Improved method for measuring small optical rotation angle of chiral medium

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Improved method for measuring small optical rotation angle

of chiral medium

Jiun-You Lin, Kun-Huang Chen, Der-Chin Su

*

Institute of Electro-Optical Engineering, National ChiaoTung University, 1001 Ta-Hsueh Road, Hsin-Chu 300, Taiwan, ROC Received 19 November 2003; accepted 20 April 2004

Abstract

Based on the principle of Feng et al., an improved optical method for measuring small rotational angle in chiral medium is proposed. When a quarter-wave plate and two analyzers with proper azimuth angles are arranged in the two outputs of a Mach-Zehnder interferometer (with sample been inserted in one of the light passages), the ﬁnal phase diﬀerence of the two interference signals used to determine the rotational angle is greatly enhanced, which is found to be a about 15 the result obtained with the Feng et al.’s method. The feasibility of the measuring method was demon-strated by our experimental results. This method should bear the merits of high accuracy, short sample medium length, and simpler operational endeavor.

Keywords: Optical rotation angle; Chiral medium; Heterodyne interferometry

1. Introduction

When linearly polarized light passes through a chiral medium, its plane of polarization rotates [1,2]. The amount of angular rotation per unit thickness is known as the rotatory power of the chiral medium, which can be measured with a polarimeter [3] constructed on the base of Malus’s law. Several methods such as the high accuracy universal polarimetry (HAUP) [4,5] and the quasi-heterodyne/heterodyne interferometry [6–10] were proposed to improve the rotatory power measurement. The HAUP method measures the output intensities at several different azimuthal angles of the polarizer and the analyzer by a conventional polarimetry setup. The exact relation between the output intensity and the angular positions of the pola-rizer and the analyzer are derived with fitting procedure. The desired rotational angle is obtained by re-moving systematic errors with either the reference crystal method or the method of Meekes et al. [5]. In the quasi-heterodyne interferometry [6–8], the rotational angle is determined by an intensity variation of an associated light beam. In the heterodyne interferometry [9,10], on the other hand, the rotational angle is inferred from the phase difference between two interference signals. All these methods did result in better

*

Corresponding author. Tel.: +886-3-573-1951; fax: +886-3-571-6631. E-mail address:[email protected](D.-C. Su).

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resolution and higher accuracy. However, they require thicker chiral media to achieve their derived results. Our method utilized Feng et al.’s [10] consideration enhances the phase diﬀerence of interference signals to improve resolution. Experimentally, the sample medium is situated in one arm of a Mach-Zehnder inter-ferometer and the two outputs from the interinter-ferometer separately pass through polarization components. When the azimuth angles of these components are chosen properly, the phase diﬀerence determined with heterodyne interferometric technique of the interference signal is greatly enhanced to result in accurate rotational angle. A solution of 6 105 _{deg from our experimental method is 15 times better than that}

obtained with Feng et al. 2. Principle

The experimental setup pertinent to this measurement is shown in Fig. 1. Laser light passing through a half-wave plate H has the Jones vector

Ei¼

cos h sin h

; ð1Þ

where h measures the fast axis at h=2 from the horizontal x-axis. The Jones vector after emerging from an electro-optic modulator (EO) driven with angular frequency x then becomes [11]

E_i0¼ e ixt=2 ₀ 0 eixt=2 cos h sin h ¼ cos h e ixt=2 sin h eixt=2 : ð2Þ

Light in Mach-Zehnder interferometer (Fig. 1) after been split by PBS travels in two paths: (a) PBS! Ma! S ! BS and (b) PBS ! Mb! BS, with the sample in the (a) path. The transmitted p- polarized

light and reﬂected s- polarized light at BS superimposed to produce the amplitude Etwhich is to end at Dt:

Et¼ Ep1þ Es1¼ cos h cos a sin a ei½ðxt=2Þ/Ma_{þ sin h} 0 1 ei½ðxt=2Þþkdþ/Mbþð/BS=2Þ_; _ð3Þ

whereas, the transmitted s-polarized light and the reﬂected p-polarized light also summed to bear the amplitude Erwhich is to end at Dr:

Er¼ Ep2þ Es2¼ cos h cos a e i/BS=2 sin a ei/BS=2 ei½ðxt=2Þ/Ma_{þ sin h} 0 1 ei½ðxt=2Þþkdþ/Mb_; _ð4Þ y z E'r ANr(β2) ANt(β1) Q(0˚) Et Er It Ir Dt Dr E't Driver H EO Lock-in Amp BS PBS Ma Mb S Laser

Fig. 1. Schematic diagram for measuring the optical rotation angle of a chiral medium, H: half-wave plate; EO: electro-optic mod-ulator; PBS: polarizing beam-splitter; Maand Mb: mirrors; S: sample medium; BS: beam-splitter; Q: quarter-wave plate; AN: analyzer;

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where subscripts p and s denote p- and s-polarizations, 1 and 2 mark the outputs from BS and a is the rotational angle of the incident light inflicted by the chiral medium. Let d be the optical path difference between these two paths, /BSbe the phase difference between p- and s-polarizations of the reflection at BS,

and /Ma and /Mb be that at Ma and Mb, respectively.

After passing through a quarter-wave plate Q (fast axis lying at the x-axis), and an analyzer ANt(with

the transmission axis being at b1 to the x-axis), Etbecomes E0t and is detected by Dt:

E0_t¼ ANtðb1Þ Qð0°Þ Et

¼ cos

2_b

1 sinb1cosb1

sinb1cosb1 sin 2 b1 _{1 0} 0 i cosh cosa sina ei½ðxt=2Þ/Ma þ sinh 0 1 ei½ðxt=2Þþkdþ/Mbþð/BS=2Þ ¼ A1cosh ei½ðxt=2Þþ/t/Ma þ A2sinh ei½ðxt=2Þþkdðp=2Þþ/Mbþð/BS=2Þ cosb1 sinb₁ : ð5Þ

The intensity of which hence is It¼ Et0 2 ¼ A2 1þ A 2 2þ 2A1A2cos xtð þ wtÞ; ð6Þ where w_t¼ /tþ kd þ ð/BS=2Þ þ /ð Mb /MaÞ ðp=2Þ; ð7Þ

/t¼ tan1ðtan b1tan aÞ; ð8Þ

A1¼ cos h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcos b1cos aÞ

2

þ ðsin b1sin aÞ 2

q

; ð9Þ

and

A2¼ sin h sin b1: ð10Þ

On the other hand, Erpassing through an analyzer ANr(with the transmission axis being at b2to the

x-axis) becomes E0

r and is detected by photodetector Dr with

E0_r¼ ANrðb2Þ Er

¼ cos

2_b

2 sin b2cos b2

sin b2cos b2 sin 2 b2 cos h cos a e i/BS=2 sin a ei/BS=2 ! ei½ðxt=2Þ/Ma ( þ sin h 0 1 ei½ðxt=2Þþkdþ/Mb ) ¼ B1cos h ei½ðxt=2Þþ/t/Ma þ B2sin h ei½ðxt=2Þþkdþ/Mb cos b2 sin b2 : ð11Þ

The intensity of which is Ir¼ E0r 2 ¼ B2 1þ B 2 2þ 2B1B2cos xtð þ wrÞ; ð12Þ where w_r¼ kd þ ð/Mb /MaÞ þ /r; ð13Þ /_r¼ tan1 cos bð 2þ aÞ cos bð 2 aÞ tan /ð _BS=2Þ ; ð14Þ

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B1¼ cos h ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2_b 2cos2aþ sin 2 b2sin 2 aþ1

2 sin 2a sin 2b2cos /BS r

; ð15Þ

and

B2¼ sin h sin b2: ð16Þ

The intensities It and Ir, are sent to a lock-in ampliﬁer for phase analysis. To obtain the ﬁnal phase

diﬀerence /ð¼ /t /rÞ, we calculate Dwð¼ wt wrÞ as Dw can be determined in advance. The phase

diﬀerence

Dw¼ wt wr¼ ð/t /rÞ ðp=2Þ þ ð/BS=2Þ ¼ / ðp=2Þ þ ð/BS=2Þ; ð17Þ

where /¼ Dw þ ðp=2Þ ð/BS=2Þ is obtained from Eq. (17) as /BS there can be estimated with Chiu’s

method [11].

Finally, the rotational angle a is determined with Eqs. (8) and (14). Therefore,

a¼ tan1 ½Cð1 EF Þ DðE F Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½Cð1 EF Þ DðE F Þ2þ 4CDð1 þ EF ÞðE þ F Þ q 2CDð1 þ EF Þ 8 < : 9 = ;; ð18Þ where C¼ tan b1, D¼ tan b2, E¼ tan /, and F ¼ tanð/BS=2Þ.

3. Experiments and results

We measured the rotational angles introduced by a half-wave plate with different azimuth angles, and six glucose solutions in different weight percent, namely 0.1%, 0.5%, 1%, 10%, 15%, and 20%. Each solution was held in a quartz cell of 10 mm long. A He–Ne laser 632.8 nm line modulated by an electro-optic modulator (Model 4002, New Focus) was served as the heterodyne light source. The heterodyne generates a frequency difference of 1 kHz for p- and s-polarized light. In order to have better contrast, the conditions of h¼ 3:5°, b1¼ 88:0°, and b2¼ 85:0° were chosen, while /BS¼ 25:5° was measured in advance. At first, we

inserted a half-wave plate into the interferometer to replace the sample cell. The results measured at some

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -20 -10 0 10 20 30 40 A: this method

B: the method of Feng et al.

α

φ

(deg)

= θh /2 (deg)

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chosen angles are marked as ‘‘o’’ in Fig. 2 and that of theoretical calculations in smooth curve, where the ordinate represents /, and abscissa represents the rotational angle a expressed in a half of the azimuth angle hhof the fast axis of the half-wave plate. For comparison, the theoretical curves of / versus a calculated by

this method (curve A) and that by the method of Feng et al. (curve B) are also presented where, in small angle region, the slope of curve A is seen to be almost 15 that of curve B. It is thus seen that the sensitivity of our method is more than an order of magnitude greater. The measured / and their associated a values of the samples are listed in Table 1, where the speciﬁc rotations [as] are also included. The speciﬁc rotation

quantity is deﬁned by½as ¼ a=C L, with C being the concentration of the chiral medium, and L the length

of the cell holding chiral medium. The data of ½as in each solution are obtainable from [9,12,13]. By

substituting the information into this equation, the associated reference values of rotational angle aref were

calculated and also presented in Table 1 for comparison.

4. Measurement resolution

To ﬁnd the resolution, Da, we calculate tan/ and its diﬀerential deviation. From Eqs. (8) and (14), we arrive at

tan /¼ tan b1 tan a cosðb2 aÞ tanð/BS=2Þ cosðb2þ aÞ cosðb2 aÞ þ tan b1 tanð/BS=2Þ tan a cosðb2þ aÞ

: ð19Þ

As a is small quantity (a < 1°), Eq. (19) can be cast into tan /ﬃ tan b1 tan a tanð/BS=2Þ

1þ tan b1 tanð/BS=2Þ tan a

; ð20Þ

and, thus, the errors Da and D/ can be related as follows:

Daﬃ 1 sec2_a sec2 _/ BS=2 ð Þ sec2_/ 1 tan /BStan / ð Þ tan b1 D/: ð21Þ

When the second harmonic uncertainties and the polarization-mixing uncertainties [14] were considered, the net phase diﬀerence uncertainties D/ reduced to 0.0014°. Substituting our experimental conditions b₁¼ 88:0°, b2¼ 85:0°, /BS¼ 25:5°, and D/ ¼ 0:0014° into Eq. (21), we have Da ¼ 6 105 deg.

According to the deﬁnition of rotatory power, aL¼ 2pg=k [15], we have Da ¼ ð2pDgLÞ=k, with g and Dg as the chiral parameter [16] and its error. Substituting our experimental conditions L¼ 10 mm, k ¼ 632:8

nm, and Da¼ 6105 _{deg. into this equation, we obtain Dg}_{¼ 1:110}11_{. As we had reported}

Dg¼ 2 108 _{in our previous paper [17], we also increased the resolution in the present method.} Table 1

Experimental results and the reference data

Solutions / a aref Glucose (w¼ 0.1%) )12.880 )0.00389 )0.00448 Glucose (w¼ 0.5%) )13.563 )0.02415 )0.02240 Glucose (w¼ 1%) )14.322 )0.04677 )0.04480 Glucose (w¼ 10%) )28.660 )0.47499 )0.46502 Glucose (w¼ 15%) )37.080 )0.73489 )0.71187 Glucose (w¼ 20%) )44.790 )0.98510 )0.96858

/(degree): ﬁnal phase diﬀerence; a (degree): optical rotation angle; aref(degree): calculated optical rotation angle from [9,12,13].½as:

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5. Conclusion

Based on the principle of Feng et al., an improved method for measuring small optical rotation angle in chiral medium is proposed. Two groups of output beams from a Mach-Zehender interferometer are ar-ranged to pass through several polarization components separately and recombine for interference. The phase difference was measured with the heterodyne interferometric technique and the associated optical rotation angle was estimated. When azimuth angles of polarization components were chosen appropriately, the final phase difference / between these two interference signals increased greatly. Hence, the required optical length of the sample medium can be decreased dramatically. It takes only 1/15 of the length required by the method of Feng et al. The feasibility of this method was demonstrated by its measurement resolution of 6 105 _deg.

Acknowledgements

This study was supported in part by National Science Council, Taiwan, ROC, under contract No. NSC 92-2215-E-009-052.

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