Abstract

We derive a new variational principle in optics. We first formulate the principle for paraxial waves and then generalize it to arbitrary waves. The new principle, unlike the Fermat principle, concerns both the phase and the intensity of the wave. In particular, the principle provides a method for finding the ray mapping between two surfaces in space from information on the wave’s intensity there. We show how to apply the new principle to the problem of phase reconstruction from intensity measurements.

© 2004 Optical Society of America

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References

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  1. M. R. Teague, “Deterministic phase retieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  2. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
    [CrossRef] [PubMed]
  3. T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
    [CrossRef]
  4. C. M. Lee, J. Rubinstein (manuscript in preparation; available from the author at the address on the title page).
  5. M. A. van Dam, R. G. Lane, “Wave-front sensing from defocused images by use of wave-front slopes,” Appl. Opt. 41, 5497–5502 (2002).
    [CrossRef] [PubMed]
  6. J. B. Keller, R. M. Lewis, “Asymptotic methods for partial differential equations: the reduced wave equation and Maxwell’s equations,” in Surveys in Applied Mathematics, J. B. Keller, G. C. Papanicolaou, eds. (Plenum, New York, 1993), Vol. 1, pp. 1–82.
  7. C. Villani, Topics in Optimal Transportation (American Mathematical Society, Providence, R.I., 2003).
  8. G. Wolansky, “Optimal transportation in the presence of a prescribed pressure field,” preprint, available from the author at the address on the title page.
  9. J. D. Benamou, Y. Brenier, “A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem,” Numer. Math. 84, 375–393 (2000).
    [CrossRef]
  10. J. Rubinstein, G. Wolansky, “A weighted least action principle for dispersive waves,” preprint.
  11. J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, Bristol, UK, 1999).
  12. Y. Brenier, “Polar factorization and monotone rearrangement of vector-valued functions,” Commun. Pure Appl. Math. 64, 375–417 (1991).
    [CrossRef]
  13. S. Angenent, S. Haker, A. Tannenbaum, “Minimizing flows for the Monge–Kantorovich problem,” SIAM J. Math. Anal. 35, 61–97 (2003).
    [CrossRef]
  14. A. J. Chorin, “Numerical solution of the Navier–Stokes equations,” Math. Comput. 22, 742–762 (1968).
    [CrossRef]

2003 (1)

S. Angenent, S. Haker, A. Tannenbaum, “Minimizing flows for the Monge–Kantorovich problem,” SIAM J. Math. Anal. 35, 61–97 (2003).
[CrossRef]

2002 (1)

2000 (1)

J. D. Benamou, Y. Brenier, “A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem,” Numer. Math. 84, 375–393 (2000).
[CrossRef]

1996 (1)

1991 (1)

Y. Brenier, “Polar factorization and monotone rearrangement of vector-valued functions,” Commun. Pure Appl. Math. 64, 375–417 (1991).
[CrossRef]

1988 (1)

1983 (1)

1968 (1)

A. J. Chorin, “Numerical solution of the Navier–Stokes equations,” Math. Comput. 22, 742–762 (1968).
[CrossRef]

Angenent, S.

S. Angenent, S. Haker, A. Tannenbaum, “Minimizing flows for the Monge–Kantorovich problem,” SIAM J. Math. Anal. 35, 61–97 (2003).
[CrossRef]

Benamou, J. D.

J. D. Benamou, Y. Brenier, “A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem,” Numer. Math. 84, 375–393 (2000).
[CrossRef]

Brenier, Y.

J. D. Benamou, Y. Brenier, “A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem,” Numer. Math. 84, 375–393 (2000).
[CrossRef]

Y. Brenier, “Polar factorization and monotone rearrangement of vector-valued functions,” Commun. Pure Appl. Math. 64, 375–417 (1991).
[CrossRef]

Chorin, A. J.

A. J. Chorin, “Numerical solution of the Navier–Stokes equations,” Math. Comput. 22, 742–762 (1968).
[CrossRef]

Gureyev, T. E.

Haker, S.

S. Angenent, S. Haker, A. Tannenbaum, “Minimizing flows for the Monge–Kantorovich problem,” SIAM J. Math. Anal. 35, 61–97 (2003).
[CrossRef]

Keller, J. B.

J. B. Keller, R. M. Lewis, “Asymptotic methods for partial differential equations: the reduced wave equation and Maxwell’s equations,” in Surveys in Applied Mathematics, J. B. Keller, G. C. Papanicolaou, eds. (Plenum, New York, 1993), Vol. 1, pp. 1–82.

Lane, R. G.

Lee, C. M.

C. M. Lee, J. Rubinstein (manuscript in preparation; available from the author at the address on the title page).

Lewis, R. M.

J. B. Keller, R. M. Lewis, “Asymptotic methods for partial differential equations: the reduced wave equation and Maxwell’s equations,” in Surveys in Applied Mathematics, J. B. Keller, G. C. Papanicolaou, eds. (Plenum, New York, 1993), Vol. 1, pp. 1–82.

Nugent, K. A.

Nye, J. F.

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, Bristol, UK, 1999).

Roddier, F.

Rubinstein, J.

J. Rubinstein, G. Wolansky, “A weighted least action principle for dispersive waves,” preprint.

C. M. Lee, J. Rubinstein (manuscript in preparation; available from the author at the address on the title page).

Tannenbaum, A.

S. Angenent, S. Haker, A. Tannenbaum, “Minimizing flows for the Monge–Kantorovich problem,” SIAM J. Math. Anal. 35, 61–97 (2003).
[CrossRef]

Teague, M. R.

van Dam, M. A.

Villani, C.

C. Villani, Topics in Optimal Transportation (American Mathematical Society, Providence, R.I., 2003).

Wolansky, G.

G. Wolansky, “Optimal transportation in the presence of a prescribed pressure field,” preprint, available from the author at the address on the title page.

J. Rubinstein, G. Wolansky, “A weighted least action principle for dispersive waves,” preprint.

Appl. Opt. (2)

Commun. Pure Appl. Math. (1)

Y. Brenier, “Polar factorization and monotone rearrangement of vector-valued functions,” Commun. Pure Appl. Math. 64, 375–417 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Math. Comput. (1)

A. J. Chorin, “Numerical solution of the Navier–Stokes equations,” Math. Comput. 22, 742–762 (1968).
[CrossRef]

Numer. Math. (1)

J. D. Benamou, Y. Brenier, “A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem,” Numer. Math. 84, 375–393 (2000).
[CrossRef]

SIAM J. Math. Anal. (1)

S. Angenent, S. Haker, A. Tannenbaum, “Minimizing flows for the Monge–Kantorovich problem,” SIAM J. Math. Anal. 35, 61–97 (2003).
[CrossRef]

Other (6)

C. M. Lee, J. Rubinstein (manuscript in preparation; available from the author at the address on the title page).

J. Rubinstein, G. Wolansky, “A weighted least action principle for dispersive waves,” preprint.

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, Bristol, UK, 1999).

J. B. Keller, R. M. Lewis, “Asymptotic methods for partial differential equations: the reduced wave equation and Maxwell’s equations,” in Surveys in Applied Mathematics, J. B. Keller, G. C. Papanicolaou, eds. (Plenum, New York, 1993), Vol. 1, pp. 1–82.

C. Villani, Topics in Optimal Transportation (American Mathematical Society, Providence, R.I., 2003).

G. Wolansky, “Optimal transportation in the presence of a prescribed pressure field,” preprint, available from the author at the address on the title page.

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Equations (92)

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-i uz=ku+12k Δu.
-Az=12A (A2ϕ),
ϕz=1+1k2A ΔA-12 |ϕ|2.
-Iz=(Iϕ).
ϕ=z+ψ(x, z),
ψz+12 |ψ|2=12k2A ΔA.
ψz+12 |ψ|2=0.
Iz+(Iψ)=0,
ψz+12 |ψ|2=12 (n2-1)
  P(x, z).
I1(x)dx=I2(x)dx=1,
x2Ii(x)dx<,i=1, 2.
I1(x)=I2(T(x))J(T).
T#I1=I2.
ζ(T(x))I1(x)dx=ζ(x)I2(x)dx,ζC0(R2),
M(I1, I2, T¯)  Q(x, T¯(x))I1(x)dxQ(x, T(x))I1(x)dx,T#I1=I2,
Q(x, y)  Q(x, y, Z1, Z2) minZ1Z212dxdz2+P[x(z), z]dz
Q(x, y)=|x-y|22(Z2-Z1).
|T¯(x)-x|2I1(x)dx|T(x)-x|2I1(x)dx,
T#I1=I2.
infσ,vW(I1, I2; P)=infσ,vZ1Z212 σ|v|2+Pσdxdz,
σz+(σv)=0,Z1zZ2,
σ(x, Zi)=Ii(x),i=1, 2.
σ=I,v=ψ,
σ(x, z)|v(x, z)|2dx=σ(x, z)|φ(x, z)|2dx+σ(x, z)|w(x, z)|2dx.
αz+(αψ)+(Iβ)=O().
δW=12 α|ψ|2+Iψβ+αPdxdz+O(2).
Iψβdx=-ψ(Iβ)dx.
Iψβdxdz=αz+(αψ)ψdxdz+O()=-αψz+|ψ|2dxdz+O(),
dx¯dz=ψ(x¯(z), z),x¯(Z1)=x.
TZ1z(x)  x¯(z).
I˜(x, z)=J(TZ1z)I(TZ1z(x), z).
dJ(Tz)dz=J(Tz)Δψ(Tz, z).
I˜z=J(TZ1z)IΔψ+Iψ+IzTZ1z,z=0,
I1(x)=J(TZ1z)I(TZ1z(x), z),
T¯=TZ1Z2.
infσ,vW(I1, I2; P)=Q(x, T¯(x))I1(x)dx.
ddz ψ(ξ(z), z)=ψdξdz+ψz=-12dξdz-ψ2+12dξdz2+P(ξ(z), z).
ψ(TZ1Z2(x), Z2)-ψ(x, Z1)
=Z1Z212dx¯dz2+P(x¯(z), z)dzQ(x, T(x)).
ψ(x, Z2)I2(x)dx-ψ(x, Z1)I1(x)dx
Q(x, T(x))I1(x).
E(ψ, I1, I2)  ψ(x, Z2)I2(x)dx-ψ(x, Z1)I1(x)dx=Z1Z2z(ψ(x, z)I(x, z))dxdz=Z1Z2Iz ψ+I ψzdxdz.
=Z1Z2-ψ(Iψ)-12 I|ψ|2+IPdxdz=Z1Z212 I|ψ|2+IPdxdz=W(I1, I2; P).
W(I1, I2; P)Q(x, T(x))I1(x)dxQ(x, T¯(x))I1(x)dx.
E(ζ, I1, I2)Q(x, T(x))I1(x)dx.
E(ψ, I1, I2)=maxζζ(x, Z2)I2(x)dx-ζ(x, Z1)I1(x)dx,
ζz+12 |xζ|2P(x, z).
ζ(T(x), Z2)-ζ(x, Z1)Q(x, T(x)).
ζ(T(x), Z2)I1(x)dx-ζ(x, Z1)I1(x)dx
Q(x, T(x))I1(x)dx.
TZ1Z2(x)=x+xψ(x, Z1)(Z2-Z1).
n(x, z)=1+εP(x, ε1/2z),
ϕ(x, z)=z+ε1/2ψ(x, ε1/2z).
minxyn(x)dl,
minxyn(x)dl=minZ1Z2(1+εP)1+12dxdz2dz+o(ε)=(Z2-Z1)+Q(x, y)+o(ε).
Δu+k2n2(x, z)u=0
zI ϕz+(Iϕ)=0,
ρz+ρ ϕ(n2-|ϕ|2)1/2=0,Z1<z<Z2,
ϕz2+|ϕ|2=n2,Z1<z<Z2,
Q(x, T¯(x))ρ1(x)dxQ(x, T(x))ρ1(x)dx,
T#ρ1=ρ2,
Q(x, y)=minxyndl=minZ1Z2n(x, z)1+12dxdz21/2dz
infρ,vW(I1, I2; P)infρ,vZ1Z2nρ1+v2dxdz,
ρz+(ρv)=0,Z1zZ2,
ρ(x, Zi)=ρi(x),i=1, 2.
v=ϕ(n2-|ϕ|2)1/2,
infvρn1+v2+φρz+(ρv)dxdz.
infvρn1+v2-φz-vφdxdz
+φρ2dx-φρ1dx.
n1+v2-n2-p2vp,
v=φ(n2-|φ|2)1/2.
W=ρ(n2-|φ|2)1/2-φzdxdz+φρ2dx-φρ1dx.
dx¯dz=v=ϕ(n2-|ϕ|2)1/2,x¯(Z1)=x,
ddz ψ(ξ(z), z)=ψdξdz+ψz=ψdξdz+(n2-|ψ|2)1/2n1+dξdz21/2,
infρ,vZ1Z2ρn1+v2dzdx=infTQ(x, T(x))ρ1dx=supϕϕρ2dx-ϕρ1dx,
K(m)=|x-y|2m(x, y)dxdy.
m(x, y)dx=I2(y),m(x, y)dy=I1(x).
I11Nmi(1)δxi;I21Nmi(2)δyi,
minMi=1Nj=1NMi,j|xi-yj|2,
I11NiNδxi,I21NiNδyi.
Π:{1,N}{1,N},
T(xi)=yΠ(i).
U=U(x, t) : R2×[0, )R2
U(x, t)=p(x, t)+v(x, t),v=0.
Δp=U,xD1,pν=Uν,xD1,
F(p)=|U-p|2dx.
Ut+1I1 vU=0.
U(x, t)=q(x, t)+w(x, t),(I1w)=0.
Ut+wU=0.
ddt M(I1, I2)=-|v|2dx
ddt M(I1, I2)=-I1|w|2dx.

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