Abstract

In tolerancing a lens system, it is necessary to know the effects of fabrication and alignment errors on the image quality. In this report, the wavefront aberration due to a small parameter perturbation is derived. This may be calculated from ray tracing through the nominal system only, and its linearity is extremely useful for tolerancing. The results are applied to the calculation of the variance, or rms, of the wavefront as a tolerancing criterion.

© 1970 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Maréchal, thesis, University of Paris (1948).
  2. H. H. Hopkins, Opt. Acta 13, 343 (1966).
    [CrossRef]
  3. H. H. Hopkins, H. J. Tiziani, Brit. J. Appl. Phys. 17, 33 (1966).
    [CrossRef]
  4. H. H. Hopkins, Proc. Phys. Soc. (London) B22, 449 (1957).
  5. W. B. King, Appl. Opt. 7, 489 (1968).
    [CrossRef] [PubMed]
  6. W. B. King. Appl. Opt. 7, 1193 (1968).
    [CrossRef] [PubMed]

1968 (2)

1966 (2)

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[CrossRef]

H. H. Hopkins, H. J. Tiziani, Brit. J. Appl. Phys. 17, 33 (1966).
[CrossRef]

1957 (1)

H. H. Hopkins, Proc. Phys. Soc. (London) B22, 449 (1957).

Hopkins, H. H.

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[CrossRef]

H. H. Hopkins, H. J. Tiziani, Brit. J. Appl. Phys. 17, 33 (1966).
[CrossRef]

H. H. Hopkins, Proc. Phys. Soc. (London) B22, 449 (1957).

King, W. B.

Maréchal, A.

A. Maréchal, thesis, University of Paris (1948).

Tiziani, H. J.

H. H. Hopkins, H. J. Tiziani, Brit. J. Appl. Phys. 17, 33 (1966).
[CrossRef]

Appl. Opt. (2)

Brit. J. Appl. Phys. (1)

H. H. Hopkins, H. J. Tiziani, Brit. J. Appl. Phys. 17, 33 (1966).
[CrossRef]

Opt. Acta (1)

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[CrossRef]

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

H. H. Hopkins, Proc. Phys. Soc. (London) B22, 449 (1957).

Other (1)

A. Maréchal, thesis, University of Paris (1948).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Slightly perturbed surface.

Fig. 2
Fig. 2

Change of reference sphere.

Tables (1)

Tables Icon

Table I Evaluation of a Photographic Objective a Perturbed State

Equations (54)

Equations on this page are rendered with MathJax. Learn more.

δ W = ( P Q ) ( P G ) ,
P G = S cos ( I I ) .
δ W = n P Q n P G = S cos I ( n cos I n cos I ) .
δ W = R ( n cos I n cos I ) .
δ W = ( δ r · g ) Γ
Γ = ( ni · g n i · g ) / g 2 .
y = ( y h ) cos β + ( z d ) sin β + h , z = ( y h ) sin β + ( z d ) cos β + d .
y = y + ( z d ) β , z = ( y h ) β + z ,
δ y = ( z d ) β , δ z = ( y h ) β .
δ W = Γ [ L ( z d ) M ( y h ) ] β .
δ x = Δ x , δ y = Δ y , δ z = Δ z .
δ x = 0 , δ y = Δ y , δ z = 0.
F ( x , y , z , c ) = 1 2 c ( x 2 + y 2 + x 2 ) z = 0.
F ( x + k S , y + l S , z + m S , c + Δ c ) = 0.
F x k S + F y l S + F z m S + F c Δ c = 0 ,
S = F c Δ c k F x + l F y + m F z = F c Δ c g cos I
δ W = F c Δ c Γ = 1 2 Δ c ( x 2 + y 2 + z 2 ) Γ .
δ W = n S ,
S = ( K Δ x + L Δ y + M Δ z ) / cos I ,
S = [ ( 1 2 ) Δ c ( x 2 + y 2 + z 2 ) ] / cos I .
δ W = n [ k Δ x + l Δ y + m Δ z + 1 2 Δ c ( x 2 + y 2 + z 2 ) ] .
δ W = n [ ( k Δ x + l Δ y + m Δ z + ( m 1 ) Δ r ] ,
Δ f = Δ r + Δ z
δ W = n [ k Δ x + l Δ y + Δ z + ( m 1 ) Δ f ] .
δ W = k α + l β + γ + ( m 1 ) δ
V = 1 m i = 1 m W i 2 .
V = 1 m i = 1 m [ W i k i α l i β γ ( m i 1 ) δ ] 2 ,
{ Σ k i 2 Σ k i l i Σ k i Σ k i ( m i 1 ) Σ l i k i Σ l i 2 Σ l i Σ l i ( m i 1 ) Σ k i Σ l i Σ 1 Σ ( m i 1 ) Σ ( m i 1 ) k i Σ ( m i 1 ) l i Σ ( m i 1 ) Σ ( m i 1 ) 2 } { α β γ δ } = { Σ W i k i Σ W i l i Σ W i Σ W i ( m i 1 ) } .
A x = d ,
x = A 1 d .
A x 0 = d 0
x 0 = A 1 d 0 .
A δ x = δ d ,
δ x = A 1 δ d .
d = d 0 + δ d
x = A 1 ( d 0 + δ d ) = x 0 + δ x .
W = W 0 + T δ W ,
d = d 0 + T δ d
x = x 0 + T δ x .
V ( W ) = V ( W 0 ) + [ V ( W ) V ( W 0 ) V ( δ W ) ] T + V ( δ W ) T 2 ,
V ( W ) = V ( δ W ) T 2
δ W = δ W 1 + δ W 2 + δ W 3 + .
W = W 0 + T ( δ W 1 + δ W 2 + δ W 3 + ) .
V = C + i = 1 N a i T i 2 + i = 1 N b t T i + j = i + 1 N i = 1 N c i j T i T j .
a i = V ( δ W i ) , b i = V ( W 0 + δ W i ) V ( W 0 ) V ( δ W i ) , c i j = V ( δ W i + δ W j ) a i a j .
V = 0.084 T 2 + 0.017 T + 0.034.
V = g 2 ¯ = ( g 2 d σ / d σ ) .
V = i = 1 M g i 2 Δ σ / i = 1 M Δ σ = 1 M i = 1 M g i 2 .
f = j = 1 N a j x j ,
V = 1 M i = 1 M ( g i j = 1 N a j x i j ) 2
V a j = 2 i = 1 M g i x i j 2 i = 1 M x i j k = 1 N a k x i k = 0 ,
{ y 11 y 12 y 1 n y 21 . . . . . . . y N y n n } { a 1 a 2 . . . a n } = { d 1 d 2 . . . d n } ,
y i j = k = 1 M x k i x k j , d i = k = 1 M g k x k i .
V = 1 M [ i = 1 M g i 2 j = 1 N a j d j ] .

Metrics