Abstract

At observation planes away from the image plane, an imaging lens is a nonimaging optic. We examine the variation of axial irradiance with distance in image space and highlight the following little-known observation for discussion: On a per-unit-area basis, the position of the highest concentration in image space is generally not at the focal plane. This characteristic is contrary to common experience, and it offers an additional degree of freedom for the design of detection systems. Additionally, it would also apply to lenses with negative refractive index. The position of peak concentration and its irradiance is dependent upon the location and irradiance of the image. As such, this discussion also includes a close examination of expressions for image irradiance and explains how they are related to irradiance calculations beyond the image plane. This study is restricted to rotationally symmetric refractive imaging systems with incoherent extended Lambertian sources.

© 2016 Optical Society of America

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Corrections

Ronian Siew, "Axial nonimaging characteristics of imaging lenses: erratum," J. Opt. Soc. Am. A 33, 1777-1777 (2016)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-33-9-1777

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References

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  1. J. Chavez, Introduction to Nonimaging Optics (CRC Press, 2016).
  2. R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Elsevier, 2005).
  3. R. J. Koshel, Illumination Engineering (Wiley, 2013).
  4. R. Winston and H. Ries, “Nonimaging reflectors as functionals of the desired irradiance,” J. Opt. Soc. Am. A 10, 1902–1908 (1993).
    [Crossref]
  5. E. O. Hulburt, “Optics of searchlight illumination,” J. Opt. Soc. Am. 36, 483–491 (1946).
    [Crossref]
  6. R. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983), pp. 86–89.
  7. A. Arecchi, T. Messadi, and R. J. Koshel, Field Guide to Illumination (SPIE, 2007), p. 63.
  8. The computer ray tracing simulations performed in this paper use Zemax Version 13, Release 2. Zemax is available from www.zemax.com .
  9. R. Siew, “Corrections to classical radiometry and the brightness of stars,” Eur. J. Phys. 29, 1105–1114 (2008).
    [Crossref]
  10. V. G. Veselago, “The electrodynamics of substances with simultaneous negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [Crossref]
  11. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [Crossref]
  12. R. H. Shepard, “Metamaterial lens design,” Ph.D. dissertation (The University of Arizona, 2009).
  13. M. Memarian and G. V. Eleftheriades, “Light concentration using hetero-junctions of anisotropic low permittivity metamaterials,” Light 2, e114 (2013).
  14. D. Schurig, “An aberration-free lens with zero F-Number,” in Imaging Systems (Optical Society of America, 2010), paper IMC1.
  15. Note that the equality dA′/dA = tan2 θo/tan2 θi is consistent with the so-called “Lagrange invariant” because this invariant is a paraxial quantity (e.g., see [16,17]).
  16. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986), p. 25.
  17. M. J. Kidger, Fundamental Optical Design (SPIE, 2002), pp. 25–27.
  18. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000), p. 65.
  19. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1980), p. 189.
  20. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000), pp. 225–227.
  21. H. F. Gilmore, “The determination of image irradiance in optical systems,” Appl. Opt. 5, 1812–1817 (1966).
    [Crossref]

2013 (1)

M. Memarian and G. V. Eleftheriades, “Light concentration using hetero-junctions of anisotropic low permittivity metamaterials,” Light 2, e114 (2013).

2008 (1)

R. Siew, “Corrections to classical radiometry and the brightness of stars,” Eur. J. Phys. 29, 1105–1114 (2008).
[Crossref]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[Crossref]

1993 (1)

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneous negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

1966 (1)

1946 (1)

Arecchi, A.

A. Arecchi, T. Messadi, and R. J. Koshel, Field Guide to Illumination (SPIE, 2007), p. 63.

Benitez, P.

R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Elsevier, 2005).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1980), p. 189.

Boyd, R.

R. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983), pp. 86–89.

Chavez, J.

J. Chavez, Introduction to Nonimaging Optics (CRC Press, 2016).

Eleftheriades, G. V.

M. Memarian and G. V. Eleftheriades, “Light concentration using hetero-junctions of anisotropic low permittivity metamaterials,” Light 2, e114 (2013).

Gilmore, H. F.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000), p. 65.

Hulburt, E. O.

Kidger, M. J.

M. J. Kidger, Fundamental Optical Design (SPIE, 2002), pp. 25–27.

Koshel, R. J.

R. J. Koshel, Illumination Engineering (Wiley, 2013).

A. Arecchi, T. Messadi, and R. J. Koshel, Field Guide to Illumination (SPIE, 2007), p. 63.

Memarian, M.

M. Memarian and G. V. Eleftheriades, “Light concentration using hetero-junctions of anisotropic low permittivity metamaterials,” Light 2, e114 (2013).

Messadi, T.

A. Arecchi, T. Messadi, and R. J. Koshel, Field Guide to Illumination (SPIE, 2007), p. 63.

Minano, J. C.

R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Elsevier, 2005).

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[Crossref]

Ries, H.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000), p. 65.

Schurig, D.

D. Schurig, “An aberration-free lens with zero F-Number,” in Imaging Systems (Optical Society of America, 2010), paper IMC1.

Shepard, R. H.

R. H. Shepard, “Metamaterial lens design,” Ph.D. dissertation (The University of Arizona, 2009).

Siew, R.

R. Siew, “Corrections to classical radiometry and the brightness of stars,” Eur. J. Phys. 29, 1105–1114 (2008).
[Crossref]

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000), pp. 225–227.

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneous negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986), p. 25.

Winston, R.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1980), p. 189.

Appl. Opt. (1)

Eur. J. Phys. (1)

R. Siew, “Corrections to classical radiometry and the brightness of stars,” Eur. J. Phys. 29, 1105–1114 (2008).
[Crossref]

J. Opt. Soc. Am. (1)

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

Light (1)

M. Memarian and G. V. Eleftheriades, “Light concentration using hetero-junctions of anisotropic low permittivity metamaterials,” Light 2, e114 (2013).

Phys. Rev. Lett. (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[Crossref]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneous negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

Other (14)

J. Chavez, Introduction to Nonimaging Optics (CRC Press, 2016).

R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Elsevier, 2005).

R. J. Koshel, Illumination Engineering (Wiley, 2013).

R. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983), pp. 86–89.

A. Arecchi, T. Messadi, and R. J. Koshel, Field Guide to Illumination (SPIE, 2007), p. 63.

The computer ray tracing simulations performed in this paper use Zemax Version 13, Release 2. Zemax is available from www.zemax.com .

D. Schurig, “An aberration-free lens with zero F-Number,” in Imaging Systems (Optical Society of America, 2010), paper IMC1.

Note that the equality dA′/dA = tan2 θo/tan2 θi is consistent with the so-called “Lagrange invariant” because this invariant is a paraxial quantity (e.g., see [16,17]).

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986), p. 25.

M. J. Kidger, Fundamental Optical Design (SPIE, 2002), pp. 25–27.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2000), p. 65.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1980), p. 189.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000), pp. 225–227.

R. H. Shepard, “Metamaterial lens design,” Ph.D. dissertation (The University of Arizona, 2009).

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Figures (13)

Fig. 1.
Fig. 1. Bi-convex lens imaging an incoherent disk Lambertian source emitting at 550 nm.
Fig. 2.
Fig. 2. Zemax nonsequential ray trace setup using a virtual detector placed flat along the optic axis. The aperture stop is present but not displayed.
Fig. 3.
Fig. 3. Plot of axial irradiance (relative units) versus distance to the image plane for the column of pixels along the white dotted line on the flat virtual detector of Fig. 2.
Fig. 4.
Fig. 4. Spatial irradiance distributions (absolute units) on planes A, B, and C, each separated by 10 mm. The bottom plots are their corresponding horizontal cross sections.
Fig. 5.
Fig. 5. Geometry for determining distance to the POP.
Fig. 6.
Fig. 6. Square virtual detectors placed along the optic axis.
Fig. 7.
Fig. 7. Axial irradiance (W/cm2) for source flux=1 W.
Fig. 8.
Fig. 8. Zemax model of the setup in Fig. 1 showing invariance of spatial irradiance distributions. (a) Nine 10 mm square Lambertian sources at the image (left) and at the POP (right). (b) Disk Lambertian source at the image (left) and POP (right).
Fig. 9.
Fig. 9. Generation of a small beam spot with high irradiance with an array of sources at the object plane, and an aperture slightly in front of the POP.
Fig. 10.
Fig. 10. POP irradiance for a NIM. (a) NIM at 150 mm length. (b) NIM extended to 200 mm length. (c) Irradiance distribution at the second POP plane and at the second image plane in (b).
Fig. 11.
Fig. 11. Geometry for the derivation of Eqs. (2) and (3).
Fig. 12.
Fig. 12. Supplemental dimensions for Fig. 11.
Fig. 13.
Fig. 13. Ray geometry illustrating the equality in Eq. (A12). (a) For a paraxial lens. (b) For a real lens.

Equations (24)

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ZP=ZE1+M(Do/DE).
E(z)=πLZ2[Mz(Zz)]2×[111+Do24Z2(MzZz1)2],
E(z)=πLZ2[Mz(Zz)]2×[111+DE24(MZz)2[Mz(Zz)]2],
EP=πL4M2Z2(1+MDoDE)2[1DE2+14Z2(1DoDE)2]1.
C=AA,
C=(NA)2sin2α.
1ZP=1ZE+2tanαDE.
1ZP1f1Z+2tanαDE.
E(ρ)EP(1ρρP),
ΦP2π0ρEP(ρ)ρdρ=πEP3(3ρ22ρ3).
G=ΦPπL(NA)2πρ2EP3πL(NA)2(3ρ22ρ3)ρ2.
Cef=(NA)2sin2α×G.
E(Z)=πLsin2θi,
E(Z)=πLtan2θicos2θo.
E(Z)=dAπLsin2θodA.
dE(Z)=LdAcos4θZ2dadacosθ=LdAcos4θZ21cosθ,
dE(z)=dE(Z)dacosθdacosθ=dE(Z)R2r2.
dE(z)=LdAcos4θZ2R2r2=LdAcos4θZ2Z2(Zz)2.
E(z)=LZ2Z2(Zz)2cos4θdA.
E(z)=2πLZ2(Zz)2×0ρmax(z)[1+ρ2Z2(MzZz1)2]2ρdρ,
Mx+N(A+2Bx+Cx2)pdx=NBMA+(NCMB)x2(p1)(ACB2)(A+2Bx+Cx2)p1,
EParaxial(Z)EReal(Z)=cos2θocos2θi.
dAcosθdΩ=dAcosθdΩ,
dAdA=dΩdΩ.

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