Abstract

The symmetry properties of point-spread functions of optical imaging systems with circular pupils aberrated by a Zernike-circle polynomial aberration were discussed by Nijboer [ “ The diffraction theory of aberrations,” Ph.D. dissertation ( University of Groningen, Groningen, The Netherlands, 1942)]. Although it was not pointed out by him, his analysis and some of his results are valid only for systems with large Fresnel numbers. The symmetry properties for systems with small and large Fresnel numbers are discussed. The analysis is extended to systems with annular pupils aberrated by a Zernike-annular polynomial aberration. This analysis is further extended to pupils with nonuniform but radially symmetric illumination such as Gaussian. It is shown, in particular, that whereas, for uniform pupils, the axial irradiance of the imaging-forming light cone for a primary spherical aberration is symmetric about the defocused point with respect to which the aberration variance is minimum, it is asymmetric for nonuniform pupils. The discussion is equally valid for focused beams of light. Computer-generated pictures of point-spread functions of systems with circular and annular pupils aberrated by primary aberrations illustrating their symmetry properties are given.

© 1994 Optical Society of America

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References

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  1. B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942), pp. 44–51.
  2. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), Section 9.4.
  3. V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils,” J. Opt. Soc. Am. 72, 1258–1266 (1982);errata, J. Opt. Soc. Am. A 10, 2092 (1993).
    [CrossRef]
  4. J. M. Geary, P. Peterson, “Spherical aberration: a possible new measurement technique,” Opt. Eng. 25, 286–291 (1986).
    [CrossRef]
  5. K. Nienhuis, “On the influence of diffraction on image formation in the presence of aberrations,” Ph.D. dissertation (University of Groningen, The Netherlands, 1948);some discussion is also given in K. Nienhuis, B. R. A. Nijboer, “The diffraction theory of optical aberrations, Part II,” Physica (Utrecht) 14, 590–608 (1949).
    [CrossRef]
  6. R. Kingslake, “The diffraction structure of the elementary coma image,” Proc. Phys. Soc. London 61, 147–158 (1948).
    [CrossRef]
  7. E. Collett, E. Wolf, “Symmetry properties of focused fields,” Opt. Lett. 5, 264–266 (1980).
    [CrossRef] [PubMed]
  8. E. Wolf, “Phase conjugacy and symmetries in spatially band-limited wavefields containing no evanescent components,” J. Opt. Soc. Am. 70, 1311–1319 (1980).
    [CrossRef]
  9. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981); errata 1408;J. Opt. Soc. Am. A 1, 685 (1984).
    [CrossRef]
  10. V. N. Mahajan, “Uniform versus Gaussian beams: a comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A 3, 470–485 (1986).
    [CrossRef]
  11. V. N. Mahajan, “Axial irradiance and optimum focusing of laser beams,” Appl. Opt. 22, 3042–3053 (1983).
    [CrossRef] [PubMed]

1986 (2)

J. M. Geary, P. Peterson, “Spherical aberration: a possible new measurement technique,” Opt. Eng. 25, 286–291 (1986).
[CrossRef]

V. N. Mahajan, “Uniform versus Gaussian beams: a comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A 3, 470–485 (1986).
[CrossRef]

1983 (1)

1982 (1)

1981 (1)

1980 (2)

1948 (1)

R. Kingslake, “The diffraction structure of the elementary coma image,” Proc. Phys. Soc. London 61, 147–158 (1948).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), Section 9.4.

Collett, E.

Geary, J. M.

J. M. Geary, P. Peterson, “Spherical aberration: a possible new measurement technique,” Opt. Eng. 25, 286–291 (1986).
[CrossRef]

Kingslake, R.

R. Kingslake, “The diffraction structure of the elementary coma image,” Proc. Phys. Soc. London 61, 147–158 (1948).
[CrossRef]

Mahajan, V. N.

Nienhuis, K.

K. Nienhuis, “On the influence of diffraction on image formation in the presence of aberrations,” Ph.D. dissertation (University of Groningen, The Netherlands, 1948);some discussion is also given in K. Nienhuis, B. R. A. Nijboer, “The diffraction theory of optical aberrations, Part II,” Physica (Utrecht) 14, 590–608 (1949).
[CrossRef]

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942), pp. 44–51.

Peterson, P.

J. M. Geary, P. Peterson, “Spherical aberration: a possible new measurement technique,” Opt. Eng. 25, 286–291 (1986).
[CrossRef]

Wolf, E.

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

J. M. Geary, P. Peterson, “Spherical aberration: a possible new measurement technique,” Opt. Eng. 25, 286–291 (1986).
[CrossRef]

Opt. Lett. (1)

Proc. Phys. Soc. London (1)

R. Kingslake, “The diffraction structure of the elementary coma image,” Proc. Phys. Soc. London 61, 147–158 (1948).
[CrossRef]

Other (3)

K. Nienhuis, “On the influence of diffraction on image formation in the presence of aberrations,” Ph.D. dissertation (University of Groningen, The Netherlands, 1948);some discussion is also given in K. Nienhuis, B. R. A. Nijboer, “The diffraction theory of optical aberrations, Part II,” Physica (Utrecht) 14, 590–608 (1949).
[CrossRef]

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942), pp. 44–51.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), Section 9.4.

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Figures (9)

Fig. 1
Fig. 1

Schematic of optical imaging showing the various planes and the coordinate systems for describing the irradiance distribution. P′ is the Gaussian image of a point object P. The Gaussian reference sphere of radius of curvature R has its center of curvature at P′ and passes through the center O of the exit pupil, ri is the position vector of an observation point P″ in a defocused image plane with respect to an origin O″ laying on the line joining O and P′.

Fig. 2
Fig. 2

PSF’s for spherical aberration: W(ρ) = Asρ4, with As = 1λ, in various defocused image planes, (a) = 0, (b) = 0.5. The actual irradiance values of curves marked 5×are a factor of 5 smaller.

Fig. 3
Fig. 3

Pictorial PSF’s for spherical aberration as in Fig. 2.

Fig. 4
Fig. 4

Axial irradiance for spherical aberration: W(ρ) = Asρ4, with As = 1 λ, 2λ, 3λ. (a) = 0 (b) = 0.5.

Fig. 5
Fig. 5

Pictorial PSF’s for coma: W(ρ,θ) = Acρ3 cosθ, with Ac = 1λ, in various defocused image planes, (a) = 0, (b) = 0.5.

Fig. 6
Fig. 6

Irradiance along the z axis and along an axis parallel to the z axis but passing through the diffraction focus for coma: W(ρ,θ) = Acρ2 cos θ, with Ac = 1λ. (a) = 0, (b) = 0.5.

Fig. 7
Fig. 7

Pictorial PSF’s for astigmatism: W(ρ,θ) = Aaρ2 cos2θ, with Aa = 3λ, in various defocused image planes, (a) = 0, (b) = 0.5.

Fig. 8
Fig. 8

Axial irradiance for astigmatism: W(ρ) = Aaρ2 cos2θ, with Aa = 1λ, 2λ, 3λ. (a) = 0, (b) = 0.5. The arrows indicate the points of axial symmetry.

Fig. 9
Fig. 9

Axial irradiance for a Gaussian pupil with γ = 1 aberrated by spherical aberration: W(ρ) = Asρ4. (a) = 0 and (b) = 0.5.

Tables (1)

Tables Icon

Table 1 Symmetry Properties of Aberrated PSF’s for Primary Aberrations in Systems with Annular Pupilsa

Equations (55)

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I i ( r i ; z ) = ( I p / λ 2 z 2 ) | exp { i [ Φ ( r p ) + Φ d ( r p ) ] } × exp ( 2 π i r p r i / λ z ) d r p | 2 ,
Φ d ( r p ) = π λ ( 1 z 1 R ) r p 2
I ( r , ϕ ; z ) = ( R / z ) 2 [ π ( 1 2 ) ] 2 × | 1 0 2 π exp { i [ Φ ( ρ , θ ) + Φ d ( ρ ) ] } × exp [ π i R z r ρ cos ( θ ϕ ) ] ρ d ρ d θ | 2 ,
ρ = r p / a , ρ 1 ,
r = r i / λ F ,
( x , y ) = r ( cos ϕ , sin ϕ ) ,
F = R / D ,
I ( r ) = I i ( r i ) I p S p 2 / λ 2 R 2 ,
S p = π a 2 ( 1 2 ) ,
Φ d ( ρ ) = A d ρ 2 ,
A d = π N ( R z 1 ) ,
N = a 2 / λ R .
I ( r ; R ) = 4 ( 1 2 ) 2 [ 1 J 0 ( π r ρ ) ρ d ρ ] 2
= 1 ( 1 2 ) 2 [ 2 J 1 ( π r ) π r 2 2 J 1 ( π r ) π r ] 2 ,
0 2 π exp ( i x cos θ ) d θ = 2 π J 0 ( x ) ,
0 a x J 0 ( b x ) d x = ( a / b ) J 1 ( a b ) .
I ( r ; z ) = 4 ( 1 2 ) 2 ( R z ) 2 × | 1 exp ( i A d ρ 2 ) J 0 ( π r ρ R / z ) ρ d ρ | 2 .
I ( 0 ; z ) = ( R z ) 2 { sin [ A d ( 1 2 ) / 2 ] A d ( 1 2 ) / 2 } 2 .
I ( r ; z ) = 4 ( 1 2 ) 2 | 1 exp ( i A d ρ 2 ) J 0 ( π r ρ ) ρ d ρ | 2 ,
A d = π a 2 λ R 2 ( R z ) .
I ( 0 ; z ) = { sin [ A d ( 1 2 ) / 2 ] A d ( 1 2 ) / 2 } 2 ,
Φ ( ρ , θ ) = A n m R n m ( ρ ; ) cos ( m θ ) ,
I ( r , ϕ ; z ) = ( R / z ) 2 [ π ( 1 2 ) ] 2 × | 1 0 2 π exp { i [ A n m R n m ( ρ ; ) cos ( m θ ) + Φ d ( ρ ) ] } × exp [ π i R z r ρ cos ( θ ϕ ) ] ρ d ρ d θ | 2 .
exp ( i x cos θ ) = 2 s = 0 i s J s ( x ) cos ( s θ ) ,
exp { i [ A n m R n m ( ρ ; ) cos ( m θ ) π R z r ρ cos ( θ ϕ ) ] } = 4 s = 0 s = 0 i s ( i ) s J s [ A m n R n m ( ρ ; ) ] J s ( π r ρ R / z ) × cos ( m s θ ) cos [ s ( θ ϕ ) ] .
2 π cos ( m s θ ) cos [ s ( θ ϕ ) ] d θ = π cos ( s ϕ ) δ m s , s ,
I ( r , ϕ ; z ) = ( 4 R / z ) 2 ( 1 2 ) 2 | s = 0 ( i ) ( m 1 ) s cos ( m s ϕ ) × 1 exp [ i Φ d ( ρ ) ] J s [ A n m R n m ( ρ ; ) ] J m s ( π r ρ R / z ) ρ d ρ | 2 .
cos [ m s ( ϕ + 2 π j / m ) ] = cos ( m s ϕ ) , j = 1 , 2 , m ,
I ( r , ϕ ; z ) = 16 ( 1 2 ) 2 | s = 0 ( i ) ( m 1 ) s cos ( m s ϕ ) × 1 exp ( i A d ρ 2 ) J s [ A n m R n m ( ρ ; ) ] J m s ( π r ρ ) ρ d ρ | 2 ,
I ( r ; z ) = 16 ( 1 2 ) 2 | s = 0 i s 1 exp ( i A d ρ 2 ) × J s [ A n 0 R n 0 ( ρ ; ) ] J m s ( π r ρ ) ρ d ρ | 2 .
I ( r ; z ) = 4 ( 1 2 ) 2 × | 1 exp { i [ A n 0 R n 0 ( ρ ; ) + A d ρ 2 ] } J 0 ( π r ρ ) ρ d ρ | 2 .
( i ) ( m 1 ) s = { ± 1 when s is even ± i when s is odd ,
cos [ m s ( ϕ + π / m ) ] = { cos ( m s ϕ ) when s is even cos ( m s ϕ ) when s is odd .
( i ) ( m 1 ) s cos ( m s ϕ ) = { ( i ) ( m 1 ) s cos [ m s ( ϕ + π / m ) ] } * .
I ( 0 ; z ) = 4 ( 1 2 ) 2 × | 1 exp ( i A d ρ 2 ) J 0 [ A n m R n m ( ρ ; ] ρ d ρ | 2 , m 0 .
I ( 0 ; z ) = 4 ( 1 2 ) 2 × | 1 exp ( i A d ρ 2 ) exp [ i A n 0 R n 0 ( ρ ; ] ρ d ρ | 2 , m = 0 .
R 2 n 0 ( ρ ; ) = P n [ 2 ( ρ 2 2 ) 1 2 1 ] ,
x = 2 [ ( ρ 2 2 ) / ( 1 2 ) ] 1 ,
I ( 0 ; z ) = 1 4 | 1 1 exp [ i A d x / 2 ( 1 2 ) ] exp [ i A n 0 P n / 2 ( x ) ] d x | 2 .
J s ( x ) = ( 1 ) s J s ( x ) ,
( i ) ( m 1 ) s J s [ A n m R n m ( ρ ; ) ] = { ( i ) ( m 1 ) s × J s [ A n m R n m ( ρ ; ) ] } * .
R 4 0 ( ρ ; ) = [ 6 ρ 4 6 ( 1 + 2 ) ρ 2 + ( 1 + 4 2 + 4 ) ] / ( 1 2 ) 2 .
R 3 1 ( ρ ; ) cos θ = 3 ( 1 + 2 ) ρ 3 2 ( 1 + 2 + 4 ) ρ ( 1 2 ) [ ( 1 + 2 ) ( 1 + 4 2 + 4 ) ] 1 / 2 cos θ .
r i = [ 4 ( 1 + 2 + 4 ) 3 ( 1 + 2 ) F A c , 0 ] ,
A a ρ 2 cos 2 θ = A a ρ 2 cos ( 2 θ ) ½ A a ρ 2 ,
R 2 2 ( ρ ; ) cos ( 2 θ ) = ρ 2 cos ( 2 θ ) ( 1 + 2 + 4 ) 1 / 2 .
I p ( r p ) = I 0 exp [ 2 γ ( r p / a ) 2 ]
I p ( ρ ) = I 0 exp ( 2 γ ρ 2 ) ,
γ = ( a / ω ) 2
I ( r , ϕ ; z ; γ ) = ( R / z ) 2 [ π ( 1 2 ) ] 2 | 1 0 2 π exp ( γ ρ 2 ) × exp { i [ Φ ( ρ , θ ) + Φ d ( ρ ) ] } × exp [ π i R z r ρ cos ( θ ϕ ) ] ρ d ρ d θ | 2 ,
I ( r ; z ; γ ) = 4 ( R / z ) 2 ( 1 2 ) 2 × | 1 exp ( γ ρ 2 ) exp ( i A d ρ 2 ) J 0 ( π r ρ R / z ) ρ d ρ | 2 ,
I ( 0 ; z ; γ ) = ( R / z ) 2 ( 1 2 ) 2 [ exp ( 2 γ ) / ( A d 2 + γ 2 ) ] × { 1 + exp [ 2 γ ( 1 2 ) ] 2 exp [ γ ( 1 2 ) ] cos [ ( 1 2 ) A d ] } .
I ( r ; z ; γ ) = 4 ( 1 2 ) 2 × | 1 exp ( γ ρ 2 ) exp ( i A d ρ 2 ) J 0 ( π r ρ ) ) ρ d ρ | 2 ,
I ( 0 ; z ; γ ) = ( 1 2 ) 2 [ exp ( 2 γ ) / ( A d 2 + γ 2 ) ] × { 1 + exp [ 2 γ ( 1 2 ) ] 2 exp [ γ ( 1 2 ) ] cos [ ( 1 2 ) A d ] } ,
I ( 0 ; z ; γ ) = 4 ( 1 2 ) 2 × | 1 exp ( γ ρ 2 ) exp ( i A d ρ 2 ) exp [ i A n 0 R n 0 ( ρ ; ; γ ) ] ρ d ρ | 2 .

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