Abstract

The occurrence and smoothing of speckle are studied as a function of the line width for a highly collimated illuminating source. A general theory is presented for speckling in the image of a partially diffuse, phase type of object, which has a variable number of random scattering centers per resolution element. Then, an expression is derived for the wavelength spacing required to decouple the speckle patterns arising from two monochromatic tones in an imaging system, thereby establishing that it is feasible to smooth speckle using multicolor illumination. This theory is verified in a series of experiments using both laser illumination and band-limited light from a carbon arc. With highly collimated sources, we show that speckle appears laserlike for an imaged diffuser even up to line widths of 5 Å. Then, smoothing of speckle is demonstrated in the imaging of a diffuser and for a section of an optic nerve when the illumination is provided by six narrow lines spread over 1500 Å. Since with color-blind, panchromatic viewing the speckle smooths, a direct extension of this method to holographic microscopy, using a multitone laser, should permit one to record and reconstruct holograms of diffraction-limited resolution that are essentially speckle-free.

© 1973 Optical Society of America

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References

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  1. M. v. Laue, Sitzber. Preuss. Akad.1144, (1914), trans. by H. K. V. Lotsch.
  2. J. W. Goodman, Stanford University Electronics Labs. Tech. Rept. SEL-63-140 (TR 2303-1) (Dec.1963).
  3. L. I. Goldfischer, J. Opt. Soc. Am. 55, 247 (1965).
    [CrossRef]
  4. L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).
  5. W. Martienssen, S. Spiller, Phys. Lett. 24A, 126 (1967).
  6. E. N. Leith, J. Upatnieks, Appl. Opt. 7, 2085 (1968).
    [CrossRef] [PubMed]
  7. H. J. Gerritsen, W. J. Hannan, E. G. Ramberg, Appl. Opt. 7, 2301 (1968).
    [CrossRef] [PubMed]
  8. R. F. van Ligten, Opt. Technol. 1, 71 (1969).
    [CrossRef]
  9. R. F. van Ligten, J. Opt. Soc. Am. 59, 1545 (1969).
  10. M. E. Cox, R. G. Buckles, D. Whitlow, J. Opt. Soc. Am. 59, 1545 (1969).
  11. E. Archbold, J. M. Burch, A. E. Ennos, P. A. Taylor, Nature 222, 263 (1969).
    [CrossRef]
  12. M. Young, B. Faulkner, J. Cole, J. Opt. Soc. Am. 60, 137 (1970).
    [CrossRef]
  13. J. C. Dainty, Opt. Acta 17, 761 (1970).
    [CrossRef]
  14. D. Gabor, IBM J. Res. Develop. 14, 509 (1970).
    [CrossRef]
  15. S. Lowenthal, D. Joyeux, J. Opt. Soc. Am. 61, 847 (1971).
    [CrossRef]
  16. H. H. Hopkins, H. Tiziani, Applications of Holography (Besancon Conference6–11 July 1970), viii.
  17. D. H. Close, J. Quantum Electron. QE-7, 312 (1971).
    [CrossRef]
  18. J. M. Burch, SPIE Devel. Hologr. 25, 149 (1971).
    [CrossRef]
  19. M. Elbaum, M. Greenebaum, M. King, Opt. Commun. 5, 171 (1972).
    [CrossRef]
  20. N. George, A. Jain, Opt. Commun. 6, 253 (1972).
    [CrossRef]
  21. N. George, A. Jain, Calif. Inst. of Technol. Sci. Rept. 14, AFOSR-TR-72-1308 (1972).
  22. J. Upatnieks, R. W. Lewis, J. Opt. Soc. Am. 62, 1351A (1972).
  23. Hologram volume effects and color holography are fully discussed in the following textbook and will not be treated further herein: R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic Press, New York, (1971), Chaps. 9 and 17.
  24. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.
  25. N. George, J. T. McCrickerd, Photogr. Sci. Eng. 13, 342 (1969).
  26. As we have employed it here, the Gaussian transmission function in Eq. (2) is chosen for theoretical convenience, although for electronic optics it is physically appropriate as well. In the results it will not make any qualitative difference; in fact, in deriving the form in Eq. (15) we further approximate the actual case using a simple rect function as the impulse response of the lens.
  27. In this simplification, the assumption is that the phase terms in the integrand of Eq. (3) involving the variables x′, y′ do not vary appreciably within the interval wr. For a consideration of the extent of this type of phase variation, the reader is referred to D. A. Tichenor, J. W. Goodman, J. Opt. Soc. Am. 62, 293 (1972).
    [CrossRef]
  28. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 8.
  29. M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970), Chap. 10.
  30. The 8-μm section of the optic nerve is prepared by perfusing the eyestalk in 1% K4Fe(CN)6 in crayfish saline; it is removed from the crayfish, soaked in saline saturated with picric acid, dehydrated in alcohol, embedded in paraffin, sectioned with a rotary microtome, stained in a Ponceau acid fushin solution, and mounted with Permount on glass.

1972 (4)

1971 (3)

S. Lowenthal, D. Joyeux, J. Opt. Soc. Am. 61, 847 (1971).
[CrossRef]

D. H. Close, J. Quantum Electron. QE-7, 312 (1971).
[CrossRef]

J. M. Burch, SPIE Devel. Hologr. 25, 149 (1971).
[CrossRef]

1970 (3)

J. C. Dainty, Opt. Acta 17, 761 (1970).
[CrossRef]

D. Gabor, IBM J. Res. Develop. 14, 509 (1970).
[CrossRef]

M. Young, B. Faulkner, J. Cole, J. Opt. Soc. Am. 60, 137 (1970).
[CrossRef]

1969 (5)

N. George, J. T. McCrickerd, Photogr. Sci. Eng. 13, 342 (1969).

R. F. van Ligten, Opt. Technol. 1, 71 (1969).
[CrossRef]

R. F. van Ligten, J. Opt. Soc. Am. 59, 1545 (1969).

M. E. Cox, R. G. Buckles, D. Whitlow, J. Opt. Soc. Am. 59, 1545 (1969).

E. Archbold, J. M. Burch, A. E. Ennos, P. A. Taylor, Nature 222, 263 (1969).
[CrossRef]

1968 (2)

1967 (2)

L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).

W. Martienssen, S. Spiller, Phys. Lett. 24A, 126 (1967).

1965 (1)

Archbold, E.

E. Archbold, J. M. Burch, A. E. Ennos, P. A. Taylor, Nature 222, 263 (1969).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970), Chap. 10.

Buckles, R. G.

M. E. Cox, R. G. Buckles, D. Whitlow, J. Opt. Soc. Am. 59, 1545 (1969).

Burch, J. M.

J. M. Burch, SPIE Devel. Hologr. 25, 149 (1971).
[CrossRef]

E. Archbold, J. M. Burch, A. E. Ennos, P. A. Taylor, Nature 222, 263 (1969).
[CrossRef]

Burckhardt, C. B.

Hologram volume effects and color holography are fully discussed in the following textbook and will not be treated further herein: R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic Press, New York, (1971), Chaps. 9 and 17.

Close, D. H.

D. H. Close, J. Quantum Electron. QE-7, 312 (1971).
[CrossRef]

Cole, J.

Collier, R. J.

Hologram volume effects and color holography are fully discussed in the following textbook and will not be treated further herein: R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic Press, New York, (1971), Chaps. 9 and 17.

Cox, M. E.

M. E. Cox, R. G. Buckles, D. Whitlow, J. Opt. Soc. Am. 59, 1545 (1969).

Dainty, J. C.

J. C. Dainty, Opt. Acta 17, 761 (1970).
[CrossRef]

Elbaum, M.

M. Elbaum, M. Greenebaum, M. King, Opt. Commun. 5, 171 (1972).
[CrossRef]

Enloe, L. H.

L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).

Ennos, A. E.

E. Archbold, J. M. Burch, A. E. Ennos, P. A. Taylor, Nature 222, 263 (1969).
[CrossRef]

Faulkner, B.

Gabor, D.

D. Gabor, IBM J. Res. Develop. 14, 509 (1970).
[CrossRef]

George, N.

N. George, A. Jain, Opt. Commun. 6, 253 (1972).
[CrossRef]

N. George, J. T. McCrickerd, Photogr. Sci. Eng. 13, 342 (1969).

N. George, A. Jain, Calif. Inst. of Technol. Sci. Rept. 14, AFOSR-TR-72-1308 (1972).

Gerritsen, H. J.

Goldfischer, L. I.

Goodman, J. W.

Greenebaum, M.

M. Elbaum, M. Greenebaum, M. King, Opt. Commun. 5, 171 (1972).
[CrossRef]

Hannan, W. J.

Hopkins, H. H.

H. H. Hopkins, H. Tiziani, Applications of Holography (Besancon Conference6–11 July 1970), viii.

Jain, A.

N. George, A. Jain, Opt. Commun. 6, 253 (1972).
[CrossRef]

N. George, A. Jain, Calif. Inst. of Technol. Sci. Rept. 14, AFOSR-TR-72-1308 (1972).

Joyeux, D.

King, M.

M. Elbaum, M. Greenebaum, M. King, Opt. Commun. 5, 171 (1972).
[CrossRef]

Laue, M. v.

M. v. Laue, Sitzber. Preuss. Akad.1144, (1914), trans. by H. K. V. Lotsch.

Leith, E. N.

Lewis, R. W.

J. Upatnieks, R. W. Lewis, J. Opt. Soc. Am. 62, 1351A (1972).

Lin, L. H.

Hologram volume effects and color holography are fully discussed in the following textbook and will not be treated further herein: R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic Press, New York, (1971), Chaps. 9 and 17.

Lowenthal, S.

Martienssen, W.

W. Martienssen, S. Spiller, Phys. Lett. 24A, 126 (1967).

McCrickerd, J. T.

N. George, J. T. McCrickerd, Photogr. Sci. Eng. 13, 342 (1969).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 8.

Ramberg, E. G.

Spiller, S.

W. Martienssen, S. Spiller, Phys. Lett. 24A, 126 (1967).

Taylor, P. A.

E. Archbold, J. M. Burch, A. E. Ennos, P. A. Taylor, Nature 222, 263 (1969).
[CrossRef]

Tichenor, D. A.

Tiziani, H.

H. H. Hopkins, H. Tiziani, Applications of Holography (Besancon Conference6–11 July 1970), viii.

Upatnieks, J.

J. Upatnieks, R. W. Lewis, J. Opt. Soc. Am. 62, 1351A (1972).

E. N. Leith, J. Upatnieks, Appl. Opt. 7, 2085 (1968).
[CrossRef] [PubMed]

van Ligten, R. F.

R. F. van Ligten, Opt. Technol. 1, 71 (1969).
[CrossRef]

R. F. van Ligten, J. Opt. Soc. Am. 59, 1545 (1969).

Whitlow, D.

M. E. Cox, R. G. Buckles, D. Whitlow, J. Opt. Soc. Am. 59, 1545 (1969).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970), Chap. 10.

Young, M.

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).

IBM J. Res. Develop. (1)

D. Gabor, IBM J. Res. Develop. 14, 509 (1970).
[CrossRef]

J. Opt. Soc. Am. (7)

J. Quantum Electron. (1)

D. H. Close, J. Quantum Electron. QE-7, 312 (1971).
[CrossRef]

Nature (1)

E. Archbold, J. M. Burch, A. E. Ennos, P. A. Taylor, Nature 222, 263 (1969).
[CrossRef]

Opt. Acta (1)

J. C. Dainty, Opt. Acta 17, 761 (1970).
[CrossRef]

Opt. Commun. (2)

M. Elbaum, M. Greenebaum, M. King, Opt. Commun. 5, 171 (1972).
[CrossRef]

N. George, A. Jain, Opt. Commun. 6, 253 (1972).
[CrossRef]

Opt. Technol. (1)

R. F. van Ligten, Opt. Technol. 1, 71 (1969).
[CrossRef]

Photogr. Sci. Eng. (1)

N. George, J. T. McCrickerd, Photogr. Sci. Eng. 13, 342 (1969).

Phys. Lett. (1)

W. Martienssen, S. Spiller, Phys. Lett. 24A, 126 (1967).

SPIE Devel. Hologr. (1)

J. M. Burch, SPIE Devel. Hologr. 25, 149 (1971).
[CrossRef]

Other (10)

H. H. Hopkins, H. Tiziani, Applications of Holography (Besancon Conference6–11 July 1970), viii.

As we have employed it here, the Gaussian transmission function in Eq. (2) is chosen for theoretical convenience, although for electronic optics it is physically appropriate as well. In the results it will not make any qualitative difference; in fact, in deriving the form in Eq. (15) we further approximate the actual case using a simple rect function as the impulse response of the lens.

M. v. Laue, Sitzber. Preuss. Akad.1144, (1914), trans. by H. K. V. Lotsch.

J. W. Goodman, Stanford University Electronics Labs. Tech. Rept. SEL-63-140 (TR 2303-1) (Dec.1963).

N. George, A. Jain, Calif. Inst. of Technol. Sci. Rept. 14, AFOSR-TR-72-1308 (1972).

Hologram volume effects and color holography are fully discussed in the following textbook and will not be treated further herein: R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic Press, New York, (1971), Chaps. 9 and 17.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 8.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970), Chap. 10.

The 8-μm section of the optic nerve is prepared by perfusing the eyestalk in 1% K4Fe(CN)6 in crayfish saline; it is removed from the crayfish, soaked in saline saturated with picric acid, dehydrated in alcohol, embedded in paraffin, sectioned with a rotary microtome, stained in a Ponceau acid fushin solution, and mounted with Permount on glass.

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Figures (9)

Fig. 1
Fig. 1

Single lens magnification with object plane (ξ,η) and image plane (x,y).

Fig. 2
Fig. 2

The idealized diffuser object (D0) in the (ξ,η) plane showing a magnified inset between S1S2 for the computation of the transmission function, Eq. (10). The diffuser has steps of width wr and random height hr.

Fig. 3
Fig. 3

The experimental arrangement for obtaining laserlike speckle from a band-limited, carbon arc source. The diffuse object D0, is magnified by the microscope OM and speckled images are photographed by the camera C. Illumination from the arc (A) is band limited by the Spex monochromater (mirrors M1, M2 are 10-cm diam and 75-cm focal length); pinholes P1 and P2 are 400 μm, P3 is 60 μm, and the microscope objective O1 is used to collimate the illumination at D0.

Fig. 4
Fig. 4

Speckle pattern with collimated laser illumination at 6328 Å incident on diffuser made from Scotch Magic Tape. Imaging is as shown in Fig. 3.

Fig. 5
Fig. 5

Speckle pattern for Scotch Magic tape diffuser, as in Fig. 4, but illuminated with band-limited light from a carbon arc (5 Å band limited at 6000 Å and 70 min of exposure using Tri-X film). Beam collimation angle is 2 × 10−3 rad.

Fig. 6
Fig. 6

Section of optic nerve at low magnification illuminated by collimated laser light.

Fig. 7
Fig. 7

Optic nerve illuminated in white light. Resolution here is much better than in the speckled image of Fig. 6. The maximum length of this specimen is approximately 1 mm (actual length).

Fig. 8
Fig. 8

Optic nerve illuminated by a collimated source at 5500 Å with 5-Å line width (180-min exposure using a high pressure mercury arc and Tri-X film). Note that the image is speckled to a slightly lesser degree than with laser illumination.

Fig. 9
Fig. 9

Optic nerve illuminated by six separate band-limited wavelengths, spanning the spectrum from 4300 Å to 5800 Å. Note that the resolution is considerably improved over that for a single tone as shown in Fig. 8. The beam collimation angle of 2 × 10−3 rad is maintained throughout the series.

Equations (49)

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λ 2 - λ 1 ( λ 0 2 / 2 π n 3 h 0 ) { [ 1 - e - ( p h 0 ) 2 ] / [ 1 + ( N - 1 ) ( p h 0 ) 2 e - ( p h 0 ) 2 ] } 1 / 2 ,
E ( x , y ) = - exp [ - ( i 2 π / λ 0 ) ( s + s ) ] λ 0 2 s s × - d ξ d η d u d v f ( ξ , η ) T ( u , v ) × exp { - i π λ 0 s [ ( u - ξ ) 2 + ( v - η ) 2 ] - i π λ 0 s [ ( x - u ) 2 + ( y - v ) 2 ] } ,
T ( u , v ) = exp [ - ( i π / λ 0 ) ( u 2 + v 2 ) / ρ ] ,
1 / ρ = - ( 1 / F ) - ( i 4 λ 0 / π D 2 ) ,
E ( x , y ) = - exp { - ( i 2 π / λ 0 ) ( M + 1 ) s - ( i π / λ 0 ) [ ( x 2 + y 2 ) / M s ] } λ 0 2 s 2 M 3 × π D 2 4 - d x d y f ( - x M , - y M ) exp { - i π λ 0 s [ ( x 2 + y 2 ) M 2 ] - ( π D 2 λ 0 s ) 2 [ ( x - x ) 2 + ( y - y ) 2 ] } .
E ( x , y ) exp [ - ( π D / 2 λ 0 s ) 2 ( x 2 + y 2 ) ] .
Δ w = ( 2 λ 0 s / π D ) or Δ w 0 = ( 2 λ 0 s / π D ) .
f ( ξ , η ) = D 0 ( ξ , η ) exp [ - i ( 2 π / λ 0 ) n 0 ξ sin θ 0 ] ,
D 0 ( ξ , η ) = D 1 ( ξ , η ) exp [ - i ψ ( ξ , η ) ] .
ψ ( ξ , η ) = ( 2 π n 0 / λ 0 ) ( Z 0 - h ) / cos θ 0 + ( 2 π n 1 / λ 0 ) ( Z 1 + h ) / cos θ 1 ;
f ( ξ , η ) = D 1 ( ξ , η ) exp [ - ( i 2 π / λ 0 ) n 0 ξ sin θ 0 ] exp [ - i 2 π λ 0 n 3 h ( ξ , η ) ] ,
h ( x , y ) = r h r rect [ ( ξ - ξ r ) / w r ] .
f ( ξ ) = exp [ - ( i 2 π / λ 0 ) n 0 ξ sin θ 0 ] exp { - ( i 2 π λ 0 ) n 3 r h r rect [ ( ξ - ξ r ) / w r ] } ,
r h r rect [ ( ξ - ξ r ) / w r ] ,
exp { - ( i 2 π / λ 0 ) n 3 r h r rect [ ( ξ - ξ r ) / w r ] } = 1 + r rect [ ( ξ - ξ r ) / w r ] { exp [ - ( i 2 π / λ 0 ) n 3 h r ] - 1 } .
f ( ξ ) = exp [ - i ( 2 π / λ 0 ) n 0 ξ sin θ 0 ] ( 1 + r rect [ ( ξ - ξ r ) / w r ] { exp [ - i ( 2 π / λ 0 ) n 3 h r ] - 1 } ) .
r rect [ ( ξ - ξ r ) / w r ] ,
f ( ξ ) = exp [ - ( i 2 π λ 0 ) n 0 ξ sin θ 0 ] ( 1 + r w r δ ( ξ - ξ r ) { exp [ - ( i 2 π / λ 0 ) n 3 h r ] - 1 } ) .
E 1 ( x ) = exp [ - ( i π n 0 / λ 0 ) ( x 2 / M s ) ] - d x f ( - x M ) exp { - i π n 0 ( x ) 2 λ 0 M 2 s - [ ( x - x ) Δ w ] 2 } .
E 1 ( x ) = Δ w ( π ) 1 / 2 exp [ - i π n 0 λ 0 ( x 2 M s - 2 x sin θ 0 M ) - 4 π n 0 2 x 2 sin 2 θ 0 λ 2 M 2 ] + exp [ - i π n 0 λ 0 ( x 2 M s ) ] × r exp [ - ( x - x r Δ w ) 2 + i π n 0 λ 0 ( 2 x r sin θ 0 M - ( x r ) 2 M 2 s ) ] × M w r [ exp ( - i 2 π λ 0 n 3 h r - 1 ) - 1 ] .
I ( x ) = π ( Δ w ) 2 e - 2 α + 2 ( π ) 1 / 2 Δ w e - α r M w r exp [ - ( x - x r Δ w ) 2 ] { cos ( χ - ϕ 1 r + ϕ 2 r ) - cos ( χ - ϕ 1 r ) } + 2 r exp [ - 2 ( x - x r ) 2 / ( Δ w ) 2 ] ( M w r ) 2 { 1 - cos ϕ 2 r } + m r m r exp [ - ( x - x m ) 2 / ( Δ w ) 2 - ( x - x r ) 2 / ( Δ w ) 2 ] × exp [ i ( ϕ 1 m - ϕ 1 r ) ] × M 2 w m w r ( e - i ϕ 2 m - 1 ) ( e + i ϕ 2 r - 1 ) , in which α = 4 π ( n 0 x sin θ 0 ) 2 / ( λ 0 M ) 2 , χ = ( 2 π / λ 0 ) ( n 0 x sin θ 0 / M ) , ϕ 1 r = ( π n 0 / λ 0 ) { 2 x r sin θ 0 / M - ( x r ) 2 / ( M 2 s ) } , and ϕ 2 r = ( 2 π / λ 0 ) n 3 h r .
E 1 ( x ) = Δ w ( π ) 1 / 2 + r exp [ - ( x - x r Δ w ) 2 - i π n 0 ( x r ) 2 λ 0 M 2 s ] M w r { exp ( - i 2 π λ 0 n 3 h r ) - 1 } .
E 1 ( x ) = A - B N + B r = 1 N exp ( + i p h r ) .
B = M w c exp [ - ( i π n 0 x 2 ) / ( λ 0 M 2 s ) ] and p as p = - ( 2 π n 3 / λ 0 ) .
e i p h = F ( p ) ,
σ 2 ( e i p h ) = 1 - F ( p ) F * ( p ) .
E 1 ( x ) = A - N B + B N F ( p ) .
σ 2 [ E 1 ( x ) ] = N ( 1 - F F * ) B B * ,
Δ E 1 ( x ) Δ E 1 ( x ) * σ 2 [ E 1 ( x ) ] .
Δ E 1 ( x ) = [ E 1 ( x ) / λ ] Δ λ .
Δ E 1 ( x ) = ( - Δ B N + B r = 1 N i h r e i p h r 2 π n 3 λ 0 2 + Δ B r = 1 N e i p h r ) Δ λ ,
Δ B = i π n 0 x 2 M w c / ( λ 0 2 M 2 s ) exp [ i π n 0 x 2 / ( λ 0 M 2 s ) ] .
Δ E 1 ( x ) = B r = 1 N i h r 2 π n 3 λ 0 2 e i p h r Δ λ .
Δ E 1 ( x ) Δ E 1 ( x ) * = ( 2 π n 3 Δ λ / λ 0 2 ) 2 { h 0 2 N + ( N 2 - N ) [ ( d / d p ) F ( p ) ] [ ( d / d p ) F ( p ) ] * } B B * .
λ 2 - λ 1 = λ 0 2 2 π n 3 { 1 - F ( p ) F * ( p ) h 0 2 + ( N - 1 ) [ ( d / d p ) F ( p ) ] [ ( d / d p ) F ( p ) ] * } ½ [ decoupled case ] .
Δ λ 1 10 ( λ 2 - λ 1 ) . λ 0 2 20 π n 3 { 1 - F ( p ) F * ( p ) h 0 2 + ( N - 1 ) [ ( d / d p ) F ( p ) ] [ ( d / d p ) F ( p ) ] * } ½ [ l a s e r l i k e c a s e ] .
f ( h r ) = exp [ - h r 2 / ( 2 h 0 2 ) ] / h 0 ( 2 π ) 1 / 2
F ( p ) = exp ( - ½ p 2 h 0 2 ) .
λ 2 - λ 1 = λ 0 2 2 π n 3 h 0 [ 1 - e - p 2 h 0 2 1 + ( N - 1 ) ( h 0 p ) 2 e - p 2 h 0 2 ] ½ .
λ 2 - λ 1 = λ 0 2 / ( 2 π n 3 h 0 ) , when ( p h 0 ) 2 1.
λ 2 - λ 1 = λ 0 [ 1 + ( N - 1 ) ( p h 0 ) 2 ] 1 / 2 , when ( p h 0 ) 2 1.
R = ( N ) 1 / 2 ( 1 - e - p 2 h 0 2 ) 1 / 2 B A - N B + N B e - p 2 h 0 2 / 2 .
d n 3 / d θ 0 = ( 1 / 2 ) ( n 0 2 / n 1 ) ( sin 2 θ 0 / cos 3 θ 1 ) - ( n 0 sin θ 0 / cos 2 θ 0 ) .
d p / d θ 0 = ( - 2 π / λ 0 ) ( d n 3 / d θ 0 ) .
Δ θ 0 = ( λ 0 { 1 - F ( p ) F * ( p ) h 0 2 + ( N - 1 ) [ ( d / d p ) F ( p ) ] [ ( d / d p ) F ( p ) ] * } ½ ) / { π n 0 [ 2 sin θ 0 / cos 2 θ 0 - n 0 sin 2 θ 0 / ( n 1 cos 3 θ 1 ) ] } .
Δ θ 0 = ( λ 0 { ( 1 - e - p 2 h 0 2 ) / [ 1 + ( N - 1 ) ( h 0 p ) 2 e - p 2 h 0 2 ] } 1 / 2 ) / [ π n 0 h 0 ( 2 sin θ 0 cos 2 θ 0 - n 0 sin 2 θ 0 n 1 cos 3 θ 1 ) ] .
Δ λ λ = λ 2 π n 3 h 0 ( { 1 - exp [ - ( 2 π n 3 h 0 / λ ) 2 ] } / { 1 + [ ( Δ w 0 / w r ) - 1 ] [ h 0 ( 2 π n 3 / λ ) ] 2 exp [ - ( 2 π n 3 h 0 / λ ) 2 ] } ) ,
λ 2 - λ 1 = 80 Å
Δ θ = 1 10 { λ cos 2 θ 0 π h 0 [ n 0 - ( n 0 2 / n 1 ) ] } ½ ,

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