Spectroscopic Mueller matrix polarimeter using four-channeled spectra

Optics Communications 281 (2008) 5725–5730

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Spectroscopic Mueller matrix polarimeter using four-channeled spectra Yukitoshi Otani a, Toshitaka Wakayama a,*, Kazuhiko Oka b, Norihiro Umeda a a b

Department of Mechanical Systems of Engineering, Tokyo University of Agriculture and Technology, Koganei, Tokyo, Japan Division of Applied Physics, Hokkaido University, Sapporo, Hokkaido, Japan

a r t i c l e

i n f o

Article history: Received 18 January 2008 Received in revised form 24 July 2008 Accepted 14 August 2008

Keywords: Mueller matrix Spectropolarimeter Birefringence Channeled spectrum

a b s t r a c t A Mueller matrix polarimeter acquired for four-channeled spectra is proposed. Both the polarizing and analyzing optics of this system consist of a linear polarizer and a high-order retarder. The polarizing elements can modulate the polarization states in the wavenumber space. By applying a Fourier transform method to a single-channeled spectrum, nine elements of the Mueller matrix can be deconvoluted without modifying the configuration of either the polarizing or analyzing optics. It is thus possible to determine the wavelength dependence of all the Mueller matrix elements from four-channeled spectra obtained using four different configurations for the polarizing and analyzing optics. The performance of this method is evaluated by measuring polarization properties, such as retardance, azimuthal angle, and linear diattenuation, from the obtained Mueller matrix in wavenumber space. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction Mueller matrices describe the polarization properties of objects, including linear and circular retardance, diattenuation, and depolarization. A conventional Mueller matrix polarimeter determines Mueller matrices by using two quarter-wave plates as the polarizing and analyzing optics; these quarter-wave plates rotate synchronously at angular speeds of x and 5x [1–5]. As a result of the recent development of electro-optical modulators, Pockels cells, photoelastic modulators, and liquid-crystal variable retarders have been employed in the polarizing and analyzing optics of Mueller matrix polarimeters [6,7]. Although these components of polarimeters have limited spectral ranges, they have several advantages, including an absence of moving parts and fast measurement times. Determining the wavelength dependence of Mueller matrices is extremely important in the fields of material technology and biological science. Against this background, several methods have been proposed for calibrating the wavelength dependence of Mueller matrices [2,8,9]. In this paper, we present a method that differs from these previously proposed calibration techniques for determining the wavelength dependence of a Mueller matrix. Our group has been engaged in developing polarization measurement methods; in particular, we have developed a polarimeter that acquires channeled spectrum and a method for measuring birefringence dispersion [10–13]. A channeled spectrum is generated when white light passes through a material that has a high de* Corresponding author. Present address: Yamane 1397-1, Hidaka, Saitama Medical University, Japan. Tel./fax: +81 42 984 4819. E-mail address: [email protected] (T. Wakayama). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.08.017

gree of birefringence and that is located between a pair of polarizers. A snapshot spectroscopic Mueller matrix polarimeter was proposed in 2007 [14]. It can measure the Mueller matrix using only a single-channeled spectrum. However, there currently appear to be some difficulties in using this polarimeter in practical applications for industry and biological science, because of the trade-off in spectral resolution. The instrument is also too complicated to determine the polarization properties of a sample from the encoded channeled spectrum. On the other hand, we have demonstrated a single-shot birefringence dispersion measurement method that has the potential to determine a Mueller matrix [12,13]. In this method, a single-channeled spectrum can be used to determine the nine elements of a Mueller matrix without moving the polarizing or analyzing optics. Based on this method, we describe a spectroscopic Mueller matrix polarimeter that acquires fourchanneled spectra.

2. Principle of spectroscopic Mueller matrix polarimeter using four-channeled spectra Fig. 1 shows a simplified schematic of a spectroscopic Mueller matrix polarimeter that has polarizing and analyzing optics. Light from a white light source is collimated using a lens and an iris. This light is polarized by the polarizing optics, which consists of a polarizer P and a retarder R1 that are oriented at angles of h1 and h2, respectively. After passing through a sample that has a Mueller matrix M, the polarized light passes through the analyzing optics, which consists of a second retarder R2 and an analyzer A that are oriented at angles of h3 and h4, respectively. The first and second retarders have retardances of d1(k) and d2(k), respectively, where

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Y. Otani et al. / Optics Communications 281 (2008) 5725–5730

white light source

Lens Iris P(θ1) R1(δ1,θ2)

Polarizing Optics (PO)

R2(δ2,θ3) M

A(θ4) Iris

Analyzing Optics (AO)

Lens spectrometer

Fig. 1. Optical configuration.

k is the wavenumber. From Mueller calculus, the Stokes vector Sout of the output light is given by

S out ¼ Aðh4 Þ  R2 ðd2 ðkÞ; h3 Þ  M  R1 ðd1 ðkÞ; h2 Þ  Pðh1 Þ  S in ;

ð1Þ

where Sin indicates the Stokes vector of the incident light. When the optical configuration of the polarimeter (h1, h2, h3, h4) has values (0°, 45°, 0°, 45°), (0°, 45°, 45°, 0°), (45°, 0°, 0°, 45°), and (45°, 0°, 45°, 0°), we obtain the four-channeled spectra shown in Table 1. From Mueller calculus, the intensities of the channeled spectra can be expressed as follows:

b1þ ðkÞ ¼ m23 ðkÞ þ m31 ðkÞ;

ð3:22Þ

b2þ ðkÞ ¼ m13 ðkÞ  m31 ðkÞ; b3þ ðkÞ ¼ m32 ðkÞ  m23 ðkÞ;

ð3:23Þ ð3:24Þ

b4þ ðkÞ ¼ ðm13 ðkÞ þ m32 ðkÞÞ:

ð3:25Þ

Here mij are the 16 elements of the Mueller matrix M. Rearranging Eq. (2) to assist the analysis of the channeled spectra gives

IN ðkÞ ¼ 1=8½biasN ðkÞ þ ampN1 ðkÞ cosfd1 ðkÞ  /N1 ðkÞg þ ampN ðkÞ cosfd1 ðkÞ  d2 ðkÞ  /N ðkÞg þ ampN2 ðkÞ  cosfd2 ðkÞ  /N2 ðkÞg þ ampNþ ðkÞ cosfd1 ðkÞ þ d2 ðkÞ

IN ðkÞ ¼1=8½aN0 ðkÞ þ aN1 ðkÞ cos d1 ðkÞ þ bN1 ðkÞ sin d1 ðkÞ

 /Nþ ðkÞg;

þ aN ðkÞ cosfd1 ðkÞ  d2 ðkÞg þ bN ðkÞ sinfd1 ðkÞ where,

 d2 ðkÞg þ aN2 ðkÞ cos d2 ðkÞ þ bN2 ðkÞ sin d2 ðkÞ þ aNþ ðkÞ cosfd1 ðkÞ þ d2 ðkÞg þ bNþ ðkÞ sinfd1 ðkÞ þ d2 ðkÞg

ð4Þ

ð2Þ

where N is the acquisition number of four-channeled spectra. aN0(k), aN1(k), bN1(k), aN(k), bN(k), aN2(k), bN2(k), aN+(k), and bN+(k) are the Fourier coefficients that appear in the Mueller matrix. The Fourier coefficients from aN0 to bN+ are given by

a10 ðkÞ ¼ a20 ðkÞ ¼ a30 ðkÞ ¼ a40 ðkÞ ¼ 2m00 ðkÞ;

ð3:1Þ

a11 ðkÞ ¼ a21 ðkÞ ¼ 2m01 ðkÞ;

ð3:2Þ

b11 ðkÞ ¼ b21 ðkÞ ¼ 2m03 ðkÞ;

ð3:3Þ

a31 ðkÞ ¼ a41 ðkÞ ¼ 2m02 ðkÞ;

ð3:4Þ

b31 ðkÞ ¼ b41 ðkÞ ¼ 2m03 ðkÞ;

ð3:5Þ

a1 ðkÞ ¼ m21 ðkÞ þ m33 ðkÞ;

ð3:6Þ

a2 ðkÞ ¼ m11 ðkÞ  m33 ðkÞ; a3 ðkÞ ¼ m22 ðkÞ  m33 ðkÞ;

ð3:7Þ ð3:8Þ

a4 ðkÞ ¼ m12 ðkÞ þ m33 ðkÞ;

ð3:9Þ

b1 ðkÞ ¼ m23 ðkÞ  m31 ðkÞ;

ð3:10Þ

b2 ðkÞ ¼ m13 ðkÞ þ m31 ðkÞ;

ð3:11Þ

biasN ðkÞ ¼ aN0 ðkÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ampN1 ðkÞ ¼ a2N1 ðkÞ þ bN1 ðkÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ampN ðkÞ ¼ a2N ðkÞ þ bN ðkÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ampN2 ðkÞ ¼ a2N2 ðkÞ þ bN2 ðkÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ampNþ ðkÞ ¼ a2Nþ ðkÞ þ bNþ ðkÞ;

ð5:1Þ ð5:2Þ ð5:3Þ ð5:4Þ ð5:5Þ

1

/N1 ðkÞ ¼ tan ðbN1 ðkÞ=aN1 ðkÞÞ;

ð5:6Þ

/N ðkÞ ¼ tan1 ðbN ðkÞ=aN ðkÞÞ;

ð5:7Þ

1

/N2 ðkÞ ¼ tan ðbN2 ðkÞ=aN2 ðkÞÞ;

ð5:8Þ

/Nþ ðkÞ ¼ tan1 ðbNþ ðkÞ=aNþ ðkÞÞ

ð5:9Þ

and biasN(k), ampN1(k), ampN(k), ampN2(k), ampN+(k), /N1(k), /N(k), /N2(k), and /N+(k) are the bias component, the amplitude components of each term and the arguments of the a and b components in Eq. (2), respectively. The intensities of the channeled spectra are modulated by the phase retardances d1(k), d1(k)  d2(k), d2(k), and d1(k) + d2(k); Using the thickness of retarder di and birefringence Dn(k), these phase retardances Ci(k) are approximately proportional to the wavenumber k [11].

b3 ðkÞ ¼ ðm23 ðkÞ þ m32 ðkÞÞ;

ð3:12Þ

b4 ðkÞ ¼ m32 ðkÞ  m13 ðkÞ;

ð3:13Þ

Ci ðkÞ ¼ 2pdi DnðkÞ  k ¼ 2pLi  k þ Ui ðkÞ;

a12 ðkÞ ¼ a32 ðkÞ ¼ 2m20 ðkÞ;

ð3:14Þ

where

a22 ðkÞ ¼ a42 ðkÞ ¼ 2m10 ðkÞ;

ð3:15Þ

b12 ðkÞ ¼ b32 ðkÞ ¼ 2m30 ðkÞ; b22 ðkÞ ¼ b42 ðkÞ ¼ 2m30 ðkÞ;

ð3:16Þ ð3:17Þ

 1 dCi  Li ¼ ¼ 2p dk k0

a1þ ðkÞ ¼ m21 ðkÞ  m33 ðkÞ;

ð3:18Þ

a2þ ðkÞ ¼ m11 ðkÞ þ m33 ðkÞ;

ð3:19Þ

a3þ ðkÞ ¼ m22 ðkÞ þ m33 ðkÞ;

ð3:20Þ

a4þ ðkÞ ¼ m12 ðkÞ  m33 ðkÞ;

ð3:21Þ

!  Dn Dnðk0 Þ þ  k0  di ; dk k0  2 1 d Ci  2 Ui ðkÞ ¼ fCi ðkÞ  2pLi k0 g þ  ðk  k0 Þ þ    ; 2 dk2  k

ð6:1Þ

ð6:2Þ ð6:3Þ

0

C1 ðkÞ ¼ d1 ðkÞ; C ðkÞ ¼ d1 ðkÞ  d2 ðkÞ; C2 ðkÞ ¼ d2 ðkÞ and Cþ ðkÞ ¼ d1 ðkÞ þ d2 ðkÞ

ð6:4Þ

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Y. Otani et al. / Optics Communications 281 (2008) 5725–5730 Table 1 Combination of optical configuration, channeled spectrum, Fourier spectrum, and Muller matrix

By using Euler’s theorem, the intensity given in Eq. (4) can be rearranged as

F1 ½AN0 ðmÞ ¼ aN0 ðkÞ; F ½C N1 ðmÞ ¼ cN1 ðkÞ;

ð9:2Þ

1 IN ðkÞ ¼ aN0 ðkÞ þ cN1 ðkÞ þ cN1 ðkÞ þ cN ðkÞ þ cN ðkÞ þ cN2 ðkÞ 8  þ cN2 ðkÞ þ cNþ ðkÞ þ cNþ ðkÞ ð7:1Þ

F1 ½C N ðmÞ ¼ cN ðkÞ;

ð9:3Þ

where

aN0 ðkÞ ¼ biasN ðkÞ; 1 ampN1 ðkÞ  exp½ifC1 ðkÞ  /N1 ðkÞg; 2 1 cN ðkÞ ¼ ampN ðkÞ  exp½ifC ðkÞ  /N ðkÞg; 2 1 cN2 ðkÞ ¼ ampN2 ðkÞ  exp½ifC2 ðkÞ  /N2 ðkÞg; 2 1 cNþ ðkÞ ¼ ampNþ ðkÞ  exp½ifCþ ðkÞ  /Nþ ðkÞg; 2 cN1 ðkÞ ¼

ð7:2Þ ð7:3Þ ð7:4Þ ð7:5Þ ð7:6Þ

Here, cN1 ðkÞ; cN ðkÞ; cN2 ðkÞ; and cNþ ðkÞ are the complex conjugates of each component. By taking the Fourier transform [15] with respect to the wavenumber k, the Fourier spectrum Î(m) can be obtained By taking the Fourier transform [15] with respect to the wavenumber k, the Fourier spectrum Î(m) can be obtained as

^IN ðmÞ ¼ F½IN ðkÞ 1 ¼ fANO ðmÞ þ C N1 ðmÞ þ C N1 ðmÞ þ C N ðmÞ þ C N ðmÞ 8 þ C N2 ðmÞ þ C N2 ðmÞ þ C Nþ ðmÞ þ C Nþ ðmÞg;

ð8Þ

where F indicates the Fourier transform operator and AN0(m), CN1(m), CN(m), CN2(m), and CN+(m) are the Fourier spectra of each component in Eq. (7.1). cN1 ðmÞ; cN ðmÞ; cN2 ðmÞ; and cNþ ðmÞ are the complex conjugates of the Fourier spectra. Each spectrum is selected and filtered using a Hanning window [13]. The five terms from aN0(k) to cN+(k) shown in Eqs. (7.2), (7.4)–(7.6) can be determined by taking the inverse Fourier transform F1

ð9:1Þ

1

F1 ½C N2 ðmÞ ¼ cN2 ðkÞ;

ð9:4Þ

F1 ½C Nþ ðmÞ ¼ cNþ ðkÞ:

ð9:5Þ

We can obtain biasN(k), ampNi(k), and Ci(k)/Ni(k) by considering real R and imaginary parts I.

biasN ðkÞ ¼ R½aN0 ðkÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ampNi ðkÞ ¼ 2 fR½cNi ðkÞ2 þ I½cNi ðkÞ2 g;

Ci ðkÞ  /Ni ðkÞ ¼ tan1

I½cNi ðkÞ : R½cNi ðkÞ

ð10:1Þ ð10:2Þ ð10:3Þ

Here, the subscript i represents 1, , 2, or +. If the phase retardation C1(k), C(k), C2(k), and C+(k) are measured in advance, we can obtain /Ni(k). Fourier coefficients aN0(k), aNi(k), and bNi(k) shown in Eq. (2) can be expressed as

aN0 ðkÞ ¼ biasN ðkÞ;

ð11:1Þ

aNi ðkÞ ¼ ampNi ðkÞ  cos /Ni ðkÞ;

ð11:2Þ

bNi ðkÞ ¼ ampNi ðkÞ  sin /Ni ðkÞ:

ð11:3Þ

Finally, we can obtain the nine Mueller matrix elements from a single-channeled spectrum by solving the simultaneous equations defined by Eqs. (3.1)–(3.25). By using four-channeled spectra (i.e., N = 4) it is possible to determine all the elements of the Mueller matrix. The advantage of this method is it has a simple optical configuration and algorithm compared with those for the snapshot technique [14]. Fig. 2 shows the Fourier spectrum after calculating channeled spectrum based on simulated data. It shows nine Fourier spectra that contain information about the polarization properties of the sample. A(m) is the bias component of Fourier spectrum, Ci(m) are the first, second, third, and fourth components of the Fourier

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Y. Otani et al. / Optics Communications 281 (2008) 5725–5730

3. Experimental setup and results

100

A(ν) amplitude spectrum [a.u.]

80

60

C2* (ν)

C1*(ν) C1(ν)

C2(ν)

40

C+* (ν)

C+(ν)

20

C-* (ν)

0 -30

-20

C-(ν)

0 -10 10 optical path difference[ µm]

20

30

Fig. 2. The amplitude spectra in Fourier space.

spectra. C i ðmÞ is the complex conjugate of each Fourier spectrum. The subscript i represents 1, , 2, and + of the Fourier spectra. Table 1 shows the four-channeled spectra obtained by the spectrometer and the four Fourier spectra; they represent unprocessed data. The symbol ‘‘” in Table 1 indicates those elements that cannot be determined for that particular optical configuration. An essential aspect of this method is the utilization of four-channeled spectra to determine the wavelength dependence of all the elements of the Mueller matrix, in addition to independently determining nine Mueller matrix elements from only a singlechanneled spectrum. If we use the obtained Mueller matrix, we can determine multiple polarization properties. The Mueller matrix for both a linear retardance and a linear diattenuation can be written as [3]

2

aðkÞ

bðkÞC

6 6 bðkÞC MðkÞ ¼ 6 6 4 bðkÞS 0

0

bðkÞS

C ¼ cos 2h;

bðkÞ ¼ qðkÞ  rðkÞ;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð12:3Þ

h¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m210 ðkÞ þ m202 ðkÞ

2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m00 ðkÞ  m210 ðkÞ þ m202 ðkÞ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m213 ðkÞ þ m232 ðkÞ m33 ðkÞ

1 m13 ðkÞ tan1 2 m32 ðkÞ

m10 ðkÞ ¼ m01

or h ¼

ð12:1Þ

ð12:2Þ

S ¼ sin 2h

m00 ðkÞ þ

DðkÞ ¼ tan1

7

cðkÞ ¼ 2 qðkÞrðkÞ;

where q(k), r(k), D(k), and h are the intensity transmittances of the fast and slow axis, the retardance, and the azimuthal angle, respectively. The polarization properties are given by

rðkÞ ¼

3

aðkÞC 2 þ S2 cðkÞ cos DðkÞ CSðaðkÞ  cðkÞ cos DðkÞÞ cðkÞS sin DðkÞ 7 7; 7 CSðaðkÞ  cðkÞ cos DðkÞÞ aðkÞS2 þ C 2 cðkÞ cos DðkÞ cðkÞC sin DðkÞ 5 cðkÞS sin DðkÞ cðkÞC sin DðkÞ cðkÞ cos DðkÞ

aðkÞ ¼ qðkÞ þ rðkÞ;

qðkÞ ¼

We performed an experiment to demonstrate the operation of this spectroscopic Mueller matrix polarimeter. We used a tungsten–halogen lamp as the white light source. The white light is collimated as a point source of u3 mm after passing through an optical fiber and an objective lens. We used Glan–Thompson prisms as polarizers and quartz retarders. The first and second retarders had 23 wavelengths and 7 wavelengths of retardances, respectively, and they could generate channeled spectrum of different frequencies with phase retardances of d1(k), d1(k)  d2(k), d2(k), and d1(k) + d2(k). We obtained channeled spectra over the visible wavelength range from 400 nm (k = 2.50  106/m) to 800 nm (k = 1.25  106/m), and the number of data points per spectrum was 1213 (with a resolution of approximately 0.35 nm). The spectral data were least square fitted at wavenumber intervals of Dk = 2.44  103/m. The sample is made from mica; they were designed to have a retardance of 90° at 575 nm. After calculating the Fourier transform, we obtained nine spectral peaks in Fourier space. These peaks are separated from each other, and each peak contains information concerning the Mueller elements. Due to the distortion generated by the Fourier transform operation, the actual measurement range in this experiment was from 475 nm to 675 nm (i.e., from k = 2.11  106/m to k = 1.48  106/m). This distortion is a result of the discontinuities at the end points of the channeled spectrum. Fig. 3 shows the wavelength dependence of the Mueller matrix elements of the sample. These Mueller matrix elements were normalized using the m00 component. The Mueller matrices in Fig. 3a, f, k, l, o, and p are shown within the range ±1.0, and the other in Fig. 3 is shown within the range ±0.1. With the exception of m00 and m11  m13, m21  m23, and m31  m33 that include information concerning birefringence, the other elements are approximately zero. However, mica plates are known to exhibit linear diattenuation; according to our results, the linear diattenuation of the retar-

;

ð13:1Þ

;

ð13:2Þ

;

1 m02 ðkÞ tan1 ; 2 m10 ðkÞ

or m20 ðkÞ ¼ m02 ðkÞ

ð13:3Þ

der that has retardance 82.3° was 0.006 at a wavelength of 632 nm. Diattenuation of mica plate that have retardance 61.3° at a wavelength of 632 nm is 0.004 according to the value obtained in Ref. [6]. When the retardances and/or thicknesses are taken into consideration, these values corresponded very well. Fig. 4 shows the wavelength dependence of the retardance and the azimuthal angle after calculating m13, m32, and m33. The measured retardance is in good agreement with the calibrated data. In addition to the retardance, the azimuthal angle corresponded to the setting angle (h = 0°) and remained constant within the approximate range of ±0.1°. 4. Discussion

ð13:4Þ

4.1. Optical resolution of this system

ð13:5Þ

The spectroscopic Mueller matrix polarimeter can measure the 16 Mueller matrix elements using four-channeled spectra. In this

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Y. Otani et al. / Optics Communications 281 (2008) 5725–5730

m0j [a.u.]

1.0 0

-1.0 600 (a)

m1j [a.u.]

0.1

0

0

0

[nm]

(b)

0

-0.1 500

600

(e)

[nm]

-1.0

(c)

m01

600

(i)

0

0

-1.0 500

[nm]

600

-0.1 500

600

(m)

500

m11

600

(g)

[nm]

-0.1

(h) 1.0

0

0

0

500

600

[nm]

-1.0

m21 1.0

0

0

-0.1

m30

[nm]

600

(n)

600

[nm]

-1.0

500

(k) m22

0.1

500

500

500

m12

1.0

(j)

[nm]

-0.1

0.1

m20

0

(d)

0

[nm]

0.1

500

m02 0.1

-0.1 500

[nm]

0.1

(f)

0

600

500

1.0

m10

0.1

-0.1

-0.1 [nm]

600

500

m00

0.1

m2j [a.u.]

0.1

-0.1 500

m3j [a.u.]

0.1

600 m03

[nm]

600

[nm]

m13

600

[nm]

(l) m23 1.0

0

-1.0 500

m31

600

(o)

[nm]

-1.0

500

m32

600

(p)

[nm]

m33

Fig. 3. Muller matrix of mica plate.

experiment, waveplates with 23 wavelengths and 7 wavelengths of retardances were employed as retarders, and the optical resolution

10

135

retardance azimuthal angle 5

90

0

-5

45 475

575 wavelength [nm]

-10 675

Fig. 4. Wavelength dependence of birefringence dispersion.

azimuthal angle [ ]

retardance [ ]

calibrated data

of the spectrometer was about 0.3 nm. A high-resolution spectrometer is required to achieve a measurement accuracy as high as those obtained using other spectroscopic Mueller matrix polarimeters. If a high-resolution spectrometer is used, the retarders having greater retardances can be employed. As a result, both the measured accuracy and the optical resolution are improved, since the nine Fourier spectra shown in Fig. 2 are expected to become completely separate from each other. Currently, the Fourier spectra AN0(m) and CN1(m) shown in Table 1 partially overlap each other. This causes small errors in the measurement results. 4.2. Estimation of the error and calibration Using the result shown in Fig. 3, we estimated the error of Mueller matrices in this system. Mica plate is known to have both retardance and linear diattenuation, but have neither optical rotation nor circular diattenuation. Therefore, Eq. (12.1) can be used as the Mueller matrix of a mica plate, and ideally the Mueller matrices m03 and m30 become zero. Compared with the results obtained using Eq. (12.1), the error of Mueller matrices in this system is estimated to be ±1%. We can consider the influence of two errors, such as the azimuth error for misalignment and the retardance error that is caused by the wavenumber dependence of the birefringence of the retarders. In the case of misalignment, the errors of Mueller matrices estimated by Mueller calculus were about ±0.5% when the azimuthal

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errors in the polarization optics were given by about 0.5°, respectively. If we apply a calibration technique such as the ones described in Refs. [2,8], the error will decrease, although the algorithm will become very complex. The retardance error with the wavenumber dependence is ignored in this system because the retardance is approximately proportional to wavenumber according to Eqs. (6.1)–(6.3). The wavenumber dependence cannot be ignored if high accuracy is required. In this case, the calibration technique given in Ref. [16] is very useful for reducing the retardance error. 5. Conclusions We have developed a spectroscopic Mueller matrix polarimeter that acquires four-channeled spectra. The polarizing and the analyzing optics consist of polarizers and retarders with high-order retardance. This measurement utilizes a polarization modulation method in wavenumber space. Four-channeled spectra are required to determine the wavelength dependence of the Mueller matrix elements. Using this polarimeter, it is simple to measure all the elements of Mueller matrix as a function of wavenumber, although this method is not a snapshot technique but requires four-channeled spectra. In addition, we can independently determine nine Mueller matrix elements from only a single-channeled spectrum. The determination of the nine elements of a Mueller matrix using single-channeled spectrum is a very powerful technique. For example, linear birefringence and linear diattenuation are re-

quired to evaluate polymer materials in the field of polymer science. When this method is used to evaluate the optical properties of a film, the retardance, azimuthal angle, and linear diattenuation given by Eqs. (13.1)–(13.4) can be simultaneously derived from six elements of a Mueller matrix. The affiliation of one of the authors, T. Wakayama, is now Saitama Medical University. References [1] R.M.A. Azzam, Opt. Lett. 2 (1978) 148. [2] P.S. Hauge, J. Opt. Soc. Am. 68 (1978) 1519. [3] R.A. Chipman, Polarimetry, in: M. Bass (Ed.), Handbook of Optics, second ed., vol. 2, McGraw-Hill, New York, 1995 (Chapter 2). [4] D.H. Goldstein, Mueller matrix dual-rotating retarder polarimeter, Appl. Opt. 31 (1992) 6676. [5] H. Kogelnik, R. Jopson, L. Nelson, Polarization-Mode Dispersion, in: I. Kaminow, T. Li (Eds.), Optical Fiber Telecommunications IVB, Academic, San Diego, CA, 2002. [6] A.D. Martino, Y. Kim, E. Garcia-Caurel, B. Laude, B. Drèvillon, Opt. Lett. 28 (2003) 616. [7] G.E. Jellison, C.O. Griffiths, D.E. Holcomb, C.M. Rouleau, Appl. Opt. 41 (2002) 6555. [8] L. Jin, K. Takizawa, Y. Otani, N. Umeda, Opt. Rev. 12 (2005) 281. [9] P. Westbrook, L. Nelson, S. Wielandy, J. Fini, Opt. Lett. 29 (2004) 2593. [10] J.W. Ellis, L. Glatt, J. Opt. Soc. Amer. 40 (1950) 141. [11] K. Oka, T. Kato, Opt. Lett. 24 (1999) 1475. [12] T. Wakayama, Y. Otani, N. Umeda, Proc. SPIE 5888 (2005) 55. [13] T. Wakayama, Y. Otani, N. Umeda, Opt. Commun. 281 (2008) 3668. [14] N. Hagen, K. Oka, E.L. Dereniak, Opt. Lett. 32 (2007) 2100. [15] M. Takeda, H. Ina, S. Kobayashi, J. Opt. Soc. Amer. 72 (1982) 156. [16] A. Taniguchi, K. Oka, H. Okabe, M. Hayakawa, Opt. Lett. 31 (2006) 3279.