Abstract

The geometrical vector flux, a quantity related to measures of illumination, is defined; some of its properties are explained and used to develop new forms of nonimaging concentrators.

© 1979 Optical Society of America

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References

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  1. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974).
  2. W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).
  3. W. B. Joyce, Phys. Rev. D,  9, 3234–3256 (1974).
    [Crossref]
  4. R. Winston, J. Opt. Soc. Amer.,  60, 245–247 (1970).
    [Crossref]
  5. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).
  6. S. Chandrasekhar, Radiative Transfer (Dover, New York1960).
  7. E. M. Sparrow and R. D. Cess, Radiation Heat Transfer augmented edition (McGraw Hill, New York, 1978).
  8. R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer (McGraw Hill, New York, 1972).
  9. H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Tome III (Dover, New York, 1957).
  10. By introducing a suitable lens at the entry aperture the source can be placed at infinity, but then there are complications because the lens will have aberrations and because the input cone will vary in angle over the aperture.

1974 (1)

W. B. Joyce, Phys. Rev. D,  9, 3234–3256 (1974).
[Crossref]

1970 (1)

R. Winston, J. Opt. Soc. Amer.,  60, 245–247 (1970).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

Cess, R. D.

E. M. Sparrow and R. D. Cess, Radiation Heat Transfer augmented edition (McGraw Hill, New York, 1978).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York1960).

Howell, J. R.

R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer (McGraw Hill, New York, 1972).

Joyce, W. B.

W. B. Joyce, Phys. Rev. D,  9, 3234–3256 (1974).
[Crossref]

Poincaré, H.

H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Tome III (Dover, New York, 1957).

Siegel, R.

R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer (McGraw Hill, New York, 1972).

Sparrow, E. M.

E. M. Sparrow and R. D. Cess, Radiation Heat Transfer augmented edition (McGraw Hill, New York, 1978).

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974).

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

Winston, R.

R. Winston, J. Opt. Soc. Amer.,  60, 245–247 (1970).
[Crossref]

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

J. Opt. Soc. Amer. (1)

R. Winston, J. Opt. Soc. Amer.,  60, 245–247 (1970).
[Crossref]

Phys. Rev. D (1)

W. B. Joyce, Phys. Rev. D,  9, 3234–3256 (1974).
[Crossref]

Other (8)

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974).

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

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

S. Chandrasekhar, Radiative Transfer (Dover, New York1960).

E. M. Sparrow and R. D. Cess, Radiation Heat Transfer augmented edition (McGraw Hill, New York, 1978).

R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer (McGraw Hill, New York, 1972).

H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Tome III (Dover, New York, 1957).

By introducing a suitable lens at the entry aperture the source can be placed at infinity, but then there are complications because the lens will have aberrations and because the input cone will vary in angle over the aperture.

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Figures (9)

FIG. 1
FIG. 1

Geometry for calculating the vector flux from a strip in two dimensions.

FIG. 2
FIG. 2

Lines of flow and loci of constant |J| for the two-dimensional strip: the width is taken as two units. The axes of circles are labeled with the values of |J|. The two sets of lines are not orthogonal because J is not derived from a scalar potential.

FIG. 3
FIG. 3

Lines of flow and loci of constant |J| for a semi-infinite strip.

FIG. 4
FIG. 4

Lines of flow and loci of constant |J| for a luminous infinite wedge.

FIG. 5
FIG. 5

Geometry for calculating J for a luminous disk.

FIG. 6
FIG. 6

Lines of flow and loci of constant |J| for a luminous disk.

FIG. 7
FIG. 7

Placing a mirror in a vector flux field.

FIG. 8
FIG. 8

The hyperboloid of revolution as a concentrator.

FIG. 9
FIG. 9

Geometry of hyperboloid concentrator for skew rays.

Equations (12)

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d U = d p x d p y dx dy
U = ( ) d p x d p y dx dy
d U = n 2 cos θ z d Ω dx dy
d U = cos θ z d J dx dy ,
J z = d p x d p y
J x = d p y d p z
J y = d p z d p x .
| J | = 2 sin θ
y = ± a [ 1 + ( z 2 / c 2 a 2 ) ] 1 / 2
r = 2 c | J | ( 4 | J | 2 ) 1 / 2 .
J y = z [ ( y c ) 2 + z 2 ) 1 / 2 z [ ( y + c ) 2 + z 2 ] 1 / 2 J z = y + c [ ( y + c ) 2 + z 2 ] 1 / 2 y c [ ( y c ) 2 + z 2 ] 1 / 2 | J | 2 = 2 2 ( z 2 + y 2 c 2 ) [ ( z 2 + y 2 + c 2 ) 2 4 c 2 y 2 ] 1 / 2
J y = π 2 z y ( c 2 + y 2 + z 2 [ ( c 2 + y 2 + z 2 ) 4 c 2 y 2 ] 1 / 2 1 ) J y = π 2 ( c 2 y 2 z 2 [ ( c 2 + y 2 + z 2 ) 4 c 2 y 2 ] 1 / 2 + 1 )