A new method for measuring the chiral parameter and the average refractive index of a chiral liquid

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(1)Optics Communications 218 (2003) 317–323 www.elsevier.com/locate/optcom. A new method for measuring the chiral parameter and the average refractive index of a chiral liquid Jiun-You Lin, Der-Chin Su * Institute of Electro-Optical Engineering, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsin-Chu 300, Taiwan, ROC Received 10 July 2002; received in revised form 15 January 2003; accepted 23 January 2003. Abstract We designed a method for measuring chiral parameter and average refractive index of an isotropic chiral medium using critical angle phenomena. Linearly polarized light is guided to project onto the interface of a semi-spherical glass and a chiral liquid. The reflected light passes through an analyzer to extract the interference signal of s- and p-polarization lights. Their phase difference is first optimized by a suitable optical arrangement and subsequently measured by a heterodyne interferometer. The result, then, is fed into equations to estimate the chiral parameter. We determine the average index of refraction from the critical angle occurred at the discontinuity of the phase difference of two polarized lights. Our method of measurement has the implicational merits of both common-path interferometry and heterodyne interferometry. Ó 2003 Elsevier Science B.V. All rights reserved.. 1. Introduction Optical activity (gyrotropy) is a property of the non-centrosymmetric media of chiral structural elements. An isotropic chiral liquid is described by Drude–Born–Fedorov constitutive relations [1–4] (in Gaussian units) ~ ¼ e½~ D E þ br ~ E;. ð1aÞ. ~ ~ þ br H ~ ; B ¼ l½H. ð1bÞ. where b is the gyrotropy, e the average dielectric constant and l the permeability. Through use of MaxwellÕs equations, and assumption of plane*. Corresponding author. Tel.: +886-3-573-1951; fax: +886-3571-6631. E-mail address: t7503@faculty.nctu.edu.tw (D.-C. Su).. wave solutions of frequency x0 and wave vector ~ k ¼ k k^, Eqs. (1a) and (1b) can be expressed as follows: h i ~ D¼e ~ E þ if ð~ k ~ EÞ=nk0 ; ð2aÞ i h ~ ~ þ if ~ ~ =nk0 ; B¼l H kH. ð2bÞ. pffiffiffiffiffi where k0 ¼ x0 =c, n ¼ el is the average refractive index and f ¼ ðelÞ 1=2 k0 b provides a convenient measure of the gyrotropy of the medium. If nþ and n are the refractive indices of the leftand the right-circularly polarized lights in the chiral liquid, then we have n ¼ n g. Here, g is the chiral parameter. In addition, g can be expressed as g ¼ nf . The chiral parameter is ordinarily very much less than unity (g

(2) 10 4 10 7 Þ for many organic and inorganic materials.. 0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01183-0.

(3) 318. J.-Y. Lin, D.-C. Su / Optics Communications 218 (2003) 317–323. A chiral parameter and an average refractive index are important characteristic constants of a chiral liquid [5]. The chiral parameter is so smaller (1) that its effect on the amplitude and the phase of the light reflected from the chiral liquid are nearly unnoticeable. Therefore, almost all of the previous methods measured the light transmitted through a chiral liquid to estimate its chiral parameter [6–9]. These transmission methods required relative larger amount of liquid to make the chiral parameter measurable. Although many methods were proposed for measuring the average refractive index, to the best of our knowledge, there is no single optical setup that can measure the chiral parameter and the average refractive index simultaneously. To get around of this point we present, an alternative method for measuring the chiral parameter and the average refractive index of a chiral liquid in one single optical configuration. Linearly polarized light is incident on the interface between a glass semi-sphere and a chiral liquid. The reflected light passes through an analyzer and the desired polarizing components are extracted to interfere. If the incident angle is slightly smaller than the critical angle and the polarization plane of the incident light is nearly orthogonal to the transmission axis of the analyzer, then the phase difference between s- and ppolarizations of the reflected light will be enhanced from 2 to 4 orders of magnitude, and thus becomes measurable by a heterodyne interferometry [10,11]. Substituting the measured result into specially derived equation Eq. (15) (infra vide), the chiral parameter can be estimated. In addition, this phase difference changes abruptly as the incident angle is approaching the critical angle. This limiting angle then is taken to be the critical angle. Consequently, the average refractive index can be calculated. In addition, our method is benefited from the advantages of the common-path interferometry and the heterodyne interferometry.. Fig. 1. Schematic diagrams for measuring (a) the phase difference owing to reflection at the interface between a glass hemisphere and a chiral liquid and (b) the initial phase difference of the reference signal. H, half-wave plate; EO, electro-optic modulator; LVA, linear voltage amplifier; FG, function generator; AN, analyzer; D, detector.. the x-axis goes into the paper plane perpendicularly. A beam of linearly polarized laser light passes through a half-wave plate H, whose polarization plane is hp to the x-axis. Its Jones vector can be written as ! cos hp Ei ¼ : ð3Þ sin hp. 2. Principle The schematic diagram of this method is shown in Fig. 1(a). For convenience, the +z-axis is chosen to be along the direction of light propagation and. This linearly polarized light passes through an electro-optic modulator (EO) with the fast axis lying along the x-axis. An external sawtooth voltage signal from a driver consisting of a linear.

(4) J.-Y. Lin, D.-C. Su / Optics Communications 218 (2003) 317–323. voltage amplifier (LVA) and a function generator (FG) is applied to the EO. Its angular frequency and amplitude are x and Vk=2 (half-voltage of EO), respectively. The phase retardation produced by the EO can be expressed as xt, and the Jones vector of the light becomes Ei0 ¼ EOðxtÞ Ei ixt=2 . cos hp e 0 ¼ sin hp 0 e ixt=2 . cos hp eixt=2 ¼ : sin hp e ixt=2. ð6aÞ r22 ffi ð cos h q1 Þ=ðcos h þ q1 Þ;. ð6bÞ. r12 ¼ r21. h i i ðn=n0 Þ2 ðzþ z Þ cos h n oi ; ¼h 2 ðcos h þ q1 Þ ðn=n0 Þ cos h þ q1. ð4Þ. The light penetrates into a glass semi-sphere of refractive index n0 with incident angle h onto the interface between the semi-sphere and the chiral liquid. The light reflected from this interface passes through an analyzer AN with the transmission axis being at a with respect to the x-axis, and enters a photodetector D. Consequently, the Jones vector of the light becomes Et ¼ ANðaÞTag ð0 Þ SðRÞ Tga ð0 Þ Ei0. 0 t0 0 sin a cos a cos2 a ¼ 0 t00 sin a cos a sin2 a . r11 r12 t0 0 r21 r22 0 t0 . cos hp eixt=2 sin hp e ixt=2 . cos a ¼ Aeiððxt=2Þþd1 Þ þ Be iððxt=2Þ d2 Þ ; sin a. 319. h i h i 2 2 r11 ffi ðn=n0 Þ cos h q1 = ðn=n0 Þ cos h þ q1 ;. ð5Þ. where SðRÞ is the Jones matrix of the chiral liquid as the light is reflected from it, and Tga ð0°Þ and Tag ð0°Þ are Jones matrices, t0 and t00 are transmission coefficients, as the light propagates from the air to the glass semi-sphere and vice versa, respectively. Although the chiral liquid has a weak light-induced effect of magnetization, it may be treated as an intrinsically non-magnetic material. Hence its permeability is nearly equivalent to unity. Based on Drude–Born–Fedorov constitutive relations, MaxwellÕs equations and electrodynamic boundary conditions, the chiral reflection coefficients r11 ; r12 ; r21 and r22 can be expressed as [12,13]. ð6cÞ. respectively; and i1=2 h 2 q1 ¼ ðn=n0 Þ sin2 h ;. ð6dÞ. z ¼ cos h ;. ð6eÞ. zþ z ffi 2n0 g sin2 h=n2 q1 ;. ð6fÞ. A ¼ t00 t0 cos hp . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðr11 cos aÞ ðr21 sin aÞ ;. ð7aÞ. B ¼ t00 t0 sin hp . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2. ðr12 cos aÞ þ ðr22 sin aÞ ;. ð7bÞ. d1 ¼ Argðr11 þ r21 tan aÞ;. ð7cÞ. d2 ¼ Argðr22 þ r12 cot aÞ;. ð7dÞ. where h represent the refractive angles of the leftand the right-circularly lights in the chiral liquid [9] as shown in Fig. 2. Hence, the intensity measured by D is It ¼ jEt j2 ¼ A2 þ B2 þ 2AB cosðxt þ /Þ;. ð8Þ. where / is given as / ¼ d1 d2 ¼ Argðr11 þ r21 tan aÞ Argðr22 þ r12 cot aÞ: ð9Þ Here, It is the test signal. On the other hand, the electrical signal generated by the function generator FG is filtered and becomes the reference signal. So, the reference signal has the form of Ir ¼ I ½1 þ cosðxt þ /r Þ;. ð10Þ.

(5) 320. J.-Y. Lin, D.-C. Su / Optics Communications 218 (2003) 317–323. M2 ¼. sin 2h sin h h i : 2 n0 q1 ðcos h q1 Þ ðn=n0 Þ cos h q1 . ð13bÞ As h is slightly smaller than the critical angle hc , we obtain M1 ffi M2 . In general, the parameter g of a chiral liquid is smaller than 10 4 , and the values M1 and M2 are in the range of 10–103 . So we have the condition M1 M2 g2 1. Hence Eq. (12) can be written as. Fig. 2. Geometry of chiral reflection. kt : wave vector of predicted transmitted ray in the absence of chirality.. where /r is the initial phase. These two sinusoidal signals are sent to a lock-in amplifier, as shown in Fig. 1(a). The phase difference between the reference signal and test signal, 0. / ¼ / /r. ð11Þ. can be obtained. In the second measurement, the test beam is allowed to enter the photodetector D directly without the reflection at the interface between the glass semi-sphere and the chiral liquid, as shown in Fig. 1(b). The test signal still has the form of Eq. (8), but in this time with / ¼ 0. Therefore the lock-in amplifier in Fig. 1(b) represents /r . Substituting /r into Eq. (11), we can obtain the phase difference /. Then, substituting Eqs. (6a)–(6f) into Eq. (9), we have / ffi tan 1 ðM1 g tan aÞ tan 1 ðM2 g cot aÞ . ðM1 tan a þ M2 cot aÞg ¼ tan 1 ; 1 M1 M2 g 2. ð12Þ. sin 2h sin h h i ; n0 q1 ðcos h þ q1 Þ ðn=n0 Þ2 cos h q1 ð13aÞ. and. 1. 0. 1. ð14Þ Under this condition, if a is so chosen near 0 or 90°, then j/j has an extreme value. Although / is independent of /p , it is obvious from Eqs. (7a) and (7b) that the contrast of the test signal depends on /p . To enhance the contrast of the test signal, either the following two conditions should be chosen: (i) a is near 0° and hp is near 90°; (ii) a is near 90° and /p is near 0°. Consequently, Eq. (14) can be rewritten as 2 n0 q1 tan / 4 tan a gffi. sin 2h sin h ðcos h þ q1 Þ ðn=n0 Þ2 cos h q1 þ. where M1 ¼. / ffi tan 1 ½ ðM1 tan a þ M2 cot aÞg 8 < sin 2h sin h g ¼ tan 1. : n0 q 1 2 tan a 4 2 ðcos h þ q1 Þ ðn=n0 Þ cos h q1 39 = cot a 5 : þ ; ðcos h q Þ ðn=n Þ2 cos h þ q. cot a. 3 1. 5 : 2 ðcos h q1 Þ ðn=n0 Þ cos h þ q1 ð15Þ. It is seen from Eq. (15) that if n and / are specified, then g can be calculated with the measurement of /. According to SnellÕs law, we obtain nþ ¼ n0 sin hcþ , and n ¼ n0 sin hc , where hcþ and hc represent the critical angles of left- and right-circularly polarized lights. Since the difference ðnþ n Þ is.

(6) J.-Y. Lin, D.-C. Su / Optics Communications 218 (2003) 317–323. 321. extremely small, we have the condition hcþ ffi hc ffi hc , where hc is the critical angle associated to the average refractive index. According to Eq. (14), we can see that / will be changed abruptly as h ffi hc . Based on this fact, hc can be measured accurately. Substituting this data into the following equation: ð16Þ. n ¼ n0 sin hc ; n can be obtained.. 3. Experiments and results In order to show the validity of this method, a He–Ne laser with 632.8 nm wavelength and an EO (Model 4002 Broadband, Newfocus) with a halfwave voltage 156 V were used to measure a 50% (wt%) glucose solution and a 50% (wt%) saccharose solution. The frequency of the sawtooth signal applied to the EO was 1 kHz. A BK7 glass semi-sphere with 1.51509 refractive index and the tested chiral liquid were mounted together on a high-precision rotation stage (Model MURM100PP, Newfocus) with an angular resolution 0.001°. A locked-in amplifier (Model SR850, Stanford) with an angular resolution 0.001° was used to measure the phase difference. The experiments were operated at hp ¼ 4°, a ¼ 88° and with at room temperature 21 °C. First, the tested chiral liquid was rotated slowly to identify the critical angle hc . Next, / was measured with a 0.001° angular step from the incident angle h, which is slightly small than the critical angle hc . Their measurement results and theoretical results shown in Figs. 3(a) and (b), respectively; where denotes the measured data, and the solid lines are depicted by substituting the reference values nref and gref into Eq. (14). The reference values nref are obtained from [14], and gref can also be calculated with the following equation [15]:. Fig. 3. Measurement results and theoretical curves of / versus h of (a) a 50% glucose solution and (b) a 50% saccharose solution, respectively.. h ðdegÞ ð2p=kÞ gref L ðmÞ ð180=pÞ ¼ Cv L ðdmÞ Cv L ðdmÞ ð36=kÞgref ¼ ; ð17Þ Cv. ½as ¼. where Cv ðg=cm3 Þ is the volume concentration, ½as (deg/(dm g/cm3 )) the specific rotation (taken. Table 1 Experimental results and the corresponding reference data Solution. h. /. gð107 Þ. gref ð107 Þ. hc. n. nref. Saccharose ðCw ¼ 50%Þ Glucose ðCw ¼ 50%Þ. 69.548° 69.340°. )0.160° )0.130°. 6.4 4.9. 6.2 4.8. 69.600° 69.381°. 1.42007 1.41804. 1.4201 1.4181. Note. 1. Cw : wt% concentration; 2. ½as ¼ 44:8 for glucose at 632.8 nm; ½as ¼ 55:85 for saccharose at 632.8 nm, which is obtained with curve-fitting technique..

(7) 322. J.-Y. Lin, D.-C. Su / Optics Communications 218 (2003) 317–323. from [7,14]) and L the optical path length of the chiral liquid. Table 1 lists essential experimental and calculated parameters of the two chiral samples, together with the ½as values from [7,14] for reference. From Fig. 3, it is obvious that the value j/j changes abruptly as h increases to approach hc .. 4. Measurement resolution of the method If the conditions h ¼ a ¼ 45° are chosen as that used in the general polarization interferometer, then j/j is in the range 0–0:001° as h < hc . / thus is too small to be detected with a commercial high-resolution phase meter. In our experiments, we have achieved the measurement of j/j to be about 0:1–0:2°. It is obvious that the value of j/j has been enhanced from 2 to 4 orders of magnitude. While h > hc in our experimental conditions, / occurs due to the total internal reflection and it is independent of the chiral parameter. Under this condition the average refractive index can be estimated from the data of /, as reported in [10]. From Eq. (16) we get Dn ¼ jn0 cos hc j Dh;. ð18Þ. where Dn and Dh are the uncertainties in n and h, respectively. Substituting the angular resolution Dh ¼ 0:001° of the rotation stage and hc into Eq. (18), we have Dn ffi 1 10 5 for both two tested solutions in our experiments. Under our experimental conditions, Eq. (15) can be simplified as h i n0 q1 ð cos h þ q1 Þ ðn=n0 Þ2 cos h q1 / gffi. : 2 sin2 h cos h tan a ð19Þ So we have dg dg dq1 dg D/ þ Dh þ Dg ¼ d/ dq1 dh dh . dg dq1 dg þ Dn; þ dq1 dn dn. where Dg; D/; Dh and Dn are the uncertainties in g, /; h and n, respectively. Considering the second harmonic uncertainties and the polarization-mixing uncertainties, the total phase difference uncertainty D/ can decrease to 0.002° in our experiments [11]. Substituting these data, Dh ¼ 0:001°; Dn ffi 1 10 5 ; a ¼ 88°; n0 ¼ 1:51509; the estimated values of n and g into Eq. (20), we can depict the relation curve of Dg versus h, as shown in Fig. 4. As h approaches hc , the last two terms in Eq. (20) approach infinite and Dg changes abruptly. This fact can be seen that Dg in Fig. 4. Both of them have the same resolution Dg ffi 2 10 8 as h is in the ranges of 69:3–69:35° and 69:45–69:55°, respectively.. (a). (b). ð20Þ. Fig. 4. Calculated curves of Dg versus h of (a) a 50% glucose solution and (b) a 50% saccharose solution, respectively..

(8) J.-Y. Lin, D.-C. Su / Optics Communications 218 (2003) 317–323. 323. 5. Conclusion. References. We have here presented an alternative method for measuring the chiral parameter and the average refractive index of a chiral liquid using only one set of optical configuration. The phase difference between s- and p-polarizations of the reflected light is enhanced and measured under appropriate conditions. Because of the experiment common-path configuration and heterodyne phase measurement, it has such merits as high stability and high resolution. We measured both 50% glucose solution and 50% saccharose solution with an accuracy of 1 10 5 for the average refractive index and 2 10 8 for the chiral parameter, respectively.. [1] A. Lakhtakia, V.K. Varadan, V.V. Varadan, J. Mater. Res. 8 (1993) 917. [2] M.P. Silverman, J. Badoz, J. Opt. Soc. Am. A 7 (1990) 1163. [3] M.P. Silverman, J. Opt. Am. A 3 (1986) 830. [4] M.P. Silverman, J. Badoz, B. Briat, Opt. Lett. 17 (1992) 886. [5] M.P. Silverman, N. Ritchie, G.M. Cushman, B. Fisher, J. Opt. Soc. Am. A 5 (1988) 1852. [6] H.J. King, C. Chou, H. Chang, Y.C. Huang, Opt. Commun. 110 (1994) 259. [7] T.W. King, G.L. Cote, R. McNichols, M.J. Goetz Jr., Opt. Eng. 33 (1994) 2746. [8] C. Chou, Y.C. Huang, C.M. Feng, M. Chang, Jpn. J. Appl. Phys. 36 (1997) 356. [9] G.L. Cote, M.D. Fox, R.B. Northrop, IEEE Trans. Biomed. Eng. 39 (1992) 752. [10] M.H. Chou, J.Y. Lee, D.C. Su, Appl. Opt. 36 (1997) 2936. [11] M.H. Chiu, J.Y. Lee, D.C. Su, Appl. Opt. 38 (1999) 4047. [12] M.P. Silverman, J. Badoz, J. Electromagn. Waves Appl. 6 (1992) 587. [13] M.P. Silverman, J. Badoz, Opt. Commun. 74 (1989) 129. [14] R.C. West (Ed.), Handbook of Chemistry and Physics, CRC Press, 61st ed., 1981, pp. D227-270 and E-418. [15] N. Berova, K. Nakanishi, R.W. Woody (Eds.), Circular Dichroism: Principles and Applications, second ed., Wiley-VCH, 2000 (Chapter 1).. Acknowledgements This study was supported in part by National Science Council, Taiwan, ROC, under contract NO. NSC 90-2215-E-009-077..

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