Abstract

To simulate nondiffraction-limited laser beams numerically, one needs to understand their nature better. For a phase aberration the correlation length of which is much less than the width of the beam, one can show that the far-field irradiance distribution can be written as the sum of two beams. One beam is the attenuated diffraction-limited beam; the other is a much wider beam the exact shape of which is closely related to the power spectrum of the phase fluctuations. Computer simulations of random phase aberrations are shown to agree well with the analytic predictions. Attempts at simulation of nondiffraction-limited beams for nonlinear wave propagation problems are discussed.

© 1974 Optical Society of America

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References

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  1. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  2. J. Ruze, Proc. IEEE 54, 633 (1966).
    [CrossRef]
  3. W. B. Davenport, W. L. Root, Random Signals and NoiseMcGraw-Hill, New York, 1958.
  4. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, New York, 1963).
  5. R. Barakat, Optica Acta 18, 683 (1971).
    [CrossRef]
  6. W. L. Root, T. S. Pitcher, Ann. Math. Statist. 26(2), P313 (1955).
    [CrossRef]
  7. C. B. Hogge, M. Burlakoff, AFWL TR 73–77 (1973).

1971 (1)

R. Barakat, Optica Acta 18, 683 (1971).
[CrossRef]

1966 (2)

1955 (1)

W. L. Root, T. S. Pitcher, Ann. Math. Statist. 26(2), P313 (1955).
[CrossRef]

Barakat, R.

R. Barakat, Optica Acta 18, 683 (1971).
[CrossRef]

Burlakoff, M.

C. B. Hogge, M. Burlakoff, AFWL TR 73–77 (1973).

Davenport, W. B.

W. B. Davenport, W. L. Root, Random Signals and NoiseMcGraw-Hill, New York, 1958.

Hogge, C. B.

C. B. Hogge, M. Burlakoff, AFWL TR 73–77 (1973).

Kogelnik, H.

Li, T.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, New York, 1963).

Pitcher, T. S.

W. L. Root, T. S. Pitcher, Ann. Math. Statist. 26(2), P313 (1955).
[CrossRef]

Root, W. L.

W. L. Root, T. S. Pitcher, Ann. Math. Statist. 26(2), P313 (1955).
[CrossRef]

W. B. Davenport, W. L. Root, Random Signals and NoiseMcGraw-Hill, New York, 1958.

Ruze, J.

J. Ruze, Proc. IEEE 54, 633 (1966).
[CrossRef]

Ann. Math. Statist. (1)

W. L. Root, T. S. Pitcher, Ann. Math. Statist. 26(2), P313 (1955).
[CrossRef]

Appl. Opt. (1)

Optica Acta (1)

R. Barakat, Optica Acta 18, 683 (1971).
[CrossRef]

Proc. IEEE (1)

J. Ruze, Proc. IEEE 54, 633 (1966).
[CrossRef]

Other (3)

W. B. Davenport, W. L. Root, Random Signals and NoiseMcGraw-Hill, New York, 1958.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, New York, 1963).

C. B. Hogge, M. Burlakoff, AFWL TR 73–77 (1973).

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Figures (6)

Fig. 1
Fig. 1

Computer simulations of a phase-aberrated beam with σϕ = 0.083 wavelength and l0 ≃ 5 cm. Dashed circle is centered in optical axis and represents the 10% irradiance contour of ideal beam. Innermost solid contours are 90% relative irradiance. Subsequent contours are spaced with a 20% relative energy decrement.

Fig. 2
Fig. 2

Computer simulations of a phase-aberrated beam with σϕ = 0.125 wavelength and l0 ≃ 5 cm. Dashed circle is centered in optical axis and represents the 10% irradiance contour of ideal beam. Innermost solid contours are 90% relative irradiance. Subsequent contours are spaced with a 20% relative energy decrement.

Fig. 3
Fig. 3

Computer simulations of a phase-aberrated beam with σϕ = 0.166 wavelength and l0 ≃ 5 cm. Dashed circle is centered in optical axis and represents the 10% irradiance contour of ideal beam. Innermost solid contours are 90% relative irradiance. Subsequent contours are spaced with a 20% relative energy decrement.

Fig. 4
Fig. 4

Dashed curve is irradiance profile of ideal beam. Solid curves are irradiance profile traces selected from two of the computer realizations shown in Fig. 2.

Fig. 5
Fig. 5

Solid curves are the same as in Fig. 4. Dashed curve shows the predicted analytic result of Sec. II.

Fig. 6
Fig. 6

Comparison of pseudowavelength-scaled, nondiffraction-limited beam and the predicted result of Sec. II shown in Fig. 5.

Equations (20)

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U i ( x , y ) = A 0 exp [ - ( x 2 + y 2 ) / ω 1 2 ] · exp [ i ϕ ( x , y ) ] ,
U 0 ( x 0 , y 0 ) = A 0 λ f + - exp [ - ( x 1 2 + y 1 2 ) / ω 1 2 ] · × exp [ + i ϕ ( x 1 , y 1 ) ] · exp [ - i k f ( x 0 x 1 + y 0 y 1 ] d x 1 d y 1 ,
I ( x 0 , y 0 ) = A 0 2 λ 2 f 2 · + - × exp [ - ( x 1 2 + y 1 2 + x 2 2 + y 2 2 ) / ω 1 2 ] · × exp [ - { σ ϕ 2 - C ϕ ( [ ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 ] 1 / 2 ) } ] · × exp { - ( i k / f ) [ x 0 ( x 1 - x 2 ) + y 0 ( y 1 - y 2 ) ] } × d x 1 d y 1 d x 2 d y 2 .
I ( r 0 ) = ( π ω 1 ) 2 A 0 2 λ 2 f 2 e - σ ϕ 2 · 0 × exp ( - r 2 2 ω 1 2 ) exp [ C ϕ ( r ) ] J 0 ( k r 0 r f ) r d r .
I ( r 0 ) = e - σ ϕ 2 [ I d ( r 0 ) + I 0 ω 1 2 · 0 × exp ( - r 2 2 ω 1 2 ) C ϕ ( r ) J 0 ( k r 0 r f ) r d r ] ,
P incho / Total power = σ ϕ 2 / ( 1 + σ ϕ 2 ) .
I ( 0 ) = exp ( - σ ϕ 2 ) [ I 0 + I 0 ( l 0 / ω 1 ) 2 · σ ϕ 2 ] .
( l 0 / ω 1 ) 2 · σ ϕ 2 1.
I ( r 0 ) = exp ( - σ ϕ 2 ) [ I d ( r 0 ) + ( I 0 / 2 π ω 1 2 ) · Ω ( k r 0 / 2 π f ) ] ,
Ω ( ρ ) = 2 π 0 C ϕ ( r ) J 0 ( 2 π ρ r ) r d r .
I ( r ) = exp ( - σ ϕ 2 ) { I 0 exp [ - 2 r 2 / ω 2 f ] + I 0 exp [ - 2 r 2 / ( ω f ) 2 ] } ,
I 0 = I 0 [ σ ϕ 2 l 0 2 / ( l 0 2 + 2 ω 1 2 ) ] ,
ω f = ω f · [ ( l 0 2 + 2 ω 1 2 ) 1 / 2 / l 0 ]
C ϕ ( r ) = 2 σ ϕ 2 [ J 1 ( 3.8 r / l 0 ) / ( 3.8 r / l 0 ) ] ,
I ( r 0 ) = exp ( - σ ϕ 2 ) [ I d ( r 0 ) + I 0 ] for r ω f , = exp ( - σ ϕ 2 ) I d ( r 0 ) for r > ω f ,
I 0 = I 0 · 2 ( σ ϕ l 0 / 3.8 ω 1 ) 2 ,
ω f = ω f · ( 1.9 ω 1 / l 0 ) .
a n m = 1 T 2 ( T / 2 ) ( - T / 2 ) C ϕ ( τ x , τ y ) exp [ - i ( n τ x + m τ y ) w 0 ] d τ x d τ y ,
ϕ ( x , y ) = n m n m = - = - = = b n m Z n m exp [ i ( n x + m y ) w 0 ] .
a n m = 2.7 ( σ ϕ 2 l 0 / T 2 π ) for w 0 ( n 2 + m 2 ) 1 / 2 _ 1.22 π / l 0 , = 0 otherwise ,

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