Abstract

By direct numerical-integration comparisons, it is established that the Fresnel approximation for collimated propagation is quite good (within about 2% in amplitude and 0.02 rad in phase) in every case, including that with the limit of a high Fresnel number. Moreover, the Fresnel approximation begins to break down in phase for spherical-wave propagation for beams faster than about f/12. It has been discovered, however, that if one also invokes the paraxial approximation, that is, replaces the spherical wave by a quadratic phase front, then the Fresnel approximation becomes valid for expanding (or diverging) beams as well. This result is substantiated through the use of stationary-phase arguments.

© 1981 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 45–60.
  2. P. Horwitz, “Asymptotic theory of unstable resonator modes,” J. Opt. Soc. Am. 63, 1528–1543 (1973).
    [Crossref]
  3. P. Horwitz, “Modes in misaligned unstable resonators,” Appl. Opt. 15, 167–178 (1976).
    [Crossref] [PubMed]
  4. W. H. Southwell, “Asymptotic solution of the Huygens–Fresnel integral in circular coordinates,” Opt. Lett. 3, 100–102 (1978).
    [Crossref]
  5. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
    [Crossref]
  6. F. D. Feiock, “Wave propagation in optical systems with large apertures,” J. Opt. Soc. Am. 68, 485–489 (1978).
    [Crossref]
  7. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123 (1962).
    [Crossref]
  8. E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am. 58, 1235–1237 (1968).
    [Crossref]
  9. M. Abramowity and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 302.
  10. E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1976 (1975).
    [Crossref]

1979 (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[Crossref]

1978 (2)

1976 (1)

1975 (1)

E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1976 (1975).
[Crossref]

1973 (1)

1968 (1)

1962 (1)

Abramowity, M.

M. Abramowity and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 302.

Feiock, F. D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 45–60.

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[Crossref]

Horwitz, P.

Lalor, E.

Leith, E. N.

Siegman, A. E.

E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1976 (1975).
[Crossref]

Southwell, W. H.

Stegun, I. A.

M. Abramowity and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 302.

Sziklas, E. A.

E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1976 (1975).
[Crossref]

Upatnieks, J.

Am. J. Phys. (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[Crossref]

Appl. Opt. (2)

P. Horwitz, “Modes in misaligned unstable resonators,” Appl. Opt. 15, 167–178 (1976).
[Crossref] [PubMed]

E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1976 (1975).
[Crossref]

J. Opt. Soc. Am. (4)

Opt. Lett. (1)

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 45–60.

M. Abramowity and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 302.

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Figures (13)

Fig. 1
Fig. 1

Fractional error in the amplitude produced by the Fresnel approximation and the asymptotic solution on an incident uniform plane wave for a field point x = 0. The propagation parameters used here, as well as for the results shown in Figs. 210, are a = 6.75 cm, λ = 0.001 cm, and z, depending on the Fresnel number NF [Nf = a2/(λz)].

Fig. 2
Fig. 2

Phase error in radians for an on-axis field point x = 0.

Fig. 3
Fig. 3

Fractional error in the amplitude produced by the Fresnel approximation and the asymptotic solution on an incident uniform plane wave at a field point x = 0.9a.

Fig. 4
Fig. 4

Phase error in radians for a field point x = 0.9a.

Fig. 5
Fig. 5

Error due to the Fresnel approximation for an on-axis field point x = 0.

Fig. 6
Fig. 6

Error due to the Fresnel approximation for an on-axis field point x = 0.

Fig. 7
Fig. 7

Error due to the Fresnel approximation for an on-axis field point x = 0.

Fig. 8
Fig. 8

Error due to the Fresnel approximation for an on-axis field point x = 0.

Fig. 9
Fig. 9

Error due to the Fresnel approximation for an off-axis field point x = 0.9a.

Fig. 10
Fig. 10

Error due to the Fresnel approximation for a field point in the shadow region x = 1.1a.

Fig. 11
Fig. 11

Phase error due to the Fresnel approximation for an expanding spherical wave with a2/(λz) = 450 and x = 0.9a.

Fig. 12
Fig. 12

Propagation through an afocal beam-expander telescope with a = 6.75, λ = 0.001, z = 101.25, and M = 3. The expanding beam has an f number equal to 3.75. In (a) the exact plane-wave-spectrum propagator with a spherical wave was used. In (b) the Fresnel approximation with a spherical wave was used.

Fig. 13
Fig. 13

Propagation through the same beam expander described in Fig. 12, except that in (a) the exact propagator with a quadratic expanding wave (paraxial approximation) was used and in (b) both the Fresnel approximation and the paraxial approximation were used.

Equations (30)

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U ( x , y , z ) = 1 2 π A U ( x , y , 0 ) × exp ( - i k r ) r z r ( i k + 1 r ) d x d y ,
r = [ z 2 + ( x - x ) 2 + ( y - y ) 2 ] 1 / 2
r z + ( x - x ) 2 + ( y - y ) 2 2 z - [ ( x - x ) 2 + ( y - y ) 2 ] 2 8 z 3 + ·
U ( x , y , z ) = i exp ( - i k z ) λ z U ( x , y , 0 ) × exp { - i k [ ( x - x ) 2 + ( y - y ) 2 ] 2 z } d x d y ,
( 2 + k 2 ) U = 0 ,
( 2 x 2 + 2 y 2 - 2 i k z ) U = 0.
U ( x , y , z ) = A ( α , β , z ) exp [ i k ( α x + β y ) ] d α d β ,
H = A ( α , β , z ) A 0 ( α , β , 0 ) = exp [ - i k z ( 1 - α 2 - β 2 ) 1 / 2 ] ,
A 0 ( α , β , 0 ) = U ( x , y , 0 ) exp [ - i k ( α x + β b ) ] d x d y .
F - 1 { exp [ - i k z ( 1 - α 2 - β 2 ) 1 / 2 ] } = 1 2 π ( i k + 1 R ) z R exp ( - i k R ) R ,
R = ( x 2 + y 2 + z 2 ) 1 / 2 .
A ( α , β , z ) A 0 ( α , β , 0 ) = exp ( - i k z ) exp [ i k z ( α 2 + β 2 ) 2 ] .
F - 1 { exp [ i k z ( α 2 + β 2 ) / 2 } = k i 2 π z exp [ - i k ( x 2 + y 2 ) 2 z ] .
U e ( x , z ) = ( i λ ) 1 / 2 - a a U ( x , 0 ) exp ( - i k r ) r 1 / 2 z r d x ,
U f ( x , z ) = ( i λ z ) 1 / 2 exp ( - i k z ) × - a a U ( x , 0 ) exp [ - i k ( x - x ) 2 2 z ] d x ,
r = [ z 2 + ( x - x ) 2 ] 1 / 2 .
Δ x = 9 a / ( 2 N F ) ,
N F = a 2 / ( λ Z ) .
U A ( x , z ) = ( i 2 ) 1 / 2 exp ( - i k z ) { E [ ( 2 λ z ) 1 / 2 ( a - x ) ] - E [ - ( 2 λ z ) 1 / 2 ( a + x ) ] } ,
E ( y ) = ( 1 2 i ) 1 / 2 sgn ( y ) + ϕ ( y ) exp ( - i π 2 y 2 ) ,
ϕ ( y ) = sgn ( y ) [ if ( y ) - g ( y ) ] .
f ( y ) = 1 + 0.926 y 2 + 1.792 y + 3.104 y 2 ,
g ( y ) = 1 2 + 4.142 y + 3.492 y 3 + 6.67 y 3 .
U ( x , 0 ) = exp { - i k F [ 1 - ( 1 + x 2 / F 2 ) 1 / 2 ] } ,
U ( x , 0 ) = exp [ - i k x 2 / ( 2 F ) ] ,
U ( x , z ) = ( i λ ) 1 / 2 - a a exp [ - i Φ ( x , x ) ] r d x ,
Φ e ( x , x ) = k ( F [ 1 - ( 1 + x 2 F 2 ) 1 / 2 ] + [ z 2 + ( x - x ) 2 ] 1 / 2 + ( z - F ) { 1 - [ 1 + x 2 ( z - F ) 2 ] 1 / 2 } ) ,
Φ p ( x , x ) = k [ - x 2 2 F + ( x - x ) 2 2 z + x 2 2 ( z - F ) ] - k z
Φ p ( x , x ) = k ( x - M x ) 2 2 M z .
Φ x = 0.