Abstract

To characterize the linear birefringence of a multiple-order wave plate (MWP), an oblique incidence is one of the methods available. Multiple reflections in the MWP are produced, and oscillations in the phase retardation measurement versus the oblique incident angle are then measured. Therefore, an antireflection coated MWP is required to avoid oscillation of the phase retardation measurement. In this study, we set up a phase-sensitive heterodyne ellipsometer to measure the phase retardations of an uncoated MWP versus the oblique incident angle, which was scanned in the x–z plane and y–z plane independently. Thus, the effect on multiple reflections by the MWP is reduced by means of subtracting the two measured phase retardations from each other. As a result, a highly sensitive and accurate measurement of retardation parameters (RPs), which includes the refractive indices of the extraordinary ray ne and ordinary ray no, is obtained by this method. On measurement, a sensitivity (ne,no) of 106 was achieved by this experiment setup. At the same time, the spatial shifting of the P and S waves emerging from the MWP introduced a deviation between experimental results and the theoretical calculation.

© 2007 Optical Society of America

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References

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1992 (2)

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1980 (1)

Appl. Opt. (7)

J. Opt. Soc. Am. A (2)

Other (4)

E. D. Palik, Handbook of Optical Constants of Solids III (Academic, 1998), p. 729.

E. Hecht, Optics (Addison-Wesley, 1998), pp. 409-412.

Optics Guide (CASIX, Inc., 1995), p. 1 [no = 1.5427, no = 1.5518 (633 nm)].

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1979), p. 99.

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Figures (9)

Fig. 1
Fig. 1

Experimental setup: B S 1 , B S 2 : beam splitters, A O M 1 , A O M 2 : acousto-optic modulators, M 1 , M 2 : mirrors, A 1 , A 2 : analyzers, S: test sample on rotation stage, PBS: polarization beam splitter, D p , D s : photo detectors, BPF: band-pass filter, LIA: lock-in amplifier, DSC: digital stepping controller, PC: personal computer.

Fig. 2
Fig. 2

(a) Schematic diagram of normal incidence. (b) Schematic diagram of the oblique incident angle in the x–z plane. (c) Schematic diagram of the oblique incident angle in the y–z plane.

Fig. 3
Fig. 3

(Color online) Computer simulations of phase retardations due to multiple reflections of an uncoated QWP tilted in the x–z plane (a) Δ t y and (b) Δ t x tilted in the y–z plane where n e = 1.5518 , n o = 1.5428 , and d = 0.506   mm with tilted angle ϕ t y at λ = 632.8   nm .

Fig. 4
Fig. 4

(Color online) Experimental results (dots) for phase retardation δ ( o ) ( θ ) versus rotation angle θ of an uncoated QWP at normal incidence. The theoretical calculation (solid curve) is δ ( o ) ( θ θ min ) after an angle being shifted.

Fig. 5
Fig. 5

(a) δ t y ( ϕ t y ) δ 0 and (b) δ t x ( ϕ t x ) δ 0 versus ϕ t y and ϕ t x of an uncoated QWP where ϕ ty = ϕ tx = ϕ t is arranged.

Fig. 6
Fig. 6

Phase retardations of (a) δ t y ( ϕ t y ) δ 0 and (b) δ t x ( ϕ t x ) δ 0 of an antireflection QWP under the condition of ϕ t y = ϕ t x = ϕ t .

Fig. 7
Fig. 7

Oscillations are eliminated in δ t y δ t x of an uncoated QWP at a condition of ϕ t y = ϕ t x = ϕ t .

Fig. 8
Fig. 8

Measured (a) δ t y ( ϕ t y ) δ 0 versus ϕ t y and (b) δ t x ( ϕ t x ) δ 0 versus ϕ t x of an antireflection QWP where ϕ t y = ϕ t x = ϕ t . Dots are the measured data while the solid curve shows the calculated data based on n e = 1.5518 , n o = 1.5428 , and d = 0.506   mm , at λ = 632.8   nm . (c) Result of δ t y δ t x .

Fig. 9
Fig. 9

Spatial shifting of emerging laser beams from an anisotropic wave plate ( d p > d s ) and an isotropic glass plate ( d p = d s ) of (a) and (c), respectively, where the angle is tilted in the x–z plane and of (b) and (d), where the angle is tilted in the y–z plane. P 0 and S 0 are the position of P and S waves at ϕ t y = 0 and ϕ t x = 0 , respectively.

Equations (14)

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I p ( Δ ω t ) = | E p 1 + E p 2 | 2 = I p 1 + I p 2 + 2 I p 1 I p 2  cos ( Δ ω t + δ p ) ,
I s ( Δ ω t ) = | E s 1 + E s 2 | 2 = I s 1 + I s 2 + 2 I s 1 I s 2  cos ( Δ ω t + δ s ) ,
X = E s E p  exp [ i ( δ s δ p ) ] = | X | exp [ i δ ] ,
X ( o ) = T 22 + X ( i ) T 21 T 12 + X ( i ) T 11 ,
T ε = [ T 11 T 12 T 21 T 22 ] = [ cos   γ 2 + i   cos   2 ε   sin   γ 2   cos   2 ( θ + π 2 ) sin   γ 2   sin   2 ε + i   cos   2 ε   sin   γ 2   sin   2 ( θ + π 2 ) sin   γ 2   sin   2 ε + i   cos   2 ε   sin   γ 2   sin   2 ( θ + π 2 ) cos   γ 2 i   cos   2 ε   sin   γ 2   cos   2 ( θ + π 2 ) ] .
X ( o ) = T 22 + T 21 T 12 + T 11 .
δ ε ( o ) = tan 1 ( { 2   sin   γ 2  cos   2 ε [ sin   γ 2   sin   2 ε   sin   2 ( θ + π 2 ) cos   γ 2   cos   2 ( θ + π 2 ) ] } / [ cos 2 γ 2 sin 2 γ 2 sin 2   2 ε 1 2 sin 2 γ 2 cos 2   2 ε   cos   4 ( θ + π 2 ) ] ) ,
δ ε ( o ) = tan 1 ( { 2 sin γ 2 [ cos   γ 2   cos   2 ( θ + π 2 ) ] } / [ cos 2 γ 2 1 2 sin 2 γ 2   cos   4 ( θ + π 2 ) ] ) .
δ t y = δ s δ p = 2 π λ ( n o A C ¯ ) 2 π λ ( n ˜ e A D ¯ + D E ¯ ) = 2 π λ d ( n o   cos   ϕ o n ˜ e   cos   ϕ ˜ e ) = 2 π λ d ( n o 2 sin 2 ϕ t y n e 2 n e 2 n o 2 sin 2 ϕ t y ) ,
δ t y δ 0 = 2 π λ d [ ( n o 2 sin 2 ϕ t y n e 2 n e 2 n o 2 sin 2 ϕ t y ) ( n o n e ) ] ,
δ t x δ 0 = 2 π λ d [ ( n o 2 sin 2 ϕ t x n e 2 sin 2 ϕ t x ) ( n o n e ) ] ,
2 π λ d ( n e n o ) = 2 m π + Γ .
Δ = tan 1 [ r s 2   sin   Γ s r s 2  cos   Γ s 1 ] tan 1 [ r p 2  sin   Γ p r p 2  cos   Γ p 1 ] ,
( δ t y δ t x ) / ( 2 m π + Γ ) = ( n e 2 sin 2 ϕ t n e 2 n e 2 sin 2 ϕ t / n o 2 ) / ( n e n o ) .

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