsurface but also caused a weak scattering. Because thisscattering is at a large angle, only a small part of the scat-tered light can hit the detector, which is far from the BRfilm and occupies a very small stereoangle (,10–6). Ac-cordingly, the scattered light has very little effect on thedetection of the reflected light from the interface of theprism and the BR.

4. RESULTS AND DISCUSSIONFigure 9 shows the measured reflection curve (solid)of this setup as a function of the incident angle a forthe wavelength at 514.5 nm. The derivative is shownby the dotted curve. We obtain the maximum deriva-tive (point M) by comparing the differences betweenadjacent measured data. In this measurement the il-lumination intensity is kept below 30 mWycm2. Thisguarantees that the absorption band does not changeappreciably during the measurement to ensure that themeasured refractive index is the one for BR at the groundstate.

The dispersion curve of the BR film obtained throughthe procedure described above for some important wave-lengths is shown in Fig. 10. It is similar to that ofglasses in the visible region. According to the theory ofdispersion in dielectrics, there should be a variation of therefractive index in the region of a single absorption band:the refractive index increases (decreases) at wavelengthsshorter (longer) than that of the absorption peak. Thereis an absorption peak located at l 568 nm for BR film,but no such change in the refractive index is observed.This phenomenon can be explained as follows. There area number of weak absorption bands located in the range350–450 nm and a strong absorption band at 280 nm, be-sides the absorption band at 568 nm.14 These absorptionbands change the shape of the single absorption band near568 nm. The strong absorption band at 280 nm has the

same effect as that of the absorption bands of glasses atwavelengths shorter than 300 nm. Hence the dispersioncurve of BR is similar to that of glasses. Another reasonfor this phenomena is that the change in the refractiveindex induced by the absorption change is smaller than60.003,15,16 close to the measurement error in our case.

The error in the measurement can be evaluated bydifferentiating Eq. (9):

dn2 dn2s bd 1 dn2sdd 1 dn2sn1d

É≠n2

≠b

Édb 1

ØØØØØØ ≠n2

≠a0c

ØØØØØØda0c 1

É≠n2

≠n1

Édn1 , (10)

where db is the error in the measurement of the vertexangle of the prism, dn1 is the error of the refractive indexof the prism, and da is the error on the measurement ofthe critical illumination angle.

From Eq. (9) the error contributions in the determina-tion of the refractive index of the sample, resulting fromthe errors of the vertex angle sdbd, the critical illumi-nation angle sdad, and the refractive index of the prismsdn1d, are

dn2s bd

ØØØØØØØn1 cos

24b 6 arcsin

√1n1

sin a0c

!35ØØØØØØØdb , (11)

Fig. 9. Measured reflection of BR film as a function of theincident angle and the maximum derivative of the reflectioncurve.

Fig. 10. Dispersion of BR film versus wavelength.


dn2sad n1

0B@0B@0B@ 2 ssin bd

sin a

n12 cos

8<: a

f1 2 sinsayn1d2g1/2

9=;1 cos b cos

√a

n1

!1CA1CA1CAda , (12)

dn2sn1d sinf b 6 arcsins1yn1dsin ag

1 n1 cosf b 6 arcsins1yn1dgdn1 . (13)

In the case of this measurement, a , 5± and n1 rangesfrom 1.72 to 1.74 for the wavelengths used. If we sub-stitute these values into Eqs. (11)–(13), they can be ap-proximated as

dn2sbd ø db , (14)

dn2sad ø 0.5da , (15)

dn2sn1d ø dn1 . (16)

Hence the total error in the measurement of the refractiveindex of the sample is given as

dn2 db 1 0.5da 1 dn1 . (17)

The error da consists of two parts: one is the measurederror of the angle ac,M , and the other is the error inapproximating the critical illumination angle a0

c by ac,M .In our case, db 0.0002 rad, da ø 0.0015 rad, and dn1 ø0.0002. Hence the total error dn2 is 60.002.

We believe that the data on the measured refractive in-dex in Fig. 10 are the refractive index of pure BR film, al-though the BR film used in our experiment contains 1.5%host bovine skin. There are a number of theories andmethods to estimate the properties of mixtures or inho-mogeneous media.17–19 The simplest method to estimatethe refractive index of the BR film is given by17

n2 qn 1 q3n3 , (18)

where n2 is the refractive index of the BR film (mixture),n and n3 are the refractive indices of the pure BR andthe host bovine skin gelatin, respectively, and q and q3

are the volume fractions, with q 1 q3 1. By definingn3 n 2 Dn, we can rewrite Eq. (18) as

n2 qsn3 1 Dnd 1 q3n3 n3 1 qDn (19)

or

Dn sn2 2 n3dyq . (20)

From the measured refractive index of the host bovineskin gelatin (,1.54) in the visible region and the data inFig. 10 we obtain Dn ø 0.02. From Eqs. (19) and (20) therefractive index of the pure BR film, n, can be written as

n n2 1 q3Dn . (21)

Substituting q3 0.015 and Dn ø 0.02 into Eq. (21),we obtain Dn0 n2 2 n , 0.00015. This value is muchsmaller than the error of the measured refractive index.In other words, in our case the effect of the host bovineskin gelatin inside the BR film on the measured resultsof the refractive index of pure BR film can be ignored.

We also analyzed the effect of the host bovine skingelatin by using the effective-medium theory.18,19 Thecalculated error is smaller than 0.00005. Hence we takethe conservative estimation by Eq. (21) that the error issmaller than 0.00015.

5. SUMMARYThe critical-angle method is modified for the measure-ment of the refractive indices of bio-optical materials,which are typically absorptive and are difficult to makeinto good freestanding optical surfaces. This methodgenerally can be used for measuring the refractive in-dices of either solid or liquid materials. The refractiveindices of BR are given for a number of common laserwavelengths in the visible region. The accuracy of theresults is shown to be smaller than 0.002. The disper-sive behavior of BR film and the effect of the host bovineskin gelatin on the measured results are also discussed.

ACKNOWLEDGMENTSThe research is supported in part by the W. M. KeckFoundation, the New York Science and Technology Foun-dation through the New York State Center for ComputerApplications and Software Engineering, and the NationalInstitutes of Health under grant GM-34548.

Chunping Zhang is on leave from the Department ofPhysics, Nankai University, Tianjin, China.

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