Abstract

It is shown that the Helmholtz wave equation follows from a new uncertainty principle: Given, as data, the position of a photon in an unknown diffraction pattern, the estimated position of the centroid of the pattern will suffer minimum precision. This implies a maximally spread out diffraction pattern, obeying a principle of minimum Fisher information. The minimum is constrained by knowledge of the refractive-index function n(x, y, z) of the medium through a requirement that the mean-square spatial phase gradient across the medium should be generally nonzero. Operationally the principle works directly with intensities and not complex amplitudes. As a practical matter the numerical use of the intensity-based principle might permit a widening of the known scope of solutions to diffraction problems.

© 1989 Optical Society of America

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References

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  1. P. J. Huber, Ann. Math. Stat. 35, 73 (1964).
    [CrossRef]
  2. P. J. Huber, Ann. Math. Stat. 43, 1041 (1972).
    [CrossRef]
  3. P. J. Huber, Ann. Stat. 2, 1029 (1974).
    [CrossRef]
  4. B. R. Frieden, J. Mod. Opt. 35, 1297 (1988).
    [CrossRef]
  5. H. L. van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Part I, p. 66.
  6. B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).
    [CrossRef] [PubMed]
  7. See, e.g., J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 34.
  8. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 633.
  9. Ref. 5, pp. 79–81.
  10. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, pp. 306–307.
  11. Ref. 8, p. 355.
  12. Ref. 8, p. 356.
  13. R. Eisberg, R. Resnick, Quantum Physics (Wiley, New York, 1974), p. 167.
  14. More discussion of these points is given in A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982), pp. 7–8.
  15. B. R. Frieden, “Fisher information as the basis for the Schrödinger wave equation,” Am. J. Phys. (to be published).

1988 (1)

B. R. Frieden, J. Mod. Opt. 35, 1297 (1988).
[CrossRef]

1974 (1)

P. J. Huber, Ann. Stat. 2, 1029 (1974).
[CrossRef]

1972 (2)

1964 (1)

P. J. Huber, Ann. Math. Stat. 35, 73 (1964).
[CrossRef]

Eisberg, R.

R. Eisberg, R. Resnick, Quantum Physics (Wiley, New York, 1974), p. 167.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, pp. 306–307.

Frieden, B. R.

B. R. Frieden, J. Mod. Opt. 35, 1297 (1988).
[CrossRef]

B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).
[CrossRef] [PubMed]

B. R. Frieden, “Fisher information as the basis for the Schrödinger wave equation,” Am. J. Phys. (to be published).

Goodman, J. W.

See, e.g., J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 34.

Huber, P. J.

P. J. Huber, Ann. Stat. 2, 1029 (1974).
[CrossRef]

P. J. Huber, Ann. Math. Stat. 43, 1041 (1972).
[CrossRef]

P. J. Huber, Ann. Math. Stat. 35, 73 (1964).
[CrossRef]

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 633.

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 633.

Marathay, A. S.

More discussion of these points is given in A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982), pp. 7–8.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, pp. 306–307.

Resnick, R.

R. Eisberg, R. Resnick, Quantum Physics (Wiley, New York, 1974), p. 167.

van Trees, H. L.

H. L. van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Part I, p. 66.

Ann. Math. Stat. (2)

P. J. Huber, Ann. Math. Stat. 35, 73 (1964).
[CrossRef]

P. J. Huber, Ann. Math. Stat. 43, 1041 (1972).
[CrossRef]

Ann. Stat. (1)

P. J. Huber, Ann. Stat. 2, 1029 (1974).
[CrossRef]

J. Mod. Opt. (1)

B. R. Frieden, J. Mod. Opt. 35, 1297 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (10)

H. L. van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Part I, p. 66.

See, e.g., J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 34.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 633.

Ref. 5, pp. 79–81.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, pp. 306–307.

Ref. 8, p. 355.

Ref. 8, p. 356.

R. Eisberg, R. Resnick, Quantum Physics (Wiley, New York, 1974), p. 167.

More discussion of these points is given in A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982), pp. 7–8.

B. R. Frieden, “Fisher information as the basis for the Schrödinger wave equation,” Am. J. Phys. (to be published).

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Figures (1)

Fig. 1
Fig. 1

Diffraction scenario. Given an inhomogeneous medium of known refractive-index function n(x, y, z) within boundary surface B, find the intensity i(x, y, z) within B. The answer is the i(x, y, z) pattern that is maximally spread out spatially, in the sense of a Fisher smoothness criterion, about the centroid point θ in the pattern.

Equations (22)

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a b d x p ( x ) 2 / p ( x ) + n = 1 N μ n [ a b d x k n ( x ) p ( x ) d n ] = minimum
e 2 1 / I , I = a b d x p ( x ) 2 / p ( x ) .
y i = θ i + x i , i = 1 , 2 , 3
h = 1 / ( 2 e ) .
h total 2 = h 1 2 + h 2 2 + h 3 2 = 1 2 ( 1 e 1 2 + 1 e 2 2 + 1 e 3 2 ) = minimum .
v = [ θ ˆ 1 ( y ) θ 1 ln p / θ 1 ln p / θ 2 ln p / θ 3 ] T ,
v T v = [ e 1 2 1 0 0 1 J 11 J 12 J 13 0 J 21 J 22 J 23 0 J 31 J 32 J 33 ]
J i j d y ( ln p / θ i ) ( ln p / θ j ) p .
det [ e 1 2 1 1 J 11 ] 0 , or e 1 2 1 / J 11 .
e 1 2 1 / J i i , i = 1 , 2 , 3 ,
J i i = d x [ p ( x ) / x i ] 2 / p ( x ) .
d x p x 2 + p y 2 + p z 2 p = minimum ,
B d x d y d z i x 2 + i y 2 + i z 2 i + n = 1 N μ n ( B d x d y d z k n i d n ) = minimum ,
ϕ 2 ( 2 π / λ ) 2 B d x d y d z n 2 ( x , y , z ) i ( x , y , z ) > 0 , ϕ 2 ϕ ϕ .
B d x d y d z i x 2 + i y 2 + i z 2 i 4 ( 2 π / λ ) 2 B d x d y d z n 2 i = minimum .
i = q 2 .
B d x d y d z ( q x 2 + q y 2 + q z 2 ) ( 2 π λ ) 2 B d x d y d z n 2 q 2 = minimum ,
q x x + q y y + q z z + ( 2 π λ ) 2 n 2 q = 0 .
x ( L q x ) + y ( L q y ) + z ( L q z ) = L q .
2 L / q x 2 > 0 , 2 L / q y 2 > 0 , 2 L / q z 2 > 0 ,
L = q x 2 + q y 2 + q z 2 ( 2 π λ ) 2 n 2 q 2 .
ln i ln i 4 ϕ ϕ = minimum .

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