Abstract

An optical compensator is described which is achromatized in a broad spectral range. It consists of two pairs of birefringent plates of different materials under oblique incidence rotated around the normal to the plates. The phase shift is achromatized by adjusting the plate thicknesses. An electrooptic modulator based on this design is also described. Instead of mechanically rotating the system to introduce phase shifts the optic axis is rotated by applying an electric field.

© 1989 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. W. G. Driscoll, W. Vaughan, Eds., Handbook of Optics (McGraw-Hill, New York, 1978), p. 10–234.
  2. Ref. 1, p. 10–115.
  3. W. Shurcliff, S. Ballard, Polarized Light (Van Nostrand, New York, 1964).

Ballard, S.

W. Shurcliff, S. Ballard, Polarized Light (Van Nostrand, New York, 1964).

Shurcliff, W.

W. Shurcliff, S. Ballard, Polarized Light (Van Nostrand, New York, 1964).

Other (3)

W. G. Driscoll, W. Vaughan, Eds., Handbook of Optics (McGraw-Hill, New York, 1978), p. 10–234.

Ref. 1, p. 10–115.

W. Shurcliff, S. Ballard, Polarized Light (Van Nostrand, New York, 1964).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Compensator system consisting of two pairs of plates in optical contact. Each plate pair is of the same material, but with their optical axes crossed. φ is the angle between the plane of incidence (x-y plane) and the optical axis of the top plate. When the system is rotated around its normal (x-axis) the phase shift changes between ordinary and extraordinary waves in the plates.

Fig. 2
Fig. 2

Compensator phase shift p as a function of rotation angle φ (see Fig. 1). This example corresponds to that in Fig. 3(e) (0.50-mm KDP plates and 4.45-mm sapphire plates achromatized at wavelengths of 0.5 and 0.8 μm shown for a wavelength of 0.55 μm).

Fig. 3
Fig. 3

Compensator phase shift p as a function of wavelength for different systems. All the systems are achromatized at wavelengths of 0.55 and 0.8 μm, except (c) with 0.35 and 0.8 μm. The seven curves in each part of the figure correspond to the angles φ = 42, 43, 44, 45, 46, 47, 48° (see Fig. 2). φ = 45° always yields zero phase shift when the plates of the individual pairs have the same thickness: (a) 10-mm quartz plates and 6.04-mm MgF2 plates, (b) 2.5-mm quartz plates and 0.439-mm KDP plates, (c) 10-mm quartz plates and 0.555-mm calcite plates, (d) 0.441-mm KDP plates and 1.50-mm MgF2 plates, (e) 0.50-mm KDP plates and 4.45-mm sapphire plates, (f) 0.75-mm KDP plates and 0.233-mm calcite plates.

Fig. 4
Fig. 4

(a) Phase shift for quartz plates with d1 = 12.9 μm in the Brewster angle (56°) and rotation angles φ = 40, 45, and 50°. (b) Phase shift for a pair of quartz plates with the same thickness of d = 0.65 mm. Their optical axes lie on the surfaces and are crossed. Angles of rotation φ are indicated.

Fig. 5
Fig. 5

Phase shift for a pair of 100-μm quartz plates and a 59-μm MgF2 plate. Achromatization was obtained, but φ is far from 45°.

Fig. 6
Fig. 6

Phase shift for a zero-order λ/4 plate pair of quartz (d1 = 0.63469 mm and d2 = 0.64969 mm) in the Brewster angle (56°) rotated the angle φ around its normal.

Fig. 7
Fig. 7

Thickness of individual plates of a zero-order λ/4 plate pair vs angle φ0 where the phase shift of the system almost vanishes.

Fig. 8
Fig. 8

Electrooptic modulator system consisting of two electrooptic crystals (LiNbO3) of the same thickness d in optical contact with their optical axes c and c′ crossed. The electric field is applied perpendicular to the optical axes. The effect is just as in the system in Fig. 1, i.e., the optical axes are turned slightly away from the 45° angle where zero phase shift exists.

Fig. 9
Fig. 9

Phase shift for the modulator system in Fig. 8. A plate pair of LiNbO3 with a thickness of d = 10 mm in the Brewster angle (66°). The applied electric field is indicated.

Equations (42)

Equations on this page are rendered with MathJax. Learn more.

p = 2 π λ { [ d 1 ( ω 1 cos r ω 1 n 1 cos r n 1 ) d 1 ( ω 1 cos r ω 1 n 1 cos r n 1 ) ] ± [ d 2 ( ω 2 cos r ω 2 n 2 cos r n 2 ) d 2 ( ω 2 cos r ω 2 n 2 cos r n 2 ) ] } .
1 n i 2 = sin 2 θ i ε i 2 + cos 2 θ i ω i 2 ,
cos θ 1 = 1 n 1 sin i cos ϕ and cos θ 2 = 1 n 2 sin i sico ϕ ,
sico ϕ = { sin ϕ for case 1 , cos ϕ for case 2 .
n 1 2 = ε 1 2 [ ( ε 1 ω 1 ) 2 1 ] sin 2 i cos 2 ϕ ,
n 2 2 = ε 2 2 [ ( ε 2 ω 2 ) 2 1 ] sin 2 i sico 2 ϕ .
n i cos r n i = ( n i 2 sin 2 i ) 1 / 2 = ε i [ 1 sin 2 i ν i 2 ( ϕ ) ] 1 / 2 ,
1 ν 1 2 ( ϕ ) = 1 ε 1 2 + ( 1 ω 1 2 1 ε 1 2 ) cos 2 ϕ and 1 ν 2 2 ( ϕ ) = 1 ε 2 2 + ( 1 ω 2 2 1 ε 2 2 ) sico 2 ϕ .
p = 2 π λ [ ( d 1 { ω 1 ( 1 sin 2 i ω 1 2 ) 1 / 2 ε 1 [ 1 sin 2 i ν 1 2 ( ϕ ) ] 1 / 2 } d 1 { ω 1 ( 1 sin 2 i ω 1 2 ) 1 / 2 ε 1 [ 1 sin 2 i ν 1 2 ( ϕ π 2 ) ] 1 / 2 } ) ± ( d 2 { ω 2 ( 1 sin 2 i ω 2 2 ) 1 / 2 ε 2 [ 1 sin 2 i ν 2 2 ( ϕ ) ] 1 / 2 } d 2 { ω 2 ( 1 sin 2 i ω 2 2 ) 1 / 2 ε 2 [ 1 sin 2 i ν 2 2 ( ϕ π 2 ) ] 1 / 2 } ) ] .
d 1 = d 1 and d 2 = d 2 .
p = 2 π λ ( d 1 ε 1 { [ 1 sin 2 i ν 1 2 ( ϕ π 2 ) ] 1 / 2 [ 1 sin 2 i ν 1 2 ( ϕ ) ] 1 / 2 } ± d 2 ε 2 { [ 1 sin 2 i ν 2 2 ( ϕ π 2 ) ] 1 / 2 [ 1 sin 2 i ν 2 2 ( ϕ ) ] 1 / 2 } ) .
p = cos 2 ϕ [ B 1 ( λ ) d 1 λ ± B 2 ( λ ) d 2 λ ] ,
B i ( λ ) = 2 π ε i sin 2 i [ 1 sin 2 i ν i 2 ( ϕ ) ] 1 / 2 + [ 1 sin 2 i ν i 2 ( ϕ π 2 ) ] 1 / 2 ( 1 ω i 2 i ε i 2 ) .
α = 2 [ B 1 ( λ ) d 1 λ ± B 2 ( λ ) d 2 λ ] .
d 1 d 2 = ± λ 1 B 2 ( λ 2 ) λ 2 B 2 ( λ 1 ) λ 2 B 1 ( λ 1 ) λ 1 B 1 ( λ 2 ) ,
p = 2 π λ d 1 { ω 1 ( 1 sin 2 i ω 1 2 ) 1 / 2 ε 1 [ 1 sin 2 i ν 1 2 ( ϕ ) ] 1 / 2 }
p = 2 π λ ( d 1 { ω 1 ( 1 sin 2 i ω 1 2 ) 1 / 2 ε 1 [ 1 sin 2 i ν 1 2 ( ϕ ) ] 1 / 2 } d 2 [ ω 2 ( 1 sin 2 i ω 2 2 ) 1 / 2 ε 2 ( 1 sin 2 i ν 2 ) 1 / 2 ] )
ν 2 = { ν 2 2 ( ϕ π 2 ) axes crossed , ν 2 2 ( ϕ ) axes parallel .
Δ p 2 [ B 1 ( λ ) d 1 λ ± B 2 ( λ ) d 2 λ ] Δ ϕ
x 2 / ω 2 + y 2 / ω 2 + z 2 / ε 2 + 2 r 42 E y y z = 1 ,
1 ω 2 = 1 ω 2 + r 12 E y ,
1 n 1 , 2 2 = 1 2 ( 1 ω 2 + 1 ε 2 ) ± 1 2 [ ( 1 ω 2 1 ε 2 ) 2 + 4 r 42 2 E y 2 ] 1 / 2 .
tan ( Δ ϕ ) = 2 r 42 E y ( 1 ε 2 1 ω 2 ) + [ ( 1 ε 2 1 ω 2 ) 2 + 4 r 41 2 E y 2 ] 1 / 2 r 42 E y ( 1 ε 2 1 ω 2 ) .
p = 4 π λ ε r 42 E y d sin 2 i [ 1 sin 2 i ν 2 ( ϕ ) ] 1 / 2 + [ 1 sin 2 i ν 2 ( ϕ π 2 ) ] 1 / 2 ,
p 2 π λ [ ( ω ε ) + 1 2 ( ε 3 r 33 ω 3 r 13 ) E z ] 2 d ,
q ε r 42 ε 3 r 33 ω 3 r 13 ,
j = exp ( i p / 2 ) { cos p / 2 2 i n sin ( p / 2 ) 1 + n 2 cos 2 ε 2 i n sin ( p / 2 ) 1 + n 2 cos 2 ε cos p / 2 } + exp ( i p / 2 ) { i sin ( p / 2 ) sin 2 ε 0 0 i sin ( p / 2 ) sin 2 ε } ,
ε = ϕ π 4 .
j = exp ( i p / 2 ) { cos p / 2 i sin p / 2 i sin p / 2 cos p / 2 } .
{ 1 i } .
{ 1 i sin 2 ε i ( 2 n cos 2 ε 1 + n 2 ) } .
{ 1 0 . 9357 exp ( i δ ) } δ = 95 . 967 ° .
( a ) I = I 0 ( cos 2 p 2 + sin 2 p 2 sin 2 2 ε ) ,
( b ) I = I 0 4 n 2 ( 1 + n 2 ) 2 sin 2 p 2 cos 2 2 ε .
( a ) I = I 0 cos 2 p 2 ,
( b ) I = I 0 sin 2 p 2 .
j = { exp ( i p ) cos 2 ϕ + sin 2 ϕ 1 2 sin 2 ϕ [ exp ( i p ) 1 ] 1 2 sin 2 ϕ [ exp ( i p ) 1 ] exp ( i p ) sin 2 ϕ + cos 2 ϕ } ,
cos 2 ϕ = 1 2 ( 1 sin 2 ε ) ,
sin 2 ϕ = 1 2 ( 1 + sin 2 ε ) .
j = 1 2 { [ exp ( i p ) + 1 ] [ exp ( i p ) 1 ] sin 2 ε [ exp ( i p ) 1 ] cos 2 ε [ exp ( i p ) 1 ] cos 2 ε [ exp ( i p ) + 1 ] + [ exp ( i p ) 1 ] sin 2 ε } ,
j = exp ( i p / 2 ) { cos p 2 i sin p 2 cos 2 ε i sin p 2 cos 2 ε cos p 2 } + exp ( i p 2 ) { i sin p 2 sin 2 ε 0 0 i sin p 2 sin 2 ε } .
j total = { n 0 0 2 n 2 1 + n 2 } j { 1 n 0 0 2 1 + n 2 } = exp ( i p 2 ) { cos p 2 i sin p 2 sin 2 ε 2 i n 1 + n 2 sin p 2 cos 2 ε 2 i n 1 + n 2 sin p 2 cos 2 ε 4 n 2 ( 1 + n 2 ) 2 ( cos p 2 + i sin p 2 sin 2 ε ) } .

Metrics