Abstract

By use of Jones’ matrix techniques, nine-element achromatic waveplates are developed. These combination plates are achromatic to within 1° throughout the visible (3,500–10,000 Å). In addition, a two-section general retarder rotator is demonstrated. The retardation of the combination is twice the complement of the angle between the halves, while the rotation is equal to the angle between the halves. For a 90° retardation, the dual five-element combination is also achromatic to within 1° throughout the visible.

© 1975 Optical Society of America

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References

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  1. A. Title, Solar Phys. in press.
  2. S. Pancharatnam, Proc. Indian Acad. Sci. A41, 137 (1955).
  3. S. E. Harris, C. M. McIntyre, J. Opt. Soc. Am. 58, 1575 (1968).
    [CrossRef]
  4. R. C. Jones, J. Opt. Soc. Am. 31500 (1941).
    [CrossRef]
  5. C. J. Koester, J. Opt. Soc. Am. 49, 560 (1959).
    [CrossRef]

1968 (1)

1959 (1)

1955 (1)

S. Pancharatnam, Proc. Indian Acad. Sci. A41, 137 (1955).

1941 (1)

Harris, S. E.

Jones, R. C.

Koester, C. J.

McIntyre, C. M.

Pancharatnam, S.

S. Pancharatnam, Proc. Indian Acad. Sci. A41, 137 (1955).

Title, A.

A. Title, Solar Phys. in press.

J. Opt. Soc. Am. (3)

Proc. Indian Acad. Sci. (1)

S. Pancharatnam, Proc. Indian Acad. Sci. A41, 137 (1955).

Other (1)

A. Title, Solar Phys. in press.

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Figures (10)

Fig. 1
Fig. 1

Retardation difference vs relative frequency difference for 180°, 90°, and 60° three-element combinations. For 180° retardation 2γ1 = 180°, 2γ2 = 180°, Δ = 60°; for 90° retardation 2γ1 = 115.5°, 2γ2 = 180°, Δ = 70.6°; for 60° retardation 2γ1 = 101.75°, 2γ2 = 180°, Δ = 76.096°.

Fig. 2
Fig. 2

Tilt difference vs relative frequence difference for the above three-element combinations.

Fig. 3
Fig. 3

Retardation difference vs relative frequency difference for three-element combination half waveplates for several adjustments of the central plate. The dashed curves indicate negative values of the relative retardation differences. All three waveplates have 180° retardation. The angle from the central plate is 60° minus δ.

Fig. 4
Fig. 4

Tilt difference vs relative frequency difference for three-element combination half waves for several adjustments of the central plate.

Fig. 5
Fig. 5

Tolerance β vs adjustment angle δ for three-element combination half waveplates.

Fig. 6
Fig. 6

Tolerance β vs the range of the relative frequency difference β for (a) three-element and (b) nine-element combination half waveplates.

Fig. 7
Fig. 7

Retardation difference vs relative frequency difference for nine-element combination half waveplates for several adjustments of the central group of three elements.

Fig. 8
Fig. 8

Tilt difference vs relative frequency difference for the S and O configurations of the central group of three-elements.

Figure 9
Figure 9

Retardation difference versus relative frequency difference for 3(a), 4(b), and 10(c), element combination quarter waveplates.

Figure 10
Figure 10

Rotation difference (a), and tilt difference for positive (b), and negative (c), values of α versus relative frequency difference, .

Equations (74)

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M ( γ , θ ) = R ( θ ) G ( γ ) R ( θ ) .
M ( γ , θ ) = cos γ I + i sin γ E R ( 2 θ ) ,
R ( θ ) = ( cos θ sin θ sin θ cos θ ) ,
I = ( 1 0 0 1 ) ,
E = ( 1 0 0 1 ) ,
G ( γ ) = ( exp ( + i γ ) 0 0 exp ( i γ ) .
E R ( θ ) = R ( θ ) E ,
E 2 = I .
i = 1 μ M ( γ i , θ i ) = R ( ω ) M ( γ ¯ , Ω ) ,
i = 1 μ M ( γ i , θ i ) = cos γ ¯ R ( ω ) + i sin γ ¯ E R ( 2 Ω ω ) ,
γ ( ) = γ 0 ( 1 + ) ,
Δ ν = ν ν 0 ,
= Δ ν / ν 0 .
M 3 = M ( γ 3 , θ 3 ) M ( γ 2 , θ 2 ) M ( γ 1 , θ 1 ) = cos γ 3 cos γ 2 cos γ 1 I { cos γ 3 sin γ 2 sin γ 1 R [ 2 ( θ 1 θ 2 ) ] + cos γ 2 sin γ 3 sin γ 1 R [ 2 ( θ 1 θ 3 ) ] + cos γ 1 sin γ 3 sin γ 2 R [ 2 ( θ 2 θ 3 ) ] } + i E { sin γ 3 sin γ 2 sin γ 1 R [ 2 ( θ 3 θ 2 + θ 1 ) ] + sin γ 3 cos γ 2 cos γ 1 R ( 2 θ 3 ) + sin γ 2 cos γ 3 cos γ 1 R ( 2 θ 2 ) + sin γ 1 cos γ 3 cos γ 2 R ( 2 θ 1 ) } .
Real ( M 3 12 ) = Real ( M 3 21 ) = 0.
sin 2 ( θ 2 θ 3 ) = ( tan γ 1 / tan γ 3 ) sin 2 θ 2 + ( tan γ 1 / tan γ 2 ) sin 2 θ 3 .
cos ( 2 θ 3 ) = tan γ 1 / tan γ 3 ,
cos ( 2 θ 2 ) = ( tan γ 1 / tan γ 2 ) .
γ 1 = ± γ 3 ,
γ 1 = ± γ 2 ,
θ 1 = θ 3 .
M 3 = ( cos 2 γ 1 cos γ 2 sin 2 γ 1 sin γ 2 cos 2 Δ ) R ( 0 ) + i E [ sin 2 γ 1 sin γ 2 R ( 2 Δ ) + cos 2 γ 1 sin γ 2 R ( 2 Δ ) + sin 2 γ 1 cos γ 2 R ( 0 ) ]
γ ¯ | ¯ 0 = 0.
cos γ ¯ = cos 2 γ 1 cos γ 2 sin 2 γ 1 sin γ 2 cos 2 Δ ,
γ ¯ | t ¯ 0 = 1 sin γ ¯ [ sin 2 γ 1 cos γ 2 ( γ 2 cos 2 Δ + 2 γ 1 ) + cos 2 γ 1 sin γ 2 ( 2 γ 1 cos 2 Δ + γ 2 ) ] = 0.
( 1 ) : ( 1 ) cos 2 Δ = 2 γ 1 0 / γ 2 0 ,
( 2 ) cos 2 γ 1 0 sin γ 2 0 = 0 ;
( 2 ) : ( 1 ) cos 2 Δ = ( γ 2 0 / 2 γ 1 0 ) ,
( 2 ) sin 2 γ 1 0 cos γ 2 0 = 0 ;
( 3 ) : ( 1 ) 2 γ 1 0 = π / 2 ,
( 2 ) γ 2 0 = π / 2 .
Ω | = 0 = 0.
tan 2 Ω = Imag ( M 3 21 ) / Imag ( M 3 11 ) ,
Ω | = 0 = cos 2 2 Ω 2 M 3 11 [ γ 2 cos γ 2 sin 2 Δ M 3 21 M 3 11 × [ sin 2 γ 1 sin γ 2 ( 2 γ 1 cos 2 Δ + γ 2 ) + cos 2 γ 1 cos γ 2 ( γ 2 cos 2 Δ + 2 γ 1 ) ] ] = 0.
γ 2 0 = π / 2
γ ¯ = γ ¯ 0 1 2 ( 1 tan γ ¯ 0 ) [ ( 2 γ 1 0 ) 2 ( π / 2 ) 2 ] 2 ,
Ω = Ω 0 + ( sin 4 Δ 8 ) ( cos 2 γ 0 sin 2 γ ¯ 0 ) [ ( 2 γ 1 0 ) 2 2 ( π / 2 ) 2 ] 2
β = { β tan ( γ ¯ 0 + β / 2 ) / [ ( 2 γ 1 ) 2 ( π / 2 ) 2 ] } 1 / 2 ,
cos ( γ ¯ 0 + β / 2 ) = ( π / 2 ) ( sin 2 γ ¯ 1 0 / 2 γ ¯ 1 0 ) ,
cos 2 Δ = ( π / 2 ) [ 1 / ( 2 γ 1 0 ) ] .
cos γ = sin ( π / 2 ) [ A cos 2 ( π / 2 ) sin 2 ( π / 2 ) ] ,
A = ( 1 + 2 cos 2 Δ ) .
cos γ ¯ = ( π / 2 ) { A [ 1 + ( 7 / 6 ) A ] [ ( π / 2 ) ] 2 } .
Δ = ( π / 3 ) δ ,
A = 2 3 δ .
A = 0.061.
cos γ ¯ = ( π / 2 ) { A [ ( π / 2 ) ] 2 } .
| π / 2 = ± A 1 / 2 ( 2 / π ) = ± ( 2 / π ) ( 2 3 δ ) 1 / 2 = ± 1.185 δ 1 / 2 .
e x = ± ( 2 / π ) [ ( 1 / 3 ) A 1 / 2 ] = ± 0.684 δ 1 / 2
cos γ ¯ e x = ± 2 5 / 2 3 3 / 4 δ 3 / 2 = ± 2.482 δ 3 / 2 .
β / 2 = cos 1 [ ( 2 / 3 3 ) ( 2 3 δ ) 3 / 2 ] ,
δ = 0.55 ( β / 2 ) 2 / 3 .
> e x + β / 4 A .
β 1.04 ( β / 2 ) 1 / 3 .
β ( 2 / π ) ( β / 2 ) 1 / 3 β 0.637 ( β / 2 ) 1 / 3 .
M 3 = cos 3 γ 2 1 + i sin γ 2 E [ R ( 2 Δ + cos 2 γ 2 R ( 0 ) ] ,
M 9 S = M 3 R [ ( π / 3 ) ] M 3 R ( π / 3 ) ] M 3 ,
M 90 = M 3 R ( π / 3 ) M 3 R [ ( π / 3 ) ] M 3.
M 9 S = cos 9 γ 2 1 + i sin γ 2 E [ R ( 0 ) cos 2 γ 2 R ( 120 ) ( 1 + cos 6 γ 2 ) + cos 8 γ 2 R ( 120 ) ] ,
M 90 = cos 9 γ 2 1 + i sin γ 2 E [ R ( 0 ) cos 2 γ 2 R ( 120 ) + cos 6 γ 2 R ( 0 ) + cos 8 γ 2 R ( 0 ) ] .
cos γ ¯ = sin 9 ( π / 2 ) .
β = ( 2 / π ) sin 1 [ sin ( β / 2 ) ] 1 / 9 .
M 3 = M ( π / 2 , 0 ) M ( π / 2 , π / 3 ) M ( π / 2 , 0 ) .
M 4 = R ( α ) M ( π / 2 , 0 ) M ( π / 4 , π / 3 ) R ( α ) × M ( π / 4 , π / 3 ) M ( π / 2 , 0 ) .
M 4 = sin α R ( 90 α ) sin 3 ( π / 2 ) cos α R ( α ) + ( i / 2 ) sin π cos α E { R [ α 2 ( π / 3 ) ] + sin 2 ( π / 2 ) R ( α ) } .
sin ω cos γ ¯ = Real ( M 4 21 ) , cos ω cos γ ¯ = Real ( M 4 11 ) .
γ ¯ = cos 1 [ sin 2 α + cos 2 α sin 6 ( π / 2 ) ] .
sin 6 ( π / 2 ) sin 2 α ,
cos γ ¯ cos ( 90 α ) { 1 + 1 2 [ sin 6 ( π / 2 ) / tan 2 α ] } ,
tan ω tan ( 90 α ) { 1 + [ sin 3 ( π / 2 ) / sin 2 α ] } .
tan [ 2 Ω ( 90 α ) ] tan [ ( 2 π / 3 ) + α ] × [ 1 + 3 sin 2 ( π / 2 ) / sin 2 ( 2 π 3 + α ) ]
M 10 = R ( α ) M 3 M ( π / 2 , π / 3 ) M ( π / 4 , 2 π / 3 ) R ( α ) × M ( π / 4 , 2 π / 3 ) M ( π / 2 , ( π / 3 ) M 3 .
cos γ ¯ = cos ( 90 α ) { 1 + [ sin 18 ( π / 2 ) / 2 tan 2 α ] } ,
tan ω tan ( 90 α ) { 1 + [ sin 9 ( π / 2 ) / sin 2 α ] } .

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