Abstract

The local principal curvatures (shape) of a wavefront propagating through an optical system can be calculated from geometrical optics. Formulas are derived for the change in shape during transfer, refraction, reflection, and diffraction by a grating in a system of homogeneous isotropic media. The results not only have immediate application to design, but also provide necessary data for treating points where the geometric approximation fails.

© 1964 Optical Society of America

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References

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  1. C. S. Hastings, New Methods in Geometrical Optics (The Macmillan Company, New York, 1927).
  2. A. Gullstrand, Svenska Vetensk. Handl. 41, 1 (1906).
  3. See for example, C. E. Weatherburn, Differential Geometry of Three Dimensions (Cambridge University Press, New York, 1955), pp. 51–62, 66–74, 90–91.
  4. See, for example, M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York, 1958), p. 296.
  5. C. E. Weatherburn, Ref. 3, p. 66.
  6. M. Herzberger, Ref. 4, pp. 6, 7.
  7. G. H. Spencer and M. V. R. K. Murty, J. Opt. Soc. Am. 52, 676 (1962).
    [Crossref]
  8. O. Stavroudis, J. Opt. Soc. Am. 52, 187 (1962).
    [Crossref]
  9. See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), pp. 38–40.

1962 (2)

G. H. Spencer and M. V. R. K. Murty, J. Opt. Soc. Am. 52, 676 (1962).
[Crossref]

O. Stavroudis, J. Opt. Soc. Am. 52, 187 (1962).
[Crossref]

1906 (1)

A. Gullstrand, Svenska Vetensk. Handl. 41, 1 (1906).

Born, M.

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), pp. 38–40.

Gullstrand, A.

A. Gullstrand, Svenska Vetensk. Handl. 41, 1 (1906).

Hastings, C. S.

C. S. Hastings, New Methods in Geometrical Optics (The Macmillan Company, New York, 1927).

Herzberger, M.

M. Herzberger, Ref. 4, pp. 6, 7.

See, for example, M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York, 1958), p. 296.

Murty, M. V. R. K.

G. H. Spencer and M. V. R. K. Murty, J. Opt. Soc. Am. 52, 676 (1962).
[Crossref]

Spencer, G. H.

G. H. Spencer and M. V. R. K. Murty, J. Opt. Soc. Am. 52, 676 (1962).
[Crossref]

Stavroudis, O.

Weatherburn, C. E.

C. E. Weatherburn, Ref. 3, p. 66.

See for example, C. E. Weatherburn, Differential Geometry of Three Dimensions (Cambridge University Press, New York, 1955), pp. 51–62, 66–74, 90–91.

Wolf, E.

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), pp. 38–40.

J. Opt. Soc. Am. (2)

G. H. Spencer and M. V. R. K. Murty, J. Opt. Soc. Am. 52, 676 (1962).
[Crossref]

O. Stavroudis, J. Opt. Soc. Am. 52, 187 (1962).
[Crossref]

Svenska Vetensk. Handl. (1)

A. Gullstrand, Svenska Vetensk. Handl. 41, 1 (1906).

Other (6)

See for example, C. E. Weatherburn, Differential Geometry of Three Dimensions (Cambridge University Press, New York, 1955), pp. 51–62, 66–74, 90–91.

See, for example, M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York, 1958), p. 296.

C. E. Weatherburn, Ref. 3, p. 66.

M. Herzberger, Ref. 4, pp. 6, 7.

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), pp. 38–40.

C. S. Hastings, New Methods in Geometrical Optics (The Macmillan Company, New York, 1927).

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Figures (3)

Fig. 1
Fig. 1

Wavefront W(u,v,t) (negative curvature).

Fig. 2
Fig. 2

Vectors in plane of refraction (P into paper).

Fig. 3
Fig. 3

Grating coordinate system.

Equations (107)

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W t = c Q ,
W ( u , v , t 0 + τ ) = W ( u , v , t 0 ) + τ c Q ( u , v ) ,
Q = W u × W v ,
E = 1 ,             F = 0 ,             G = 1 ,             M = 0 ,
W u ( t 0 + τ ) = W u ( t 0 ) + d Q u .
Q u = - L ( t 0 ) W u ( t 0 ) ,
W u ( t 0 + τ ) = [ 1 - d L ( t 0 ) ] W u ( t 0 ) ;
W v ( t 0 + τ ) = [ 1 - d N ( t 0 ) ] W v ( t 0 ) .
M ( t 0 + τ ) = - W v ( t 0 + τ ) · Q u = L ( t 0 ) [ 1 - d N ( t 0 ) ] W v ( t 0 ) · W u ( t 0 ) = 0 ,
L ( t 0 + τ ) = - W u ( t 0 + τ ) · Q u = L ( t 0 ) [ 1 - d L ( t 0 ) ] ,
E ( t 0 + τ ) = W u 2 ( t 0 + τ ) = [ 1 - d L ( t 0 ) ] 2 .
L = L ( t 0 + τ ) / E ( t 0 + τ ) = L ( t 0 ) / [ 1 - d L ( t 0 ) ] ,
N = N ( t 0 ) / [ 1 - d N ( t 0 ) ] .
P = ( Q × S ) / Q × S = ( Q × S ) / sin i
T = Q × P T ¯ = S × P , T = Q × P ,
P = W x T = W y , P = R x T ¯ = R y ¯ , P = W x T = W y .
α = W x · W u ,             β = W x · W v ,
W u = α W x - β W y , W v = β W x + α W y ;
W x = α W u + β W v , W y = - β W u + α W v ,
u x = α , u y = - β , v x = β , v y = α ,
Q x = α Q u + β Q v ,             Q y = - β Q u + α Q v .
L r = - ( Q x · W x ) = α 2 L + β 2 N , M r = - ( Q x · W y ) = α β ( N - L ) , N r = - ( Q x · W y ) = β 2 L + α 2 N .
L = α 2 L r - 2 α β M r + β 2 N r ,
M = α β ( L r - N r ) + ( α 2 - β 2 ) M r = 0 ,
N = β 2 L r + 2 α β M r + α 2 N r ,
W [ x , y , ϕ ( x , y ) ] = R ( x , y ) = W [ x , y , ϕ ( x , y ) ] .
W x + ϕ x W t = R x = W x + ϕ x W t ,
P + c ϕ x Q = R x = W x + c ϕ x Q ,
c ϕ x ( Q · S ) = 0.
P = R x = R x ¯ = W x = W x .
W y + c ϕ y Q = R y ,
c ϕ y = - ( T · S ) / ( Q · S ) = sin i / cos i .
T + ( sin i / cos i ) Q = R y .
cos i T + sin i Q = R y ¯ ,
R y ¯ = cos i R y .
R y ¯ = cos r R y .
R y = ( cos i / cos r ) R y .
d x = d x ¯ = d x d y = cos i d y ¯ = ( cos i / cos r ) d y .
Q x = - L r W x - M r W y , Q y = - M r W x - N r W y ,
S x = S x ¯ = - L ¯ r R x ¯ - M ¯ r R y ¯ , S y = S y ¯ ( d y ¯ / d y ) = ( 1 / cos i ) ( - M ¯ r R x ¯ - N ¯ r R y ¯ ) ,
Q x = Q x , Q y = Q y ( d y / d y ) = ( cos i / cos r ) Q y .
Q = μ Q + ( { 1 - μ 2 [ 1 - ( S · Q ) 2 ] } 1 2 - μ ( S · Q ) ) S = μ Q + ( cos r - μ cos i ) S ,
Q x = μ Q x + γ x S + γ S x , Q y = ( cos i / cos r ) ( μ Q y + γ y S + γ S y ) .
T = R y - ( sin r / cos r ) Q = ( cos i / cos r ) R y - ( sin r / cos r ) Q ,
L r = - ( Q x · W x ) = - ( Q x · P ) = μ L r + γ L ¯ r ,
M r = - ( Q y · W x ) = - ( Q y · P ) = ( cos i / cos r ) [ μ M r + ( γ / cos i ) M ¯ r ] ,
N r = - ( Q y · W y ) = - ( Q y · T ) = - ( cos i / cos r ) ( Q y · R y ) = ( cos 2 i / cos 2 r ) [ μ N r + ( γ / cos 2 i ) N ¯ r ] .
L ¯ r = N ¯ r = k ,             and             M ¯ r = 0.
1 s s = μ s s + γ R s ,             cos 2 r s m = μ cos 2 i s m + γ R m .
M = α ( 1 - α 2 ) 1 2 ( L r - N r ) + ( 2 α 2 - 1 ) M r 0.
D = L r - N r
S = ( D 2 + 4 M r 2 ) 1 2 ,
α = + [ 1 2 ( 1 - D / S ) ] 1 2             or             - [ 1 2 ( 1 + D / S ) ] 1 2 ,
α = + [ 1 2 ( 1 + D / S ) ] 1 2             or             - [ 1 2 ( 1 - D / S ) ] 1 2 .
Q = Q - 2 ( S · Q ) S = Q - 2 cos i S ,
L r = L r - 2 cos i L ¯ r , M r = - M r + 2 M ¯ r , N r = N r - ( 2 / cos i ) N ¯ r .
L r = L r ,             M r = - M r ,             N r = N r .
( Q - μ Q ) × S = μ p G ,
A = S × G ,
s = Q · S ,             g = Q · G ,             and             a = Q · A .
Q = μ Q + ( { 1 - μ 2 [ g 2 + ( a + p ) 2 ] } 1 2 - μ s ) S + μ p A
= { 1 - μ 2 [ g 2 + ( a + p ) 2 ] } 1 2 S + μ g G + μ ( a + p ) A ;
Q = Q - { [ 1 - g 2 - ( a + p ) 2 ] 1 2 + s } S + p A
= - [ 1 - g 2 - ( a + p ) 2 ] 1 2 S + g G + ( a + p ) A ,
P = Q × S / ( 1 - s 2 ) 1 2 = ( a G - g A ) / ( a 2 + g 2 ) 1 2 ,
P = Q × S / Q × S = [ ( a + p ) G - g A ] / [ ( a + p ) 2 + g 2 ] 1 2 ;
T = Q × P = [ ( s 2 - 1 ) S + s g G + s a A ] / ( a 2 + g 2 ) 1 2 T ¯ = S × P = ( g G + a A ) / ( a 2 + g 2 ) 1 2 T = Q × P = [ ( s 2 - 1 ) S / μ + s g G + s a A ] / [ ( a + p ) 2 + g 2 ] 1 2 T = S × P = [ g G + ( a + p ) A ] / [ ( a + p ) 2 + g 2 ] 1 2 ,
P = W x , T = W y ; P = R x ¯ , T ¯ = R y ¯ ; P = W x , T = W y ; P = R x ¯ , T ¯ = R y ¯ .
W [ x , y , ϕ ( x , y ) ] = R ( x , y ) , W [ x , y , ϕ ( x , y ) ] = R ( x , y ) ;
ϕ x = 0 ,             ϕ x = 0 ; c ϕ y = - ( T · S ) / ( Q · S ) = ( 1 - s 2 ) 1 2 / s , c ϕ y = - ( T · S ) / ( Q · S ) = ( 1 - s 2 ) 1 2 / s ; W x = R x = R x ¯ ,             W x = R x = R x ¯ ,
s W y + ( 1 - s 2 ) 1 2 Q = R y ¯ ,             s W y + ( 1 - s 2 ) 1 2 Q = R y ¯ ;
R y ¯ = s R y ,             R y ¯ = s R y ;
d x = d x ¯ , d x = d x ¯ , d y = s d y ¯ , d y = s d y ¯ .
θ = R x ¯ · R x ¯ = P · P ,             η = R x ¯ · R y ¯ = P · T ¯ ;
P = R x ¯ = θ P - η T ¯ = θ R x ¯ - η R y ¯ , T ¯ = R y ¯ = η P + θ T ¯ = η R x ¯ + θ R y ¯ ,
x ¯ x ¯ = θ ,             x ¯ y ¯ = η ,             y ¯ x ¯ = - η ,             y ¯ y ¯ = θ .
θ = a ( a + p ) + g 2 { ( a 2 + g 2 ) [ ( a + p ) 2 + g 2 ] } 1 2 = μ a ( a + p ) + g 2 [ ( 1 - s 2 ) ( 1 - s 2 ) ] 1 2 , η = - g p { ( a 2 + g 2 ) [ ( a + p ) 2 + g 2 ] } 1 2 = - μ g p [ ( 1 - s 2 ) ( 1 - s 2 ) ] 1 2 .
d = D / ( B · A ) ,
G = R u ¯ ,             A = R v ¯ .
α ¯ = P · G = a / ( g 2 + a 2 ) 1 2 ,             β ¯ = P · A = - g / ( g 2 + a 2 ) 1 2 .
α ¯ = P · G = ( a + p ) / [ g 2 + ( a + p ) 2 ] 1 2 , β ¯ = P · A = - g / [ g 2 + ( a + p ) 2 ] 1 2 .
θ = α ¯ α ¯ + ( 1 - α ¯ 2 ) 1 2 ( 1 - α ¯ 2 ) 1 2 = β ¯ β ¯ + ( 1 - β ¯ 2 ) 1 2 ( 1 - β ¯ 2 ) 1 2 ,
Q x = Q x x x + Q y y x Q y = Q x x y + Q y y y .
x x = θ ,             y x = - s η ,             x y = η / s ,             y y = s θ / s .
A x = A x ¯ = A u ¯ u ¯ x ¯ + A v ¯ v ¯ x ¯ = α ¯ R v ¯ u ¯ + β ¯ R v ¯ v ¯ = ( α ¯ M ¯ + β ¯ N ¯ ) S = β ¯ N ¯ S ,
A y = ( 1 / s ) ( α ¯ N ¯ S ) .
d ū 0 d v ¯ = - D ( B · A v ) / ( B · A ) 2 = - N ¯ d ( B · S ) / ( B · A ) = - N ¯ d b ,
p x = ( m λ / n d 2 ) d x = p β ¯ N ¯ b , p y = p α ¯ N ¯ b / s .
Q x = θ ( μ Q x + γ x S + γ S x + μ p x A + μ p A x ) - s η ( μ Q y + γ y S + γ S y + μ p y A + μ p A y ) = [ μ ( - θ L r + s η M r ) + γ ( - θ L ¯ r + η M ¯ r ) ] P + μ ( - θ M r + s η N r ) T + γ ( - θ M ¯ r + η N ¯ r ) T ¯ + [ θ γ x - s η γ y - μ p N ¯ ( θ β ¯ - η α ¯ ) ] S + μ ( θ p x - s η p y ) A ,
Q y = ( 1 / s ) { [ μ ( - η L r - s θ M r ) + γ ( - η L ¯ r - θ M ¯ r ) ] P + μ ( - η M r - s θ N r ) T + γ ( - η M ¯ r - θ N ¯ r ) T + [ η γ x + s θ γ y + μ p N ¯ ( θ α ¯ + η β ¯ ) ] S + μ ( η p x + s θ p y ) A } .
T = ( 1 / s ) { T ¯ - μ [ g 2 + ( a + p ) 2 ] 1 2 Q }
L r = - ( Q x · W x ) = - ( Q x · P ) = μ ( θ 2 L r - 2 s θ η M r - s 2 η 2 N r ) + γ ( θ 2 L ¯ r - 2 θ η M ¯ r + η 2 N ¯ r ) - g 2 μ p N ¯ b / [ g 2 + ( a + p ) 2 ] ,
M r = - ( Q y · W x ) = - ( Q y · P ) = ( 1 / s ) { μ [ θ η ( L r - s 2 N r ) + ( θ 2 - η 2 ) s M r ] + γ [ θ η ( L ¯ r - N ¯ r ) + ( θ 2 - η 2 ) M ¯ r ] + g ( a + p ) μ p N ¯ b / [ g 2 + ( a + p ) 2 ] } ,
N r = - ( Q y · W y ) = - ( Q y · T ) = - ( 1 / s ) ( Q y · T ¯ ) = ( 1 / s 2 ) { μ [ η 2 L r + 2 s θ η M r + s 2 θ 2 N r ] + γ [ η 2 L ¯ r + 2 θ η M ¯ r + θ 2 N ¯ r ] - ( a + p ) 2 μ p N ¯ b / [ g 2 + ( a + p ) 2 ] } .
L ¯ r = μ ( θ 2 L r - 2 s θ η M r + s 2 η 2 N r ) , M r = ( μ / s ) [ θ η ( L r - s 2 N r ) + s ( θ 2 - η 2 ) M r ] , N r = ( μ / s 2 ) ( η 2 L r + 2 s θ η M r + s 2 θ 2 N r ) .
L = L / ( 1 - d L ) , N = N / ( 1 - d N ) .
L r = α 2 L + β 2 N , M r = α β ( N - L ) , N r = β 2 L + α 2 N , L ¯ r = α ¯ 2 L ¯ + β ¯ 2 N ¯ , etc .
L r = μ L r + γ L ¯ r , M r = ( cos i / cos r ) [ μ M r + ( γ / cos i ) M ¯ r ] , N r = ( cos 2 i / cos 2 r ) [ μ N r + ( γ / cos 2 i ) N ¯ r ] ,
α = ± ( 1 2 { 1 - L r - N r [ ( L r - N r ) 2 + 4 M r 2 ] 1 2 } ) 1 2 ,
β = + ( 1 - α 2 ) 1 2 .
W u = α P - β ( Q × P ) , W v = β P + α ( Q × P ) .             as in
L = α 2 L r - 2 α β M r + β 2 N r , N = β 2 L r + 2 α β M r + α 2 N r .             as in
L r = L r - 2 cos i L ¯ r M r = - M r + 2 M ¯ r N r = N r - 2 N ¯ r / cos i .
θ = P · P             η = P · ( S × S )
L r = μ ( θ 2 L r - 2 s θ η M r - s 2 η 2 N r ) + γ ( θ 2 L ¯ r - 2 θ η M ¯ r + η 2 N ¯ r ) - g 2 μ p N ¯ b / [ g 2 + ( a + p ) 2 ] , M r = ( 1 / s ) { μ [ θ η ( L r - s 2 N r ) + ( θ 2 - η 2 ) s M r ] + γ [ θ η ( L ¯ r - N ¯ r ) + ( θ 2 - η 2 ) M ¯ r ] + g ( a + p ) μ p N ¯ b / [ g 2 + ( a + p ) 2 ] } , N r = ( 1 / s ) 2 { μ ( η 2 L r + 2 s θ η M r + s 2 θ 2 N r ) + γ ( η 2 L ¯ r + 2 θ η M ¯ r + θ 2 N ¯ r ) - ( a + p ) 2 μ p N ¯ b / [ g 2 + ( a + p ) 2 ] } ,
p = m λ / n d ,             b = ( B · S ) / ( B · A ) ,
γ = { { 1 - μ 2 [ g 2 + ( a + p ) 2 ] } 1 2 - μ s ( transmission ) - { 1 - [ g 2 + ( a + p ) 2 ] } 1 2 - s ( reflection ) .