Abstract

Given a 3D reflecting surface G, which is not necessarily rotationally symmetric, away from a point source, and a direction e, we design a reflecting surface F so that the system (G,F) reflects all rays into the direction e. We also solve the near-field problem when the direction e is replaced by a given point R. The surface F satisfies a system of first-order partial differential equations that can be explicitly solved in terms of G and e. We also apply the same ideas to designing refracting plates, of which the Schmidt corrector plate is an example.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. E. Gutiérrez, “Aspherical lens design,” J. Opt. Soc. Am. A 30, 1719–1726 (2013).
    [CrossRef]
  2. R. Kingslake and R. B. Johnson, Lens Design Fundamentals, 2nd ed. (Academic/SPIE, 2010).
  3. C. A. Balanis, Antenna Theory Analysis and Design, 3rd ed. (Wiley, 2005).
  4. R. N. Wilson, Reflecting Telescope Optics I, 2nd ed., Astronomy and Astrophysics Library (Springer, 2007).
  5. J. Chen, The Principles of Astronomical Telescope Design, Vol. 360 of Astrophysics and Space Science Library (Springer, 2009).
  6. A. W. Love, “Some highlights in reflector antennas development,” Radio Sci. 11, 671–684 (1976).
    [CrossRef]
  7. B. van-Brunt, “Mathematical possibility of certain systems in geometrical optics,” J. Opt. Soc. Am. A 11, 2905–2914 (1994).
    [CrossRef]
  8. See the file far-field-concave.cdf at https://math.temple.edu/~gutierre/pair-reflectors-cdf/ .
  9. See the file far-field-convex.cdf at https://math.temple.edu/~gutierre/pair-reflectors-cdf/ .
  10. See the file cardioid-condenser.cdf at https://math.temple.edu/~gutierre/pair-reflectors-cdf/ .
  11. See the file near-field-reflectors.cdf at https://math.temple.edu/~gutierre/pair-reflectors-cdf/ .
  12. See the file mirror-and-corrector-plate.cdf at https://math.temple.edu/~gutierre/pair-reflectors-cdf/ .
  13. R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).
  14. V. Vodyanoy, A. Vainrub, and O. Pustovyy, “High resolution optical microscope with cardioid condenser for brightfield and darkfield illumination,” U.S. patent7,551,349 B2 (June23, 2009).
  15. P. Erdös, “Mirror anastigmatic with two concentric spherical surfaces,” J. Opt. Soc. Am. 49, 877–886 (1959).
    [CrossRef]
  16. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory, Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 2006).
  17. E. H. Linfoot, “Modern developments in telescope optics,” Vistas Astron. 1, 351–371 (1955).
    [CrossRef]
  18. L. G. Cook, “Flat field Schmidt telescope with extended field of view,” U.S. patent7,933,067 B1 (April26, 2011).

2013

1994

1976

A. W. Love, “Some highlights in reflector antennas development,” Radio Sci. 11, 671–684 (1976).
[CrossRef]

1959

1955

E. H. Linfoot, “Modern developments in telescope optics,” Vistas Astron. 1, 351–371 (1955).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Antenna Theory Analysis and Design, 3rd ed. (Wiley, 2005).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory, Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 2006).

Chen, J.

J. Chen, The Principles of Astronomical Telescope Design, Vol. 360 of Astrophysics and Space Science Library (Springer, 2009).

Cook, L. G.

L. G. Cook, “Flat field Schmidt telescope with extended field of view,” U.S. patent7,933,067 B1 (April26, 2011).

Erdös, P.

Gutiérrez, C. E.

Johnson, R. B.

R. Kingslake and R. B. Johnson, Lens Design Fundamentals, 2nd ed. (Academic/SPIE, 2010).

Kingslake, R.

R. Kingslake and R. B. Johnson, Lens Design Fundamentals, 2nd ed. (Academic/SPIE, 2010).

Linfoot, E. H.

E. H. Linfoot, “Modern developments in telescope optics,” Vistas Astron. 1, 351–371 (1955).
[CrossRef]

Love, A. W.

A. W. Love, “Some highlights in reflector antennas development,” Radio Sci. 11, 671–684 (1976).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).

Pustovyy, O.

V. Vodyanoy, A. Vainrub, and O. Pustovyy, “High resolution optical microscope with cardioid condenser for brightfield and darkfield illumination,” U.S. patent7,551,349 B2 (June23, 2009).

Vainrub, A.

V. Vodyanoy, A. Vainrub, and O. Pustovyy, “High resolution optical microscope with cardioid condenser for brightfield and darkfield illumination,” U.S. patent7,551,349 B2 (June23, 2009).

van-Brunt, B.

Vodyanoy, V.

V. Vodyanoy, A. Vainrub, and O. Pustovyy, “High resolution optical microscope with cardioid condenser for brightfield and darkfield illumination,” U.S. patent7,551,349 B2 (June23, 2009).

Wilson, R. N.

R. N. Wilson, Reflecting Telescope Optics I, 2nd ed., Astronomy and Astrophysics Library (Springer, 2007).

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory, Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 2006).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Radio Sci.

A. W. Love, “Some highlights in reflector antennas development,” Radio Sci. 11, 671–684 (1976).
[CrossRef]

Vistas Astron.

E. H. Linfoot, “Modern developments in telescope optics,” Vistas Astron. 1, 351–371 (1955).
[CrossRef]

Other

L. G. Cook, “Flat field Schmidt telescope with extended field of view,” U.S. patent7,933,067 B1 (April26, 2011).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory, Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 2006).

See the file far-field-concave.cdf at https://math.temple.edu/~gutierre/pair-reflectors-cdf/ .

See the file far-field-convex.cdf at https://math.temple.edu/~gutierre/pair-reflectors-cdf/ .

See the file cardioid-condenser.cdf at https://math.temple.edu/~gutierre/pair-reflectors-cdf/ .

See the file near-field-reflectors.cdf at https://math.temple.edu/~gutierre/pair-reflectors-cdf/ .

See the file mirror-and-corrector-plate.cdf at https://math.temple.edu/~gutierre/pair-reflectors-cdf/ .

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).

V. Vodyanoy, A. Vainrub, and O. Pustovyy, “High resolution optical microscope with cardioid condenser for brightfield and darkfield illumination,” U.S. patent7,551,349 B2 (June23, 2009).

R. Kingslake and R. B. Johnson, Lens Design Fundamentals, 2nd ed. (Academic/SPIE, 2010).

C. A. Balanis, Antenna Theory Analysis and Design, 3rd ed. (Wiley, 2005).

R. N. Wilson, Reflecting Telescope Optics I, 2nd ed., Astronomy and Astrophysics Library (Springer, 2007).

J. Chen, The Principles of Astronomical Telescope Design, Vol. 360 of Astrophysics and Space Science Library (Springer, 2009).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

Obstruction example: Σ1 is a circle and C=1.

Fig. 2.
Fig. 2.

Here C=2.2, R=1.1, and a=0.3.

Fig. 3.
Fig. 3.

Here C=1, R=0.38, and a=1.5.

Fig. 4.
Fig. 4.

Here Σ2 (in blue) is given.

Fig. 5.
Fig. 5.

Near-field example.

Fig. 6.
Fig. 6.

Corrector plate with a=1, R=1.74, n1=1, n2=1.5, and C=0.6.

Equations (89)

Equations on this page are rendered with MathJax. Learn more.

x×ν=m×ν,
xm=λν.
m=x2(x·ν)ν.
x=m2(m·ν)ν.
x:=x(u,v)=(x1(u,v),x2(u,v),x3(u,v)),
m1(u,v)=x(u,v)2(x(u,v)·ν(P1(u,v)))ν(P1(u,v)),
P2(u,v)=P1(u,v)+d(u,v)m1(u,v).
(x(u,v)m1(u,v))·(P1)u=0,(x(u,v)m1(u,v))·(P1)v=0,
m1(u,v)·(P1)u=x·(P1)u=x·(ρu(u,v)x(u,v)+ρ(u,v)xu(u,v))=ρu(u,v),
m1(u,v)·(P1)v=ρv(u,v).
(m1(u,v)e1)·(P2)u=0,(m1(u,v)e1)·(P2)v=0,
0=m1(u,v)·((P1)u+du(u,v)m1(u,v)+d(u,v)(m1)u(u,v))e1·(P2)u=ρu(u,v)+du(u,v)e1·(P2)u,
0=ρv(u,v)+dv(u,v)e1·(P2)v.
ρ(u,v)+d(u,v)e1·P2(u,v)=C,
C=ρ(u,v)+d(u,v)e1·(ρ(u,v)x(u,v)+d(u,v)m1(u,v))=ρ(u,v)(1e1·x(u,v))+d(u,v)(1e1·m1(u,v)),
d(u,v)=Cρ(u,v)(1e1·x(u,v))1e1·m1(u,v).
max(u,v)De1·m1(u,v)<1.
Cmax(u,v)Dρ(u,v)(1e1·x(u,v)).
min(u,v)De1·m1(u,v)1+ϵ,
P1(u0,v0)P1(u,v)d(u,v)m1(u,v)=λe1.
(P1(u0,v0)P1(u,v))·e1d(u,v)(m1(u,v)·e1)=λ;(P1(u0,v0)P1(u,v))·m1(u,v)d(u,v)=λ(e1·m1(u,v)).
d(u,v)(1(e1·m1(u,v))2)=[P1(u0,v0)P1(u,v)]·m1(u,v)[(P1(u0,v0)P1(u,v))·e1](e1·m1(u,v)),
(C(1e1·x(u,v))ρ(u,v))(1+e1·m1(u,v))=[P1(u0,v0)P1(u,v)]·m1(u,v)[(P1(u0,v0)P1(u,v))·e1](e1·m1(u,v)).
ϵ(C(1e1·x(u,v))ρ(u,v))[P1(u0,v0)P1(u,v)]·m1(u,v)[(P1(u0,v0)P1(u,v))·e1](e1·m1(u,v)):=Δ(u0,v0;u,v).
M=max(u,v),(u0,v0)D{Δ(u0,v0;u,v)ϵ+(1e1·x(u,v))ρ(u,v)},
ρ(θ)=acosθ+R2a2sin2θ,
ν1(θ)=ρ(θ)(cosθ,sinθ)(a,0)R,
m1(θ)=(cosθ,sinθ)2((cosθ,sinθ)·ν1)ν1=(cosθ,sinθ)2(ρ(θ)acosθR2)(ρ(θ)cosθa,sinθ).
d(θ)=Cρ(θ)(1cosθ)1(cosθ2R2(ρ(θ)cosθa)(ρ(θ)acosθ)),
ν(θ)=12(1+cosθ)(cosθ+cos2θ,sinθ+sin2θ),
(cosθ,sinθ)2((cosθ,sinθ)·ν(θ))ν(θ)=(cos2θ,sin2θ).
d(θ)=Crsin2θ1+cos2θ,
P2(θ)=r(1+cosθ)(cosθ,sinθ)+d(θ)(cos2θ,sin2θ).
P1(u,v)=P2(u,v)d(u,v)m1(u,v),
|P1|u=(Z·P1)u=Zu·P1+Z·(P1)u=(Zu·Z)|P1|+Z·(P1)u=Z·(P1)u,
|P1|v=Z·(P1)v.
(m1(u,v)e1)·(P2)u=0,(m1(u,v)e1)·(P2)v=0,
m1(u,v)·(P2)u=e1·(P2)u,
m1(u,v)·(P2)v=e1·(P2)v.
(Z(u,v)m1(u,v))·(P1)u=0,(Z(u,v)m1(u,v))·(P1)v=0.
0=|P1|um1(u,v)·((P2)udu(u,v)m1(u,v)d(u,v)(m1)u)=|P1|u+due1·(P2)u,
0=|P1|v+dve1·(P2)v.
|P1(u,v)|+d(u,v)e1·P2(u,v)=C,
(e1·P2(u,v)+Cd(u,v))2=|P2(u,v)d(u,v)m1|2.
d(u,v)=|P2(u,v)|2(e1·P2(u,v)+C)22(P2(u,v)·(m1(u,v)e1)C)).
P2(u,v)=P1(u,v)+d(u,v)m1(u,v).
|RP2(u,v)|u=Z(u,v)·(RP2(u,v))u=Z(u,v)·(P2)u,
|RP2(u,v)|v=Z(u,v)·(P2)v.
(m1(u,v)Z(u,v))·(P2)u=0,(m1(u,v)Z(u,v))·(P2)v=0.
ρu(u,v)+du(u,v)+|RP2(u,v)|u=0.
ρv(u,v)+dv(u,v)+|RP2(u,v)|v=0.
ρ(u,v)+d(u,v)+|RP2(u,v)|=C.
|RP2(u,v)|2=|RP1(u,v)d(u,v)m1|2=(Cρ(u,v)d(u,v))2.
d(u,v)=(Cρ(u,v))2|RP1(u,v)|22(Cρ(u,v)m1·(RP1(u,v))),
A(u,v)=Cρ(u,v)|RP1(u,v)|,B(u,v)=Cρ(u,v)+|RP1(u,v)|,H(u,v)=Cρ(u,v)m1·(RP1(u,v)).
A(u,v)=d(u,v)+|RP2(u,v)||RP1(u,v)|=d(u,v)+|RP1(u,v)d(u,v)m1(u,v)||RP1(u,v)|d(u,v)+|RP1(u,v)||d(u,v)||m1(u,v)||RP1(u,v)|=d(u,v)|d(u,v)|.
d(u,v)|d(u,v)|A(u,v)H(u,v)B(u,v),
Cmax(u,v)D(ρ(u,v)+|RP1(u,v)|).
1|R|((CRρ(u,v))2|RP1(u,v)|2)2(ρ(0,0)+d(0,0)e1·P2(0,0)ρ(u,v)(1e1·x)).
1|R|(2(CRρ(u,v)m1·(RP1(u,v))))2(1e1·m1(u,v)).
P1(u0,v0)P1(u,v)d(u,v)m1(u,v)=λeR(u0,v0).
(P1(u0,v0)P1(u,v))·eR(u0,v0)d(u,v)(m1(u,v)·eR)=λ;(P1(u0,v0)P1(u,v))·m1(u,v)d(u,v)=λ(eR(u0,v0)·m1(u,v)).
d(u,v)(1(eR(u0,v0)·m1(u,v))2)=[P1(u0,v0)P1(u,v)]·m1(u,v)[(P1(u0,v0)P1(u,v))·eR(u0,v0)](eR(u0,v0)·m1(u,v)).
d(u,v)=|RP1(u,v)|×(Cρ(u,v)|RP1(u,v)|1)(Cρ(u,v)|RP1(u,v)|+1)2(Cρ(u,v)|RP1(u,v)|m1(u,v)·eR(u,v))|RP1(u,v)|×(Cρ(u,v)|RP1(u,v)|1)(Cρ(u,v)|RP1(u,v)|+1)2(Cρ(u,v)|RP1(u,v)|+1)=12(Cρ(u,v)|RP1(u,v)|).
1+ϵeR(u0,v0)·m1(u,v)1δ,(u,v),(u0,v0)D,
d(u,v)(1(eR·m1(u,v))2)12(Cρ(u,v)|RP1(u,v)|)ϵδ.
Γ(u0,v0;u,v):=[P1(u0,v0)P1(u,v)]·m1(u,v)[(P1(u0,v0)P1(u,v))·eR(u0,v0)](eR(u0,v0)·m1(u,v))12(Cρ(u,v)|RP1(u,v)|)ϵδ.
M=max(u,v),(u0,v0)D{2Γ(u0,v0;u,v)ϵδ+ρ(u,v)+|RP1(u,v)|},
P2(u,v)=ρ(u,v)x(u,v)+d(u,v)m1(u,v).
(m1κe1)·(P2)u=0,(m1κe1)·(P2)v=0.
0=(m1(u,v)κe1)·(P2)u=m1(u,v)·(P2)uκe1·(P2)u=m1(u,v)·((P1)u+d(u,v)(m1)u+(d)um1(u,v))κe1·(P2)u=ρu(u,v)+duκe1·(P2)u,
1κρu(u,v)+1κdue1·(P2)u=0.
1κρv(u,v)+1κdve1·(P2)v=0.
1κρ(u,v)+1κd(u,v)e1·P2(u,v)=C.
C=1κρ(u,v)+1κd(u,v)e1·(P1(u,v)+d(u,v)m1(u,v))=ρ(u,v)(1κe1·x(u,v))+d(u,v)(1κe1·m1(u,v)).
d(u,v)=Cρ(u,v)(n1n2e1·x(u,v))n1n2e1·m1(u,v),
C<min(u,v)Dρ(u,v)(n1n2e1·x(u,v)).
m1(θ)·e1=cosθ2(ρ(θ)cosθa)R2R2a2sin2θ>n1n2.
C<ρ(θ)(n1n2cosθ),
ρ(θ)(cosθ,sinθ)+d(θ)m1(θ),
d(θ)=Cρ(θ)(n1n2cosθ)n1n2(cosθ2R2(ρ(θ)acosθ)(ρ(θ)cosθa)).
ρ(θ)R=λcosθ+1λ2sin2θ=λcosθ+1λ2+λ2cos2θ,
m1(θ)·e1=cosθ2(ρ(θ)Rcosθλ)1λ2+λ2cos2θ=2λ2cos3θ+(2λ21)cosθ+2λ(1cos2θ)1λ2+λ2cos2θ.
gλ(X)=(2λ2)X3+(2λ21)X+2λ(1X2)1λ2+λ2X2,
C=ρ(π)(n1n2+1)+d(π)(n1n21).
C=(Ra)(n1n2+1)+(Ra+b)(n1n21).
hλ(X)=(λX+1λ2+λ2X2)(n1n2X),
hλ(X)=(λX+1λ2+λ2X2)×(λn1n2(λX+1λ2+λ2X2))1λ2+λ2X2.
hλ(1)hλ(0)=(1λ)(n1n2+1)n1n21λ2=n1n2(1λ1λ2)+1λ.

Metrics