Abstract

The consequences of Liouville’s theorem that are known to be valid in the geometrical-optics limit are shown to hold when wavelength-dependent effects, for example, diffraction and scattering, are taken into account.

© 1982 Optical Society of America

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References

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  1. W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).
  2. W. T. Welford and R. Winston, “Nonconventional optical systems and the brightness theorem,” Appl. Opt.1982;“An upper bound on the efficiency of certain nonimaging concentrators in the physical-optics model,” J. Opt. Soc. Am. 72, 1244–1248 (1982).
    [Crossref] [PubMed]
  3. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622–1623 (1973).
    [Crossref]
  4. D. Gabor, “Light and information,” in Progress in Optics, Vol. I, E. Wolf, ed. (North-Holland, Amsterdam, 1961), pp. 111–152;H. Gamo, “Matrix treatment of parital coherence,” in Progress in Optics, Vol. III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 189–326.
  5. A. Walther, “Gabor’s theorem and energy transfer through lenses,” J. Opt. Soc. Am. 57, 639–644 (1967).
    [Crossref]
  6. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I (Interscience, New York, 1953), pp. 429 et. seq.

1982 (1)

W. T. Welford and R. Winston, “Nonconventional optical systems and the brightness theorem,” Appl. Opt.1982;“An upper bound on the efficiency of certain nonimaging concentrators in the physical-optics model,” J. Opt. Soc. Am. 72, 1244–1248 (1982).
[Crossref] [PubMed]

1973 (1)

1967 (1)

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I (Interscience, New York, 1953), pp. 429 et. seq.

Gabor, D.

D. Gabor, “Light and information,” in Progress in Optics, Vol. I, E. Wolf, ed. (North-Holland, Amsterdam, 1961), pp. 111–152;H. Gamo, “Matrix treatment of parital coherence,” in Progress in Optics, Vol. III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 189–326.

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I (Interscience, New York, 1953), pp. 429 et. seq.

Walther, A.

Welford, W. T.

W. T. Welford and R. Winston, “Nonconventional optical systems and the brightness theorem,” Appl. Opt.1982;“An upper bound on the efficiency of certain nonimaging concentrators in the physical-optics model,” J. Opt. Soc. Am. 72, 1244–1248 (1982).
[Crossref] [PubMed]

W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).

Winston, R.

W. T. Welford and R. Winston, “Nonconventional optical systems and the brightness theorem,” Appl. Opt.1982;“An upper bound on the efficiency of certain nonimaging concentrators in the physical-optics model,” J. Opt. Soc. Am. 72, 1244–1248 (1982).
[Crossref] [PubMed]

W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).

Appl. Opt. (1)

W. T. Welford and R. Winston, “Nonconventional optical systems and the brightness theorem,” Appl. Opt.1982;“An upper bound on the efficiency of certain nonimaging concentrators in the physical-optics model,” J. Opt. Soc. Am. 72, 1244–1248 (1982).
[Crossref] [PubMed]

J. Opt. Soc. Am. (2)

Other (3)

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I (Interscience, New York, 1953), pp. 429 et. seq.

D. Gabor, “Light and information,” in Progress in Optics, Vol. I, E. Wolf, ed. (North-Holland, Amsterdam, 1961), pp. 111–152;H. Gamo, “Matrix treatment of parital coherence,” in Progress in Optics, Vol. III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 189–326.

W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).

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Figures (2)

Fig. 1
Fig. 1

(a) Schematic optical concentrator, not necessarily imaging.

Fig. 2
Fig. 2

(a) Propagating beam (quasi-monochromatic) of cross-sectional area 4ab, (b) the direction cosine space subdivided into tiny rectangles of area (λ/2a)(λ/2b).

Equations (22)

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

u ( x ) = u ¯ ( L ) e ikLx d L ,
B = Re u * ( x ) u ¯ ( L ) N e ikLx ,
H = 4 B a sin θ 0 .
Γ ( L , L ) = u ¯ ( L ) u ¯ * ( L ) = 2 a B λ N sinc k a ( L L ) .
L 1 L 1 ϕ m * ( L ) ϕ n ( L ) N d L = δ m n ,
Γ ( L , L ) = m n A m n ϕ m ( L ) ϕ n * ( L ) ,
A m n = ϕ m * ( L ) ϕ n ( L ) N N Γ ( L , L ) d L d L .
A m n = B ϕ m * ( L ) ϕ n ( L ) N d L = B δ m n
ϕ i * ( L ¯ ) ϕ i ( L ) E ( L , L ¯ ) d L d L ¯ = τ i ϕ i * ( L ) ϕ i ( L ) N d L ,
N ¯ ϕ i ( L ¯ ) = 1 τ i E ( L , L ¯ ) ϕ i ( L ) d L .
ψ i ( L ) = K ( L , L ) ϕ i ( L ) d L .
ψ i ( L ) ψ j * ( L ) N d L = E ( L , L ¯ ) ϕ i ( L ) ϕ j * ( L ¯ ) d L d L ¯ .
E ( L , L ¯ ) ϕ i ( L ) d L = τ i ϕ i ( L ¯ ) N ¯ ,
ψ i ( L ) ψ j * ( L ) N d L = τ i ϕ i ( L ¯ ) ϕ j * ( L ¯ ) N ¯ d L ¯ = τ i δ i j .
B i ψ i ( L ) ψ i * ( L ) ,
λ B i | ψ i ( L ) | 2 N d L = λ B i τ i ( f terms ) ,
Γ ( L , M , L , M ) = B λ 2 N S exp { i k [ ( L L ) x + ( M M ) y ] } d x d y ,
g [ ( L L ) , ( M M ) ] = 1 λ 2 S exp { i k [ ( L L ) x + ( M M ) y ] } d x d y .
f ( L , M ) = S f ¯ ( x , y ) exp [ i k ( L x + M y ) ] d x d y ,
f ( L , M ) = 1 λ 2 f ( L , M ) exp { i k ( L L ) x + ( M M ) y } d x d y d L d M = f ( L , M ) g [ ( L L ) , ( M M ) ] d L d M .
A m n = ϕ m * ( L , M ) ϕ n ( L , M ) N N × B N g [ ( L L ) , ( M M ) ] d L d M d L d M = B ϕ m * ( L , M ) ϕ n ( L , M ) N d L d M = B δ n m .
u ¯ ( L , M ) = p q u ¯ ( p λ 2 a , q λ 2 b ) sinc π ( 2 a λ L p ) sinc π ( 2 b λ M q ) ,