Abstract

The field of view characteristics of wide field birefringent (WFB) elements are compared with those of the Fabry-Perot (F.P.), Michelson (MI), and wide field Michelson (WFM) interferometers. Throughput gains of 50 to 300 or more with respect to the F.P. or MI are demonstrated. Further, it is shown that by proper choice of material WFB elements can have angular characteristics identical with WFM interferometers. The properties of misaligned and mismatched half-length WFB elements are calculated. It is shown that properly misadjusted WFB elements can exhibit throughput gains with respect to properly adjusted systems. Finally, a catalog of fringe patterns for WFB elements of different materials and differing angular misadjustment is presented.

© 1979 Optical Society of America

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References

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  1. B. Lyot, C. R. Acad. Sci. Ser. A 197, 1593 (1933).
  2. B. Lyot, Ann. Astrophys. 7(1–2), 31 (1944).
  3. J. W. Evans, J. Opt. Soc. Am. 39, 229 (1949).
    [CrossRef]
  4. A. M. Title, T. P. Pope, M. E. Ramsey, S. A. Schoolman, NASA Report CR 156674 (1976).
  5. R. L. Hilliard, G. G. Shepherd, J. Opt. Soc. Am. 56, 362 (1966).
    [CrossRef]
  6. M. Francon, S. Mallick, Polarization Interferometers (Wiley, Interscience, New York, 1971).
  7. H. H. Zwick, G. G. Shepherd, Appl. Opt. 10, 2569 (1971).
    [CrossRef] [PubMed]
  8. A. M. Title, Appl. Opt. 14, 445 (1975).
    [CrossRef] [PubMed]
  9. A. M. Title, Solar Phys. 33, 521 (1973).
  10. A. M. Title, Appl. Opt. 14, 229 (1975).
    [CrossRef] [PubMed]
  11. A. M. Title, Appl. Opt. 15, 2871 (1976).
    [CrossRef] [PubMed]

1976 (1)

1975 (2)

1973 (1)

A. M. Title, Solar Phys. 33, 521 (1973).

1971 (1)

1966 (1)

1949 (1)

1944 (1)

B. Lyot, Ann. Astrophys. 7(1–2), 31 (1944).

1933 (1)

B. Lyot, C. R. Acad. Sci. Ser. A 197, 1593 (1933).

Evans, J. W.

Francon, M.

M. Francon, S. Mallick, Polarization Interferometers (Wiley, Interscience, New York, 1971).

Hilliard, R. L.

Lyot, B.

B. Lyot, Ann. Astrophys. 7(1–2), 31 (1944).

B. Lyot, C. R. Acad. Sci. Ser. A 197, 1593 (1933).

Mallick, S.

M. Francon, S. Mallick, Polarization Interferometers (Wiley, Interscience, New York, 1971).

Pope, T. P.

A. M. Title, T. P. Pope, M. E. Ramsey, S. A. Schoolman, NASA Report CR 156674 (1976).

Ramsey, M. E.

A. M. Title, T. P. Pope, M. E. Ramsey, S. A. Schoolman, NASA Report CR 156674 (1976).

Schoolman, S. A.

A. M. Title, T. P. Pope, M. E. Ramsey, S. A. Schoolman, NASA Report CR 156674 (1976).

Shepherd, G. G.

Title, A. M.

A. M. Title, Appl. Opt. 15, 2871 (1976).
[CrossRef] [PubMed]

A. M. Title, Appl. Opt. 14, 229 (1975).
[CrossRef] [PubMed]

A. M. Title, Appl. Opt. 14, 445 (1975).
[CrossRef] [PubMed]

A. M. Title, Solar Phys. 33, 521 (1973).

A. M. Title, T. P. Pope, M. E. Ramsey, S. A. Schoolman, NASA Report CR 156674 (1976).

Zwick, H. H.

Ann. Astrophys. (1)

B. Lyot, Ann. Astrophys. 7(1–2), 31 (1944).

Appl. Opt. (4)

C. R. Acad. Sci. Ser. A (1)

B. Lyot, C. R. Acad. Sci. Ser. A 197, 1593 (1933).

J. Opt. Soc. Am. (2)

Solar Phys. (1)

A. M. Title, Solar Phys. 33, 521 (1973).

Other (2)

M. Francon, S. Mallick, Polarization Interferometers (Wiley, Interscience, New York, 1971).

A. M. Title, T. P. Pope, M. E. Ramsey, S. A. Schoolman, NASA Report CR 156674 (1976).

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Figures (6)

Fig. 1
Fig. 1

Optical schematic of a simple birefringent element and a normal Michelson interferometer.

Fig. 2
Fig. 2

Optical schematic of a wide field birefringent element and a wide field Michelson interferometer.

Fig. 3
Fig. 3

Top half of the figure (a) shows angle of incidence and sine of the angle of incidence vs wavelength tolerance and bandpass for quartz, calcite, and optimal q wide field birefringent elements. The bottom half of the figure (b) shows the gain in incidence angle and throughput, wrt, F.P., or MI for quartz, calcite, and optimal q wide field birefringent elements. Also shown are the plots for the second-and fourth-order approximations for quartz and calcite.

Fig. 4
Fig. 4

Shown in Figs. 4, 5, and 6 are the positions of the bright fringes in (i,θ) space for m = ±1–±5 of a 1-Å wide field element. Positive and negative fringes are shown as solid and dotted lines, respectively. The vertical (y) axis of the plot corresponds to the optic axis. The value of the radius vector to a point on a fringe corresponds to the incident angle and the azimuth of the radius vector to theta. The tick marks are at 5° intervals. For Fig. 4 the full field is 40°. Figures (a),(b),(c),(d), and (e) represent misalignments of 0, 0.4α0, 0.8α0, α0, and 1.5α0, respectively, of a quartz element.

Fig. 5
Fig. 5

These figures follow the conventions of Fig. 4, but the full field of view is 75°. Figures (a),(b),(c),(d), and (e) represent a wide field 1-Å calcite element with misalignments of 0, 0.2q, 0.4q, 0.5q, and 0.75q, respectively.

Fig. 6
Fig. 6

These figures are similar to those of Fig. 5 for a 1-Å reverse calcite wide field birefringent element.

Tables (1)

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Table I Solutions for x2

Equations (124)

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Δ 0 = d c ( n e - n o ) ,
I ( λ ) = cos 2 ( π Δ 0 c λ ) ,
Δ 0 = 2 c ( d M - d M )
I ( λ ) = cos 2 ( π Δ 0 c λ ) .
Δ = Δ 0 [ 1 - ( a 2 + b 2 n 2 ) ] 1 / 2 ,
a = sin i cos θ ,             b = sin i sin θ ,
Δ = Δ 0 ( 1 - sin 2 i n 2 ) 1 / 2 .
Δ = Δ 0 ( 1 - 1 2 sin 2 i n 2 ) .
( δ Δ ) / Δ 0 = ( δ λ ) / λ 0 ,
δ λ λ 0 = - 1 2 sin 2 i n 2
δ λ λ 0 - 1 2 ( i n ) 2 .
Δ = d c { n e [ 1 - ( a 2 n o 2 + b 2 n e 2 ) ] 1 / 2 - n o [ 1 - ( a 2 + b 2 n o 2 ) ] 1 / 2 } ,
Δ = Δ 0 { 1 - sin 2 i 2 n o 2 [ cos 2 θ - ( n o n e ) sin 2 θ ] } ,
cos 2 θ = ( n o n e ) sin 2 θ
tan θ = ± ( n e n o ) 1 / 2 ,
( δ Δ ) / Δ 0 = ( δ λ ) / λ 0 = 0 ;
λ λ 0 | θ = 0 = - sin 2 i 2 n o 2 ,
λ λ 0 | θ = 90 ° = + sin 2 i 2 n o n e .
Δ = d 2 c { n e [ 1 - ( a 2 n o 2 + b 2 n e 2 ) ] 1 / 2 - n o [ 1 - ( a 2 + b 2 n o 2 ) ] 1 / 2 } + d 2 c { n e [ 1 - ( a 2 n o 2 + b 2 n e 2 ) ] 1 / 2 - n o [ 1 - ( a 2 + b 2 n o 2 ) ] 1 / 2 } ,
a - sin i cos ( θ + 90 ) ,             b = sin i sin ( θ + 90 ) .
Δ = Δ 0 [ 1 - 1 4 n o 2 ( n e - n o n e ) sin 2 i ]
δ λ λ 0 = - 1 4 n o 2 ( n e - n o n e ) sin 2 i .
Δ = 2 c [ n d M ( 1 - sin 2 i n 2 ) 1 / 2 - n d M ( 1 - sin 2 i n 2 ) 1 / 2 ] .
Δ = 2 c [ ( n d M - n d M ) - sin 2 i 2 ( d M n - d M n ) - sin 4 i 8 ( d M n 3 - d M n 3 ) ] .
d M n n d M ,
Δ = Δ 0 ( 1 + sin 4 i 8 n 2 n 2 ) ,
Δ 0 = 2 c ( n d M - n d M ) ,
δ λ λ 0 = sin 4 i 8 n 2 n 2 .
d M = n n d M + c Δ 0 2 n ,
δ λ λ 0 sin 2 i 2 n n + sin 4 i 8 n 2 n 2 .
Δ = Δ 0 ( 1 q { 1 - x 2 [ 1 - ( 1 - p 2 ) z ] } 1 / 2 - p q ( 1 - x 2 ) 1 / 2 )
Δ 0 = ( d / c ) ( n e - n 0 ) ( a ) p = n 0 / n e ( b ) q = 1 - p ( c ) x = ( sin i ) / n o ( d ) z = sin 2 θ ( e ) } .
Δ = Δ 0 f ( x , p , q , z )
Δ = Δ 0 ( 1 + C 1 x 2 0 ( 1 ) + C 2 x 4 0 ( 2 ) + + C n x 2 n 0 ( n ) + ..
C n = - 2 ( 2 n - 2 ) ! 4 n n ! ( n - 1 ) ! .
0 ( n ) = 1 q { [ 1 - ( 1 - p 2 ) z ] n - p } .
C 1 = - 1 / 2 0 ( 1 ) = 1 - ( 1 + p ) z ( a ) C 2 = - 1 / 8 0 ( 2 ) = 1 - 2 ( 1 + p ) z + q ( 1 + p ) 2 z 2 ( b ) C 3 = - 1 / 16 0 ( 3 ) = 1 - 3 ( 1 + p ) z + 3 q ( 1 + p ) 2 z 2 - q 2 ( 1 + p ) 3 z 3 ( c ) } .
z = 1 / ( 1 + p )
tan θ = ± ( n e / n o ) 1 / 2 .
0 ( 2 ) = - p .
δ λ λ 0 = sin 4 i 8 n o 4 p .
Δ = Δ 0 2 [ f ( x , p , q , z ) + f ( x , p , q , z ) ] ,
z = sin 2 θ ,
z = sin 2 ( θ + 90 ° ) = cos 2 θ .
Δ = Δ 0 ( 1 + C 1 x 2 0 WF ( 1 ) + C 2 x 4 0 WF ( 2 ) + + C n x 2 n 0 WF ( 2 n ) + ..
0 WF ( n ) = 1 2 q { [ 1 ( 1 - p 2 ) z ] n + [ 1 - ( 1 - p 2 ) z ] n - 2 p } .
z = sin 2 θ = 0 ,             z = cos 2 θ = 1 ,
0 WF ( n ) θ = 0 = p 2 n - 2 p + 1 2 q .
δ λ λ 0 = - 1 4 q sin 2 i n o 2 - 1 16 [ q - p 2 ( 1 + p ) ] sin 4 i n o 4 - 1 32 ( p 6 - 2 p + 1 q ) sin 6 i n o 6 .
0 WF ( n ) θ = 0 { q / 2 n = 1 1 - n n > 1.
δ λ λ 0 = - q 4 sin 2 i n o 2 + 1 8 sin 4 i n o 4 + 1 8 sin 6 i n 0 6 .
C n 0 WF ( n ) θ = 0 = 2 ( 2 n - 2 ) ! 4 n n ! ( n - 1 ) ! ( n - 1 ) = 2 4 n ( 2 n - 2 2 ) n > 1.
C n 0 WF ( n ) θ = 0 = 1 8 = 0.125 n = 2 = 1 8 = 0.125 n = 3 = 15 128 = 0.117 n = 4 } .
0 WF ( n ) = 1 + 1 2 q i = 1 n ( - 1 ) i ( 1 - p 2 ) i ( n i ) × ( sin 2 i θ + cos 2 i θ )             n > 1.
S i = sin 2 i θ + cos 2 i θ
S i = 1 - k = 1 i / 2 A i ( k ) sin 2 k 2 θ
A i ( 1 ) = i / 4 i > 2 A i + 1 ( k ) = A i ( k ) - 1 4 A i - 1 ( k - 1 ) k > 1 , i > 2.
S 1 = sin 2 θ + cos 2 θ = 1 S 2 = sin 4 θ + cos 4 θ = 1 - 1 2 sin 2 2 θ S 3 = sin 6 θ + cos 6 θ = 1 - 3 4 sin 2 2 θ S 4 = sin 8 θ + cos 8 θ = 1 - sin 2 2 θ + 1 8 sin 4 2 θ } .
0 WF ( 1 ) = q / 2 0 WF ( 2 ) = p 4 - 2 p + 1 2 q - ( 1 + p ) 2 q 4 sin 2 2 θ 0 WF ( 3 ) = p 6 - 2 p + 1 2 q - 3 8 ( 1 + p ) 2 q ( 1 + p 2 ) sin 2 2 θ 0 WF ( 4 ) = p 8 - 2 p + 1 2 q - 1 2 ( 1 + p ) 2 q ( 1 + p 2 + p 4 ) × sin 2 2 θ + 1 16 ( 1 + p ) 4 p 3 sin 4 2 θ } .
0 WF ( 1 ) = q / 2 0 WF ( 2 ) - 1 - q sin 2 2 θ 0 WF ( 3 ) - 2 - 3 q sin 2 2 θ 0 WF ( 4 ) - 3 - 6 q sin 2 2 θ + q 3 sin 4 2 θ } .
( - 1 ) n ( 1 + p ) 2 n q 2 n - 1 4 n .
( - 1 ) n q 2 n - 1 ,
Δ = Δ 0 { 1 + Δ ( i ) θ = 0 + Δ ( i , θ ) } .
δ λ λ 0 = δ λ λ 0 | θ = 0 + δ λ λ 0 ( θ ) .
δ λ λ 0 ( θ ) = q 8 sin 4 i n o 4 sin 2 2 θ + 3 q 8 sin 6 i n o 6 sin 2 2 θ .
δ λ λ 0 = - q 4 sin 2 i n o 2 + 1 8 sin 4 i n o 4 ( 1 + q sin 2 2 θ ) .
sin 2 i e = 2 q n o 2 ,
i e = sin - 1 ( 2 q n o 2 ) 1 / 2 .
δ λ i e = λ 0 2 q 2 .
z = sin 2 ( θ + 90 + α ) = cos 2 ( θ + α ) ,
0 WF ( n ) = 1 + 1 2 q i = 1 n ( - 1 ) i ( 1 - p 2 ) i ( n i ) × [ sin 2 i θ + cos 2 i ( θ + α ) ]             n > 1.
S i ( α ) = sin 2 i θ + cos 2 i ( θ + α ) .
R = sin 2 α cos 2 θ + 1 2 sin 2 α sin 2 θ ,
cos 2 ( θ + α ) = cos 2 θ - R ,
cos 2 i ( θ + α ) = cos 2 i θ + k = 1 i ( - 1 ) k ( n k ) R k cos 2 i - 2 k θ .
S i ( α ) = S i + k = 1 i ( - 1 ) k ( n k ) R k cos 2 i - 2 k θ .
0 WF ( n ; α ) = 1 2 q i = 1 n ( - 1 ) i ( 1 - p 2 ) i ( n i ) × k = 1 i ( - 1 ) k ( i k ) R k cos 2 i - 2 k θ .
R α sin 2 θ ,
0 WF ( n ; α ) α sin 2 θ 2 q i = 1 n ( - 1 ) i ( 1 - p 2 ) i ( n i ) i ,
n α sin 2 θ ( 1 + p ) 2 p 2 n - 2 .
0 WF ( n ; α n α sin 2 θ ) .
0 WF ( 1 ) q 2 + α sin 2 θ 0 WF ( 2 ) - 1 - q sin 2 2 θ + 2 α sin 2 θ 0 WF ( 3 ) - 2 - 3 q sin 2 2 θ + 3 α sin 2 θ 0 WF ( 4 ) - 3 - 6 q sin 2 2 θ + q 3 sin 4 2 θ + 4 α sin 2 θ } .
δ λ λ 0 - 1 4 sin 2 i n o 0 ( q + 2 α sin 2 θ ) + 1 8 sin 4 i n o 4 ( 1 - 2 α sin 2 θ + q sin 2 2 θ ) .
d 1 = d 0 2 + T d 0 ,             d 2 = d 0 2 - T d 0 .
Δ = Δ 0 2 [ f ( x , p , q , z ) + f ( x , p , q , z ) ] + T Δ 0 [ f ( x , p , q , z ) - f ( x , p , q , z ) ] .
0 WF ( n ; T ) = 1 2 q { [ 1 - ( 1 - p 2 ) z ] n - [ 1 - ( 1 - p 2 ) z ] n } .
0 WF ( n ; T ) = 1 2 q i = 1 n ( - 1 ) i ( 1 - p 2 ) i ( n i ) ( sin 2 i θ - cos 2 i θ ) .
D i = sin 2 i θ - cos 2 i θ
D i = - cos 2 θ [ 1 + k = 1 ( i - 1 ) 2 B i ( k ) sin 2 k 2 θ ] ,
B i ( 1 ) = 2 - i 4             i > 2 B i ( k ) = 1 4 B i - 2 ( k - 1 ) - A i - 1 ( k )             i > 3 , k > 1
0 WF ( n ; T ) n cos 2 θ .
Δ ( T ) = Δ 0 T cos 2 θ ( C 1 x 2 + 2 C 2 x 4 + + n C n x 2 n + ) .
δ λ λ 0 = - 1 4 sin 2 i n o 2 ( q + 2 T cos 2 θ ) + 1 8 sin 4 i n o 4 ( 1 + q sin 2 2 θ - 2 T cos 2 θ ) .
δ λ / λ < q 2 / 2 ,
sin i = 2 n o q 1 / 2 ( δ λ λ ) 1 / 2 .
I 2 = ( n o n F . P . ) 2 q 1 / 2 .
G 2 = I 2 2 = 2 q ( n o n F . P . ) 2 .
sin i = 2 n o ( 1 2 δ λ λ ) 1 / 4 = n o ( 8 δ λ λ ) 1 / 4 ,
I 4 = ( n o n F . P . ) ( 2 λ δ λ ) 1 / 4 ,
G 4 = ( n o n F . P . ) 2 ( 2 λ δ λ ) 1 / 2 .
( sin i ) ( δ λ λ ) = 0
sin i = n o q .
δ λ λ = - q 2 / 8.
sin i = 2 n o ( 2 δ λ λ ) 1 / 4 ,
I 0 = 2 ( n o n F . P . ) ( 1 2 λ δ λ ) 1 / 4 ,
G 0 = 4 ( n o n F . P . ) 2 ( 1 2 λ δ λ ) 1 / 2 ,
sin i major = 2 n o ( 1 2 δ λ λ 0 ) 1 / 4 ,
sin i minor = ( 2 1 / 2 n o ) ( δ λ λ ) 1 / 2 .
W = q 1 / 2 2 ( λ 0 2 δ λ ) 1 / 4 ,
m = R [ - sin 2 i 4 n o 2 ( q + 2 α sin 2 θ + 2 T cos 2 θ ) + sin 4 i 8 n o 4 ] ,
R = ( Δ 0 c ) / λ 0 .
x 4 = 2 q x 2 - 8 m R = 0 ,
q = q + 2 α sin 2 θ + 2 T cos 2 θ .
Q = [ ( 2 q ) 2 + 32 m / R ] 1 / 2 .
q 2 < 8 m / R ,
q 2 > 8 m / R ,
sin i = n o ( 8 m / R ) 1 / 4 .
sin i = 2 n o ( m R q ) 1 / 2 .
sin i = 2 n o ( m R q ) 1 / 2 ,
sin i = n o [ 2 q ( 1 + 2 m R q 2 ) ] 1 / 2 .
sin i = n o [ 2 q ( 1 - 2 m R q 2 ) ] 1 / 2 .
i = sin - 1 ( 2 q n o 2 ) 1 / 2
m / R = ( 1 - 2 α sin 2 θ ) x 4 - 2 α sin 2 θ x 2 .
α 0 = 1 / R .

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