Abstract

Using Fresnel diffraction integrals, calculations have been made of the irradiance and power distributions in the near field and in the vicinity of the focus for Gaussian beams focused through annular apertures. Universal curves have been plotted which display the calculations in terms of dimensionless parameters. For very large focal length infrared systems it has been found that the irradiance distribution is not symmetrical about the geometrical focal plane as is commonly assumed. Gaussian and sinusoidal phase aberrations in the aperture field have been included.

© 1972 Optical Society of America

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References

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  1. A. L. Buck, Proc. IEEE 55, 448 (1967).
    [CrossRef]
  2. J. P. Campbell, L. G. DeShazer, J. Opt. Soc. Am. 59, 1427 (1969).
    [CrossRef]
  3. G. O. Olaofe, J. Opt. Soc. Am. 60, 1654 (1970).
    [CrossRef]
  4. R. G. Schell, G. Tyras, J. Opt. Soc. Am. 61, 31 (1971).
    [CrossRef]
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4.
  6. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 439.
  8. L. D. Dickson, Appl. Opt. 9, 1854 (1970).
    [CrossRef] [PubMed]
  9. J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), pp. 205–206.
  10. J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), p. 184 (problem 9–3).

1971 (1)

1970 (2)

1969 (1)

1967 (1)

A. L. Buck, Proc. IEEE 55, 448 (1967).
[CrossRef]

1966 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 439.

Buck, A. L.

A. L. Buck, Proc. IEEE 55, 448 (1967).
[CrossRef]

Campbell, J. P.

DeShazer, L. G.

Dickson, L. D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4.

Kogelnik, H.

Li, T.

Olaofe, G. O.

Schell, R. G.

Stone, J. M.

J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), pp. 205–206.

J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), p. 184 (problem 9–3).

Tyras, G.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 439.

Appl. Opt. (2)

J. Opt. Soc. Am. (3)

Proc. IEEE (1)

A. L. Buck, Proc. IEEE 55, 448 (1967).
[CrossRef]

Other (4)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4.

J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), pp. 205–206.

J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), p. 184 (problem 9–3).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 439.

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Figures (8)

Fig. 1
Fig. 1

Normalized focal-point irradiance (λR)2If/πb2P0 vs b/w with a/b as a changing parameter. The curves are labeled with the value α. For all values of α, the curves show that the focal-point irradiance is maximized for β → 0 or w → ∞.

Fig. 2
Fig. 2

Normalized on-axis irradiance Ī(0,ζ) as a function of normalized distance ζ from the aperture with α, β, and γ as changing parameters. For each figure, the curves sharpen as γ increases. Note that by comparing (a) with (c) and (b) with (d), the change from β = 0 to β = 1 does not greatly change the general shape of the on-axis irradiance curves; this behavior is indicated in Fig. 1. The β = 0 solution for Ī(0,ζ) is Ī(0,ζ) = (1/ζ2) sinc2[γ(1 − α2)(ζ − 1)/2ζ], which shows clearly that Ī(0,ζ) becomes a more sharply peaked function of ζ as γ increases.

Fig. 3
Fig. 3

The Ī(ξ,ζ) surface. The normalized irradiance is plotted on the vertical axis. The [ξ,ζ,Ī(ξ,ζ)] coordinate axes form a right-hand coordinate system. The axes cover the ranges 0 ≤ Ī(ξ,ζ) ≤ 1.6, 0 ≤ ξ ≤ 2.5, and 0.5 ≤ ζ ≤ 1.5. In each figure, Φ = 0.

Fig. 4
Fig. 4

The P ¯ ( ξ, ζ ) surface. Each plot here corresponds to a plot in Fig. 3. The normalized power is plotted on the vertical axis, The [ ξ, ζ , P ¯ ( ξ, ζ ) ] coordinate axes form a left-hand coordinate system. The axes cover the ranges 0 P ¯ ( ξ, ζ ) 1, 0 ≤ ξ ≤ 2.5, and 0.5 ≤ ξ ≤ 1.5. In each case, Φ = 0.

Fig. 5
Fig. 5

Focal-plane profiles. In (a) and (b) we show P ¯ ( ξ, 1 ) as a function of ξ with α and β as changing parameters. Each curve is labeled by the corresponding value of α. In (a), β = 0, and in (b), β = 1. In (c) and (d), we show the normalized power as a function of β for selected values of ξ and α. In (d), the curves are labeled by the corresponding value of α; the curves in (c) correspond to the same values of α. In (e) and (f) we show constant power loci; each curve is labeled by the corresponding constant value of normalized power. Graphs (e) and (f) would be useful for quickly estimating the required radius for a focal-plane aperture that is to transmit, say, 50% of the total focal-plane power.

Fig. 6
Fig. 6

Ī(R/z) = |ū(R/z)|2 vs R/z, calculated from Eqs. (27) and (28) with γ′ and α as changing parameters. For all graphs we have β = 1. Each curve is labeled by the corresponding value of Δ. The Δ = 0 curves peak at R/z = 1 as was discussed in the text. For 0 ≤ Δ ≤ 1, it is seen that a change of the system focus will restore most of the peak irradiance at a fixed receiver site. The Gaussian phase distortion is thus seen to contain a significant change-of-focus component.

Fig. 7
Fig. 7

The Ī(ξ,ζ) surface for a sinusoidal phase distortion in the aperture field. The parameter that changes is δ; for all graphs α = β = 0, γ = 20, and ν = 1. The right-hand coordinate system axes are identified as in Fig. 3 except that we now have 0 ≤ ξ ≤ 3.0. Note that the Ī(ξ,ζ) surface contours depend strongly upon the sign of δ. This is because of the low value of the frequency ν, for small values of ν, the sinusoidal phase distortion contains a significant change-of-focus component. For ν = 5 the change-of-focus component is almost negligible and the Ī(ξ,ζ) surface contours (not shown here) are virtually independent of the sign of the parameter δ.

Fig. 8
Fig. 8

Focal-point irradiance Ī(0,1) as a function of the sinusoidal phase amplitude δ. Neither of the curves departs greatly from the α = β = 0 curve, which goes as J02(2πδ).

Equations (42)

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υ ( r ) = υ 0 exp [ ( r / w ) 2 ( i π r 2 / λ R ) i Φ ] .
P 0 = 2 π a b d r r | υ ( r ) | 2 .
υ 0 2 = ( 2 P 0 / π w 2 ) / { exp [ 2 ( a / w ) 2 ] exp [ 2 ( b / w ) 2 ] } .
u ( r , z ) = ( 2 π / λ z ) a b d ρ ρ υ ( ρ ) J 0 ( 2 π ρ r / λ z ) exp ( i π ρ 2 / λ z ) ,
P ( r , z ) = 2 π 0 r d ρ ρ | u ( ρ , z ) | 2 .
I ( 0 , z ) = υ 0 2 NUM / DEN ,
NUM = { exp [ ( a / w ) 2 ] exp [ ( b / w ) 2 ] } 2 + 4 exp [ ( a 2 + b 2 ) / w 2 ] sin 2 [ π ( b 2 a 2 ) ( z R ) / 2 λ z R ] , DEN = ( λ z / π w 2 ) 2 + ( z R ) 2 / R 2 .
I ( 0 , z 0 ; z 0 , w , a , b ) > I ( 0 , z 0 ; R , w , a , b )
I ( 0 , z ; z 0 , w , a , b ) > I ( 0 , z 0 ; z 0 R , w , a , b )
I f = { 2 π w 2 P 0 exp [ ( b 2 a 2 ) / w 2 ] 1 } / ÷ { ( λ R ) 2 exp [ ( b 2 a 2 ) / w 2 ] + 1 } .
lim w I f = π ( b 2 a 2 ) P 0 / ( λ R ) 2 .
α = a / b ,
β = b / w .
( λ R ) 2 I f / π b 2 P 0 = ( 2 / β 2 ) { exp [ β 2 ( 1 α 2 ) ] 1 } / { exp [ β 2 ( 1 α 2 ) ] + 1 } .
I ( 0 , z ) = ( 2 P 0 / π w 2 ) / { ( λ z / π w 2 ) 2 + [ ( z R ) 2 / R 2 ] } .
γ = π b 2 / λ R ,
ζ = z / R ,
I ¯ ( 0 , ζ ) = I ( 0 , z ) / I f .
ξ = 2 r b / λ R ,
u ( ξ , ζ ) = 2 γ υ 0 ζ α 1 J 0 ( π ξ x / ζ ) exp [ ( β x ) 2 i x 2 γ ( ζ 1 ) / ζ i Φ ] x d x .
u ¯ ( ξ , ζ ) = α 1 J 0 ( π ξ x / ζ ) exp [ ( β x ) 2 i x 2 γ ( ζ 1 ) ] / ζ i Φ ] x d x / ( ζ / 2 β 2 ) [ exp ( α 2 β 2 ) exp ( β 2 ) ] .
I ¯ ( ξ , ζ ) = | u ¯ ( ξ , ζ ) | 2 = | u ( ξ , ζ ) | 2 / I f ,
P ¯ ( ξ , ζ ) = P ( r , z ) / P 0 ,
P ¯ ( ξ , ζ ) = ( π / β ) 2 exp ( α 2 β 2 ) exp ( β 2 ) exp ( α 2 β 2 ) + exp ( β 2 ) 0 ξ η I ¯ ( η , ζ ) d η .
P ¯ ( ξ , 1 ) 1.25 ( ξ 0.14 ) ,
lim ξ P ¯ ( ξ , 1 ) = 1.
P ¯ ( ξ , 1 ) = f 1 ( α , β ) f 2 ( ξ , α , β ) ,
f 1 ( α , β ) = ( 2 β ) 2 [ exp ( 2 α 2 β 2 ) exp ( 2 β 2 ) ] 1 ,
f 2 ( ξ , α , β ) = 0 π ξ y d y [ a 1 d x x J 0 ( y x ) exp ( β 2 x 2 ) ] 2 .
lim ξ f 2 = α 1 d η η exp [ ( β η ) 2 ] α 1 d σ σ × exp [ ( β σ ) 2 ] 0 d ρ ρ J 0 ( ρ η ) J 0 ( ρ σ ) .
0 d p ρ J 0 ( ρ η ) J 0 ( ρ σ ) = δ ( σ η ) / σ .
lim ξ f 2 = α 1 d η η exp [ 2 ( β η ) 2 ] = 1 / f 1 ,
Φ = ( 2 π Δ ) exp [ 2 ( r / w ) 2 ] .
Φ = ( 2 π Δ ) [ 1 2 ( r / w ) 2 + higher order terms ] .
1 / R = 1 / R 4 Δ λ / w 2 .
u ¯ ( R / z ) = 2 β 2 α 1 d x x g ( x , R / z ) exp [ ( β x ) 2 2 π i Δ exp ( 2 β 2 x 2 ) ] ,
g ( x , R / z ) = exp [ i x 2 γ ( 1 R / z ) / ( R / z ) ] exp ( α 2 β 2 ) exp ( β 2 ) ,
γ = π b 2 / λ z .
Φ = 2 π δ cos ( 2 π r / T ) .
Φ = 2 π δ cos ( 2 π ν x ) ,
exp ( i z cos θ ) = k = + i k J k ( z ) exp ( i k θ ) ,
u ¯ ( 0,1 ) = J 0 ( 2 π δ ) .

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