Abstract

The conservation of etendue for general 2-D bundles of rays (not necessarily coplanar) is examined (a 2-D bundle of rays is that whose rays are distinguishable by giving each one two parameters). This is one of the integral invariants of Poincaré and it is directly related to the Lagrange invariant. The application of this theorem to selected 2-D bundles of rays crossing an arbitrary cylindrical concentrator gives us a relationship between the maximum geometrical concentration of a cylindrical concentrator and the angular field of view which is more restrictive than the general one (i.e., the relationship is valid for an arbitrary concentrator) when the collector is surrounded by a refractive medium.

© 1984 Optical Society of America

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References

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  1. R. Winston, “Light Collection Within the Framework of Geometrical Optics,” J. Opt. Soc. Am. 60, 245 (1970).
    [CrossRef]
  2. W. T. Welford, R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).
  3. H. Poincaré, Les méthodes nouvelles de la mécanique céleste (Dover, New York, 1957), Vol. 3.
  4. R. Weinstock, Calculus of Variations (McGraw-Hill, New York, 1952).
  5. L. D. Landau, E. M. Lifshitz, Spanish translation: Mecánica (Reverté, Barcelona, 1965).
  6. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974).
  7. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964).
  8. A generating function of this transformation can be G(p,q,ρ,θ) = −(pρ cosθ + qρ sinθ).
  9. The Hamiltonian characteristic function can also be defined for the cases where the terminal points of the rays lie on a curved surface. See H. A. Buchdall, An Introduction to Hamiltonian Optics (Cambridge U.P., London, 1970); G. W. Forbes, “New Class of Characteristic Functions in Hamiltonian Optics,” J. Opt. Soc. Am. 72, 1698 (1982).
    [CrossRef]
  10. J. C. Minano, A. Luque, “Limit of concentration under Nonhomogeneous Extended Light Sources,” Appl. Opt. 22, 2751 (1983).
    [CrossRef] [PubMed]
  11. For the sake of simplicity we have assumed in this work that the collector is monofacial (only collects rays coming from one face of the collector surface). If it is assumed that the collector is bifacial, as is done in Ref. 10, the right-hand side of Eq. (10) should be multiplied by 2 and the definitions of Cgm(â) and g suffer the corresponding changes. Note that, with the treatment used here, a concentrator with a monofacial collector can have g = 1, while in the treatment of Ref. 10g cannot, in this case, be >1/2. From an optical point of view, a bifacial collector can be considered as having two monofacial faces.
  12. See, for example, M. Herzberger, “Mathematics and Geometrical Optics,” Supplementary Note III in Ref. 7.
  13. M. Herzberger, Modern Geometrical Optics (Wiley Interscience, New York, 1958).
  14. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).
  15. R. Winston, W. T. Welford, “Geometrical Vector Flux and Some New Nonimaging Concentrators,” J. Opt. Soc. Am. 69, 532 (1979).
    [CrossRef]
  16. J. C. Minano, A. Luque, “Limit of Concentration for Cylindrical Concentrators Under Extended Light Sources,” Appl. Opt. 22, 2437 (1983).
    [CrossRef] [PubMed]

1983 (2)

1979 (1)

1970 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Buchdall, H. A.

The Hamiltonian characteristic function can also be defined for the cases where the terminal points of the rays lie on a curved surface. See H. A. Buchdall, An Introduction to Hamiltonian Optics (Cambridge U.P., London, 1970); G. W. Forbes, “New Class of Characteristic Functions in Hamiltonian Optics,” J. Opt. Soc. Am. 72, 1698 (1982).
[CrossRef]

Herzberger, M.

See, for example, M. Herzberger, “Mathematics and Geometrical Optics,” Supplementary Note III in Ref. 7.

M. Herzberger, Modern Geometrical Optics (Wiley Interscience, New York, 1958).

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Spanish translation: Mecánica (Reverté, Barcelona, 1965).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Spanish translation: Mecánica (Reverté, Barcelona, 1965).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964).

Luque, A.

Minano, J. C.

Poincaré, H.

H. Poincaré, Les méthodes nouvelles de la mécanique céleste (Dover, New York, 1957), Vol. 3.

Weinstock, R.

R. Weinstock, Calculus of Variations (McGraw-Hill, New York, 1952).

Welford, W. T.

R. Winston, W. T. Welford, “Geometrical Vector Flux and Some New Nonimaging Concentrators,” J. Opt. Soc. Am. 69, 532 (1979).
[CrossRef]

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

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

Winston, R.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

Other (12)

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

H. Poincaré, Les méthodes nouvelles de la mécanique céleste (Dover, New York, 1957), Vol. 3.

R. Weinstock, Calculus of Variations (McGraw-Hill, New York, 1952).

L. D. Landau, E. M. Lifshitz, Spanish translation: Mecánica (Reverté, Barcelona, 1965).

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

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964).

A generating function of this transformation can be G(p,q,ρ,θ) = −(pρ cosθ + qρ sinθ).

The Hamiltonian characteristic function can also be defined for the cases where the terminal points of the rays lie on a curved surface. See H. A. Buchdall, An Introduction to Hamiltonian Optics (Cambridge U.P., London, 1970); G. W. Forbes, “New Class of Characteristic Functions in Hamiltonian Optics,” J. Opt. Soc. Am. 72, 1698 (1982).
[CrossRef]

For the sake of simplicity we have assumed in this work that the collector is monofacial (only collects rays coming from one face of the collector surface). If it is assumed that the collector is bifacial, as is done in Ref. 10, the right-hand side of Eq. (10) should be multiplied by 2 and the definitions of Cgm(â) and g suffer the corresponding changes. Note that, with the treatment used here, a concentrator with a monofacial collector can have g = 1, while in the treatment of Ref. 10g cannot, in this case, be >1/2. From an optical point of view, a bifacial collector can be considered as having two monofacial faces.

See, for example, M. Herzberger, “Mathematics and Geometrical Optics,” Supplementary Note III in Ref. 7.

M. Herzberger, Modern Geometrical Optics (Wiley Interscience, New York, 1958).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

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Figures (4)

Fig. 1
Fig. 1

Coordinate systems used for the rays at the entry aperture x,y,p,q and the exit aperture x′,y′,p′,q′ of an arbitrary optical system. The two rays drawn have coordinates x = 0, y = 0 and x′ = 0, y′ = 0.

Fig. 2
Fig. 2

Coordinate system used for the rays at the entry aperture ∑e and at the exit aperture ∑c of an arbitrary cylindrical concentrator.

Fig. 3
Fig. 3

Upper bound of degree of isotropy for cylindrical concentrators whose collector is surrounded by a medium of index of refraction 1.5.

Fig. 4
Fig. 4

Upper bound of geometrical concentration vs average acceptance area a for cylindrical concentrators whose collector is surrounded by a medium of index of refraction nc = 1.5. The dashed line shows the general upper bound for arbitrary concentrators of nc = 1.5.

Equations (29)

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δ z 1 z 2 n ( x , y , z ) ( 1 + x ˙ 2 + y ˙ 2 ) d z = 0 ,
p = L x ˙ = n x ˙ ( 1 + x ˙ 2 + y ˙ 2 ) 1 / 2 ,
q = L y ˙ = n y ˙ ( 1 + x ˙ 2 + y ˙ 2 ) 1 / 2 .
d x d y d p d q = d x d y d p d q .
j = p cos θ + q sin θ ,
h = ρ ( q cos θ - p sin θ ) .
d x d y d p d q = d ρ d θ d j d h = d x d y d p d q = d ρ d θ d j d h ,
E = D l X l Y d X d Y d P d Q = D l X l Y d X d Y d P d Q ,
E = A e a ^
E A c π n c 2 ,
C g π n c 2 a ^ = C g m ( α ^ ) ,
C g n c 2 sin 2 ϕ a .
d x d p + d y d q = d x d p + d y d q .
x = x ( u , v ) , x = x ( u , v ) , y = y ( u , v ) , y = y ( u , v ) , p = p ( u , v ) , p = p ( u , v ) , q = q ( u , v ) , q = q ( u , v ) .
D ( x , p ) D ( u , v ) + D ( y , q ) D ( u , v ) = D ( x , p ) D ( u , v ) + D ( y , q ) D ( u , v ) .
( x u , y u ) · ( p v , q v ) - ( p u , q u ) · ( x v , y v ) = ( x u , y u ) · ( p v , q v ) - ( p u , q u ) · ( x v , y v ) ,
d ρ d j + d θ d h = d ρ d j + d θ d h .
d Y d Q = d Y d Q .
E 2 - D ( X , P ) = D S ( X , P ) d Y d Q = D 2 - D ( X , P ) d Y d Q ,
X = X + f ( Y , P , Q ) .
E 2 - D ( X , P ) 2 A e ( 1 - P 2 ) 1 / 2
E 2 - D ( X , P ) 2 A c ( n c 2 - P 2 ) 1 / 2
E = X = 0 X = 1 P = - 1 P = 1 E 2 - D ( P ) d X d P .
g g M ( C g , n c ) = 2 π n c 2 { C g sin - 1 ( n c 2 - 1 C g 2 - 1 ) 1 / 2 + n c 2 sin - 1 [ 1 n c ( C g 2 - n c 2 C g 2 - 1 ) 1 / 2 ] } ,
C g = g C g m ( a ^ ) ,
C o = g γ C o M ( I ) .
C g g M ( C g , n c ) C g m ( a ^ ) .
E 2 - D ( h ) ρ = h ρ = R j = - [ 1 - ( h / ρ ) 2 ] 1 / 2 j = [ 1 - ( h / ρ ) 2 ] 1 / 2 d ρ d j ,
E 2 - D ( h ) ρ = h / n c ρ = R j = - [ n c 2 - ( h / ρ ) 2 ] 1 / 2 j = [ n c 2 - ( h / ρ ) 2 ] 1 / 2 d ρ d j ;

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