Abstract

The concept of “throughput” is used in traditional radiometry of Lambertian sources for computing and estimating the radiant flux passed through a pair of stops, in particular through the window and the pupil of an optical system. It is shown that in a more general case of quasi-homogeneous sources for energetic calculations of the perfect optical system, one must use instead of the throughput a functional that is similar to the famous “Dirac bra-ket.” This functional takes into account the radiation pattern of the source. As the Dirac bra-ket satisfies the axioms of the inner product, powerful mathematical tools of functional analysis for the energy calculation of the optical systems are used. The main equations and principles of radiometry (the principle of reversibility and Maxwell’s principle) are reformulated from the concept “throughput” into the concept “Dirac bra-ket.” For generalization of Maxwell’s principle to the class of quasi-homogeneous sources the concept of “effective stops” is introduced.

© 2011 Optical Society of America

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References

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  13. D. S. Volosov and M. V. Tsivkin, Theory and Design of Photo-Optical Systems (Moskow, 1960), in Russian.
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  16. A. V. Gitin, “Optical systems for measuring the Wigner function of a laser beam by the method of phase-spatial tomography,” Quantum Electron. 37, 85–91 (2007).
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  17. T. Jannson, S. Kupiec, A. Kostrzewski, K. Spariosu, D. Minyzer, M. Rud, I. Tengara, and A. Vasiliev, “Phase-space formalism and ray-tracing modeling of photometric quantities in photometric engineering,” Proc. SPIE 3140, 36–47 (1997).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. A. Arkangy, “Liouville’s theorem and the intensity of beam,” Am. J. Phys. 25, 519–525 (1957).
    [CrossRef]
  22. A. V. Gitin, “Effect of the radiation pattern of a quasihomogeneous source on the point-spread function of an isoplanar optical system,” J. Opt. Technol. 60, 372–374 (1993).
  23. A. V. Gitin and Ur. A. Fligontov, “Equation of radiation transfer through optical media in the Hamiltonian approximation,” Opt. Spectrosc. 66, 371–374 (1989).
  24. A. V. Gitin, “Radiometry. A complex approach,” J. Opt. Technol. 65, 132–140 (1998).
  25. A. V. Gitin, “Radiometry of optical systems with quasihomogeneous sources: a linear systems approach,” Optik 122, 1713–1718 (2011).
    [CrossRef]
  26. A. V. Gitin, “Energy calculation of optical systems by a methods of the harmonious analysis,” in Pulse Photometry: Collection of Papers (Mashinostroenie, 1986) (in Russian).
  27. A. V. Gitin, “Radiometry as a section of optical system theory,” Opt. Spectrosc. 63, 106–109 (1987).
  28. A. V. Gitin, “Method of an energy calculation of the viewfinder of a reflex camera having a focusing screen,” J. Opt. Technol. 56, 440–442 (1989).
  29. A. V. Gitin, “Laser beam pumping homogenizator calculation,” Photonics No. 2, 26–29 (2009) (in Russian), http://www.photonics.su/issue/2009/2/6.
  30. A. V. Gitin, “Laser pulses compressor,” Photonics No. 5, 8–13 (2009) (in Russian), http://www.photonics.su/issue/2009/5/2.
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  32. M. M. Gurevich, Photometry (Theory of the Methods and Apparatus) (Energoatomizdat, Leningrad).
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  36. W. Brouwer and A. Walther, “Geometrical optics,” in Advanced Optical Techniques (North-Holland, 1967).
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  41. J. C. Maxwell, “On the general laws of optical instruments,” in The Scientific Papers of James Clerk Maxwell, Vol.  1, W.D.Niven, ed. (The University Press, 1890).
  42. V. Guillemin and S. Sternberg, Geometric Asymptotics(American Mathematical Society, 1977).
  43. R. H. Akin and J. M. Hood, “Photometry,” in Display System Engineering, H.R.Luxenberg and R.L.Kuehn, ed. (McGraw-Hill, 1968).
  44. S. Liebes, Jr., “On the ray invariance of B/n2,” Am. J. Phys. 37, 932–934 (1969).
    [CrossRef]
  45. R. Clausius, “Ueber die Concentration vom Wa¨rme und Lichtstrahlen und die Grenzen ihrer Wirkung,” Annalen der Physik und Chemie 197, 1–44 (1864).
    [CrossRef]
  46. J. C. Maxwell, “On the theory of compound colours and the relation of the colours of the spectrum,” in The Scientific Papers of James Clerk Maxwell, Vol.  1, W.D.Niven, ed. (University Press, 1890).
  47. P. M. Tikhodeev, Light Measurements in Illumination Engineering (Gos. Energo. Izd., 1962) (in Russian).
  48. J. M. Lloyd, Thermal Imaging Systems (Plenum, 1975).
  49. A. V. Gitin, “The effective point source,” Opt. Spectrosc. 76 (1), 157–158 (1994).
  50. A. V. Gitin, “Technique for checking the correctness of computer programs intended for calculating the energy of optical systems,” J. Opt. Technol. 67, 844–845 (2000).
    [CrossRef]
  51. A. V. Gitin, “Radiometry of the light-scattering characteristics of liquid crystal elements,” J. Opt. Technol. 61, 131–135(1994).
  52. A. A. Gershun, “Telecentric method of measuring the intensity of light,” in Selected Works on Photometry and Illumination Engineering (Phys. Math. Gos. Izd., 1958) (in Russian).
  53. F. E. Nicodemua, “Radiance,” Am. J. Phys. 31, 368–377(1963).
    [CrossRef]

2011 (1)

A. V. Gitin, “Radiometry of optical systems with quasihomogeneous sources: a linear systems approach,” Optik 122, 1713–1718 (2011).
[CrossRef]

2007 (1)

A. V. Gitin, “Optical systems for measuring the Wigner function of a laser beam by the method of phase-spatial tomography,” Quantum Electron. 37, 85–91 (2007).
[CrossRef]

2004 (1)

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).
[CrossRef]

2000 (1)

1998 (1)

A. V. Gitin, “Radiometry. A complex approach,” J. Opt. Technol. 65, 132–140 (1998).

1997 (1)

T. Jannson, S. Kupiec, A. Kostrzewski, K. Spariosu, D. Minyzer, M. Rud, I. Tengara, and A. Vasiliev, “Phase-space formalism and ray-tracing modeling of photometric quantities in photometric engineering,” Proc. SPIE 3140, 36–47 (1997).
[CrossRef]

1994 (2)

A. V. Gitin, “Radiometry of the light-scattering characteristics of liquid crystal elements,” J. Opt. Technol. 61, 131–135(1994).

A. V. Gitin, “The effective point source,” Opt. Spectrosc. 76 (1), 157–158 (1994).

1993 (1)

A. V. Gitin, “Effect of the radiation pattern of a quasihomogeneous source on the point-spread function of an isoplanar optical system,” J. Opt. Technol. 60, 372–374 (1993).

1989 (2)

A. V. Gitin and Ur. A. Fligontov, “Equation of radiation transfer through optical media in the Hamiltonian approximation,” Opt. Spectrosc. 66, 371–374 (1989).

A. V. Gitin, “Method of an energy calculation of the viewfinder of a reflex camera having a focusing screen,” J. Opt. Technol. 56, 440–442 (1989).

1987 (1)

A. V. Gitin, “Radiometry as a section of optical system theory,” Opt. Spectrosc. 63, 106–109 (1987).

1981 (1)

G. K. Grau, “Comments on index profile measurement of fibers and their evaluation,” Proc. IEEE 69, 753–754 (1981).
[CrossRef]

1980 (1)

D. Marcuse and H. M. Presby, “Index profile measurements of fibers and their evaluation,” Proc. IEEE 68, 1198–1203(1980).
[CrossRef]

1978 (1)

I. N. Tarnakin, “Determination of illumination and light flux by an optical-system eikonal,” Opt. Spectrosc. 44, 463–465(1978).

1975 (1)

1974 (1)

1969 (1)

S. Liebes, Jr., “On the ray invariance of B/n2,” Am. J. Phys. 37, 932–934 (1969).
[CrossRef]

1968 (1)

1963 (1)

F. E. Nicodemua, “Radiance,” Am. J. Phys. 31, 368–377(1963).
[CrossRef]

1957 (1)

A. Arkangy, “Liouville’s theorem and the intensity of beam,” Am. J. Phys. 25, 519–525 (1957).
[CrossRef]

1948 (1)

1945 (1)

1864 (1)

R. Clausius, “Ueber die Concentration vom Wa¨rme und Lichtstrahlen und die Grenzen ihrer Wirkung,” Annalen der Physik und Chemie 197, 1–44 (1864).
[CrossRef]

1860 (1)

G. Kirchhoff, “On the relation between the radiating and the absorbing powers of different bodies for light and heat,” Philos. Mag. Series 4 20, 1–21 (1860).

Akin, R. H.

R. H. Akin and J. M. Hood, “Photometry,” in Display System Engineering, H.R.Luxenberg and R.L.Kuehn, ed. (McGraw-Hill, 1968).

Arkangy, A.

A. Arkangy, “Liouville’s theorem and the intensity of beam,” Am. J. Phys. 25, 519–525 (1957).
[CrossRef]

Becherer, R. J.

F. Grum and R. J. Becherer, “Radiometry,” in Optical Radiation Measurement (Academic, 1979), Vol.  1.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Brouwer, W.

W. Brouwer and A. Walther, “Geometrical optics,” in Advanced Optical Techniques (North-Holland, 1967).

Cengel, Y. A.

Y. A. Cengel and A. J. Ghajar, Heat and Mass Transfer: Fundamentals and Applications (McGraw-Hill, 2011).

Clausius, R.

R. Clausius, “Ueber die Concentration vom Wa¨rme und Lichtstrahlen und die Grenzen ihrer Wirkung,” Annalen der Physik und Chemie 197, 1–44 (1864).
[CrossRef]

Datla, R. U.

A. C. Parr, R. U. Datla, and J. Gardner, Optical Radiometry (Elsevier, 2005).

Dirac, P.

P. Dirac, Principles of Quantum Mechanics (Clarendon, 1930).

Fincham, W. H. A.

W. H. A. Fincham, and M. H. Freeman, Optics(Butterworths, 1980).

Fligontov, Ur. A.

A. V. Gitin and Ur. A. Fligontov, “Equation of radiation transfer through optical media in the Hamiltonian approximation,” Opt. Spectrosc. 66, 371–374 (1989).

Freeman, M. H.

W. H. A. Fincham, and M. H. Freeman, Optics(Butterworths, 1980).

Gardner, J.

A. C. Parr, R. U. Datla, and J. Gardner, Optical Radiometry (Elsevier, 2005).

Gershun, A. A.

A. A. Gershun, “Telecentric method of measuring the intensity of light,” in Selected Works on Photometry and Illumination Engineering (Phys. Math. Gos. Izd., 1958) (in Russian).

Ghajar, A. J.

Y. A. Cengel and A. J. Ghajar, Heat and Mass Transfer: Fundamentals and Applications (McGraw-Hill, 2011).

Gitin, A. V.

A. V. Gitin, “Radiometry of optical systems with quasihomogeneous sources: a linear systems approach,” Optik 122, 1713–1718 (2011).
[CrossRef]

A. V. Gitin, “Optical systems for measuring the Wigner function of a laser beam by the method of phase-spatial tomography,” Quantum Electron. 37, 85–91 (2007).
[CrossRef]

A. V. Gitin, “Technique for checking the correctness of computer programs intended for calculating the energy of optical systems,” J. Opt. Technol. 67, 844–845 (2000).
[CrossRef]

A. V. Gitin, “Radiometry. A complex approach,” J. Opt. Technol. 65, 132–140 (1998).

A. V. Gitin, “Radiometry of the light-scattering characteristics of liquid crystal elements,” J. Opt. Technol. 61, 131–135(1994).

A. V. Gitin, “The effective point source,” Opt. Spectrosc. 76 (1), 157–158 (1994).

A. V. Gitin, “Effect of the radiation pattern of a quasihomogeneous source on the point-spread function of an isoplanar optical system,” J. Opt. Technol. 60, 372–374 (1993).

A. V. Gitin and Ur. A. Fligontov, “Equation of radiation transfer through optical media in the Hamiltonian approximation,” Opt. Spectrosc. 66, 371–374 (1989).

A. V. Gitin, “Method of an energy calculation of the viewfinder of a reflex camera having a focusing screen,” J. Opt. Technol. 56, 440–442 (1989).

A. V. Gitin, “Radiometry as a section of optical system theory,” Opt. Spectrosc. 63, 106–109 (1987).

A. V. Gitin, “Laser beam pumping homogenizator calculation,” Photonics No. 2, 26–29 (2009) (in Russian), http://www.photonics.su/issue/2009/2/6.

A. V. Gitin, “Laser pulses compressor,” Photonics No. 5, 8–13 (2009) (in Russian), http://www.photonics.su/issue/2009/5/2.

A. V. Gitin, “Energy calculation of optical systems by a methods of the harmonious analysis,” in Pulse Photometry: Collection of Papers (Mashinostroenie, 1986) (in Russian).

Goodman, D. S.

D. S. Goodman, “General principles of geometrical optics,” in Handbook of Optics, Vol.  1, 2nd ed., M.Bass, ed. (McGraw-Hill, 1995).

Grau, G. K.

G. K. Grau, “Comments on index profile measurement of fibers and their evaluation,” Proc. IEEE 69, 753–754 (1981).
[CrossRef]

Gross, H.

H. Gross, “Radiometry,” in Handbook of Optical Systems, Vol.  1, Fundamentals of Technical Optics (Wiley-VCH, 2005).
[CrossRef]

Grum, F.

F. Grum and R. J. Becherer, “Radiometry,” in Optical Radiation Measurement (Academic, 1979), Vol.  1.

Guillemin, V.

V. Guillemin and S. Sternberg, Geometric Asymptotics(American Mathematical Society, 1977).

Gurevich, M. M.

M. M. Gurevich, Photometry (Theory of the Methods and Apparatus) (Energoatomizdat, Leningrad).

Hood, J. M.

R. H. Akin and J. M. Hood, “Photometry,” in Display System Engineering, H.R.Luxenberg and R.L.Kuehn, ed. (McGraw-Hill, 1968).

Jannson, T.

T. Jannson, S. Kupiec, A. Kostrzewski, K. Spariosu, D. Minyzer, M. Rud, I. Tengara, and A. Vasiliev, “Phase-space formalism and ray-tracing modeling of photometric quantities in photometric engineering,” Proc. SPIE 3140, 36–47 (1997).
[CrossRef]

Kingslake, R.

R. Kingslake, “Illumination in Optical Images,” in Applied Optics and Optical Engineering, Vol.  2, R.Kingslake, ed. (Academic, 1965).

Kirchhoff, G.

G. Kirchhoff, “On the relation between the radiating and the absorbing powers of different bodies for light and heat,” Philos. Mag. Series 4 20, 1–21 (1860).

Kostrzewski, A.

T. Jannson, S. Kupiec, A. Kostrzewski, K. Spariosu, D. Minyzer, M. Rud, I. Tengara, and A. Vasiliev, “Phase-space formalism and ray-tracing modeling of photometric quantities in photometric engineering,” Proc. SPIE 3140, 36–47 (1997).
[CrossRef]

Kupiec, S.

T. Jannson, S. Kupiec, A. Kostrzewski, K. Spariosu, D. Minyzer, M. Rud, I. Tengara, and A. Vasiliev, “Phase-space formalism and ray-tracing modeling of photometric quantities in photometric engineering,” Proc. SPIE 3140, 36–47 (1997).
[CrossRef]

Lambert, J. H.

J. H. Lambert, Photometria, Sive de Mensura et Gradibus Luminis, Colorum et Umbrae (Augsburg, 1760).

Liebes, S.

S. Liebes, Jr., “On the ray invariance of B/n2,” Am. J. Phys. 37, 932–934 (1969).
[CrossRef]

Lloyd, J. M.

J. M. Lloyd, Thermal Imaging Systems (Plenum, 1975).

Marchand, E. W.

Marcuse, D.

D. Marcuse and H. M. Presby, “Index profile measurements of fibers and their evaluation,” Proc. IEEE 68, 1198–1203(1980).
[CrossRef]

Maxwell, J. C.

J. C. Maxwell, “On the theory of compound colours and the relation of the colours of the spectrum,” in The Scientific Papers of James Clerk Maxwell, Vol.  1, W.D.Niven, ed. (University Press, 1890).

J. C. Maxwell, “On the general laws of optical instruments,” in The Scientific Papers of James Clerk Maxwell, Vol.  1, W.D.Niven, ed. (The University Press, 1890).

Minyzer, D.

T. Jannson, S. Kupiec, A. Kostrzewski, K. Spariosu, D. Minyzer, M. Rud, I. Tengara, and A. Vasiliev, “Phase-space formalism and ray-tracing modeling of photometric quantities in photometric engineering,” Proc. SPIE 3140, 36–47 (1997).
[CrossRef]

Nicodemua, F. E.

F. E. Nicodemua, “Radiance,” Am. J. Phys. 31, 368–377(1963).
[CrossRef]

Parr, A. C.

A. C. Parr, R. U. Datla, and J. Gardner, Optical Radiometry (Elsevier, 2005).

Planck, M.

M. Planck, The Theory of Heat Radiation (Dover, 1959).

Potton, R. J.

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).
[CrossRef]

Presby, H. M.

D. Marcuse and H. M. Presby, “Index profile measurements of fibers and their evaluation,” Proc. IEEE 68, 1198–1203(1980).
[CrossRef]

Reiss, M.

Rud, M.

T. Jannson, S. Kupiec, A. Kostrzewski, K. Spariosu, D. Minyzer, M. Rud, I. Tengara, and A. Vasiliev, “Phase-space formalism and ray-tracing modeling of photometric quantities in photometric engineering,” Proc. SPIE 3140, 36–47 (1997).
[CrossRef]

Smith, W. J.

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

Spariosu, K.

T. Jannson, S. Kupiec, A. Kostrzewski, K. Spariosu, D. Minyzer, M. Rud, I. Tengara, and A. Vasiliev, “Phase-space formalism and ray-tracing modeling of photometric quantities in photometric engineering,” Proc. SPIE 3140, 36–47 (1997).
[CrossRef]

Steele, W. H.

Sternberg, S.

V. Guillemin and S. Sternberg, Geometric Asymptotics(American Mathematical Society, 1977).

Tarnakin, I. N.

I. N. Tarnakin, “Determination of illumination and light flux by an optical-system eikonal,” Opt. Spectrosc. 44, 463–465(1978).

Tengara, I.

T. Jannson, S. Kupiec, A. Kostrzewski, K. Spariosu, D. Minyzer, M. Rud, I. Tengara, and A. Vasiliev, “Phase-space formalism and ray-tracing modeling of photometric quantities in photometric engineering,” Proc. SPIE 3140, 36–47 (1997).
[CrossRef]

Tikhodeev, P. M.

P. M. Tikhodeev, Light Measurements in Illumination Engineering (Gos. Energo. Izd., 1962) (in Russian).

Tsivkin, M. V.

D. S. Volosov and M. V. Tsivkin, Theory and Design of Photo-Optical Systems (Moskow, 1960), in Russian.

Vasiliev, A.

T. Jannson, S. Kupiec, A. Kostrzewski, K. Spariosu, D. Minyzer, M. Rud, I. Tengara, and A. Vasiliev, “Phase-space formalism and ray-tracing modeling of photometric quantities in photometric engineering,” Proc. SPIE 3140, 36–47 (1997).
[CrossRef]

Volosov, D. S.

D. S. Volosov and M. V. Tsivkin, Theory and Design of Photo-Optical Systems (Moskow, 1960), in Russian.

Walther, A.

A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
[CrossRef]

W. Brouwer and A. Walther, “Geometrical optics,” in Advanced Optical Techniques (North-Holland, 1967).

Wetherell, W. B.

W. B. Wetherell, “The calculation of image quality,” in Applied Optics and Optical Engineering, vol.  8, R.R.Shannon and J.C.Wyant, ed. (Academic, 1980).

Wolf, E.

Wolfe, W. L.

W. L. Wolfe, Introduction to Radiometry (SPIE Optical Press, 1998).
[CrossRef]

W. L. Wolfe, “Radiometry,” in Applied Optics and Optical Engineering, vol.  8, R.R.Shannon, and J.C.Wyant, ed. (Academic, 1980).

Am. J. Phys. (3)

A. Arkangy, “Liouville’s theorem and the intensity of beam,” Am. J. Phys. 25, 519–525 (1957).
[CrossRef]

S. Liebes, Jr., “On the ray invariance of B/n2,” Am. J. Phys. 37, 932–934 (1969).
[CrossRef]

F. E. Nicodemua, “Radiance,” Am. J. Phys. 31, 368–377(1963).
[CrossRef]

Annalen der Physik und Chemie (1)

R. Clausius, “Ueber die Concentration vom Wa¨rme und Lichtstrahlen und die Grenzen ihrer Wirkung,” Annalen der Physik und Chemie 197, 1–44 (1864).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

J. Opt. Technol. (5)

A. V. Gitin, “Radiometry of the light-scattering characteristics of liquid crystal elements,” J. Opt. Technol. 61, 131–135(1994).

A. V. Gitin, “Technique for checking the correctness of computer programs intended for calculating the energy of optical systems,” J. Opt. Technol. 67, 844–845 (2000).
[CrossRef]

A. V. Gitin, “Effect of the radiation pattern of a quasihomogeneous source on the point-spread function of an isoplanar optical system,” J. Opt. Technol. 60, 372–374 (1993).

A. V. Gitin, “Radiometry. A complex approach,” J. Opt. Technol. 65, 132–140 (1998).

A. V. Gitin, “Method of an energy calculation of the viewfinder of a reflex camera having a focusing screen,” J. Opt. Technol. 56, 440–442 (1989).

Opt. Spectrosc. (4)

A. V. Gitin and Ur. A. Fligontov, “Equation of radiation transfer through optical media in the Hamiltonian approximation,” Opt. Spectrosc. 66, 371–374 (1989).

I. N. Tarnakin, “Determination of illumination and light flux by an optical-system eikonal,” Opt. Spectrosc. 44, 463–465(1978).

A. V. Gitin, “The effective point source,” Opt. Spectrosc. 76 (1), 157–158 (1994).

A. V. Gitin, “Radiometry as a section of optical system theory,” Opt. Spectrosc. 63, 106–109 (1987).

Optik (1)

A. V. Gitin, “Radiometry of optical systems with quasihomogeneous sources: a linear systems approach,” Optik 122, 1713–1718 (2011).
[CrossRef]

Philos. Mag. Series 4 (1)

G. Kirchhoff, “On the relation between the radiating and the absorbing powers of different bodies for light and heat,” Philos. Mag. Series 4 20, 1–21 (1860).

Proc. IEEE (2)

D. Marcuse and H. M. Presby, “Index profile measurements of fibers and their evaluation,” Proc. IEEE 68, 1198–1203(1980).
[CrossRef]

G. K. Grau, “Comments on index profile measurement of fibers and their evaluation,” Proc. IEEE 69, 753–754 (1981).
[CrossRef]

Proc. SPIE (1)

T. Jannson, S. Kupiec, A. Kostrzewski, K. Spariosu, D. Minyzer, M. Rud, I. Tengara, and A. Vasiliev, “Phase-space formalism and ray-tracing modeling of photometric quantities in photometric engineering,” Proc. SPIE 3140, 36–47 (1997).
[CrossRef]

Quantum Electron. (1)

A. V. Gitin, “Optical systems for measuring the Wigner function of a laser beam by the method of phase-spatial tomography,” Quantum Electron. 37, 85–91 (2007).
[CrossRef]

Rep. Prog. Phys. (1)

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).
[CrossRef]

Other (28)

D. S. Volosov and M. V. Tsivkin, Theory and Design of Photo-Optical Systems (Moskow, 1960), in Russian.

W. H. A. Fincham, and M. H. Freeman, Optics(Butterworths, 1980).

W. B. Wetherell, “The calculation of image quality,” in Applied Optics and Optical Engineering, vol.  8, R.R.Shannon and J.C.Wyant, ed. (Academic, 1980).

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Figures (11)

Fig. 1
Fig. 1

An elementary beam of light rays restricted by two differential area elements d A 1 and d A 2 (a), or their corresponding projections d A 1 d A 1 cos θ 1 and d A 2 d A 2 cos θ 2 (b).

Fig. 2
Fig. 2

Optical radiation passing (a) from the luminous surface A 1 to the illuminated surface A 2 ; (b) from the extended uniform Lambertian source through two openings Σ 1 and Σ 2 (in the forward direction); (c) from the extended uniform Lambertian source through two openings Σ 2 and Σ 1 (in the opposite direction), ( Σ 1 = A 1 and Σ 2 = A 2 ).

Fig. 3
Fig. 3

Arrangements of stops (a) in object space of a perfect optical system with the Lambertian source-object; (b) in image space of a perfect optical system with the Lambertian source- object; (c) in optical homogeneous medium with the Lambertian source (Maxwell’s principal).

Fig. 4
Fig. 4

Teleradiometer (a), Eye as a teleradiometer (b).

Fig. 5
Fig. 5

The shift invariance (a) and the inverse square law (b).

Fig. 6
Fig. 6

A planar quasi-homogeneous source { M ( q ) , g ( q ˙ ) } is equivalent to a planar homogeneous source { M , g ( q ˙ ) } with the source transparency τ ( q ) .

Fig. 7
Fig. 7

Radiant flux from homogeneous source { M , g ( q ˙ ) } through a layer of optically homogeneous medium with thickness Z restricted by stops with transparencies τ ( q ) and τ o ( q z ) in the forward direction (a), in the opposite direction (b).

Fig. 8
Fig. 8

Geometrical interpretation of the correlation function of the transparencies τ ( q ) and τ o ( q z ) (a) or the stops with circular openings (b) and also an axial section of this function (c).

Fig. 9
Fig. 9

Radiant flux from homogeneous source { M , g ( q ˙ ) } through a layer of optically homogeneous medium with thickness Z and restricted by small openings in screens localized in the neighborhoods of points q * P and q Z * P Z .

Fig. 10
Fig. 10

Arrangement of stops in the object and the image spaces of a perfect optical system (a). Corresponding radiometric characteristics for calculating the radiation flux in the object (b) and the image (c) spaces.

Fig. 11
Fig. 11

(a) Real and effective stops in the image space of a perfect optical system for the conditions n = n , m o = 2 / 3 , m = 1 / 2 , (b) Effectively point stop in the image space of a perfect optical system for the conditions n = n , m o = 1 , m 0 .

Equations (55)

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d 2 F = L d A 1 cos θ 1 · d A 2 cos θ 2 d 2 = L d 2 d A 1 · d A 2 ,
F = L · G ,
G Σ 1 Σ 2 cos θ 1 cos θ 2 d 2 d Σ 1 d Σ 2
q = q m , q o = q o m o ,
m · m o · Z = n n Z ,
F = L · G .
G Σ Σ o cos 4 θ Z 2 d Σ d Σ o ,
F = L · G .
G Σ Σ o cos 4 θ Z 2 d Σ d Σ o .
F = F .
Δ A o = Δ A o / m o 2 ,
Δ A = Δ A / m 2 .
F L A · A o Z 2 ,
F L A / · A o / Z 2 .
F L A / · A o / ( m m o Z ) 2 .
F L ( n n ) 2 A · A o Z 2 .
L n 2 = L n 2 .
d 2 F = L ( q 1 ; sin θ 1 sin φ 1 , sin θ 1 sin φ ) d A 1 · d Ω 1 ,
L ( q ; sin θ sin φ , sin θ sin φ ) = M ( q ) · β ( sin θ sin φ , sin θ sin φ ) ,
0 π 2 0 2 π β ( sin θ sin φ , sin θ sin φ ) sin θ d ( sin θ ) d φ = 1 .
E ( q z ) = L { M ( q ) } .
M ( q ) = R 2 M ( ξ ) δ ( q ξ ) d ξ .
E ( q z ) = L { M ( q ) } = L { R 2 M ( ξ ) δ ( q ξ ) d ξ } = R 2 M ( ξ ) L { δ ( q ξ ) } d ξ .
L { tr ξ { δ ( q ) } } = tr ξ { L { δ ( q ) } }
E ( q Z ) = ( g Z * M ) ( q Z ) R 2 M ( q ) · g Z ( q Z q ) d q ,
g Z ( q Z q ) = 1 Z 2 g ( q Z q Z ) ,
g ( q ˙ ) = L ( q ˙ 1 + q ˙ 2 ) cos 4 θ ( q ˙ ) .
cos θ ( q ˙ ) = 1 1 + q ˙ 2 ,
q ˙ d q d z = q Z q Z = tan θ
R 2 g Z ( q Z q ) d q z = R 2 g ( q ˙ ) d q ˙ = 1 for any Z .
g Lambert ( q ˙ ) = 1 π cos 4 θ ( q ˙ ) .
M ( q ) = M · τ ( q ) .
F = E | τ o ,
F = M τ | g Z | τ o ,
τ | g z | τ o R 4 τ ( q ) g Z ( q Z q ) τ o ( q Z ) d q d q Z
τ | g Z | τ o = cov ( τ , τ o ) | g R 4 cov { τ , τ o } ( q ˙ ) g ( q ˙ ) d q ˙ .
cov { τ , τ o } ( Z q ˙ ) R 4 τ ( q ) τ o ( q + Z q ˙ ) d q
cov { τ , τ o } ( Z q ˙ ) g ( q ˙ ) .
τ ( q ) = circ ( q R ) and τ o ( q Z ) = circ ( q z r ) ,
circ ( q R ) { 1 , for | q | R 1 0 , for | q | R > 1 ,
F = M τ | g Z | τ o M · g Z ( q Z * q * ) · A · A o ,
F = M π G
F = M τ | g Z | τ o .
F = M τ | g Z | τ o / .
τ / ( q Z / ) = τ ( q Z / m ) .
τ o / ( q Z / ) = τ o ( q / m o ) .
{ M = M g Z ( q Z / q / ) = g n n Z ( q / m q Z / m o ) ,
τ m o / ( q Z / ) = 1 m o 2 τ / ( q Z / m o ) and τ m o / ( q / ) = 1 m 2 τ o / ( q / m ) ,
R 2 τ m o / ( q Z / ) d q Z / A = const for any m o ,
R 2 τ m o / ( q / ) d q / A o / = const for any m .
F = M τ m o / | g n n Z | τ m o / ,
lim m 0 { τ m o / ( q ) } = A o / · δ ( q ) .
F = lim m 0 M τ m o | g n n Z | τ m o / = M τ m o | g n n Z | lim m 0 τ m o / = M · A o / · τ m o | g n n f ,
F = M · A o · τ m o | g f .
E ( q / ) = M · A o / f 2 · g ( q / f ) ,

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