Dual-frequency heterodyne ellipsometer for ... - OSA Publishing

Dual-frequency heterodyne ellipsometer for characterizing generalized elliptically birefringent media Chih-Jen Yu1, Chu-En Lin1, Ying-Chang Li1, Li-Dek Chou2, Jheng-Syong Wu1, Cheng-Chung Lee1, and Chien Chou1,2,* 1

Department of Optics and Photonics, National Central University, Jhongli, Taiwan 320 Graduate Institute of Electro-Optical Engineering, Chang Gung University, Taoyuan, Taiwan 333 *[email protected]

2

Abstract: This research proposed a dual-frequency heterodyne ellipsometer (DHE) in which a dual-frequency collinearly polarized laser beam with equal amplitude and zero phase difference between p- and s-polarizations is setup. It is based on the polarizer-sample-analyzer, PSA configuration of the conventional ellipsometer. DHE enables to characterize a generalized elliptical phase retarder by treating it as the combination of a linear phase retarder and a polarization rotator. The method for measuring elliptical birefringence of an elliptical phase retarder based on the equivalence theorem of an unitary optical system was derived and the experimental verification by use of DHE was demonstrated too. The experimental results show the capability of DHE on characterization of a generalized phase retardation plate accurately. ©2009 Optical Society of America OCIS codes: (040.2840) Heterodyne; (120.2130) Ellipsometry and polarimetry; (120.3180) Interferometry; (260.1440) Birefringence

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

J. F. Nye, Physical Properties of Crystals (Oxford University Press, Oxford, 1957), pp. 261–268. F. Ratajczyk, and P. Kurzynowski, “Phase difference superposition law for elliptically birefringent media,” Optik (Stuttg.) 99, 92–94 (1995). R. C. Jones, “A new calculus for the treatment of optical systems. I. description and discussion of the calculus,” J. Opt. Soc. Am. 31(7), 488–493 (1941). G. D. VanWiggeren, and R. Roy, “Transmission of linearly polarized light through a single-mode fiber with random fluctuations of birefringence,” Appl. Opt. 38(18), 3888–3892 (1999). T. Chartier, A. Hideur, C. Özkul, F. O. Sanchez, and G. M. Stéphan, “Measurement of the elliptical birefringence of single-mode optical fibers,” Appl. Opt. 40(30), 5343–5353 (2001). S. Berezhna, I. Berezhnyy, and M. Takashi, “Integrated photoelasticity through imaging Fourier polarimetry of an elliptic retarder,” Appl. Opt. 40(5), 644–651 (2001). K. Pietraszkiewicz, W. A. Woźniak, and P. Kurzynowski, “Effect of multiple reflections in retardation plates with elliptical birefringence,” J. Opt. Soc. Am. A 12(2), 420–424 (1995). S. Berezhna, I. Berezhnyy, and M. Takashi, “Determination of the normalized Jones matrix of elliptical retarder,” Proc. SPIE 4317, 129–134 (2001). P. Kurzynowski, “Senarmont compensator for elliptically birefringent media,” Opt. Commun. 197(4-6), 235–238 (2001). P. Kurzynowski, W. A. Woźniak, and S. Drobczyński, “A new phase difference compensation method for elliptically birefringent media,” Opt. Commun. 267(1), 44–49 (2006). C. Chou, Y. C. Huang, and M. Chang, “Effect of elliptical birefringence on the measurement of the phase retardation of a quartz wave plate by an optical heterodyne polarimeter,” J. Opt. Soc. Am. A 14(6), 1367–1372 (1997). C. Chou, Y. C. Huang, and M. Chang, “Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz wave plate,” Jpn. J. Appl. Phys. 35(Part 1, No. 10), 5526–5529 (1996). H. Hurwitz, Jr., and R. C. Jones, “A new calculus for the treatment of optical systems. II. Proof of three general equivalence theorems,” J. Opt. Soc. Am. 31, 493–499 (1941). S. T. Tang, and H. S. Kwok, “3 × 3 Matrix for unitary optical systems,” J. Opt. Soc. Am. A 18(9), 2138–2145 (2001).

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Received 3 Sep 2009; revised 1 Oct 2009; accepted 1 Oct 2009; published 8 Oct 2009

12 October 2009 / Vol. 17, No. 21 / OPTICS EXPRESS 19213

15. V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Cell parameter determination of a twisted-nematic liquid crystal display by single-wavelength polarimetry,” J. Appl. Phys. 97(4), 043101 (2005). 16. V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. 99(11), 113101 (2006). 17. R. M. A. Azzam, and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), pp. 98–99. 18. P. Yeh, and C. Gu, Optics of liquid crystal displays (Wiley, New York, 1999), pp. 120–122. 19. C. Chou, C. W. Lyu, and L. C. Peng, “Polarized differential-phase laser scanning microscope,” Appl. Opt. 40(1), 95–99 (2001). 20. H. F. Chang, C. Chou, H. K. Teng, H. T. Wu, and H. F. Yau, “The use of polarization and amplitude-sensitive optical heterodyne interferometry for linear birefringence parameters measurement,” Opt. Commun. 260(2), 420–426 (2006). 21. D. C. Su, M. H. Chiu, and C. D. Chen, “Simple two-frequency laser,” Precis. Eng. 18(2-3), 161–163 (1996). 22. C. J. Yu, C. E. Lin, L. P. Yu, and C. Chou, “Paired circularly polarized heterodyne ellipsometer,” Appl. Opt. 48(4), 758–764 (2009). 23. W. Mao, S. Zhang, L. Cui, and Y. Tan, “Self-mixing interference effects with a folding feedback cavity in Zeeman-birefringence dual frequency laser,” Opt. Express 14(1), 182–189 (2006).

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(ns − nf)t/λ0. Where in the equation, t is the thickness of the quartz crystal plate; λ0 is the wavelength of light wave in vacuum; nf and ns 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: γ2= 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]. 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].

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

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 dualfrequency 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].

 a + ib JU =   −c + id

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c + id  , a − ib 

(1)



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.

Jones matrix elements of J U

Optical properties Circular birefringence Linear birefringence Elliptical birefringence

a, c ≠ 0, and b, d = 0 a, b, d ≠ 0, and c = 0 a, b, c, d ≠ 0

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:

J eq = J CB ( Φ eq ) J LB ( Γ eq , ψ eq )  aeq + ibeq =  −ceq + ideq

ceq + id eq  , aeq − ibeq 

(2)

and aeq = cos beq = sin

Γ eq 2

cos Φ eq ,

(3)

cos ( 2ψ eq − Φ eq ) ,

(4)

ceq = cos d eq = sin

Γ eq 2

Γ eq 2

Γ eq

sin Φ eq ,

(5)

sin ( 2ψ eq − Φ eq ) .

(6)

2

(b) Equivalently linear phase retarder:

Iɶ3

(7)

 cos Φ eq sin Φ eq  JCB ( Φ eq ) =  .  − sin Φ sin Φ  eq eq  

(8)

(c) Equivalent polarization rotator:

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 LB ), respectively; Φ eq is the circular phase retardation of an equivalent polarization rotator ( J CB ); these three parameters, Γ eq , ψ eq and Φ eq are the characteristic parameters of an unitary optical system ( J U ) [14].

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2.2 The fundamental parameters of an elliptical phase retarder According to the optical property of an elliptical phase retarder, the Jones matrix J EB is expressed by [8,17]

J EB

γ γ  cos + i sin cos 2ε cos 2θ  2 2 = γ  − sin sin 2ε + i sin γ cos 2ε sin 2θ  2 2 

γ γ  sin sin 2ε + i sin cos 2ε sin 2θ  2 2  . (9) γ γ cos − i sin cos 2ε cos 2θ  2 2 

Thus, the relationship between an elliptical phase retarder ( J EB ) and its optically equivalent system ( J eq ) can be expressed as J EB ( γ , θ , ε ) = J eq ( Γ eq , ψ eq , Φ eq ) .

(10)

We found that cos

γ 2

= cos

Γ eq 2

cos Φ eq ,

(11)

Γ eq γ sin cos 2ε cos 2θ = sin cos ( 2ψ eq − Φ eq ) , 2 2

(12)

Γ eq γ sin sin 2ε = cos sin Φ eq , 2 2

(13)

Γ eq γ sin cos 2ε sin 2θ = sin sin ( 2ψ eq − Φ eq ) . 2 2

(14)

Thus, elliptical phase retardation γ is calculated as



γ = 2 cos −1  cos 

Γ eq

 cos Φ eq  , 2 

(15)

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

θ = ψ eq − ( Φ eq 2 ) ,

(16)

Γ eq γ sin cos 2ε = sin , 2 2

(17)

and

finally, the ellipticity angle ε is shown by

ε=

Γ eq   1 tan −1  cot sin Φ eq  . 2 2  

(18)

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

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axis orientations are ρ, 2ρ, 3ρ,…, Nρ sequentially. From the above properties, and using Jones calculus, a TNLCD is derived as [18] N

J 0LC = ∏ R ( −m ρ )J 0LB ( −i β 2 N ) R ( m ρ ) ,

(19)

m =1

 cos m ρ R ( mρ ) =   − sin m ρ

sin m ρ  , cos m ρ 

(20)

 exp ( −i β 2 N ) 0  J 0LB ( −i β 2 N ) =  , 0 exp ( i β 2 N )  

(21)

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

J 0LC

β  cos χ − i sin χ  2χ  cos Ω − sin Ω   =  Ω  sin Ω cos Ω   − sin χ  χ 

Ω

  χ .  β cos χ + i sin χ  2χ  sin χ

(22)

In the above equations, R is the rotation matrix and it has the identical form to J CB ; J 0LB 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 = 2π L∆n λ0 , where L∆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 0 J LC = R ( −D ) J LC R (D )

 a + ibLC =  LC  −cLC + id LC

cLC + id LC  , aLC − ibLC 

(23)

and aLC = cos χ cos Ω + ( Ω χ ) sin χ sin Ω,

(24)

bLC = − ( β 2 χ ) sin χ cos ( 2D + Ω ) ,

(25)

cLC = − cos χ sin Ω + ( Ω χ ) sin χ cos Ω,

(26)

d LC = − ( β 2 χ ) sin χ sin ( 2D + Ω ) ,

(27)

where χ 2 = Ω 2 + ( β 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). 2

Γ eq = cos −1  2 ( aLC 2 + c LC 2 ) − 1 ,

ψ eq

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 2  a c ( bLC 2 − d LC 2 ) + bLC d LC ( aLC 2 − cLC 2 )   1  , −1   LC LC = tan   2 2 2 2 4  ( aLC − cLC )( bLC − d LC ) − 4aLC bLC cLC d LC 

(28)

(29)



 (30) .  In general, the Jones matrix elements of a TNLCD, aLC , bLC , cLC 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. Φ eq =

 2a c 1 tan −1  2LC LC 2 2  aLC − cLC

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 ∆ϕ = ϕ p − ϕs 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  Ep exp i (ωp t + ϕ p )    , EZL =   Es exp i ( ωs t + ϕs )     

(31)

where ( Ep , Es ) and (ϕ p , ϕs ) are the amplitudes and phases with respect to p-polarized and spolarized waves, respectively. We define ωp = ω0 + (ω 2 ) , ωs = ω0 − (ω 2 ) , ϕ0 = ϕ p + ϕs , and ∆ϕ = ϕ p − ϕs , where ω0 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  Ep exp i (ω t + ∆ϕ ) 2    exp i ω0 t + (ϕ0 2 )  . EZL =   Es exp  −i (ωt + ∆ϕ ) 2     

{

}

(32)

After the Zeeman He-Ne laser beam EZL 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 EDCLL becomes

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EDCLL = J PL (α ) J H ( Θ ) EZL  cos 2 α sin α cos α   cos 2Θ sin 2Θ  =   sin 2 α   sin 2Θ − cos 2Θ   sin α cos α  Ep exp i ( ωt + ∆ϕ ) 2    exp i ω0 t + (ϕ 0 2 )  ×  Es exp  −i (ωt + ∆ϕ ) 2      E′  =  p  exp i ω0 t + (ϕ0 2 )  ,  Es′ 

{

{

}

(33)

}

where Ep′ = Ep cos ( 2Θ − α ) cos α exp i (ω t + ∆ϕ ) 2  + Es sin ( 2Θ − α ) cos α exp  −i (ωt + ∆ϕ ) 2 ,

(34)

Es′ = Ep cos ( 2Θ − α ) sin α exp i (ω t + ∆ϕ ) 2 + Es sin ( 2Θ − α ) sin α exp  −i (ωt + ∆ϕ ) 2  ,

(35)

and

when half wave plate is rotated so that the condition of Ep cos ( 2Θ − α ) = Es sin ( 2Θ − α ) = E0 ,

(36)

 cos α   ϕ       ωt + ∆ϕ  exp i ω0 t +  0    , EDCLL (α ) =   ( 2 E0 ) cos   2      2     sin α 

(37)

is satisfied, then

or

     cos α   ω ω  ϕ0 + ∆ϕ    ϕ − ∆ϕ    EDCLL (α ) =  + exp i  ω0 −  t + i  0  E0 exp i  ω0 +  t + i      2 2 2 2        sin α        cos α   cos α  =  E0 exp i ( ωp t + ϕ p )  +   E0 exp i (ωs t + ϕs )  .  sin α   sin α  (38) According to Eq. (38), EDCLL 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. 1  ϕ       ωt + ∆ϕ  EDCLL ( 45° ) =   2 E0 cos  exp i ω0 t +  0    .  2      2    1

(39)

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

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

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

En = J A ( A ) J eq J P (P ) EDCLL ( 45° ) ,

(40)

I n = En * .E n = Fn I 0 [1 + cos(ω t + ∆ϕ ) ] .

(41)

and the measured intensity is

Where I 0 = 2 E02 , 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 Iɶn = Fn I 0 cos (ω t + ∆ϕ ) is measured by selecting a narrow-bandwidth filter to match the beat frequency in this experiment. And according to previous analysis, Fn in the intensity equation is a function of the Jones matrix elements aeq , beq , ceq , and d eq as shown in Table 2.

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

Measured intensities Iɶ

Azimuth angles P A

n

Jones matrix elements dependent factor Fn

Iɶ1

0°

0°

(a

+ beq 2 ) 2

Iɶ2

0°

45°

(1 − 2a

Iɶ3

0°

90°

Iɶ4

0°

135°

(1 + 2a

c − 2beq deq ) 4

Iɶ5

45°

180°

(1 + 2a

c + 2beq d eq ) 2

Iɶ6

45°

225°

Iɶ7

45°

270°

Iɶ8

45°

315°

2

eq

c + 2beq deq ) 4

eq eq

(c

+ deq 2 ) 2

2

eq

eq eq

eq eq

aeq 2 + d eq 2

(1 − 2a

c − 2beq d eq ) 2

eq eq

beq 2 + ceq 2

We defined four experimentally measured quantities as follows: A=

Iɶ1 − Iɶ3 , B= Iɶ1 + Iɶ3

Iɶ6 − Iɶ8 Iɶ − Iɶ , C = 5 7 , and Iɶ6 + Iɶ8 Iɶ5 + Iɶ7

D=

Iɶ2 − Iɶ4 . Iɶ2 + Iɶ4

(42)

Thus, the characteristic parameters of a specimen are

{

}

(43)

 ,  ( A2 − C 2 ) − ( B 2 − D 2 ) 

(44)

12

2 2 Γ eq = cos −1 ( A + B ) + ( C − D )   

1 4



ψ eq = tan −1 

Φ eq =

2 ( AC + BD )

1 C −D tan −1  , 2  A+ B 

−1 ,

(45)

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° ≤ Γeq ≤ 180°,

−45°≤ ψeq ≤ 45°, and – 90° ≤ Φeq ≤ 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.

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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 UD , 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 UD ( 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.

Parameters Theoretical Measured

Quarter wave plate ΓQ ψQ 90.00° 90.18°

15.00° 15.04°

Faraday rotator ΦF 45.00° 45.53°

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∆n ∼ 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 (Γeq, ψeq, Φeq) 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.

Theoretical Measured

Equivalently linear birefringence and circular birefringence Гeq ψeq Φeq 54.49° −32.84° −65.69° 55.84° −32.63° −65.63°

Equivalently elliptical birefringence γ ε θ 137.15° 0.00° −30.29° 137.23° 0.18° −29.23°

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 (Γeq, ψeq, Φeq) of an elliptical phase retarder. Since Γeq, ψeq, and Φeq 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 Φeq ≪ Γeq and hence can be applied to measure the EB of an elliptically birefringent medium in general case.

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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 HeNe 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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