Abstract

The efficiency of a Lyot stop in reducing diffraction effects in a simple optical system is analyzed. The transformation rules of Fourier optics are used to follow the optical signal through the system. This analysis reduces the problem to the evaluation of a two-dimensional integral. An approximate analytic technique for evaluating this integral is presented. The final result is a simple formula for calculating the diffraction-reduction efficiency of the Lyot stop. A comparison of this work with other published work is also presented.

© 1987 Optical Society of America

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References

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  1. B. Lyot, “A study of the solar corona and prominences without eclipses,” Mon. Not. R. Astron. Soc. 99, 580–594 (1939).
  2. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 8.9.
  3. R. J. Noll, “Reduction of diffraction of use of a Lyot stop,”J. Opt. Soc. Am. 63, 1399–1402 (1973).
    [CrossRef]
  4. W. A. Kleinhans, “Diffraction from a sequence of apertures and disks,”J. Opt. Soc. Am. 65, 1451–1456 (1975).
    [CrossRef]
  5. Guerap II Users Guide, (Perkin-Elmer Corporation, Norwalk, Conn., February1974); pp. 56–71.
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  7. Ref. 6, Chaps. 4 and 5.
  8. Ref. 6, Chap. 5, Sec. 5.3.
  9. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stand. (US) Appl. Math. Ser. 55 (U.S. Government Printing Office, Washington, D.C., 1964; Dover, New York, 1965).
  10. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

1975 (1)

W. A. Kleinhans, “Diffraction from a sequence of apertures and disks,”J. Opt. Soc. Am. 65, 1451–1456 (1975).
[CrossRef]

1973 (1)

R. J. Noll, “Reduction of diffraction of use of a Lyot stop,”J. Opt. Soc. Am. 63, 1399–1402 (1973).
[CrossRef]

1939 (1)

B. Lyot, “A study of the solar corona and prominences without eclipses,” Mon. Not. R. Astron. Soc. 99, 580–594 (1939).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stand. (US) Appl. Math. Ser. 55 (U.S. Government Printing Office, Washington, D.C., 1964; Dover, New York, 1965).

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 8.9.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

Kleinhans, W. A.

W. A. Kleinhans, “Diffraction from a sequence of apertures and disks,”J. Opt. Soc. Am. 65, 1451–1456 (1975).
[CrossRef]

Lyot, B.

B. Lyot, “A study of the solar corona and prominences without eclipses,” Mon. Not. R. Astron. Soc. 99, 580–594 (1939).

Noll, R. J.

R. J. Noll, “Reduction of diffraction of use of a Lyot stop,”J. Opt. Soc. Am. 63, 1399–1402 (1973).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stand. (US) Appl. Math. Ser. 55 (U.S. Government Printing Office, Washington, D.C., 1964; Dover, New York, 1965).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 8.9.

J. Opt. Soc. Am. (2)

R. J. Noll, “Reduction of diffraction of use of a Lyot stop,”J. Opt. Soc. Am. 63, 1399–1402 (1973).
[CrossRef]

W. A. Kleinhans, “Diffraction from a sequence of apertures and disks,”J. Opt. Soc. Am. 65, 1451–1456 (1975).
[CrossRef]

Mon. Not. R. Astron. Soc. (1)

B. Lyot, “A study of the solar corona and prominences without eclipses,” Mon. Not. R. Astron. Soc. 99, 580–594 (1939).

Other (7)

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Sec. 8.9.

Guerap II Users Guide, (Perkin-Elmer Corporation, Norwalk, Conn., February1974); pp. 56–71.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Ref. 6, Chaps. 4 and 5.

Ref. 6, Chap. 5, Sec. 5.3.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stand. (US) Appl. Math. Ser. 55 (U.S. Government Printing Office, Washington, D.C., 1964; Dover, New York, 1965).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

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Figures (4)

Fig. 1
Fig. 1

A simple optical system consisting of an entrance aperture lens L1, a field lens L2, a transfer lens L3, and a detector D. The quantities in parentheses are the focal length and the radius of each lens. The solid lines are rays from a distant source that is outside the field of view of the system. The dashed lines are the diffracted rays that emanate from the rim of L1 and are blocked by the Lyot stop at L3.

Fig. 2
Fig. 2

A graphical representation that shows the relation of the coordinates r and θ to the geometry of the problem. The dashed–dotted circle is the boundary between regions I and II; region I is interior to this circle, region II is between this circle and the circle of radius B. The inner dashed circle is a typical angular integration path in region I; the outer dashed circle is a typical angular integration path in region II.

Fig. 3
Fig. 3

Diffraction-reduction ratio as a function of position on the detector. The curves are calculated using Eq. (54) in the text and are labeled by the parameter α. The other parameters used in this example problem are given in the text.

Fig. 4
Fig. 4

Diffraction-reduction ratio at the center of the detector as a function of the source angle. The curves are calculated by using Eq. (54) and are labeled by the parameter α. The dots are calculated by using Noll’s formula given by Eq. (58). The dashed line is drawn at the source angle γ = 1.9°, which pertains to the results shown in Fig. 3.

Equations (110)

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U ( x , y , z ) = U 0 exp { i k [ cos ( γ ) z - sin ( γ ) x ] } ,
k = 2 π / λ .
U ( x , y ) = U 0 exp ( - i k γ x ) .
U ( x , y ) = U 0 exp ( - i k γ x ) circ ( ρ / a ) exp ( - i k ρ 2 2 f ) ,
ρ = ( x 2 + y 2 ) 1 / 2
circ ( ρ ) = { 0 ρ > 1 1 0 ρ 1 .
U ( x , y ) = U 0 π a 2 λ f exp ( i k 2 f ρ 2 ) A { k a f [ ( x + γ f ) 2 + y 2 ] 1 / 2 } ,
A ( ρ ) = 2 J 1 ( ρ ) ρ ,
U ( x , y ) = U 0 π a 2 λ f exp ( - i k ρ 2 2 f ) circ ( ρ / b ) × A { k a f [ ( x + γ f ) 2 + y 2 ] 1 / 2 } .
U g ( x , y ) = U 0 π a 2 λ f exp ( - i k ρ 2 2 f ) circ ( ρ / b ) × A { k a f [ ( x - γ f ) 2 + y 2 ] 1 / 2 } .
h ( x 1 , y 1 ; x , y ) = π c 2 λ 2 f 2 exp [ i k 2 f ( ρ 1 2 + ρ 2 ) ] × A { k c f [ ( x 1 - x ) 2 + ( y 1 - y ) 2 ] 1 / 2 } .
U ( x 1 , y 1 ) = h ( x 1 , y 1 ; x , y ) U g ( x , y ) d x d y .
U ( x 1 , y 1 ) = U 0 π 2 a 2 c 2 λ 3 f 3 exp ( i k 2 f ρ 1 2 ) d x d y circ ( ρ / b ) × A { k c f [ ( x - x 1 ) 2 + ( y - y 1 ) 2 ] 1 / 2 } × A { k a f [ ( x - γ f ) 2 + y 2 ] 1 / 2 } .
X = σ x ,
Y = σ y ,
X 1 = σ x 1 ,
Y 1 = σ y 1 ,
Σ = σ ρ = ( X 2 + Y 2 ) 1 / 2 ,
σ = k a / f .
X 0 = k a γ ,
B = σ b ,
α = c / a .
U ( X 1 , Y 1 ) = U 0 c 2 4 λ f d X d Y circ ( Σ / B ) × A { α [ X - X 1 ) 2 + ( Y - Y 1 ) 2 ] 1 / 2 } × A { [ ( X - X 0 ) 2 + Y 2 ] 1 / 2 } .
μ ( r ) = cos - 1 ( B 2 - X 1 2 - r 2 2 X 1 r ) .
z ( r , θ ) = ( r 2 + d 2 - 2 r d cos θ ) 1 / 2 ,
d = X 0 - X 1 .
U ( X 1 , 0 ) = U 0 c 2 4 γ f [ 0 B - X 1 A ( α r ) F ( r ) r d r + B - X 1 B + X 1 A ( α r ) H ( r ) r d r ] ,
F ( r ) = 2 0 π A [ z ( r , θ ) ] d θ
H ( r ) = 2 μ ( r ) π A [ z ( r , θ ) ] d θ .
z ( r , θ ) = d - r cos θ .
A ( ρ ) 2 ρ ( 2 π ρ ) 1 / 2 sin ( ρ - π / 4 ) .
F ( r ) = 4 π d ( 2 π d ) 1 / 2 sin ( d - π / 4 ) J 0 ( r ) .
F ( r ) = 4 π d J 1 ( d ) J 0 ( r ) .
F ( r ) = 4 ( d r 1 / 2 ) [ sin ( d - r ) d - r - cos ( d + r ) d + r ] ,
F ( r ) = 4 π d J 1 ( d ) J 0 ( r ) + G ( r ) .
G ( r ) = 0 ,
G ( r ) = 4 ( d r ) 1 / 2 [ sin ( d - r ) d - r - cos ( d + r ) d + r ] - 4 π d J 1 ( d ) J 0 ( r ) .
G ( r ) = 4 d ( r d ) 1 / 2 [ sin ( d - r ) d - r + cos ( d + r ) d + r ] .
H ( r ) = - 4 cos ( d + r ) ( r d ) 1 / 2 ( d + r ) .
U ( X 1 , 0 ) = 2 U 0 c 2 γ f α d ( I 1 + I 2 ) ,
I 1 = π J 1 ( d ) 0 B - X 1 J 1 ( α r ) J 0 ( r ) d r
I 2 = ( 2 π α d ) 1 / 2 { r a B - X 1 sin ( α r - π / 4 ) × [ sin ( d - r ) d - r + cos ( d + r ) d + r ] d r - d B - X 1 B + X 1 sin ( α r - π / 4 ) cos ( d + r ) r ( d + r ) d r }
I 1 = ( 2 π α d ) 1 / 2 sin ( d - π / 4 ) × { π 2 - Si [ ( 1 - α ) ( B - X 1 ) ] - Ci [ ( 1 + α ) ( B - X 1 ) ] } ,
I 2 = ( 2 π α d ) 1 / 2 [ r a B - X 1 sin ( α r - π / 4 ) sin ( d - r ) d - r d r + r a B + X 1 sin ( α r - π / 4 ) cos ( d + r ) d + r d r - B - X 1 B + X 1 sin ( α r - π / 4 ) cos ( d + 4 ) r d r ]
U ( X 1 , 0 ) = U 0 c 2 λ f α d ( 2 π α d ) 1 / 2 [ sin ( d - π / 4 ) Γ 1 + cos ( d - π / 4 ) Γ 2 + sin ( α d - π / 4 ) Γ 3 + cos ( α d - π / 4 ) Γ 4 ] ,
Γ 1 = π - Ci [ ( 1 + α ) ( B + X 1 ) ] - Ci [ ( 1 + α ) ( B - X 1 ) ] - Si [ ( 1 - α ) ( B + X 1 ) ] - Si [ ( 1 - α ) ( B - X 1 ) ] ,
Γ 2 = Ci [ ( 1 - α ) ( B + X 1 ) ] - Ci [ ( 1 - α ) ( B - X 1 ) ] - Si [ ( 1 + α ) ( B + X 1 ) ] + Si [ ( 1 + α ) ( B - X 1 ) ] ,
Γ 3 = Si [ ( 1 - α ) ( X 0 + B ) ] - Si [ ( 1 - α ) ( d + r a ) ] - Si [ ( 1 + α ) ( X 0 + B ) ] + Si [ ( 1 + α ) ( d + r a ) ] - Si [ ( 1 - α ) ( X 0 - B ) ] + Si [ ( 1 - α ) ( d - r a ) ] - Si [ ( 1 + α ) ( X 0 - B ) ] + Si [ ( 1 + α ) ( d - r a ) ] ,
Γ 4 = Ci [ ( 1 - α ) ( X 0 - B ) ] - Ci [ ( 1 - α ) ( d - r a ) ] - Ci [ ( 1 + α ) ( X 0 - B ) ] + Ci [ ( 1 + α ) ( d - r a ) ] - Ci [ ( 1 - α ) ( X 0 + B ) ] + Ci [ ( 1 - α ) ( d + r a ) ] - Ci [ ( 1 + α ) ( X 0 + B ) ] + Ci [ ( 1 + α ) ( d + r a ) ] .
X 0 B + 2 1 - α ,
X 1 B - 2 1 - α .
Γ 1 = cos [ ( 1 - α ) ( B + X 1 ) ] ( 1 - α ) ( B + X 1 ) + cos [ ( 1 - α ) ( B - X 1 ) ] ( 1 - α ) ( B - X 1 ) ,
Γ 2 = sin [ ( 1 - α ) ( B + X 1 ) ] ( 1 - α ) ( B + X 1 ) - sin [ ( 1 - α ) ( B - X 1 ) ] ( 1 - α ) ( B - X 1 ) ,
Γ 3 = cos [ ( 1 - α ) ( X 0 - B ) ] ( 1 - α ) ( X 0 - B ) - cos [ ( 1 - α ) ( X 0 + B ) ] ( 1 - α ) ( X 0 + B ) + ν 3 ,
Γ 4 = sin ( 1 - α ) ( X 0 - B ) ] ( 1 - α ) ( X 0 - B ) - sin [ ( 1 - α ) ( X 0 + B ) ] ( 1 - α ) ( X 0 + B ) + ν 4 ,
ν 3 = cos [ ( 1 - α ) ( d + r a ) ] d + r a - cos [ ( 1 - α ) ( d - r a ) ] d - r a
ν 4 = sin [ ( 1 - α ) ( d + r a ) ] d + r a - sin [ ( 1 - α ) ( d - r a ) ] d - r a .
U ( X 1 , 0 ) = U 0 c 2 λ f α d ( 2 π α d ) 1 / 2 ( Q 1 + Q 2 ) ,
Q 1 = sin ( ϕ 1 + ϕ 2 ) 1 - α [ X 0 - X 1 ( B + X 1 ) ( X 0 + B ) ] ,
Q 2 = sin ( ϕ 1 - ϕ 2 ) 1 - α [ X 0 - X 1 ( B - X 1 ) ( X 0 - B ) ] ,
ϕ 1 = X 0 - π / 4 - α X 1 ,
ϕ 2 = ( 1 - α ) B .
I ( X 1 , 0 ) = 2 U 0 2 c 4 λ 2 f 2 α 3 d 3 π ( Q 1 2 + Q 2 2 ) ,
U g ( X 1 , 0 ) = U 0 π a 2 λ f 2 J 1 ( X 0 - X 1 ) X 0 - X 1 .
I g ( X 1 , 0 ) = 4 U 0 2 π a 4 f 2 λ 2 ( X 0 - X 1 ) 3 ,
R = I ( X 1 , 0 ) / I g ( X 1 , 0 ) ,
R = α ( X 0 - X 1 ) 2 4 π 2 ( 1 - α ) 2 [ 1 ( B - X 1 ) 2 ( X 0 - B ) 2 + 1 ( B + X 1 ) 2 ( X 0 + B ) 2 ] .
β = b / f = B / ( k a ) ,
δ = x 1 / f = X 1 / ( k a ) ,
R = α ( γ - δ ) 2 4 π 2 ( k a ) 2 ( 1 - α ) 2 [ 1 ( β - δ ) 2 ( γ - β ) 2 + 1 ( β + δ ) 2 ( γ + β ) 2 ] .
I ( δ , 0 ) = R E f 2 π 2 k a ( γ - δ ) 3 ,
E = U 0 2 π a 2
0.95 α < 1 ,
δ β - 2 ( 1 - α ) k a ,
β + 2 ( 1 - α ) k a γ .
R Noll = 2 π 2 ( k a β ) 2 ( 1 - α 2 ) 2 .
R = 2 α 4 π 2 ( k a β ) 2 ( 1 - α ) 2 ,
R 1 2 π 2 ( k a β ) 2 ( 1 - α ) 2 .
R GTD = α ( π k a β ) 2 F ( α ) F ( β / γ ) ,
F ( u ) = 1 2 [ 1 ( 1 - u ) 2 + 1 ( 1 + u ) 2 ] .
R = α ( π k a β ) 2 [ 1 2 ( 1 - α ) 2 ] F ( β / γ ) .
F ( α ) 1 2 ( 1 - α ) 2 ,
R GTD R = 1 + ( 1 - α 1 + α ) 2 .
F ( r ) = 2 π 0 π [ 2 π z ( r , θ ) ] 3 / 2 cos [ z ( r , θ ) - 3 π / 4 ] d θ ,
z ( r , θ ) = a + b θ 2 ,
a = d - r
b = r d 2 ( d - r )
F 0 ( r ) = 2 π ( 2 π a ) 3 / 2 0 π Re { exp [ i ( a - 3 π 4 ) ] exp ( ω θ 2 } d θ ,
ω = - 3 2 b a + i b
ω i b .
F 0 ( r ) = 4 sin ( d - r ) ( r d ) 1 / 2 ( d - r ) .
z ( r , θ ) = a + b ( θ - π ) 2 ,
a = d + r
b = - r d 2 ( d + r ) .
F π ( r ) = 2 π ( 2 π a ) 3 / 2 0 π Re { exp [ i ( a - 3 π / 4 ) ] exp ( ω ϕ 2 ) } d ϕ ,
F π ( r ) = - 4 cos ( d + r ) ( r d ) 1 / 2 ( d + r ) .
F ( r ) = 4 ( r d ) 1 / 2 [ sin ( d - r ) ( d - r ) - cos ( d + r ) ( d + r ) ] .
F ( r ) = 4 d 3 / 2 r 1 / 2 [ sin ( d - r ) 1 - r / d - cos ( d + r ) 1 + r / d ] .
F ( r ) = 4 d 3 / 2 r 1 / 2 [ sin ( d - r ) - cos ( d + r ) ] .
F ( r ) = 4 d 3 / 2 r 1 / 2 sin ( d - π / 4 ) cos ( r - π / 4 ) .
F ( r ) = 4 π d J 1 ( d ) J 0 ( r ) .
4 π d J 1 ( d ) J 0 ( r ) 4 ( d r ) 1 / 2 [ sin ( d - r ) d - cos ( d + r ) d ] .
F ( α z ) = 0 z J 0 ( x ) J 1 ( α x ) d x ,
F ( α z ) = 0 J 0 ( x ) J 1 ( α x ) d x - z J 0 ( x ) J 1 ( α x ) d x .
0 J 0 ( x ) J 1 ( α x ) d x = α - 1 when α > 1 = 0.5 when α = 1 = 0 when α < 1.
F ( α z ) = - z J 0 ( x ) J 1 ( α x ) d x .
F ( α z ) = - 2 π α z cos ( x - π / 4 ) sin ( α x - π / 4 ) x d x ,
F ( α z ) = 1 π α { π 2 - S i [ ( 1 - α ) z ] - C i [ ( 1 + α ) z ] } ,
F ( α z ) = 1 π α { cos [ ( 1 - α ) z ] ( 1 - α ) z - sin [ ( 1 + α ) z ( 1 + α ) z } .
F ( α z ) = - cos 2 ( z - π / 4 ) α π z - cos [ ( 1 - α 2 ) z / 2 ] π α ( 1 - α 2 ) z / 2 ,

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