Dual-frequency heterodyne ellipsometer for characterizing generalized elliptically birefringent media

Chih-Jen Yu; Chu-En Lin; Ying-Chang Li; Li-Dek Chou; Jheng-Syong Wu; Cheng-Chung Lee; Chien Chou

doi:10.1364/OE.17.019213

1. Introduction

An anisotropic optically active crystal such as a quartz crystal plate, presents the properties of the elliptical birefringence (EB). As a result of the EB, the eigenstates propagating in a quartz crystal plate with different velocities are two elliptically polarized waves. The phenomenon of EB is characterized by three fundamental parameters: γ, the elliptical phase retardation between the fast and slow elliptically polarized eigenstates, θ (θ + 90°), the azimuth angle, and ε (−ε), the ellipticity angle, of the fast (slow) eigenstate of propagation in the quartz crystal plate. The last two parameters, θ and ε, are illustrated in Fig. 1 , while the elliptical phase retardation γ, is represented by γ = 2π(n_s − n_f)t/λ ₀. Where in the equation, t is the thickness of the quartz crystal plate; λ ₀ is the wavelength of light wave in vacuum; n_f and n_s are the refractive indices of the respective fast and slow elliptically eigen-polarizations. Theoretically, EB is the result of coexistence of linear birefringence (LB) and circular birefringence (CB) in an anisotropic medium [1], therefore we can treat an elliptically birefringent medium as the combination of a linearly birefringent element and a circularly birefringent element in quantitative analysis. If the CB is much smaller than LB for an elliptically birefringent medium, the principle of superposition for optical activity and ordinary birefringence: γ ²= Γ² + (2Φ)² holds [1,2]. Where Γ and Φ are linear and circular phase retardation respectively. This principle could be useful to characterize an elliptically birefringent material under the condition of Φ « Γ, however, this is not valid in general case [2].

Fig. 1 The elliptically eigen-polarizations of (a) the fast light wave and (b) the slow light wave of an elliptically birefringent medium.

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Theoretically, an unitary optical system does not alter the intensity of light wave, it merely transforms the polarization state of an incident light wave into other states due to the existence of different birefringence properties [3]. For instance, if we consider an unitary optical system composed of multiple linearly birefringent and circularly birefringent elements, the system will result in EB properties. A common example of such an unitary optical system is a single mode optical fiber, it can be seen as a series of linear phase retarders each with an arbitrary phase retardation and principal axis orientation, to result EB [4,5]. Aside from being the inherent property of unitary optical system, EB can also be induced while stress is applied to an isotropic medium, such kind of elliptically birefringent effect is termed integrated photoelasticity [6].

Whether the elliptically birefringent property of a specimen is intrinsic or induced. Generally, the properties of the polarization transformation can be illustrated explicitly by means of the transfer matrix of an elliptical phase retarder [7,8]. Current investigations of the effect of EB in terms of measuring the parameters: γ, θ, and ε of an elliptical phase retarder include various methods, such as Fourier polarimetry [6,8], Senarmont compensator method [9,10], and heterodyne interferometry [11,12]. These methods all focus on direct measurement of the parameters γ, θ, and ε of a specimen.

In accordance with the equivalence theorem of an unitary optical system, any kind of unitary optical system is optically equivalent to an optical system, consisting of a linear phase retarder and a polarization rotator [13–16]. Therefore in this study, the relationship between the EB of an unitary optical system and the system’s equivalent CB and LB is discussed. Thus, the three fundamental parameters of an elliptical phase retarder (γ, θ, ε) can be obtained in terms of the measured values of equivalent LB and CB simultaneously.

In order to verify our theoretical derivations on EB measurement experimentally, a dual-frequency heterodyne ellipsometer (DHE) is proposed and set up to measure CB, LB and EB of different specimens, where EB can be determined by measuring the equivalent CB and LB of the specimen. In other words, this method can be used to find the contributions of CB and LB in an elliptically birefringent material quantitatively. The optical setup of DHE is a common-path heterodyne interferometer with polarizer (P)-specimen (S)-analyzer (A) configuration in conventional ellipsometer. DHE is based on the detection of heterodyne signals at different polarizer and analyzer settings. Because DHE is based on the heterodyne signal detection, background noise can be reduced significantly by utilizing a proper narrow bandwidth filter.

2. Theoretical background

In theory, we define that the horizontally polarized light (p-polarized light) lies parallel to the x axis and the vertically polarized light (s-polarized light) lies parallel to the y axis, of the laboratory coordinate system (see Fig. 1), where the azimuth angles of the linear phase retarder, the polarizer, the analyzer, etc. described in the remaining article are all with respect to x axis.

2.1 Equivalence theorem of an unitary optical system

In general, the Jones matrix form of an unitary optical system of a non-absorbing medium is described by [14].

J_{U} = (\begin{array}{c} a + i b & c + i d \\ - c + i d & a - i b \end{array}),

where the Jones matrix elements, a, b, c, and d are real numbers. Because of the determinant of

J_{U}

is unity, hence

a^{2} + b^{2} + c^{2} + d^{2} = 1

and

J_{U}

is the transfer matrix of the optically birefringent system. Table 1 presents the matrix elements of different kinds of optical birefringence in

J_{U}

.

Table 1. Jones matrix elements of circularly, linearly and elliptically birefringent materials.

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As mentioned above, the optically equivalent system of an unitary optical system is composed of an equivalently linear phase retarder and an equivalent polarization rotator, therefore, the Jones matrix of an unitary optical system represented by its optical equivalence is expressed as

(a) Optically equivalent system:
$\begin{array}{l} J_{eq} = J_{C B} (Φ_{eq}) J_{L B} (Γ_{eq}, ψ_{eq}) \\ = (\begin{array}{c} a_{eq} + i b_{eq} & c_{eq} + i d_{eq} \\ - c_{eq} + i d_{eq} & a_{eq} - i b_{eq} \end{array}), \end{array}$

and

a_{eq} = \cos \frac{Γ_{eq}}{2} \cos Φ_{eq},

b_{eq} = \sin \frac{Γ_{eq}}{2} \cos (2 ψ_{eq} - Φ_{eq}),

c_{eq} = \cos \frac{Γ_{eq}}{2} \sin Φ_{eq},

d_{eq} = \sin \frac{Γ_{eq}}{2} \sin (2 ψ_{eq} - Φ_{eq}) .

(b) Equivalently linear phase retarder:
${\tilde{I}}_{3}$
(c) Equivalent polarization rotator:
$J_{CB} (Φ_{eq}) = (\begin{array}{l} \cos Φ_{eq} \\ - \sin Φ_{eq} \end{array} \begin{array}{l} \sin Φ_{eq} \\ \sin Φ_{eq} \end{array}) .$

In Eqs. (7) and (8),

Γ_{eq}

and

ψ_{eq}

are the linear phase retardation and fast axis orientation of an equivalently linear phase retarder (

J_{L B}

), respectively;

Φ_{eq}

is the circular phase retardation of an equivalent polarization rotator (

J_{C B}

); these three parameters,

Γ_{eq}

,

ψ_{eq}

and

Φ_{eq}

are the characteristic parameters of an unitary optical system (

J_{U}

) [14].

2.2 The fundamental parameters of an elliptical phase retarder

According to the optical property of an elliptical phase retarder, the Jones matrix $J_{E B}$ is expressed by [8,17]

J_{E B} = (\begin{array}{c} \cos \frac{γ}{2} + i \sin \frac{γ}{2} \cos 2 ε \cos 2 θ & \sin \frac{γ}{2} \sin 2 ε + i \sin \frac{γ}{2} \cos 2 ε \sin 2 θ \\ - \sin \frac{γ}{2} \sin 2 ε + i \sin \frac{γ}{2} \cos 2 ε \sin 2 θ & \cos \frac{γ}{2} - i \sin \frac{γ}{2} \cos 2 ε \cos 2 θ \end{array}) .

Thus, the relationship between an elliptical phase retarder (

J_{E B}

) and its optically equivalent system (

J_{eq}

) can be expressed as

J_{E B} (γ, θ, ε) = J_{eq} (Γ_{eq}, ψ_{eq}, Φ_{eq}) .

We found that

\cos \frac{γ}{2} = \cos \frac{Γ_{eq}}{2} \cos Φ_{eq},

\sin \frac{γ}{2} \cos 2 ε \cos 2 θ = \sin \frac{Γ_{eq}}{2} \cos (2 ψ_{eq} - Φ_{eq}),

\sin \frac{γ}{2} \sin 2 ε = \cos \frac{Γ_{eq}}{2} \sin Φ_{eq},

\sin \frac{γ}{2} \cos 2 ε \sin 2 θ = \sin \frac{Γ_{eq}}{2} \sin (2 ψ_{eq} - Φ_{eq}) .

Thus, elliptical phase retardation γ is calculated as

γ = 2 \cos^{- 1} (\cos \frac{Γ_{eq}}{2} \cos Φ_{eq}),

and from Eqs. (12) and (14), the azimuth angle θ can be found by

θ = ψ_{eq} - (Φ_{eq} / 2),

and

\sin \frac{γ}{2} \cos 2 ε = \sin \frac{Γ_{eq}}{2},

finally, the ellipticity angle ε is shown by

ε = \frac{1}{2} \tan^{- 1} (\cot \frac{Γ_{eq}}{2} \sin Φ_{eq}) .

2.3 Twisted nematic liquid crystal device as an elliptical phase retarder

Twisted nematic liquid crystal device (TNLCD) is known as a twisted anisotropic medium which presents EB and it is an unitary optical device. Theoretically, a TNLCD can be treated as superimposing N layers of linearly birefringent plate of equal thickness while their slow axis orientations are ρ, 2ρ, 3ρ,…, Nρ sequentially. From the above properties, and using Jones calculus, a TNLCD is derived as [18]

J_{L C}^{0} = \prod_{m = 1}^{N} R (- m ρ) J_{L B}^{0} (- i β / 2 N) R (m ρ),

R (m ρ) = (\begin{array}{c} \cos m ρ & \sin m ρ \\ - \sin m ρ & \cos m ρ \end{array}),

J_{L B}^{0} (- i β / 2 N) = (\begin{array}{c} \exp (- i β / 2 N) & 0 \\ 0 & \exp (i β / 2 N) \end{array}),

when N approaches to infinite, Eq. (19) becomes

J_{L C}^{0} = (\begin{array}{c} \cos Ω & - \sin Ω \\ \sin Ω & \cos Ω \end{array}) (\begin{array}{c} \cos χ - i \frac{β}{2 χ} \sin χ & \frac{Ω}{χ} \sin χ \\ - \frac{Ω}{χ} \sin χ & \cos χ + i \frac{β}{2 χ} \sin χ \end{array}) .

In the above equations, R is the rotation matrix and it has the identical form to

J_{C B}

;

J_{L B}^{0}

is the matrix of a linearly birefringent plate with slow axis orientation at 0°; β is the linear phase retardation and Ω is the twist angle of a TNLCD. Note that

ρ = Ω / N

and

β = 2 π L Δ n / λ_{0}

, where

L Δ n

is the optical path length difference between the ordinary ray and the extraordinary ray propagating in the liquid crystal. Consider a TNLCD in which the molecular director is along the direction at an angle D. Then

\begin{array}{l} J_{LC} = R (- D) J_{LC}^{0} R (D) \\ = (\begin{array}{c} a_{LC} + i b_{LC} & c_{LC} + i d_{LC} \\ - c_{LC} + i d_{LC} & a_{LC} - i b_{LC} \end{array}), \end{array}

and

a_{LC} = \cos χ \cos Ω + (Ω / χ) \sin χ \sin Ω,

b_{LC} = - (β / 2 χ) \sin χ \cos (2 D + Ω),

c_{LC} = - \cos χ \sin Ω + (Ω / χ) \sin χ \cos Ω,

d_{LC} = - (β / 2 χ) \sin χ \sin (2 D + Ω),

where

χ^{2} = Ω^{2} + {(β / 2)}^{2}

. Because

J_{LC}

is unitary, hence

J_{LC} = J_{eq}

[14–16]. According to Eqs. (3)-(6) and Eqs. (24)-(27), the characteristic parameters of a TNLCD can be found by Eqs. (28)-(30).

Γ_{eq} = \cos^{- 1} [2 (a_{LC}^{2} + c_{LC}^{2}) - 1],

ψ_{eq} = \frac{1}{4} \tan^{- 1} {\frac{2 [a_{LC} c_{LC} (b_{LC}^{2} - d_{LC}^{2}) + b_{LC} d_{LC} (a_{LC}^{2} - c_{LC}^{2})]}{(a_{LC}^{2} - c_{LC}^{2}) (b_{LC}^{2} - d_{LC}^{2}) - 4 a_{LC} b_{LC} c_{LC} d_{LC}}},

Φ_{eq} = \frac{1}{2} \tan^{- 1} (\frac{2 a_{LC} c_{LC}}{a_{LC}^{2} - c_{LC}^{2}}) .

In general, the Jones matrix elements of a TNLCD, $a_{LC}$ , $b_{LC}$ , $c_{LC}$ and $d_{LC}$ , are not equal to zero. Referring to the Table 1, the optical property of a TNLCD is equivalent to an elliptical phase retarder. The equivalent EB of a TNLCD can then be characterized by substituting the calculated results of Eqs. (28)-(30) into Eqs. (15), (16), and (18) accordingly.

3. Dual-frequency heterodyne ellipsometer

3.1 The dual-frequency collinearly polarized laser beam

A dual-frequency laser beam that emits a pair of linearly polarized waves of mutually orthogonal polarization states and slightly temporal frequency difference can be generated by simply using a Zeeman He-Ne laser. It also can be achieved by using a single frequency laser integrated with an electric-optic modulator or an acoustic-optic modulator [19–22]. A Zeeman He-Ne laser is commonly used as the dual-frequency laser light source in the heterodyne interferometer and ellipsometer. However, the amplitude discrepancy (approximately 5%) between the p-polarized and s-polarized waves is critical to the assurance of high accuracy performance on the ellipsometric parameters measurements [22]. In addition, the residual phase retardation $Δ φ = φ_{p} - φ_{s}$ between the p-polarized and s-polarized waves is also resulted from the nonlinearity of the birefringent effect within the laser cavity of Zeeman He-Ne laser [23]. Therefore, in order to achieve high sensitivity of heterodyne interferometer of using a Zeeman He-Ne laser light source, we proposed a method that uses a conventional Zeeman He-Ne laser as a laser light source to generate a modified dual-frequency laser beam with equal amplitude and zero phase difference between its p-polarized and s-polarized components.

In general, the Jones vector of Zeeman He-Ne laser beam, with unequal amplitudes and phase difference between the p-polarized and s-polarized waves, is expressed as

E_{Z L} = (\begin{array}{c} E_{p} \exp [i (ω_{p} t + φ_{p})] \\ E_{s} \exp [i (ω_{s} t + φ_{s})] \end{array}),

where

(E_{p}, E_{s})

and

(φ_{p}, φ_{s})

are the amplitudes and phases with respect to p-polarized and s-polarized waves, respectively. We define

ω_{p} = ω_{0} + (ω / 2)

,

ω_{s} = ω_{0} - (ω / 2)

,

φ_{0} = φ_{p} + φ_{s}

, and

Δ φ = φ_{p} - φ_{s}

, where

ω_{0}

is the central frequency and ω represents the beat frequency of the p-polarized and s-polarized waves in Zeeman He-Ne laser beam. Then Eq. (31) can be rewritten as

E_{Z L} = (\begin{array}{c} E_{p} \exp [i (ω t + Δ φ) / 2] \\ E_{s} \exp [- i (ω t + Δ φ) / 2] \end{array}) \exp {i [ω_{0} t + (φ_{0} / 2)]} .

After the Zeeman He-Ne laser beam

E_{Z L}

passing sequentially through a half wave plate with fast axis orientation Θ, and a polarizer with transmission axis orientation α, the electric field of the emergent beam

E_{D C L L}

becomes

\begin{array}{l} E_{D C L L} = J_{P L} (α) J_{H} (Θ) E_{Z L} \\ = (\begin{array}{c} \cos^{2} α & \sin α \cos α \\ \sin α \cos α & \sin^{2} α \end{array}) (\begin{array}{c} \cos 2 Θ & \sin 2 Θ \\ \sin 2 Θ & - \cos 2 Θ \end{array}) \\ \times (\begin{array}{c} E_{p} \exp [i (ω t + Δ φ) / 2] \\ E_{s} \exp [- i (ω t + Δ φ) / 2] \end{array}) \exp {i [ω_{0} t + (φ_{0} / 2)]} \\ = (\begin{array}{c} E_{p}^{'} \\ E_{s}^{'} \end{array}) \exp {i [ω_{0} t + (φ_{0} / 2)]}, \end{array}

where

\begin{array}{l} E_{p}^{'} = E_{p} \cos (2 Θ - α) \cos α \exp [i (ω t + Δ φ) / 2] \\ + E_{s} \sin (2 Θ - α) \cos α \exp [- i (ω t + Δ φ) / 2], \end{array}

and

\begin{array}{l} E_{s}^{'} = E_{p} \cos (2 Θ - α) \sin α \exp [i (ω t + Δ φ) / 2] \\ + E_{s} \sin (2 Θ - α) \sin α \exp [- i (ω t + Δ φ) / 2], \end{array}

when half wave plate is rotated so that the condition of

E_{p} \cos (2 Θ - α) = E_{s} \sin (2 Θ - α) = E_{0},

is satisfied, then

E_{D C L L} (α) = (\begin{array}{c} \cos α \\ \sin α \end{array}) (2 E_{0}) \cos (\frac{ω t + Δ φ}{2}) \exp {i [ω_{0} t + (\frac{φ_{0}}{2})]},

or

\begin{array}{l} E_{D C L L} (α) = (\begin{array}{c} \cos α \\ \sin α \end{array}) E_{0} {\exp [i (ω_{0} + \frac{ω}{2}) t + i (\frac{φ_{0} + Δ φ}{2})] + \exp [i (ω_{0} - \frac{ω}{2}) t + i (\frac{φ_{0} - Δ φ}{2})]} \\ = (\begin{array}{c} \cos α \\ \sin α \end{array}) E_{0} \exp [i (ω_{p} t + φ_{p})] + (\begin{array}{c} \cos α \\ \sin α \end{array}) E_{0} \exp [i (ω_{s} t + φ_{s})] . \end{array}

According to Eq. (38),

E_{D C L L}

acts as a pair of linearly polarized waves of mutually parallel polarization states and slightly different frequencies. If α = 45°, as shown in Eq. (39), a beam of dual-frequency collinearly polarized laser light with equal amplitude and zero phase difference between p-polarized and s-polarized components is generated in this arrangement.

E_{D C L L} (45 °) = (\begin{array}{c} 1 \\ 1 \end{array}) \sqrt{2} E_{0} \cos (\frac{ω t + Δ φ}{2}) \exp {i [ω_{0} t + (\frac{φ_{0}}{2})]} .

Accordingly, one can set α = 0° (90°) to generate dual-frequency co-horizontally (co-vertically) polarized laser beam.

3.2 Experimental setup

Figure 2 illustrates the optical setup of DHE. It uses a dual-frequency collinearly polarized laser beam in conventional polarizer-specimen-analyzer, or PSA configuration. The output power, beat frequency, and central wavelength of the Zeeman He-Ne laser (Agilent 5517A) are 2.5 mW, 1.8 MHz, and 632.8 nm, respectively. In Fig. 2, the specimen (S) is an unitary optical device, and the transmission axis of the polarizer (P) and the analyzer (A) are set at P and A, respectively. During the measurement, the initial values of P and A are both set at 0°, and then the analyzer is rotated so A changes from 0° to 135° in 45° increment to give the first four output intensities shown in Table 2 . The polarizer is then set to P = 45°, and the analyzer is rotated so A goes from 180° to 315° in 45° increment to give another four output intensities also shown in Table 2. Theoretically, the electric field of the output beam is expressed as

E_{n} = J_{A} (A) J_{eq} J_{P} (P) E_{D C L L} (45 °),

and the measured intensity is

I_{n} = E_{n} * . E_{n} = F_{n} I_{0} [1 + \cos (ω t + Δ φ)] .

Where

I_{0} = 2 E_{0}^{2}

, and the subscript n (n = 1, 2, 3,…8) denotes the specific conditions of the polarizer and the analyzer orientations (see Table 2). Because the DC term and the heterodyne term of Eq. (41) always have equal magnitude, in order to reduce noise, only the heterodyne term

{\tilde{I}}_{n} = F_{n} I_{0} \cos (ω t + Δ φ)

is measured by selecting a narrow-bandwidth filter to match the beat frequency in this experiment. And according to previous analysis,

F_{n}

in the intensity equation is a function of the Jones matrix elements

a_{eq}

,

b_{eq}

,

c_{eq}

, and

d_{eq}

as shown in Table 2.

Fig. 2 Optical setup of DHE. H: half wave plate, P_L and P: polarizers, S: specimen, A: analyzer, D: photo detector, DVM: digital voltmeter.

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Table 2. Measured intensities at different combinations of the azimuth angles of polarizer and analyzer.

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We defined four experimentally measured quantities as follows:

A = \frac{{\tilde{I}}_{1} - {\tilde{I}}_{3}}{{\tilde{I}}_{1} + {\tilde{I}}_{3}}, B = \frac{{\tilde{I}}_{6} - {\tilde{I}}_{8}}{{\tilde{I}}_{6} + {\tilde{I}}_{8}}, C = \frac{{\tilde{I}}_{5} - {\tilde{I}}_{7}}{{\tilde{I}}_{5} + {\tilde{I}}_{7}}, and D = \frac{{\tilde{I}}_{2} - {\tilde{I}}_{4}}{{\tilde{I}}_{2} + {\tilde{I}}_{4}} .

Thus, the characteristic parameters of a specimen are

Γ_{eq} = \cos^{- 1} {{[{(A + B)}^{2} + {(C - D)}^{2}]}^{1 / 2} - 1},

ψ_{eq} = \frac{1}{4} \tan^{- 1} [\frac{2 (A C + B D)}{(A^{2} - C^{2}) - (B^{2} - D^{2})}],

Φ_{eq} = \frac{1}{2} \tan^{- 1} (\frac{C - D}{A + B}),

hence the fundamental parameters γ, θ, and ε can be calculated from Eqs. (15), (16) and (18) respectively. Meanwhile, the dynamic ranges of the (

Γ_{eq}

,

ψ_{eq}

,

Φ_{eq}

) are 0° ≤ Γ_eq ≤ 180°, −45°≤ ψ _eq ≤ 45°, and – 90° ≤ Φ_eq ≤ 90°; which then resulting the dynamic ranges of (γ, θ, ε) to be 0° ≤ γ ≤ 180°, −90° ≤θ ≤ 90°, and −45°≤ ε ≤ 45°.

4. Experimental results

Two experiments were set up and demonstrated. One is to measure an unitary optical device which is a combination of a Faraday rotator (a circularly birefringent element) and a quarter wave plate (a linearly birefringent element) in order to verify the theory of DHE. The other is to characterize the equivalent CB, LB and EB of a TNLCD.

4.1 Simultaneous measurement of a circularly and a linearly birefringent elements

A Faraday rotator (Isowave I-633-2) and a quarter wave plate (CVI QWP0-633-10-4-R15) are used together to be an unitary optical device, $J_{U D}$ , exhibiting CB and LB simultaneously. In the measurement, the fast axis of the quarter wave plate was set at 15° and the polarization rotation angle of the Faraday rotator was 45°. The characteristic parameters of the unitary optical device are equal to the phase retardation ( $Γ_{Q}$ ) and fast axis orientation ( $ψ_{Q}$ ) of the quarter wave plate and polarization rotation angle of the Faraday rotator ( $Φ_{F}$ ) according to $J_{U D} (Γ_{Q}, ψ_{Q}, Φ_{F}) = J_{eq} (Γ_{eq}, ψ_{eq}, Φ_{eq}),$ i.e. $Γ_{eq} = Γ_{Q},$ $ψ_{eq} = ψ_{Q}$ and $Φ_{eq} = Φ_{F} .$ The measured results compared with the preset data are shown in Table 3 . This clearly shows that DHE has the ability to characterize the CB and LB of the respective circularly and linearly birefringent elements simultaneously and accurately.

Table 3. Linear birefringence and circular birefringence measurement of an unitary optical system composed by a quarter wave plate combining with a Faraday rotator.

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4.2 Equivalent CB, LB and EB measurement of a TNLCD

In order to verify that a TNLCD is equivalent to an elliptical phase retarder experimentally, a TNLCD provided by Chi-Mei Optoelectronics Co, Tainan, Taiwan with designed parameters was measured. These designed parameters are given as $L Δ n \sim$ 395.7 nm at 632.8 nm, twist angle Ω at 90°, and rubbing angle D is aligned along 45°. Therefore, the theoretical values of the characteristic parameters of the TNLCD (Γ_eq, ψ _eq, Φ_eq) can be obtained by substituting those designed values into Eqs. (24)-(30). The measured values of the same parameters are also available from calculation using Eqs. (42)-(45). In addition, both theoretical and measured values of equivalent EB of the TNLCD can be calculated by using Eqs. (15), (16) and (18). The values are summarized in Table. 4 where the experimental results well agree with the theoretical predictions.

Table 4. Measured and predicted results of the equivalently linear birefringence, circular birefringence, and elliptical birefringence of a TNLCD.

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5. Conclusions

An analytical method for characterizing the optical properties of an elliptically birefringent medium is proposed. The principle is to utilize the equivalence theorem of an unitary optical system to derive the relationship between the three fundamental parameters (γ, θ, ε) and the three characteristic parameters (Γ_eq, ψ _eq, Φ_eq) of an elliptical phase retarder. Since Γ_eq, ψ _eq, and Φ_eq can be measured by our developed DHE. Therefore, γ, θ, and ε can also be obtained in accordance with Eqs. (15), (16) and (18). This method does not require to satisfy the condition of Φ_eq ≪ Γ_eq and hence can be applied to measure the EB of an elliptically birefringent medium in general case.

The unequal amplitude and nonzero phase difference between two mutually orthogonal polarization states of a Zeeman He-Ne laser beam give rise to the measurement uncertainties in conventional heterodyne interferometry and ellipsometry. The drawbacks of a Zeeman He-Ne laser beam used in heterodyne interferometer can be corrected with the help of a half wave plate and a polarizer, therefore, the dual-frequency collinearly polarized laser beam with equal amplitude and zero phase difference between p-polarized and s-polarized components is generated. It can be used as an incident light source of the common-path heterodyne interferometer, ellipsometer and polarimeter. According to this optical heterodyne technique, the DHE which can simultaneously measure CB, LB, and EB of an unitary optical device is proposed and set up. The experimental verification of DHE was demonstrated by measuring the equivalent CB, LB and EB, of a TNLCD. DHE produces accurate measurement because of the heterodyne signal received via a narrow-bandwidth filter such that the background noise is reduced significantly.

Acknowledgments

We would like to acknowledge the support provided by the National Science Council of Taiwan through grant # NSC96-2221-E-010-002-MY2. TNLCD provided by Chi-Mei Optoelectronics Co., Tainan, Taiwan is also appreciated.

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Optical properties	Jones matrix elements of $J_{U}$
Circular birefringence	a, c ≠ 0, and b, d = 0
Linear birefringence	a, b, d ≠ 0, and c = 0
Elliptical birefringence	a, b, c, d ≠ 0

Measured intensities	Azimuth angles		Jones matrix elements dependent factor
${\tilde{I}}_{n}$	P	A	$F_{n}$
${\tilde{I}}_{1}$	0°	0°	$(a_{eq}^{2} + b_{eq}^{2}) / 2$
${\tilde{I}}_{2}$	0°	45°	$(1 - 2 a_{eq} c_{eq} + 2 b_{eq} d_{eq}) / 4$
${\tilde{I}}_{3}$	0°	90°	$(c_{eq}^{2} + d_{eq}^{2}) / 2$
${\tilde{I}}_{4}$	0°	135°	$(1 + 2 a_{eq} c_{eq} - 2 b_{eq} d_{eq}) / 4$
${\tilde{I}}_{5}$	45°	180°	$(1 + 2 a_{eq} c_{eq} + 2 b_{eq} d_{eq}) / 2$
${\tilde{I}}_{6}$	45°	225°	$a_{eq}^{2} + d_{eq}^{2}$
${\tilde{I}}_{7}$	45°	270°	$(1 - 2 a_{eq} c_{eq} - 2 b_{eq} d_{eq}) / 2$
${\tilde{I}}_{8}$	45°	315°	$b_{eq}^{2} + c_{eq}^{2}$

	Quarter wave plate		Faraday rotator
Parameters	$Γ_{Q}$	$ψ_{Q}$	$Φ_{F}$
Theoretical	90.00°	15.00°	45.00°
Measured	90.18°	15.04°	45.53°

	Equivalently linear birefringence and circular birefringence			Equivalently elliptical birefringence
	Г_eq	ψ _eq	Φ_eq	γ	ε	θ
Theoretical	54.49°	−32.84°	−65.69°	137.15°	−30.29°	0.00°
Measured	55.84°	−32.63°	−65.63°	137.23°	−29.23°	0.18°

Optical properties	Jones matrix elements of $J_{U}$
Circular birefringence	a, c ≠ 0, and b, d = 0
Linear birefringence	a, b, d ≠ 0, and c = 0
Elliptical birefringence	a, b, c, d ≠ 0

Dual-frequency heterodyne ellipsometer for characterizing generalized elliptically birefringent media

Abstract

1. Introduction

2. Theoretical background

2.1 Equivalence theorem of an unitary optical system

2.2 The fundamental parameters of an elliptical phase retarder

2.3 Twisted nematic liquid crystal device as an elliptical phase retarder

3. Dual-frequency heterodyne ellipsometer

3.1 The dual-frequency collinearly polarized laser beam

3.2 Experimental setup

4. Experimental results

4.1 Simultaneous measurement of a circularly and a linearly birefringent elements

4.2 Equivalent CB, LB and EB measurement of a TNLCD

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (2)

Tables (4)

Equations (45)

Optics Express