Abstract

A monochromatic analysis of the RX nonimaging concentrators as imaging optical systems is presented (R stands for refractive, X for reflective). All of them have rotational symmetry and an image-side numerical aperture of 1.46 with the use of an index of refraction n=1.5, which means a half-rim angle of illumination of 77 deg. This is equivalent to 95% of the theoretical limit of concentration. For an object-side focal length of f=17.1 mm (i.e., an entry aperture diameter of 50 mm) and a wavelength λ=950 nm, the RX has an equivalent passband above 32 mm-1 in a field of view of ±3.2 deg and above 19 mm-1 in a field of view of ±4.8 deg. This feature of RX concentrators allows one to use the same RX with receivers/emitters very different from the one of the design presented (in size and contour shape) with no loss of nonimaging quality. Moreover, the combination of simplicity, compactness, imaging capability, and high concentration makes the RX an exceptionally good optical device for high-sensitivity focal plane array applications.

© 1997 Optical Society of America

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References

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  1. W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, New York, 1989).
  2. J. C. Miñano, P. Benı́tez, J. C. González, “RX: a nonimaging concentrator,” Appl. Opt. 34, 2226–2235 (1995).
    [CrossRef] [PubMed]
  3. W. T. Welford, R. Winston, “On the problem of ideal flux concentrators: addendum,” J. Opt. Soc. Am. 69, 367 (1979).
    [CrossRef]
  4. W. T. Welford, R. Winston, “On the problem of ideal flux concentrators,” J. Opt. Soc. Am. 68, 531–534 (1978).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).
  6. C. S. Williams, O. A. Becklund, Introduction to the Optical Transfer Function (Wiley, New York, 1989).
  7. J. J. Stamnes, Waves in Focal Regions (Hilger, Boston, 1986), Chap. 15.
  8. R. Barakat, D. Lev, “Transfer functions and total illuminance of high numerical aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 53, 324–332 (1963).
    [CrossRef]
  9. R. K. Luneburg, Mathematical Theory of Optics (U. of California, Berkeley, 1964), pp. 186–187.
  10. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), Chap. 11.
  11. G. Schulz, “Higher order aplanatism,” Opt. Commun. 41, 315–319 (1982).
    [CrossRef]
  12. G. Schulz, “Aberration-free imaging of large fields with thin pencils,” Opt. Acta 32, 1361–1371 (1985).
    [CrossRef]
  13. H. Hottel, “Radiant-heat transmission,” in Heat Transmission, W. H. McAdams, ed. (McGraw-Hill, New York, 1954).

1995 (1)

1985 (1)

G. Schulz, “Aberration-free imaging of large fields with thin pencils,” Opt. Acta 32, 1361–1371 (1985).
[CrossRef]

1982 (1)

G. Schulz, “Higher order aplanatism,” Opt. Commun. 41, 315–319 (1982).
[CrossRef]

1979 (1)

1978 (1)

1963 (1)

R. Barakat, D. Lev, “Transfer functions and total illuminance of high numerical aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 53, 324–332 (1963).
[CrossRef]

Barakat, R.

R. Barakat, D. Lev, “Transfer functions and total illuminance of high numerical aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 53, 324–332 (1963).
[CrossRef]

Becklund, O. A.

C. S. Williams, O. A. Becklund, Introduction to the Optical Transfer Function (Wiley, New York, 1989).

Beni´tez, P.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

González, J. C.

Hottel, H.

H. Hottel, “Radiant-heat transmission,” in Heat Transmission, W. H. McAdams, ed. (McGraw-Hill, New York, 1954).

Lev, D.

R. Barakat, D. Lev, “Transfer functions and total illuminance of high numerical aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 53, 324–332 (1963).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. of California, Berkeley, 1964), pp. 186–187.

Miñano, J. C.

Schulz, G.

G. Schulz, “Aberration-free imaging of large fields with thin pencils,” Opt. Acta 32, 1361–1371 (1985).
[CrossRef]

G. Schulz, “Higher order aplanatism,” Opt. Commun. 41, 315–319 (1982).
[CrossRef]

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), Chap. 11.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Boston, 1986), Chap. 15.

Welford, W. T.

Williams, C. S.

C. S. Williams, O. A. Becklund, Introduction to the Optical Transfer Function (Wiley, New York, 1989).

Winston, R.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

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

R. Barakat, D. Lev, “Transfer functions and total illuminance of high numerical aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 53, 324–332 (1963).
[CrossRef]

Opt. Acta (1)

G. Schulz, “Aberration-free imaging of large fields with thin pencils,” Opt. Acta 32, 1361–1371 (1985).
[CrossRef]

Opt. Commun. (1)

G. Schulz, “Higher order aplanatism,” Opt. Commun. 41, 315–319 (1982).
[CrossRef]

Other (7)

H. Hottel, “Radiant-heat transmission,” in Heat Transmission, W. H. McAdams, ed. (McGraw-Hill, New York, 1954).

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, New York, 1989).

R. K. Luneburg, Mathematical Theory of Optics (U. of California, Berkeley, 1964), pp. 186–187.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), Chap. 11.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

C. S. Williams, O. A. Becklund, Introduction to the Optical Transfer Function (Wiley, New York, 1989).

J. J. Stamnes, Waves in Focal Regions (Hilger, Boston, 1986), Chap. 15.

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Figures (15)

Fig. 1
Fig. 1

Cross section of an RX concentrator designed with a refractive index 1.5, an acceptance angle α=±3 deg, and 95% of the maximum concentration. The half-rim angle of illumination of the receiver is β=77 deg, and the image-side NA=1.46.

Fig. 2
Fig. 2

Description of the method of design of the RX concentrator.

Fig. 3
Fig. 3

Definition of the coordinate systems in the object and image spaces. The entry aperture plane coincides with the object-side pupil plane.

Fig. 4
Fig. 4

The focal length in the RX concentrators is defined as r/sin θ, which is more constant than r/tan θ when the incidence angle θ varies.

Fig. 5
Fig. 5

Tangential spot diameter as a function of the incidence angle for the aplanatic RX concentrator and for the RX’s designed with acceptance angle α of 1.5, 3, and 4.5 deg.

Fig. 6
Fig. 6

Same as Fig. 5, but for the sagittal spot diameter.

Fig. 7
Fig. 7

Modulation transfer functions (MTF’s) for normal incidence and for an incidence angle of 3 deg of the f/4.5 plano–convex spherical lens, the aplanatic RX, and the RX’s designed with α of 1.5, 3, and 4.5 deg.

Fig. 8
Fig. 8

Tangential equivalent passband fc,T as a function of the incidence angle for the aplanatic RX concentrator and the RX’s designed with acceptance angle α of 1.5, 3, and 4.5 deg.

Fig. 9
Fig. 9

Same as Fig. 8, but for the sagittal equivalent passband fc,S.

Fig. 10
Fig. 10

Field of view (half-angle) as a function of the specified minimum equivalent passband fmin of the optimum RX concentrator designed with input parameters dA=21.4 mm and dC =17.6 mm.

Fig. 11
Fig. 11

The performance of any concentrator for imaging detection is represented by a point in the f¯cNA2 plane. f¯c is the global equivalent passband of the concentrator in the field of view of δ=±3.2 deg and indicates the imaging quality. NA2 indicates the concentrator luminosity.

Fig. 12
Fig. 12

Curve A is the angle transmission curve of the RX concentrator of Fig. 1. Each of the other transmission curves corresponds to the same concentrator with a different receiver diameter d. If the entry aperture diameter is 50 mm, then d(A)=1.79 mm, d(B)=1.33 mm, d(C)=890 μm, and d(D)=445 μm.

Fig. 13
Fig. 13

Angle transmission curves of the concentrator of Fig. 1 with receiver diameters d(A)=1.79 mm, d(E)=3.95 mm, d(F)=7.9 mm, and d(G)=11.86 mm (entry aperture diameter of 50 mm).

Fig. 14
Fig. 14

Total transmission for the RX concentrator of Fig. 1 for different receiver diameters as a function of the resulting semiacceptance angle. The upper curve considers a transparent receiver, while the other takes into account the shadow losses introduced by it. For both cases the optical losses have been neglected.

Fig. 15
Fig. 15

Flow lines transmitted through the RX concentrator of Fig. 2. The flow lines are tangent to the geometrical vector flux at every point.

Tables (1)

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Table 1 Comparison of Data on the RX and the Plano–Convex Spherical Lens

Equations (23)

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W(r, ρ, ψ)=W0(ρ, ψ)+W1(ρ, ψ)r+W2(ρ, ψ)r2+ .
W(r=rS, ρ, ψ=0)0,
W(r=rS, ρ, ψ=π)0,
fr/sin θrI/NA,
fc,T=0 MTF2(fT, 0)dfT,
fc,S=0 MTF2(0, fS)dfS.
fc(θ)fmin,
rI sin αnrR sin β,
W(r, ρ, ψ; rS)=W0(ρ, ψ; rS)+W1(ρ, ψ; rS)r+W2(ρ, ψ; rS)r2+ .
W0(ρ, ψ; rS)=a(ρ; rS),
W1(ρ, ψ; rS)=b(ρ; rS)cos ψ;
W(r, ρ, ψ; rS)=a(ρ; rS)+b(ρ; rS)(cos ψ)r+c(r, ρ, ψ; rS)r2.
a(ρ; rS)+b(ρ; rS)rS+c(rS, ρ, 0; rS)rS2=0,
a(ρ; rS)-b(ρ; rS)rS+c(rS, ρ, π; rS)rS2=0.
a(ρ; rS)=rS2 c(rS, ρ, 0; rS)+c(rS, ρ, π; rS)2,
b(ρ; rS)=rS c(rS, ρ, 0; rS)-c(rS, ρ, π; rS)2.
2(SI-SI)=2nRR,
2(SM-SM)=2n(RN-RN).
SM-SM=[(rS+h)2+ρM2]1/2-[(rS-h)2+ρM2]1/2,
SM-SM=2rS sin[γ(h, ρM)]+rS2d(rS, h, ρM),
RN-RN=2rR sin[γ(h, ρN)]+rR2e(rR, h, ρN),
2rS sin[γ(h, ρM)]+rS2d(rS, h, ρM)
=2rR sin[γ(h, ρN)]+rR2e(rR, h, ρN).

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