Abstract

The generalized radiance can never be measured at a given point of phase space, on account of the finite resolution of any real instrument; instead the instrument averages the radiance over some region of phase space. Thus a negative radiance is never measured, in spite of the fact that the generalized radiance can take on negative values. The relationship between the generalized radiance and the measurement process can be quantified by the instrument function, which is a property of the measurement apparatus and which allows one to calculate the response of the apparatus to any given incident wave field. The instrument function reveals a kind of reciprocity between the wave field being measured and the measurement apparatus. The theory of the instrument function is developed, and examples are discussed.

© 1995 Optical Society of America

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References

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  1. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [Crossref]
  2. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622–1623 (1973).
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  3. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
    [Crossref]
  4. L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk. 142, 689–711 (1984)[Sov. Phys. Usp. 27, 301–313 (1984)].
    [Crossref]
  5. W. Welford, R. Winston, “Generalized radiance and practical radiometry,” J. Opt. Soc. Am. A 4, 545–547 (1987).
    [Crossref]
  6. R. Winston, W. T. Welford, “Measurement of radiance,” Opt. Commun. 76, 191–193 (1990).
    [Crossref]
  7. R. Winston, W. T. Welford, “Non-localizability of generalized radiance,” Opt. Commun. 81, 155–156 (1991).
    [Crossref]
  8. H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1–46 (1927).
    [Crossref]
  9. T. Jannson, R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik (Stuttgart) 56, 429–441 (1980).
  10. A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
    [Crossref]
  11. R. G. Littlejohn, R. Winston, “Corrections to classical radiometry,” J. Opt. Soc. Am. A 10, 2024–2037 (1993).
    [Crossref]
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    [Crossref]
  13. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
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    [Crossref]

1993 (1)

1992 (1)

1991 (1)

R. Winston, W. T. Welford, “Non-localizability of generalized radiance,” Opt. Commun. 81, 155–156 (1991).
[Crossref]

1990 (1)

R. Winston, W. T. Welford, “Measurement of radiance,” Opt. Commun. 76, 191–193 (1990).
[Crossref]

1987 (1)

1984 (1)

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk. 142, 689–711 (1984)[Sov. Phys. Usp. 27, 301–313 (1984)].
[Crossref]

1982 (1)

1980 (1)

T. Jannson, R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik (Stuttgart) 56, 429–441 (1980).

1978 (1)

1973 (1)

1968 (1)

1932 (1)

E. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

1927 (1)

H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1–46 (1927).
[Crossref]

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Agarwal, G. S.

Apresyan, L. A.

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk. 142, 689–711 (1984)[Sov. Phys. Usp. 27, 301–313 (1984)].
[Crossref]

Foley, J. T.

Friberg, A. T.

Janicki, R.

T. Jannson, R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik (Stuttgart) 56, 429–441 (1980).

Jannson, T.

T. Jannson, R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik (Stuttgart) 56, 429–441 (1980).

Kravtsov, Yu. A.

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk. 142, 689–711 (1984)[Sov. Phys. Usp. 27, 301–313 (1984)].
[Crossref]

Littlejohn, R. G.

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Walther, A.

Welford, W.

Welford, W. T.

R. Winston, W. T. Welford, “Non-localizability of generalized radiance,” Opt. Commun. 81, 155–156 (1991).
[Crossref]

R. Winston, W. T. Welford, “Measurement of radiance,” Opt. Commun. 76, 191–193 (1990).
[Crossref]

R. Winston, W. T. Welford, “Efficiency of nonimaging concentrators in the physical-optics model,” J. Opt. Soc. Am. 72, 1564–1566 (1982).
[Crossref]

Weyl, H.

H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1–46 (1927).
[Crossref]

Wigner, E.

E. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Winston, R.

Wolf, E.

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

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

Opt. Commun. (2)

R. Winston, W. T. Welford, “Measurement of radiance,” Opt. Commun. 76, 191–193 (1990).
[Crossref]

R. Winston, W. T. Welford, “Non-localizability of generalized radiance,” Opt. Commun. 81, 155–156 (1991).
[Crossref]

Optik (Stuttgart) (1)

T. Jannson, R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik (Stuttgart) 56, 429–441 (1980).

Phys. Rev. (1)

E. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Usp. Fiz. Nauk. (1)

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk. 142, 689–711 (1984)[Sov. Phys. Usp. 27, 301–313 (1984)].
[Crossref]

Z. Phys. (1)

H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1–46 (1927).
[Crossref]

Other (1)

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

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Figures (5)

Fig. 1
Fig. 1

A simple radiometer consists of a lens filling an entrance aperture, with an exit aperture in the focal plane. Behind the focal plane is a device to measure the transversally integrated intensity.

Fig. 2
Fig. 2

Contours of the instrument function M(x, k) in phase space for N = 1.

Fig. 3
Fig. 3

Same as Fig. 2 but for N = 4.

Fig. 4
Fig. 4

Plots of the instrument function M (ξ, k) along the horizontal line κ = 0 for different values of N.

Fig. 5
Fig. 5

Same as Fig. 4 but along the vertical line ξ = 0.

Equations (54)

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( 2 + k 0 2 ) ψ = 0 ,
Γ ̂ ( z ) = | ψ ( z ) ψ ( z ) | ¯ ,
x | Γ ̂ | x = Γ ( x , z ; x , z ) = ψ ( x , z ) ψ ( x , z ) * ¯
Γ ̂ = | u 0 γ 0 u 0 | ,
Γ ̂ 2 = ( Tr Γ ̂ ) Γ ̂ ,
Γ ̂ = n | u n γ n u n | ,
γ k [ γ k ( Tr Γ ̂ ) ] | u k = 0 ,
W ( x , k ) = ds exp ( iks ) Γ ( x + s / 2 , x s / 2 ) ,
Q = + d x Γ ̂ ( x , x ) = + d x | ψ ( x ) | 2 ¯ = Tr Γ ̂ .
 ( a ) = a + a d ξ | ξ ξ | ,
x | Â ( a ) | x = a + a d ξ δ ( x ξ ) δ ( x ξ ) = { δ ( x x ) | x | , | x | a 0 otherwise .
| ψ 1 = Â ( a ) | ψ in ,
ψ 1 ( x ) = x | ψ 1 = x | Â ( a ) | ψ in = + d x x | Â ( a ) | x ψ in ( x ) = { ψ in ( x ) | x | a 0 otherwise .
x | L ̂ ( f ) | x = exp ( i k 0 x 2 2 f ) δ ( x x ) .
x | D ̂ ( l ) | x = exp ( i π / 4 ) k 0 2 π l exp [ i k 0 2 l ( x x ) 2 ] .
 ( b ) = b + b d ξ | ξ ξ | .
| ψ out = P ̂ | ψ in ,
P ̂ = Â ( b ) D ̂ ( f ) L ̂ ( f ) Â ( a ) .
Γ ̂ out = P ̂ Γ ̂ in P ̂ ,
Q = Tr Γ ̂ out = Tr ( M ̂ Γ ̂ in ) ,
M ̂ = P ̂ P ̂ .
M ̂ 2 = ( Tr M ̂ ) M ̂ .
M ̂ | ψ n = μ n | ψ n ,
| ϕ n = P ̂ | ψ n ,
ϕ m | ϕ n = μ n δ mn .
| χ n = 1 μ n | ϕ n ,
M ̂ = | ψ 0 μ 0 ψ 0 | .
Γ ̂ in = m , n | ψ m ψ m | Γ ̂ in | ψ n ψ n | ,
Γ ̂ out = | χ 0 γ 0 χ 0 | ,
γ 0 = μ 0 ψ 0 | Γ ̂ in | ψ 0 .
Γ ̂ out 2 = ( Tr Γ ̂ out ) Γ ̂ out .
Tr ( M ̂ Γ ̂ in M ̂ Γ ̂ in ) = [ Tr ( M ̂ Γ ̂ in ) ] 2 ,
Γ ̂ in = n | ψ n γ n ψ n | ,
n μ n 2 γ n 2 = ( n μ n γ n ) 2 .
μ k 2 + μ l 2 = ( μ k + μ l ) 2 ,
μ k μ l = 0 .
K = k 0 Δ θ 2 = b k 0 f .
Q = d x d k R ( x , k ) M ( x , k ) ,
M ( x , k ) = { 1 | x | a , | k | K 0 otherwise ,
M ( x , k ) = d s exp ( iks ) x + s / 2 | M ̂ | x s / 2 .
Q = Tr ( Γ ̂ in M ̂ ) = d x d k 2 π W ( x , k ) M ( x , k ) .
N = 4 ak 2 π = 2 ab k 0 π f ,
x | P ̂ | x = exp ( i π / 4 ) k 0 2 π f exp [ i k 0 2 π f exp ( x 2 2 x x ) ]
x | M ̂ | x = b + b d x x | P ̂ | x * x | P ̂ | x = 1 π ( x x ) sin [ k 0 b f ( x x ) ]
Tr M ̂ = a + a d x x | M ̂ | x = N .
M ( x , k ) = 1 π Si [ ( k 0 b f + k ) ( a | x | ) ] + 1 π Si [ ( k 0 b f k ) ( a | x | ) ]
Si x = 0 x sin t t d t .
Si x = x + O ( x 3 ) ;
lim x ± Si x = π 2 sgn x ;
Si x π [ Θ ( x ) 1 2 ] ,
ξ = x a , κ = k K ,
M ( x , k ) = 1 π Si [ N π 2 ( 1 + κ ) ( 1 | ξ | ) ] + 1 π Si [ N π 2 ( 1 κ ) ( 1 | ξ | ) ] .
M ( x , k ) = Θ [ ( 1 + κ ) ( 1 | ξ | ) ] + Θ [ ( 1 κ ) ( 1 | ξ | ) ] 1 = { 1 | ξ | , | κ | 1 0 otherwise ,
x | M ̂ | x = k 0 b π f

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