Abstract

The problem of light collection is examined from first principles within the framework of geometrical optics. From the outset, we distinguish between light collection and the usual theory of image formation. From phase-space considerations, we derive the sine inequality, a generalization of the Abbe sine law appropriate to nonimaging systems. We construct two- and three-dimensional nonimaging systems that reduce the f number to the least allowed by the sine inequality. Such systems give substantially improved light collection as compared with conventional systems.

© 1970 Optical Society of America

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References

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  1. Using time as the parameter of the light rays leads to the additional difficulty that the lagrangian vanishes identically. This is discussed by L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison–Wesley Publishing Co., Inc., Cambridge, Mass., 1951), p. 139.
  2. The stationary description of light rays that we adopt is treated by R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).
  3. This result is derived by Luneburg,2 also by J. Marshall, Phys. Rev. 86, 685 (1952).
    [CrossRef]
  4. This follows from the connection between the f number and the Abbe sine law discussed in many optics texts. See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, 1965), 3rd ed.
  5. H. Hinterberger and R. Winston, Rev. Sci. Instr. 37, 1094 (1966).
    [CrossRef]
  6. H. Hinterberger and R. Winston, Rev. Sci. Instr. 39, 1217 (1968).
    [CrossRef]
  7. D. E. Williamson, J. Opt. Soc. Am. 42, 712 (1952).
    [CrossRef]
  8. W. Witte, Infrared Phys. 5, 179 (1965).
    [CrossRef]

1968 (1)

H. Hinterberger and R. Winston, Rev. Sci. Instr. 39, 1217 (1968).
[CrossRef]

1966 (1)

H. Hinterberger and R. Winston, Rev. Sci. Instr. 37, 1094 (1966).
[CrossRef]

1965 (1)

W. Witte, Infrared Phys. 5, 179 (1965).
[CrossRef]

1952 (2)

This result is derived by Luneburg,2 also by J. Marshall, Phys. Rev. 86, 685 (1952).
[CrossRef]

D. E. Williamson, J. Opt. Soc. Am. 42, 712 (1952).
[CrossRef]

Born, M.

This follows from the connection between the f number and the Abbe sine law discussed in many optics texts. See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, 1965), 3rd ed.

Hinterberger, H.

H. Hinterberger and R. Winston, Rev. Sci. Instr. 39, 1217 (1968).
[CrossRef]

H. Hinterberger and R. Winston, Rev. Sci. Instr. 37, 1094 (1966).
[CrossRef]

Landau, L.

Using time as the parameter of the light rays leads to the additional difficulty that the lagrangian vanishes identically. This is discussed by L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison–Wesley Publishing Co., Inc., Cambridge, Mass., 1951), p. 139.

Lifshitz, E.

Using time as the parameter of the light rays leads to the additional difficulty that the lagrangian vanishes identically. This is discussed by L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison–Wesley Publishing Co., Inc., Cambridge, Mass., 1951), p. 139.

Luneburg, R. K.

The stationary description of light rays that we adopt is treated by R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

Marshall, J.

This result is derived by Luneburg,2 also by J. Marshall, Phys. Rev. 86, 685 (1952).
[CrossRef]

Williamson, D. E.

Winston, R.

H. Hinterberger and R. Winston, Rev. Sci. Instr. 39, 1217 (1968).
[CrossRef]

H. Hinterberger and R. Winston, Rev. Sci. Instr. 37, 1094 (1966).
[CrossRef]

Witte, W.

W. Witte, Infrared Phys. 5, 179 (1965).
[CrossRef]

Wolf, E.

This follows from the connection between the f number and the Abbe sine law discussed in many optics texts. See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, 1965), 3rd ed.

Infrared Phys. (1)

W. Witte, Infrared Phys. 5, 179 (1965).
[CrossRef]

J. Opt. Soc. Am. (1)

Phys. Rev. (1)

This result is derived by Luneburg,2 also by J. Marshall, Phys. Rev. 86, 685 (1952).
[CrossRef]

Rev. Sci. Instr. (2)

H. Hinterberger and R. Winston, Rev. Sci. Instr. 37, 1094 (1966).
[CrossRef]

H. Hinterberger and R. Winston, Rev. Sci. Instr. 39, 1217 (1968).
[CrossRef]

Other (3)

This follows from the connection between the f number and the Abbe sine law discussed in many optics texts. See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, 1965), 3rd ed.

Using time as the parameter of the light rays leads to the additional difficulty that the lagrangian vanishes identically. This is discussed by L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison–Wesley Publishing Co., Inc., Cambridge, Mass., 1951), p. 139.

The stationary description of light rays that we adopt is treated by R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

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Figures (4)

Fig. 1
Fig. 1

(a) The area in phase space occupied by rays at the entrance aperture of a two-dimensional light collector. In this example, θmax = 16°, n = 1. (b) The area in phase space occupied by rays at the exit aperture of the same light collector. We have assumed d2 = d1 sinθmax, the limiting case of Eq. (7).

Fig. 2
Fig. 2

Construction of an ideal light collector for the case of constant index of refraction. In this example, θmax = 16°.

Fig. 3
Fig. 3

Construction of an ideal light collector for the case n2 > n1. A typical light ray traversing the entire system is shown. In this example, n2/n1 = 1.5 and θmax = 35° at the entrance aperture.

Fig. 4
Fig. 4

The angular acceptance as a function of angle of incidence at the entrance aperture for an ideal three-dimensional light collector. Note that the angular acceptance cuts off over a region Δθ approximately 1° centered about θmax. In this example, θmax = 16°.

Equations (14)

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δ z 1 z 2 n ( 1 + x ˙ 2 + y ˙ 2 ) 1 2 d z = 0 ,
p x ( / x ˙ ) n ( 1 + x ˙ 2 + y ˙ 2 ) 1 2 = n x ˙ ( 1 + x ˙ 2 + y ˙ 2 ) - 1 2 p y ( / y ˙ ) n ( 1 + x ˙ 2 + y ˙ 2 ) 1 2 = n y ˙ ( 1 + x ˙ 2 + y ˙ 2 ) - 1 2 .
z 1 d x d p x = z 2 d x d p x ,
z 1 d x d y d p x d p y = z 2 d x d y d p x d p y .
z 1 d x d p x = 2 n 1 d 1 sin θ max ,
z 2 d x d p x 2 n 2 d 2 ,
n 2 d 2 n 1 d 1 sin θ max ,
z 1 d x d y d p x d p y = Π n 1 2 a 1 sin 2 θ max ,
z 2 d x d y d p x d p y Π n 2 2 a 2 ,
n 2 2 a 2 n 1 2 a 1 sin 2 θ max ,
( x p y - y p x ) z 1 = n 1 R 1 sin θ max ,
( x p y - y p x ) z 2 n 2 R 2 ,
n 2 R 2 n 1 R 1 sin θ max ,
l = z 2 - z 1 = ( 1 / 2 ) ( d 1 + d 2 ) cot θ max .