Abstract

We analyze the vignetting phenomenon both for optical systems with objects placed at finite distances and for systems with objects at infinity. Four of the possible definitions of the vignetting coefficient k, only two of them existing in the literature, are discussed. We propose two new definitions, i.e., a nonlinear geometric coefficient that is, in part, an analytical model of the vignetting characterization using optical software and a radiometric vignetting coefficient. The object space of each type of optical systems is studied first, defining its characteristic light circles and cones. Several simplifying assumptions are made for each of the two cases considered to derive analytical equations of the vignetting coefficient and thus to determine the best definition to be used. A geometric vignetting coefficient with two expressions, a linear classical and easy-to-use one and a nonlinear, that we propose for both types of systems is obtained. This nonlinear geometric vignetting coefficient proves to be more adequate in modeling the phenomenon, but it does not entirely fit the physical reality. We finally demonstrate that the radiometric vignetting coefficient we define and derive as a view factor for both types of optical systems is the most appropriate one. The half vignetting level, necessary in most optical design procedures to obtain a satisfactory illumination level in the image plane, is also discussed.

© 2009 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. V. F. Duma and M. Nicolov, “Neutral density filters with Risley prisms: analysis and design,” Appl. Opt. 48, 2678-2685 (2009).
    [CrossRef] [PubMed]
  14. D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Fundamental diffraction limitations in a paraxial 4f imaging system with coherent and incoherent illumination,” J. Opt. Soc. Am. A 24, 1911-1919 (2007).
    [CrossRef]
  15. J. E. Greivenkamp, Field Guide to Geometrical Optics (SPIE Press, 2004).
    [CrossRef]
  16. M. F. Modest, Radiative Heat Transfer (Academic Press, 2003).
  17. www.zemax.com.

2009

2008

V. F. Duma, “Vignetting of light beams for objects placed at a finite distance from an optical system,” Proc. SPIE 7100, 710005 (2008).
[CrossRef]

2007

2005

2004

2003

2002

2001

J. L. Rayces and M. Rosete-Aguilar, “Optics for binocular telescopes,” Proc. SPIE 4441, 1-8 (2001).
[CrossRef]

1996

1993

1991

1973

Auerhammer, J.

Bass, M.

M. Bass, Handbook of Optics (McGraw-Hill, 1995).

Bentley, J. L.

Berreman, D. W.

Crawford, M. Kate

Duma, V. F.

V. F. Duma and M. Nicolov, “Neutral density filters with Risley prisms: analysis and design,” Appl. Opt. 48, 2678-2685 (2009).
[CrossRef] [PubMed]

V. F. Duma, “Vignetting of light beams for objects placed at a finite distance from an optical system,” Proc. SPIE 7100, 710005 (2008).
[CrossRef]

Gappinger, R. O.

Gavrilyuk, A. V.

Greivenkamp, J. E.

Harkrider, C. J.

Karpova, G. V.

Kelly, D. P.

Kliment'ev, S. I.

Litvinov, A.

Liu, H.

Modest, M. F.

M. F. Modest, Radiative Heat Transfer (Academic Press, 2003).

Moore, D. T.

Nicolov, M.

Rayces, J. L.

J. L. Rayces and M. Rosete-Aguilar, “Optics for binocular telescopes,” Proc. SPIE 4441, 1-8 (2001).
[CrossRef]

Rhodes, W. T.

Rosete-Aguilar, M.

J. L. Rayces and M. Rosete-Aguilar, “Optics for binocular telescopes,” Proc. SPIE 4441, 1-8 (2001).
[CrossRef]

Rouke, J. L.

Sands, P. J.

Schechner, Y. Y.

Sheridan, J. T.

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000).

Tomkinson, T. H.

Vinogradova, O. A.

Zverev, V. A.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Technol.

Proc. SPIE

V. F. Duma, “Vignetting of light beams for objects placed at a finite distance from an optical system,” Proc. SPIE 7100, 710005 (2008).
[CrossRef]

J. L. Rayces and M. Rosete-Aguilar, “Optics for binocular telescopes,” Proc. SPIE 4441, 1-8 (2001).
[CrossRef]

Other

J. E. Greivenkamp, Field Guide to Geometrical Optics (SPIE Press, 2004).
[CrossRef]

M. F. Modest, Radiative Heat Transfer (Academic Press, 2003).

www.zemax.com.

M. Bass, Handbook of Optics (McGraw-Hill, 1995).

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000).

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Figures (13)

Fig. 1
Fig. 1

Light spot in the plane of the entrance pupil ( s ).

Fig. 2
Fig. 2

Coordinates of the intersection points M P and N P in the plane of the entrance pupil (Fig. 1): | x | ( r ) and y ( r ) functions (the s case).

Fig. 3
Fig. 3

Relative position of the circles obtained by projecting the field stop L in the plane of the entrance pupil P ( s ).

Fig. 4
Fig. 4

Geometric vignetting coefficient (state of the art).

Fig. 5
Fig. 5

Nonlinear geometric vignetting coefficient for s .

Fig. 6
Fig. 6

Radiometric vignetting coefficient k ( r ) for s .

Fig. 7
Fig. 7

Object light bundle for 0 < r < r 1 .

Fig. 8
Fig. 8

Light spot in the plane of the entrance pupil ( s = ).

Fig. 9
Fig. 9

Circles and cones of zero and total stop ( s = ).

Fig. 10
Fig. 10

Graphs of the | x | ( α ) and y ( α ) and functions ( s = ).

Fig. 11
Fig. 11

Nonlinear geometric vignetting coefficient for s = .

Fig. 12
Fig. 12

Radiometric vignetting coefficient k ( α ) for s = .

Fig. 13
Fig. 13

Intersection of the circles obtained for the s case in the plane P.

Equations (48)

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k = ρ / 2 R P .
k = N e / N i .
k = S P / π R P 2 ,
k = Φ / Φ A .
r 1 = s p R L s L R p s p s L + s R p R L s p s L = z p R L z L R p z p z L .
r 2 = s p R L + s L R p s p s L s R p + R L s p s L = z p R L + z L R p z p z L .
( z P z L ) r z L = y C P ,
C P H P = y H P y C P = z P z L R L > R P .
{ C ( P , R p ) : x 2 + y 2 = R p 2 C ( C p , C P H P ) : x 2 + ( y y C p ) 2 = C P H P 2 ,
{ x ( r ) = ± z p z L 2 z L ( r 2 r 1 2 ) ( r 2 2 r 2 ) r y ( r ) = z p z L 2 z L [ r 1 r 2 r r ] .
r m = r 1 r 2 = z p 2 R L 2 z L 2 R p 2 z p z L .
S P ( r ) / π R p 2 = 0.5 ,
k ( r ) = z p z L 2 z L R p ( r 2 r ) ,
k ( r ) = 1 π [ arccos y R p + z p 2 z L 2 · R L 2 R p 2 arcsin ( z L z p | x | R L ) z p z L z L R p 2 r | x | ] ,
k ( r ) = Φ ( r ) Φ ( 0 ) = R P 2 + z P 2 2 R P 2 [ 1 z p 2 R p 2 + r 2 [ z p 2 + ( R p r ) 2 ] · [ z p 2 + ( R p + r ) 2 ] ] .
r 1 / 2 = R P 2 z P 2 + 2 z P 3 R P 2 + 2 z P 2
α 1 = arctan R L R P d ,
α 2 = arctan R L + R P d ,
s ( 1 ) = s P R L s L R P R L R P .
s ( 2 ) = s P R L + s L R P R L R P .
{ x 2 + y 2 = R p 2 x 2 + ( y y C p ) 2 = R L 2 ,
{ x = ± [ R L 2 ( R p d tan α ) 2 ] · [ ( R p + d tan α ) 2 R L 2 ] 2 d tan α y = d 2 tan 2 α ( R L 2 R p 2 ) 2 d tan α .
α m = arctan tan α 1 · tan α 2 = arctan R L 2 R p 2 d ,
α i = arctan tan α m .
k ( α ) = 1 π [ arccos y R p + R L 2 R p 2 arcsin | x | R L | x | R p 2 d · tan α ] .
Φ | α = 0 = E 0 · π R P 2 ; Φ ( α ) = E · S P ( α ) ,
E 0 = I m z 2 and E ( α ) = I ( z / cos α ) 2 cos α , where     I = I m cos α
k ( α ) = lim z Φ ( α ) Φ | α = 0 = S p ( α ) · cos 4 α π R p 2 k ( α ) = ( arccos y R p + R L 2 R p 2 arcsin | x | R L | x | d R p 2 tan α ) cos 4 α π ,
θ P = arccos y N P R P .
θ M P = 3 π / 2 θ P , θ N P = 3 π / 2 + θ P .
S p = 1 2 [ 2 R P θ p · R p M p N p · P O + 2 φ p · C p H p 2 M p N p · C O p ] ,
φ p = arcsin z L z p O N P R L .
S p = R p 2 arccos y R p + z p 2 z L 2 R L 2 arcsin ( z L z p | x | R L ) z p z L z L r | x | ,
r 0 = ( r 1 + r 2 ) / 2 = z P R L / ( z P z L ) ,
{ | x ( r 0 ) | = R P 2 z P R L 4 z P 2 R L 2 z L 2 R P 2 y ( r 0 = z L R P 2 2 z P R L ) .
S p ( r 0 ) = R p 2 arccos z L R p z p R L + z p 2 z L 2 R L 2 arcsin [ z L R p 2 z P 2 R L 2 · 4 z p 2 R L 2 z L 2 R p 2 ] R p 2 z L 4 z p 2 R L 2 z L 2 R p 2 ,
S p ( r m ) = π R p 2 2 + z p 2 z L 2 R L 2 arcsin ( z L z p · R p R L ) R p z L z p 2 R L 2 z L 2 R p 2 .
α = E P B P Q ; θ θ B = γ ; P B p = r = x B 2 + y B 2 .
Φ = I m α = 0 2 π 0 φ sin φ · cos φ · d φ · d α .
Φ = I m 2 α = 0 2 π sin 2 φ ( α ) · d α .
t g α = B p Q / z p , B p Q sin γ = r sin ( α γ ) = R p sin α ,
B p Q = R p 2 r 2 sin 2 α r cos α .
Φ = I m 2 α = 0 2 π ( R p 2 r 2 sin 2 α r cos α ) 2 z p 2 + ( R p 2 r 2 sin 2 α r cos α ) 2 d α
Φ ( r ) = π I m 2 [ 1 z p 2 R p 2 + r 2 [ z p 2 + ( R p r ) 2 ] · [ z p 2 + ( R p + r ) 2 ] ] , Φ ( 0 ) = π I m R P 2 R P 2 + z P 2 .
S p = 1 2 [ 2 R p · arccos y R p · R p M p N p · P O + 2 R L · arcsin | x | R L · C p H p M p N p · C O p ] .
S p ( α ) = R p 2 arccos y R p + R L 2 arcsin | x | R L | x | d tan α .
S p ( α m ) = π R p 2 2 + R L 2 arcsin R p R L R p R L 2 R p 2 ,
S p ( α P ) = π [ R P 2 arccos R P 2 R L + R L 2 arcsin R P 4 R L 2 R P 2 2 R L 2 R P 4 R L 2 R P 2 ] ,

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