Abstract

The Maréchal evaluation of the Strehl definition is reexamined for a Gaussian aperture where we have included the primary aberrations and all orders of spherical aberration. The result is particularly useful for evaluating the far-field peak intensity degradation, due to aberrations, for a well-corrected optical system when the wavefront distortion at the exit pupil is known. Further, the resulting equations have the same form as Maréchal’s equations for a uniform beam with the exception of additional factors that are the consequence of the Gaussian beam. These factors approach unity as the Gaussian beam approaches a uniform beam. As a consequence, the effects of the Gaussian illumination are readily identified. It is also shown how various aberrations may be balanced against one another in order to obtain the best peak intensity in the presence of a truncated Gaussian beam.

© 1974 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 459.
  2. Rayleigh, Phil. Mag 8, 403 (1879). Reprinted in his Scientific Papers (Cambridge University Press, 1899), Vol. 1, pp. 432–435.
  3. A. Maréchal, Rev. Opt. 26, 257 (1947).
  4. W. Augustyn, R. Austin, W. Goggin, A. Slomba, Electro-Opt. Syst. Design 2, 16 (1973).
  5. J. Campbell, L. DeShazer, J. Opt. Soc. Am. 59, 1427 (1969).
    [CrossRef]
  6. G. Olaofe, J. Opt. Soc. Am. 60, 1654 (1970).
    [CrossRef]
  7. D. Holmes, P. Avizonis, Appl. Opt. 9, 2179 (1970).
    [CrossRef] [PubMed]
  8. L. Dickson, Appl. Opt. 9, 1854 (1970).
    [CrossRef] [PubMed]
  9. R. Schell, G. Tyras, J. Opt. Soc. Am. 61, 31 (1971).
    [CrossRef]
  10. D. Holmes, J. Korka, P. Avizonis, Appl. Opt. 11, 565 (1972).
    [CrossRef] [PubMed]
  11. W. King, J. Opt. Soc. Am. 58, 655 (1968).
    [CrossRef]

1973 (1)

W. Augustyn, R. Austin, W. Goggin, A. Slomba, Electro-Opt. Syst. Design 2, 16 (1973).

1972 (1)

1971 (1)

1970 (3)

1969 (1)

1968 (1)

1947 (1)

A. Maréchal, Rev. Opt. 26, 257 (1947).

1879 (1)

Rayleigh, Phil. Mag 8, 403 (1879). Reprinted in his Scientific Papers (Cambridge University Press, 1899), Vol. 1, pp. 432–435.

Augustyn, W.

W. Augustyn, R. Austin, W. Goggin, A. Slomba, Electro-Opt. Syst. Design 2, 16 (1973).

Austin, R.

W. Augustyn, R. Austin, W. Goggin, A. Slomba, Electro-Opt. Syst. Design 2, 16 (1973).

Avizonis, P.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 459.

Campbell, J.

DeShazer, L.

Dickson, L.

Goggin, W.

W. Augustyn, R. Austin, W. Goggin, A. Slomba, Electro-Opt. Syst. Design 2, 16 (1973).

Holmes, D.

King, W.

Korka, J.

Maréchal, A.

A. Maréchal, Rev. Opt. 26, 257 (1947).

Olaofe, G.

Rayleigh,

Rayleigh, Phil. Mag 8, 403 (1879). Reprinted in his Scientific Papers (Cambridge University Press, 1899), Vol. 1, pp. 432–435.

Schell, R.

Slomba, A.

W. Augustyn, R. Austin, W. Goggin, A. Slomba, Electro-Opt. Syst. Design 2, 16 (1973).

Tyras, G.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 459.

Appl. Opt. (3)

Electro-Opt. Syst. Design (1)

W. Augustyn, R. Austin, W. Goggin, A. Slomba, Electro-Opt. Syst. Design 2, 16 (1973).

J. Opt. Soc. Am. (4)

Phil. Mag (1)

Rayleigh, Phil. Mag 8, 403 (1879). Reprinted in his Scientific Papers (Cambridge University Press, 1899), Vol. 1, pp. 432–435.

Rev. Opt. (1)

A. Maréchal, Rev. Opt. 26, 257 (1947).

Other (1)

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 459.

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Figures (6)

Fig. 1
Fig. 1

Diffraction of converging spherical wave from a circular aperture illustrating geometry and notation.

Fig. 2
Fig. 2

On-axis intensity degradation for a uniformly illuminated aperture vs the spherical aberration coefficient an. The curves are plotted for values of n ranging from 2 to 12.

Fig. 3
Fig. 3

The factor F(γ), a result of the Gaussian beam, is plotted vs γ (the truncation parameter).

Fig. 4
Fig. 4

On-axis intensity degradation for a Gaussian beam vs the spherical aberration coefficient an. γ = 1.0. The curves are plotted for values of n ranging from 2 to 12.

Fig. 5
Fig. 5

The intensity degradation plotted vs the peak-to-edge distortion (Z) for the wavefront distortion given by Φ ( ρ ) = 3 4 Z ρ 2 ( 2 3 ρ 2 ). The curves show the difference between the modified Maréchal result as derived in this paper and a computer integration of the complete diffraction integral in comparison. The curves show how the accuracy of our result [Eq. (9)] falls with increasing Z.

Fig. 6
Fig. 6

Proper ratio of the focusing and third-order spherical terms for maximum peak intensity in the far-field pattern. Note that the ratio approaches −1 as the Gaussian truncation increases.

Tables (1)

Tables Icon

Table I Function sn(γ), a Result of the Gaussian Beam, Tabulated for Various Values of Truncation Parametera

Equations (57)

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i = 1 ( 2 π / λ ) 2 ( Δ Φ ) 2 .
U ( P ) 0 1 0 2 π A ( ρ ) exp [ i k ( s + Φ ) ] ρ d ρ d θ ,
U ( P ) 0 1 0 2 π A ( ρ ) exp { i k [ υ ρ cos θ + ( 1 / 2 ) u ρ 2 Φ ( ρ ) ] } ρ d ρ d θ ,
S . D . = | 0 1 0 2 π A ( ρ ) exp { i k [ υ ρ cos θ + 1 / 2 u ρ 2 Φ ( ρ ) ] } ρ d ρ d θ | 2 / | 0 1 0 2 π A ( ρ ) ρ d ρ d θ | 2 .
i = | 0 1 0 2 π A ( ρ ) ρ d ρ d θ + i k 0 1 0 2 π A ( ρ ) Φ ( ρ ) ρ d ρ d θ k 2 2 0 1 0 2 π A ( ρ ) Φ 2 ( ρ ) ρ d ρ d θ | 2 / | 0 1 0 2 π A ( ρ ) ρ d ρ d θ | 2 ,
i = 1 π 2 | 0 1 0 2 π [ 1 + i k Φ ( ρ ) 1 2 k 2 Φ 2 ( ρ ) ] ρ d ρ d θ | 2 .
Φ n ¯ = 1 π 0 1 0 2 π Φ n ( ρ ) ρ d ρ d θ ,
i = | 1 + i k Φ ¯ 1 2 k 2 Φ 2 ¯ | 2 .
i = 1 ( 2 π / λ ) 2 [ Φ 2 ¯ ( Φ ¯ ) 2 ] ,
Φ ( ρ ) = n = 0 N a n ρ 2 n , Φ 2 ( ρ ) = m = 0 M b m ρ 2 m .
Φ ¯ = n = 0 N a n / ( n + 1 ) , Φ 2 ¯ = m = 0 M b m / ( m + 1 ) .
i = 1 ( 2 π λ ) 2 [ m = 0 M b m ( m + 1 ) ( n = 0 N a n ( n + 1 ) ) 2 ] .
i n = 1 ( 2 π / λ ) 2 a n 2 { [ 1 / ( 2 n + 1 ) ] [ 1 / ( n + 1 ) ] 2 } .
i = | 0 1 0 2 π e ( ρ / γ ) 2 ρ d ρ d θ + i k 0 1 0 2 π Φ e ( ρ / γ ) 2 ρ d ρ d θ 1 2 k 2 0 1 0 2 π Φ 2 e ( ρ / γ ) 2 ρ d ρ d θ | 2 / | 0 1 0 2 π e ( ρ / γ ) 2 ρ d ρ d θ | 2 .
Δ n ( ρ ) = Φ n ( ρ ) exp [ ( ρ / γ ) 2 ] ,
Δ j ¯ = 1 π N ( γ ) 0 1 0 2 π e ( ρ / γ ) 2 Φ i ( ρ ) ρ d ρ d θ .
i = | 1 + i k Δ ¯ 1 2 k 2 Δ 2 ¯ | 2 .
i = 1 ( 2 π / λ ) 2 [ Δ 2 ¯ ( Δ ¯ ) 2 ] .
Φ ( ρ ) = n = 0 N a n ρ 2 n , Φ 2 ( ρ ) = m = 0 M b m ρ 2 m .
Δ ¯ = 1 π N ( γ ) 0 1 0 2 π e ( ρ / γ ) 2 ( n = 0 N a n ρ 2 n ) ρ d ρ d θ , Δ 2 ¯ = 1 π N ( γ ) 0 1 0 2 π e ( ρ / γ ) 2 ( m = 0 M b m ρ 2 m ) ρ d ρ d θ .
Δ ¯ = 2 N ( γ ) n = 1 N a n I n ( γ ) ,
I n ( γ ) = 0 1 e ( ρ / γ ) 2 ρ 2 n + 1 d ρ .
I n ( γ ) = 1 2 γ 2 n + 2 0 ( 1 / γ 2 ) e x x n d x ,
I n ( γ ) = 1 2 ( γ 2 n + 2 ) [ e x ( x n + n x n 1 + n ( n 1 ) x n 2 + . + n ! ) ] 0 .
I n ( γ ) = 1 2 ( γ 2 n + 2 ) { n ! e x [ 1 + x + ( x 2 / 2 ! ) + + ( x n / n ! ) ] } 0 .
I n ( γ ) = 1 2 ( γ 2 n + 2 ) { n ! e exp [ ( 1 / γ ) 2 ] [ 1 + ( 1 / γ 2 ) + 1 2 ! ( 1 / γ 2 ) 2 + + 1 n ! ( 1 / γ 2 ) n ] n ! } .
e ( 1 / γ ) 2 l = n + 1 1 l ! ( 1 γ 2 ) l ,
I n ( γ ) = 1 2 ( γ 2 n + 2 ) n ! e ( 1 / γ 2 ) l = n + 1 1 l ! ( 1 γ 2 ) l .
l = n + 1 1 l ! ( 1 γ 2 ) l = [ ( 1 / γ 2 ) n + 1 ( n + 1 ) ! + ( 1 / γ 2 ) n + 2 ( n + 2 ) ! + ] = ( 1 / γ 2 ) n + 1 ( n + 1 ) n ! [ 1 + ( 1 / γ 2 ) n + 2 + ( 1 / γ 2 ) 2 ( n + 3 ) ( n + 2 ) + ] ;
I n ( γ ) = 1 2 { exp [ ( 1 / γ 2 ) ] / ( n + 1 ) } s n ( γ ) ,
s n ( γ ) = { 1 + [ ( 1 / γ 2 ) / ( n + 2 ) ] + ( 1 / γ 2 ) 2 / [ ( n + 3 ) ( n + 2 ) ] + } .
Δ ¯ = e ( 1 / γ 2 ) N ( γ ) n = 0 N a n s n ( γ ) n + 1 .
n = 0 N C n n + 1 ,
exp [ ( 1 / γ 2 ) ] = 1 exp [ ( 1 / γ 2 ) ] / [ exp ( 1 / γ 2 ) 1 ] = N ( γ ) / γ 2 [ exp ( 1 / γ 2 ) 1 ] .
Δ ¯ = F ( γ ) n = 0 N a n s n ( γ ) / ( n + 1 ) , Δ 2 ¯ = F ( γ ) m = 0 M b m s m ( γ ) / ( m + 1 ) .
i = 1 ( 2 π λ ) 2 F ( γ ) [ m = 0 M b m s m ( γ ) m + 1 F ( γ ) ( n = 0 N a n s n ( γ ) n + 1 ) 2 ] .
i 1 ( 2 π λ ) 2 F ( γ ) s ( γ ) [ m = 0 M b m m + 1 F ( γ ) s ( γ ) ( n = 0 N a n n + 1 ) 2 ] .
n = 0 N a n n + 1 = 0
i 1 ( 2 π λ ) 2 F ( γ ) s ( γ ) ( Φ 2 ¯ ) ,
Φ ( ρ ) = a n ρ 2 n , Φ 2 ( ρ ) = a n 2 ρ 4 n ,
i = 1 [ ( 2 π / λ ) a n ] 2 { [ F ( γ ) s 2 n ( γ ) / ( 2 n + 1 ) ] [ F ( γ ) s n ( γ ) / ( n + 1 ) ] 2 } .
Error k 4 ( Φ 2 ¯ ) 2 / 4 ( 1 k 2 Φ 2 ¯ ) .
Φ ¯ = 0 , Φ 2 ¯ = z 2 ( 9 / 120 ) ,
E = m = 0 M b m s m ( γ ) m + 1 F ( γ ) [ n = 0 N a n s n ( γ ) n + 1 ] 2 .
Φ ( ρ ) = a 1 ρ 2 + a 2 ρ 4 .
Φ 2 ( ρ ) = a 1 2 ρ 4 + 2 a 1 a 2 ρ 6 + a 2 2 ρ 8 ,
b 2 = a 1 2 , b 3 = 2 a 1 a 2 , b 4 = a 2 2 .
E = [ 1 3 a 1 2 s 2 ( γ ) + 1 2 a 1 a 2 s 3 ( γ ) + 1 5 a 2 2 s 4 ( γ ) ] F ( γ ) [ 1 4 a 1 2 s 1 2 ( γ ) + 1 3 a 1 a 2 s 1 ( γ ) s 2 ( γ ) + 1 9 a 2 s 2 ( γ ) ] .
E / a 1 = { [ 2 a 1 s 2 ( γ ) / 3 ] + [ 2 a 2 s 3 ( γ ) / 4 ] } F ( γ ) { [ a 1 s 2 ( γ ) / 2 ] + ( a 3 / 3 ) s 1 ( γ ) s 2 ( γ ) } = 0 , ( a 1 / a 2 ) = { [ ( s 3 / 2 ) 1 3 s 1 ( γ ) s 2 ( γ ) F ( γ ) ] / [ 2 3 s 2 ( γ ) 1 2 s 2 ( γ ) F ( γ ) ] } .
coma : Φ C = C P 3 cos θ Astigmatism : Φ A = A ρ 2 cos ( 2 θ ) , Distortion : Φ D = D ρ cos θ ,
Φ ( ρ ) = a 0 + a 1 1 ρ 2 + n = 2 N a n ρ 2 n + ( C ρ 3 cos θ + A ρ 2 cos 2 θ + D ρ cos θ ) + υ ρ cos θ + 1 2 u ρ 2 .
Φ ( ρ ) = n = 0 N a n ρ 2 n + ( C ρ 3 cos θ + A ρ 2 cos 2 θ + D ρ cos θ ) .
Φ 2 ( ρ ) = m = 0 M b m ρ 2 m + ( cross terms ) + ( C ρ 3 cos θ + A ρ 2 cos 2 θ + D ρ cos θ ) 2 ,
m = 0 M b m ρ 2 m
Δ ¯ = 1 π N ( γ ) 0 1 0 2 π e ( ρ / γ ) 2 ( n = 0 N a n ρ 2 n + C ρ 3 cos θ + A ρ 2 cos 2 θ + D ρ cos θ ) ρ d ρ d θ Δ 2 ¯ = 1 π N ( γ ) 0 1 0 2 π e ( ρ / γ ) 2 [ n = 0 N b m ρ 2 m + ( C ρ 3 cos θ + A ρ 2 cos 2 θ + D ρ cos θ ) 2 + ( cross terms ) ] ρ d ρ d θ .
Δ ¯ = 2 N ( γ ) 0 1 e ( ρ / γ ) 2 ( n = 0 a n ρ 2 n ) ρ d ρ , Δ 2 ¯ = 2 N ( γ ) 0 1 e ( ρ / γ ) 2 ( n = 0 b m ρ 2 m + 1 2 C 2 ρ 6 + 1 2 A 2 ρ 4 + 1 2 D 2 ρ 2 + C D ρ 4 ) ρ d ρ .
i = 1 ( 2 π λ ) 2 F ( γ ) { m = 0 M b m s n ( γ ) m + 1 + C 2 s 3 ( γ ) 8 + A 2 s 2 ( γ ) 6 + D 2 s 1 ( γ ) 4 + C D s 2 ( γ ) 3 F ( γ ) [ n = 0 N a n s n ( γ ) n + 1 ] 2 } .

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