Abstract

Spot diagrams give much information about the image formation of an optical system. In order to evaluate the image qualities of lenses quantitatively, it is shown that the geometric optical response function and the intensity distribution can be calculated directly from the coordinates of the spots. Image evaluation using the geometric optical “Strehl definition” is also discussed.

Furthermore, the relation between wave optics and geometrical optics is considered in the case of partial coherence, and it is deduced that, when the wave aberration function is large compared with the wavelength (more than 2λ), the degree of coherence in the object plane scarcely affects the intensity distribution in the image plane.

© 1959 Optical Society of America

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References

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  1. M. Herzberger, J. Opt. Soc. Am. 37, 485 (1947).
    [Crossref] [PubMed]
  2. E. H. Linfoot, Recent Advances in Optics.
  3. K. Miyamoto, J. Opt. Soc. Am. 48, 57, 567 (1958).
    [Crossref]
  4. P. B. Fellgett and E. H. Linfoot, Trans. Roy. Soc. (London) A247, 369 (1955).
    [Crossref]
  5. E. H. Linfoot, J. Opt. Soc. Am. 46, 740 (1956).
    [Crossref]
  6. H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).
  7. M. De, Proc. Roy. Soc. (London) A233, 91 (1955).
  8. H. H. Hopkins, Proc. Phys. Soc. (London) B70, 449 (1957).
  9. Ingelstan, Djurle, and Sjogren, J. Opt. Soc. Am. 46, 707 (1956).
    [Crossref]
  10. K. Miyamoto, J. Opt. Soc. Am. 47, 774 (1957).
    [Crossref] [PubMed]
  11. H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

1958 (1)

1957 (2)

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 449 (1957).

K. Miyamoto, J. Opt. Soc. Am. 47, 774 (1957).
[Crossref] [PubMed]

1956 (2)

1955 (3)

H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).

M. De, Proc. Roy. Soc. (London) A233, 91 (1955).

P. B. Fellgett and E. H. Linfoot, Trans. Roy. Soc. (London) A247, 369 (1955).
[Crossref]

1953 (1)

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

1947 (1)

De, M.

M. De, Proc. Roy. Soc. (London) A233, 91 (1955).

Djurle,

Fellgett, P. B.

P. B. Fellgett and E. H. Linfoot, Trans. Roy. Soc. (London) A247, 369 (1955).
[Crossref]

Herzberger, M.

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 449 (1957).

H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

Ingelstan,

Linfoot, E. H.

E. H. Linfoot, J. Opt. Soc. Am. 46, 740 (1956).
[Crossref]

P. B. Fellgett and E. H. Linfoot, Trans. Roy. Soc. (London) A247, 369 (1955).
[Crossref]

E. H. Linfoot, Recent Advances in Optics.

Miyamoto, K.

Sjogren,

J. Opt. Soc. Am. (5)

Proc. Phys. Soc. (London) (1)

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 449 (1957).

Proc. Roy. Soc. (London) (3)

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).

M. De, Proc. Roy. Soc. (London) A233, 91 (1955).

Trans. Roy. Soc. (London) (1)

P. B. Fellgett and E. H. Linfoot, Trans. Roy. Soc. (London) A247, 369 (1955).
[Crossref]

Other (1)

E. H. Linfoot, Recent Advances in Optics.

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Figures (4)

Fig. 1
Fig. 1

The coordinates (xα,yα) and the frequency variables (l,α).

Fig. 2
Fig. 2

(a) Geometric optical response function of astigmatism ϕ = ω′(u2v2). i curves are of Ri(s) and ∞ curves are of Rg(s). (b) Geometric optical response function of coma ϕ = ωv(u2+v2) corresponding to the line image perpendicular to the sagittal direction. (c) and (d) Geometric optical response function R(t) = |R(t)| exp[−(t)] of coma corresponding to the line image perpendicular to the tangential direction. (e) Geometric optical response function of spherical aberration ϕ = ω′(u2+v2)2.

Fig. 3
Fig. 3

Optical system with partially coherent illumination. ∑, O, A, and O′ are effective source, object plane, exit pupil, and image plane, respectively.

Fig. 4
Fig. 4

Integral region of (u,v) for the frequency variables (s1,t1; s2,t2).

Equations (34)

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I g ( x , y ) = a - 1 | 2 ϕ u 2 2 ϕ v 2 - ( 2 ϕ u v ) 2 | - 1 , x = ϕ / u ,             y = ϕ / v , - I g ( x , y ) d x d y = 1 ,
R g ( s , t ) = - I g ( x , y ) exp [ - 2 π i ( s x + t y ) ] d x d y = a - 1 A exp [ - 2 π i ( s / u + t / v ) ϕ ] d u d v ,
x = X / 2 F , y = Y / 2 F , u = ξ / H , v = η / H , s = N x · 2 F , t = N y · 2 F ,
s = l cos α , t = l sin α , x α = x cos α + y sin α , y α = - x sin α + y cos α
( s / u + t / v ) ϕ = s x + t y = l x α .
R g ( s , t ) = a - 1 A exp [ - 2 π i l x α ] d u d v
= j = 0 ( - i ) j ( 2 π l ) j x α j / j !
= m = 0 [ ( - 1 ) m ( 2 π l ) 2 m / ( 2 m ) ! ] x α 2 m - i m = 1 [ ( - 1 ) m - 1 ( 2 π l ) 2 m - 1 / ( 2 m - 1 ) ! ] × x α 2 m - 1 .
x α j a - 1 A x α j d u d v N - 1 k = 1 N ( x α ) k j ,
x 2 m x 0 2 m = ( 2 ω ) 2 m / ( 2 m + 1 ) ,             x 2 m - 1 = 0 ,
x 2 m = ( 2 ω ) 2 m / ( 2 m + 1 ) 2 ,             x 2 m - 1 = 0 , y n x π / 2 n = ( ω ) n i = 0 n ( n i ) 3 i ( 2 n - 2 i + 1 ) - 1 ( 2 i + 1 ) - 1 ,
x 2 m = ( 4 ω ) 2 m i = 0 2 m ( 2 m i ) ( 6 m - 2 i + 1 ) - 1 ( 2 i + 1 ) - 1 , x 2 m - 1 = 0.
R g ( s , t ) = N - 1 j = 1 N cos ( 2 π l x α ) j - i N - 1 j = 1 N sin ( 2 π l x α ) j .
Ī g ( x , y ) = l < l 0 R g ( s , t ) exp [ 2 π i ( s x + t y ) ] d s d t = 0 l 0 l d l 0 2 π d α [ a - 1 A d u d v × exp { - 2 π i l [ x α ( u , v ) - x α ] } ] = π l 0 2 m = 0 { ( - 1 ) m ( 2 π l 0 ) 2 m / [ ( 2 m ) ! ( m + 1 ) ] } × [ x α ( u , v ) - x α ] 2 m ,
[ x α ( u , v ) - x α ] 2 m = ( 2 π ) - 1 0 2 π d α { a - 1 A d u d v [ x α ( u , v ) - x α ] 2 m } ,
Ī g ( x α ) = - l 0 l 0 R g ( l cos α , l sin α ) exp [ 2 π i l x α ] d l = 2 l 0 m = 0 [ ( - 1 ) m ( 2 π l 0 ) 2 m / ( 2 m + 1 ) ! ] × [ x α ( u , v ) - x α ] 2 m ,
[ x α ( u , v ) - x α ] 2 m = a - 1 A d u d v [ x α ( u , v ) - x α ] 2 m .
Ī g ( x , y ) = π l 0 2 { 1 - 4 - 1 ( 2 π l 0 ) 2 [ x α ( u , v ) - x α ] 2 } .
Ī g ( 0 , 0 ) = π l 0 2 { 1 - 4 - 1 ( 2 π l 0 ) 2 x α 2 ( u , v ) } ,             x α ( u , v ) = 0 ,
x α 2 ( u , v ) = ( 2 π ) - 1 0 2 π d α a - 1 A ( x cos α + y sin α ) 2 d u d v = a - 1 A ( x 2 + y 2 ) / 2 d u d v ,
I g ( 0 ) = 2 l 0 [ 1 - 6 - 1 ( 2 π l 0 ) 2 x α 2 ] = 2 l 0 [ 1 - 6 - 1 ( 2 π l 0 ) 2 × a - 1 A ( ϕ / u · cos α + ϕ / v · sin α ) 2 d u d v ] ,
x α ( u , v ) = a - 1 A ( ϕ / u · cos α + ϕ / v · sin α ) d u d v = 0.
t ( x , y ) = I w ( x , y ) e ( x - x , y - y ) d x d y T ( s , t ) = R w ( s , t ) E ( s , t ) ,
t ( 0 , 0 ) = I w ( x , y ) e ( - x , - y ) d x d y = R w ( s , t ) E ( s , t ) d s d t .
t ( 0 , 0 ) = I g ( x , y ) e ( r ) d x d y N - 1 k = 1 N e ( r k ) = m = 0 ( - 1 ) m [ ( 2 m ) ! ] - 1 x α 2 m 0 ( 2 π l ) 2 m + 1 E ( l ) d l .
e ( r ) = exp ( - r 2 / 2 ρ 2 ) / ( 2 π ρ 2 ) , E ( l ) = exp ( - 2 π 2 ρ 2 l 2 ) .
t ( 0 , 0 ) = N - 1 k = 1 N exp [ - ( x k 2 + y k 2 ) / 2 ρ 2 ] / ( 2 π ρ 2 ) = ( 2 π ρ 2 ) - 1 { m = 0 ( - 1 ) m x α 2 m × ρ - 2 m [ ( 2 m - 1 ) ( 2 m - 3 ) 3 · 1 ] - 1 }
( s , t ) = - E ( x , y ) exp [ - 2 π i ( s x + t y ) ] d x d y
Φ w ( x , y ) = - d s 1 d t 1 - d s 2 d t 2 t ( s 1 , t 1 ; s 2 , t 2 ) × ( s 1 , t 1 ) exp [ 2 π i ( s 1 x + t 1 y ) ] × * ( s 2 , t 2 ) exp [ - 2 π i ( s 2 x + t 2 y ) ]
t ( s 1 , t 1 ; s 2 , t 2 ) = a 0 - 1 A λ s 1 , λ t 1 ; λ s 2 , λ t 2 d u d v × exp { - 2 π i λ - 1 [ ϕ ( u + λ s 1 , v + λ t 1 ) - ϕ ( u + λ s 2 , v + λ t 2 ) ] } .
lim λ 0 t ( s 1 , t 1 ; s 2 , t 2 ) = a 0 - 1 A 0 d u d v × exp { - 2 π i [ ϕ / u · ( s 1 - s 2 ) + ϕ / v · ( t 1 - t 2 ) ] } = R g ( s 1 - s 2 , t 1 - t 2 ) ,
Φ g ( x , y ) - d s d t R g ( s , t ) exp 2 π i ( s x + t y ) × - d s 1 d t 1 ( s 1 , t 1 ) * ( s 1 - s , t 1 - t ) .
ϕ ( s , t ) - d s 1 d t 1 ( s 1 , t 1 ) * ( s 1 - s , t 1 - t ) = - d x d y E ( x , y ) 2 exp [ - 2 π i ( s x + t y ) ] ,
Φ g ( x , y ) = - d s d t ϕ ( s , t ) R g ( s , t ) exp [ 2 π i ( s x + t y ) ] = - d x d y Φ ( x , y ) I g ( x - x , y - y ) ,