Abstract

Aplanats were recently analyzed for the first time as maximum-performance light-transfer systems with radiative transfer approaching the thermodynamic limit. We present a fundamental categorization scheme that appears to subsume the full spectrum of aplanatic designs, illustrated for far-field dual-mirror concentrators and motivated by high-irradiance solar applications.

© 2009 Optical Society of America

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References

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  1. K. Schwarzschild, “Untersuchungen zur geometrischen Optik I-III,” Abh. Konigl. Ges. Wis.Gottingen Math-phys. Kl. 4, 1-3 (1905-1906).
  2. A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London B 70, 945-949 (1957).
    [CrossRef]
  3. D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334, 787-796 (2002).
    [CrossRef]
  4. R. V. Willstrop and D. Lynden-Bell, “Exact optics--II. Exploration of designs on- andoff-axis,” Mon. Not. R. Astron. Soc. 342, 33-49 (2003).
    [CrossRef]
  5. J. M. Gordon and D. Feuermann, “Optical performance at the thermodynamic limit with tailored imaging designs,” Appl. Opt. 44, 2327-2331 (2005).
    [CrossRef] [PubMed]
  6. R. Winston and J. M. Gordon, “Planar concentrators near the étendue limit,” Opt. Lett. 30, 2617-2619 (2005).
    [CrossRef] [PubMed]
  7. J. M. Gordon, D. Feuermann, and P. Young, “Unfolded aplanats for high-concentration photovoltaics,” Opt. Lett. 33, 1114-1116 (2008).
    [CrossRef] [PubMed]
  8. R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. Bortz, Nonimaging Optics (Elsevier, 2005).
  9. D. Feuermann, J. M. Gordon, and M. Huleihil, “Solar fiber-optic mini-dish concentrators: first experimental results and field experience,” Solar Energy 72, 459-472 (2002).
    [CrossRef]
  10. G. Conley and S. Horne, SolFocus, Inc., 510 Logue Ave., Mountain View, Calif. 94043 (personal communications, 2008).
  11. D. Nakar, D. Feuermann, and J. M. Gordon, “Aplanatic near-field optics for efficient light transfer,” Opt. Eng. 45, 030502 (2006).
    [CrossRef]
  12. D. Feuermann, J. M. Gordon, and T. W. Ng, “Photonic surgery with noncoherent light,” Appl. Phys. Lett. 88, 114104 (2006).
    [CrossRef]
  13. D. Feuermann and J. M. Gordon, “High irradiance reactors with unfolded aplanatic optics,” Appl. Opt. 47, 5722-5727(2008).
    [CrossRef]
  14. J. Bortz and N. Shatz, “Generalized functional method of nonimaging optical design,” Proc. SPIE 6338, 633805 (2006).
    [CrossRef]

2008 (2)

2006 (3)

J. Bortz and N. Shatz, “Generalized functional method of nonimaging optical design,” Proc. SPIE 6338, 633805 (2006).
[CrossRef]

D. Nakar, D. Feuermann, and J. M. Gordon, “Aplanatic near-field optics for efficient light transfer,” Opt. Eng. 45, 030502 (2006).
[CrossRef]

D. Feuermann, J. M. Gordon, and T. W. Ng, “Photonic surgery with noncoherent light,” Appl. Phys. Lett. 88, 114104 (2006).
[CrossRef]

2005 (2)

2003 (1)

R. V. Willstrop and D. Lynden-Bell, “Exact optics--II. Exploration of designs on- andoff-axis,” Mon. Not. R. Astron. Soc. 342, 33-49 (2003).
[CrossRef]

2002 (2)

D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334, 787-796 (2002).
[CrossRef]

D. Feuermann, J. M. Gordon, and M. Huleihil, “Solar fiber-optic mini-dish concentrators: first experimental results and field experience,” Solar Energy 72, 459-472 (2002).
[CrossRef]

1957 (1)

A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London B 70, 945-949 (1957).
[CrossRef]

1905 (1)

K. Schwarzschild, “Untersuchungen zur geometrischen Optik I-III,” Abh. Konigl. Ges. Wis.Gottingen Math-phys. Kl. 4, 1-3 (1905-1906).

Benítez, P.

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. Bortz, Nonimaging Optics (Elsevier, 2005).

Bortz, J.

J. Bortz and N. Shatz, “Generalized functional method of nonimaging optical design,” Proc. SPIE 6338, 633805 (2006).
[CrossRef]

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. Bortz, Nonimaging Optics (Elsevier, 2005).

Conley, G.

G. Conley and S. Horne, SolFocus, Inc., 510 Logue Ave., Mountain View, Calif. 94043 (personal communications, 2008).

Feuermann, D.

D. Feuermann and J. M. Gordon, “High irradiance reactors with unfolded aplanatic optics,” Appl. Opt. 47, 5722-5727(2008).
[CrossRef]

J. M. Gordon, D. Feuermann, and P. Young, “Unfolded aplanats for high-concentration photovoltaics,” Opt. Lett. 33, 1114-1116 (2008).
[CrossRef] [PubMed]

D. Nakar, D. Feuermann, and J. M. Gordon, “Aplanatic near-field optics for efficient light transfer,” Opt. Eng. 45, 030502 (2006).
[CrossRef]

D. Feuermann, J. M. Gordon, and T. W. Ng, “Photonic surgery with noncoherent light,” Appl. Phys. Lett. 88, 114104 (2006).
[CrossRef]

J. M. Gordon and D. Feuermann, “Optical performance at the thermodynamic limit with tailored imaging designs,” Appl. Opt. 44, 2327-2331 (2005).
[CrossRef] [PubMed]

D. Feuermann, J. M. Gordon, and M. Huleihil, “Solar fiber-optic mini-dish concentrators: first experimental results and field experience,” Solar Energy 72, 459-472 (2002).
[CrossRef]

Gordon, J. M.

J. M. Gordon, D. Feuermann, and P. Young, “Unfolded aplanats for high-concentration photovoltaics,” Opt. Lett. 33, 1114-1116 (2008).
[CrossRef] [PubMed]

D. Feuermann and J. M. Gordon, “High irradiance reactors with unfolded aplanatic optics,” Appl. Opt. 47, 5722-5727(2008).
[CrossRef]

D. Feuermann, J. M. Gordon, and T. W. Ng, “Photonic surgery with noncoherent light,” Appl. Phys. Lett. 88, 114104 (2006).
[CrossRef]

D. Nakar, D. Feuermann, and J. M. Gordon, “Aplanatic near-field optics for efficient light transfer,” Opt. Eng. 45, 030502 (2006).
[CrossRef]

J. M. Gordon and D. Feuermann, “Optical performance at the thermodynamic limit with tailored imaging designs,” Appl. Opt. 44, 2327-2331 (2005).
[CrossRef] [PubMed]

R. Winston and J. M. Gordon, “Planar concentrators near the étendue limit,” Opt. Lett. 30, 2617-2619 (2005).
[CrossRef] [PubMed]

D. Feuermann, J. M. Gordon, and M. Huleihil, “Solar fiber-optic mini-dish concentrators: first experimental results and field experience,” Solar Energy 72, 459-472 (2002).
[CrossRef]

Head, A. K.

A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London B 70, 945-949 (1957).
[CrossRef]

Horne, S.

G. Conley and S. Horne, SolFocus, Inc., 510 Logue Ave., Mountain View, Calif. 94043 (personal communications, 2008).

Huleihil, M.

D. Feuermann, J. M. Gordon, and M. Huleihil, “Solar fiber-optic mini-dish concentrators: first experimental results and field experience,” Solar Energy 72, 459-472 (2002).
[CrossRef]

Lynden-Bell, D.

R. V. Willstrop and D. Lynden-Bell, “Exact optics--II. Exploration of designs on- andoff-axis,” Mon. Not. R. Astron. Soc. 342, 33-49 (2003).
[CrossRef]

D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334, 787-796 (2002).
[CrossRef]

Miñano, J. C.

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. Bortz, Nonimaging Optics (Elsevier, 2005).

Nakar, D.

D. Nakar, D. Feuermann, and J. M. Gordon, “Aplanatic near-field optics for efficient light transfer,” Opt. Eng. 45, 030502 (2006).
[CrossRef]

Ng, T. W.

D. Feuermann, J. M. Gordon, and T. W. Ng, “Photonic surgery with noncoherent light,” Appl. Phys. Lett. 88, 114104 (2006).
[CrossRef]

Schwarzschild, K.

K. Schwarzschild, “Untersuchungen zur geometrischen Optik I-III,” Abh. Konigl. Ges. Wis.Gottingen Math-phys. Kl. 4, 1-3 (1905-1906).

Shatz, N.

J. Bortz and N. Shatz, “Generalized functional method of nonimaging optical design,” Proc. SPIE 6338, 633805 (2006).
[CrossRef]

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. Bortz, Nonimaging Optics (Elsevier, 2005).

Willstrop, R. V.

R. V. Willstrop and D. Lynden-Bell, “Exact optics--II. Exploration of designs on- andoff-axis,” Mon. Not. R. Astron. Soc. 342, 33-49 (2003).
[CrossRef]

Winston, R.

R. Winston and J. M. Gordon, “Planar concentrators near the étendue limit,” Opt. Lett. 30, 2617-2619 (2005).
[CrossRef] [PubMed]

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. Bortz, Nonimaging Optics (Elsevier, 2005).

Young, P.

Abh. Konigl. Ges. Wis.Gottingen Math-phys. Kl. (1)

K. Schwarzschild, “Untersuchungen zur geometrischen Optik I-III,” Abh. Konigl. Ges. Wis.Gottingen Math-phys. Kl. 4, 1-3 (1905-1906).

Appl. Opt. (2)

Appl. Phys. Lett. (1)

D. Feuermann, J. M. Gordon, and T. W. Ng, “Photonic surgery with noncoherent light,” Appl. Phys. Lett. 88, 114104 (2006).
[CrossRef]

Mon. Not. R. Astron. Soc. (2)

D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334, 787-796 (2002).
[CrossRef]

R. V. Willstrop and D. Lynden-Bell, “Exact optics--II. Exploration of designs on- andoff-axis,” Mon. Not. R. Astron. Soc. 342, 33-49 (2003).
[CrossRef]

Opt. Eng. (1)

D. Nakar, D. Feuermann, and J. M. Gordon, “Aplanatic near-field optics for efficient light transfer,” Opt. Eng. 45, 030502 (2006).
[CrossRef]

Opt. Lett. (2)

Proc. Phys. Soc. London B (1)

A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London B 70, 945-949 (1957).
[CrossRef]

Proc. SPIE (1)

J. Bortz and N. Shatz, “Generalized functional method of nonimaging optical design,” Proc. SPIE 6338, 633805 (2006).
[CrossRef]

Solar Energy (1)

D. Feuermann, J. M. Gordon, and M. Huleihil, “Solar fiber-optic mini-dish concentrators: first experimental results and field experience,” Solar Energy 72, 459-472 (2002).
[CrossRef]

Other (2)

G. Conley and S. Horne, SolFocus, Inc., 510 Logue Ave., Mountain View, Calif. 94043 (personal communications, 2008).

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. Bortz, Nonimaging Optics (Elsevier, 2005).

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Figures (8)

Fig. 1
Fig. 1

Aplanatic construction illustrated for one specific class (generalizable to all the fundamental classes elaborated in the text). The two mirror contours are determined by solving (a) constant optical path length L o + L 1 + L 2 , (b) Snell’s law, and (c) Abbe’s sine condition r = f sin ( ϕ ) , where r denotes radial position on the entry and ϕ is the angle at the focus. The radius of the Abbe sphere is the distance from the focus (along L 2 ) to the extension of L o , and is the focal length of the system, which establishes the length scale. The absorber faces upward in this illustration (i.e., opposite to the direction of the incident rays), but the formalism and governing Eq. (2) pertain equally well to designs with downward- facing absorbers.

Fig. 2
Fig. 2

s = 0.43 , K = 0.0385 , NA = 0.9 , an intrinsic loss of 0.03, and 0.156 ϕ arcsin ( NA ) . This particular configuration essentially achieves the compactness limit of an aspect ratio of 0.25.

Fig. 3
Fig. 3

s = 5.0 , K = 0.00477 , NA = 0.9 , an intrinsic loss of 0.0545, and π arcsin ( NA ) ϕ 2.93 . (a) Full view. (b) Magnified view near the focal plane.

Fig. 4
Fig. 4

(a) Ostensibly folded design with s = 0.61 , K = 0.23 , NA = 0.9 , 3.5% intrinsic loss, and 0.170 ϕ arcsin ( NA ) . (b) A nominally unfolded optic with s = 0.162 , K = 1.072 , NA = 0.9 , 0.157 ϕ arcsin ( NA ) , and 3.0% of incident rays not being collected due to their not intercepting the primary.

Fig. 5
Fig. 5

Varying the magnitudes of s and K reveal two apparently disparate shapes that nonetheless are unified within this basic category. (a)  s = 3.84 , K = 0.03 , NA = 0.9 , intrinsic losses of 0.039, and π arcsin ( NA ) ϕ 2.964 . (b)  s = 0.25 , K = 0.988 , NA = 0.9 , π arcsin ( NA ) ϕ 2.9674 , and a 3.8% intrinsic loss. In (b), the primary is distinguished by a thicker curve, and blocking is minimized when the exit of the primary coincides with the entry of the secondary.

Fig. 6
Fig. 6

s = 0.0445 , K = 0.152 , NA = 0.9 , and 0.425 ϕ arcsin ( NA ) . (a) Full view. (b) Zoom near the focal plane. This nominally unfolded design has 21% of the incident rays missing the primary.

Fig. 7
Fig. 7

s = 0.19 , K = 5.22 , NA = 0.9 , π arcsin ( NA ) ϕ 2.949 . and an intrinsic loss of 4.5%. The primary is distinguished by a thicker curve. The reflector vertices are not evident because of the truncation needed to minimize blocking.

Fig. 8
Fig. 8

Flux maps for the concentrator of Fig. 4a for a range of realistic θ s values. For each θ s , the net flux concentration C is normalized by its corresponding thermodynamic limit C max (Eq. (1)), and the radial position r is normalized by its corresponding equal-étendue (thermodynamic limit) value r th . The ideal limit (step-function) flux map is also plotted for comparison.

Tables (2)

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Table 1 Eight Basic Categories of Dual-Mirror Aplanats

Tables Icon

Table 2 Summary of Losses and Flux Concentration Relative to the Thermodynamic Limit C / C max at NA = 0.9 a

Equations (2)

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C max = NA 2 / sin 2 ( θ s ) ,
r p = sin ( ϕ ) , x p = s cos 2 ( ϕ / 2 ) + ( g ( ϕ ) / s ) ( 1 K f ( ϕ ) ) cos 4 ( ϕ / 2 ) , r s = 2 s K f ( ϕ ) tan ( ϕ / 2 ) K f ( ϕ ) tan 2 ( ϕ / 2 ) + g ( ϕ ) , x s = r s cot ( ϕ ) , where g ( ϕ ) = s ( 1 s ) tan 2 ( ϕ / 2 ) and f ( ϕ ) = | g ( ϕ ) / s | s / ( s 1 ) .

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