Abstract

A solution is presented for the field produced when an infinitely short pulse is focused by an infinite-aperture system. In contrast to the results obtained in the standard theory of focusing of monochromatic waves, it has proved possible to evaluate the propagation integrals in terms of known functions and generalized functions. The result provides a useful expression for the field in the focal region of small-f-number systems. The field is singular in certain space–time regions because the incident pulse was taken to be infinitely short. The singularities can be removed for pulses of finite duration by convolving the field with the time variation of the incident pulse.

© 1989 Optical Society of America

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References

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  1. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 12.1.
  2. The relevant integrals for both the finite aperture and the infinite aperture were evaluated numerically by W. H. Carter, “Band-limited angular-spectrum approximation to a spherical scalar wave field,” J. Opt. Soc. Am. 65, 1054–1058 (1975).
    [CrossRef]
  3. A. J. Devaney, G. C. Sherman, “Plane-wave representations for scalar wave fields,” Soc. Ind. Appl. Math. Rev. 15, 765–786 (1973).
  4. A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966), p. 18, Eq. (2.19).
  5. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 736, Sec. 6.677, no. 3.
  6. See Ref. 5, p. 477, Sec. 3.893, no. 2.
  7. See Ref. 5, p. 682, Sec. 6.554, no. 2.
  8. W. Heitler, Quantum Theory of Radiation, 3rd ed. (Oxford U. Press, Oxford, UK, 1954), p. 70.
  9. D. S. Jones, Generalized Functions (McGraw-Hill, New York, 1966), p. 256, example 32.
  10. See Ref. 5, p. 152, Sec. 2.562, no. 2.

1975 (1)

1973 (1)

A. J. Devaney, G. C. Sherman, “Plane-wave representations for scalar wave fields,” Soc. Ind. Appl. Math. Rev. 15, 765–786 (1973).

Baños, A.

A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966), p. 18, Eq. (2.19).

Carter, W. H.

Devaney, A. J.

A. J. Devaney, G. C. Sherman, “Plane-wave representations for scalar wave fields,” Soc. Ind. Appl. Math. Rev. 15, 765–786 (1973).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 736, Sec. 6.677, no. 3.

Heitler, W.

W. Heitler, Quantum Theory of Radiation, 3rd ed. (Oxford U. Press, Oxford, UK, 1954), p. 70.

Jones, D. S.

D. S. Jones, Generalized Functions (McGraw-Hill, New York, 1966), p. 256, example 32.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 736, Sec. 6.677, no. 3.

Sherman, G. C.

A. J. Devaney, G. C. Sherman, “Plane-wave representations for scalar wave fields,” Soc. Ind. Appl. Math. Rev. 15, 765–786 (1973).

Stamnes, J.

J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 12.1.

J. Opt. Soc. Am. (1)

Soc. Ind. Appl. Math. Rev. (1)

A. J. Devaney, G. C. Sherman, “Plane-wave representations for scalar wave fields,” Soc. Ind. Appl. Math. Rev. 15, 765–786 (1973).

Other (8)

A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966), p. 18, Eq. (2.19).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 736, Sec. 6.677, no. 3.

See Ref. 5, p. 477, Sec. 3.893, no. 2.

See Ref. 5, p. 682, Sec. 6.554, no. 2.

W. Heitler, Quantum Theory of Radiation, 3rd ed. (Oxford U. Press, Oxford, UK, 1954), p. 70.

D. S. Jones, Generalized Functions (McGraw-Hill, New York, 1966), p. 256, example 32.

See Ref. 5, p. 152, Sec. 2.562, no. 2.

J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 12.1.

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Figures (2)

Fig. 1
Fig. 1

Boundary of the region of zero field in the x = 0 plane at times (a) t = −t0 and (b) t = t0, where t0 is a positive constant. The field is zero inside the solid lines except on the dashed–dotted semicircles.

Fig. 2
Fig. 2

Boundary of the region of zero field in the x = 0 plane for the finite-aperture case. The field is zero to the right of the solid curved line.

Equations (55)

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2 V ( x , y , z , t ) - 1 c 2 2 t 2 V ( x , y , z , t ) = 0
V ( x , y , - f , t ) = δ ( t + r f / c ) r f ,
r f = ( x 2 + y 2 + f 2 ) 1 / 2 .
V ( x , y , z , t ) = δ ( t + r / c ) r ,
r = ( ρ 2 + z 2 ) 1 / 2
ρ = ( x 2 + y 2 ) 1 / 2 .
V ( x , y , z , t ) = Re { i 2 π 2 0 - - v ( k x , k y , ω ) × exp [ i ( k x x + k y y + k z z - ω t ) ] d k x d k y d ω } ,
δ ( t ) = 1 2 π - exp ( - i ω t ) d ω .
δ ( t + r / c ) r = Re { 1 π 0 exp [ - i ( k r + ω t ) ] r d ω } .
exp ( i k r ) r = i 2 π - - exp [ i ( k x x + k y y + k z z - ω t ) ] k z × d k x d k y .
exp ( - i k r ) r = - i 2 π - - exp [ i ( k x x + k y y - k z * z - ω t ) ] k z * × d k x d k y ,
δ ( t + r / c ) r = - Re { i 2 π 2 × 0 - - exp [ i ( k x x + k y y - k z * z - ω t ) ] k z * × d k x d k y d ω } .
v ( k x , k y , ω ) = - exp [ i f ( k z - k z * ) ] k z * .
V ( x , y , z , t ) = g ( x , y , z , t ) + h ( x , y , z + 2 f , t ) ,
g ( x , y , z , t ) = - Re { i 2 π 2 × 0 H exp [ i ( k x x + k y y + k z z - ω t ) ] k z × d k x d k y d ω }
h ( x , y , z , t ) = Re { i 2 π 2 × 0 I exp [ i ( k x x + k y y + k z z - ω t ) ] k z × d k x d k y d ω } .
J 0 ( λ ρ ) = 1 2 π 0 2 π exp [ i λ ρ cos ( α - θ ) ] d α
h ( x , y , z , t ) = Re { 1 π 0 k J 0 ( λ ρ ) ( λ 2 - k 2 ) 1 / 2 × exp [ - z ( λ 2 - k 2 ) 1 / 2 - i ω t ] λ d λ d ω } .
h ( x , y , z , t ) = c π 0 exp ( - z ξ ) 0 J 0 [ ρ ( ξ 2 + k 2 ) 1 / 2 ] × cos ( k c t ) d k d ξ .
h ( x , y , z , t ) = c Θ ( ρ - c t ) π ( ρ 2 - c 2 t 2 ) 1 / 2 × 0 exp ( - z ξ ) cos [ ξ ( ρ 2 - c 2 t 2 ) 1 / 2 ] d ξ ,
h ( x , y , z , t ) = Θ ( ρ - c t ) c z π ( ρ 2 - c 2 t 2 ) 1 / 2 ( r 2 - c 2 t 2 )
g ( x , y , z , t ) = δ ( t + r / c ) r - h ( x , y , z , t )
g ( x , y , 0 , t ) = - Re [ i π 0 I ( x , y , ω ) exp ( - i ω t ) d ω ] ,
I ( x , y , ω ) = 1 2 π H exp [ i ( k x x + k y y ) ] k z d k x d k y .
I ( x , y , ω ) = 0 k λ J 0 ( λ ρ ) ( k 2 - λ 2 ) 1 / 2 d λ .
I ( x , y , ω ) = sin ( k ρ ) ρ .
g ( x , y , 0 , t ) = 1 2 [ δ ( t + ρ / c ) ρ - δ ( t - ρ / c ) ρ ] .
g ( x , y , z , t ) = - δ ( t - r / c ) r + h ( x , y , z , t ) .
lim σ 0 + σ x 2 + σ 2 = π δ ( x )
δ [ f ( x ) ] = i δ ( x - x i ) f ( x i ) ,
lim z 0 h ( x , y , z , t ) = δ ( t + ρ / c ) + δ ( t - ρ / c ) 2 ρ ,
V ( x , y , z , t ) = δ ( t + r / c ) r + h ˜ ( x , y , z , t ) + h ˜ ( x , y , z + 2 f , t )
V ( x , y , z , t ) = - δ ( t - r / c ) r + h ˜ ( x , y , z , t ) + h ˜ ( x , y , z + 2 f , t )
h ˜ ( x , y , z , t ) = Θ ( ρ - c t ) c z π ( ρ 2 - c 2 t 2 ) 1 / 2 ( r 2 - c 2 t 2 )
V ( x , y , 0 , t ) = δ ( t + ρ / c ) - δ ( t - ρ / c ) 2 ρ + h ˜ ( x , y , 2 f ) .
V ( x , y , z , t ) = G ( x , y , z , t ) + H ( x , y , z , t )
G ( x , y , z , t ) = δ ( t + r / c ) r Θ ( - z ) - δ ( t - r / c ) r Θ ( z ) ,
H ( x , y , z , t ) = h ˜ ( x , y , z , t ) + h ˜ ( x , y , z + 2 f , t ) .
2 h ˜ ( x , y , z , t ) - 1 c 2 2 t 2 h ˜ ( x , y , z , t ) = δ ( t - ρ / c ) + δ ( t + ρ / c ) ρ d δ ( z ) d z .
V ˜ ( x , y , - f , t ) = a ( t + r f / c ) r f ,
V ˜ ( x , y , - f , t ) = - a ( t - t ) δ ( t + r f / c ) r f d t ,
V ˜ ( x , y , z , t ) = - a ( t - t ) V ( x , y , z , t ) d t .
V ˜ ( x , y , z , t ) = a ( t + r / c ) r Θ ( - z ) - a ( t - r / c ) r Θ ( z ) + a ( t - t ) H ( x , y , z , t ) d t .
I ( x , y , z , t ) = a ( t - t ) h ˜ ( x , y , z , t ) d t = c z π - ρ / c ρ / c a ( t - t ) d t ( ρ 2 - c 2 t 2 ) 1 / 2 ( r 2 - c 2 t 2 ) .
t = ρ c sin θ
I ( x , y , z , t ) = z π - π / 2 π / 2 a ( t - ρ sin θ / c ) d θ z 2 + ρ 2 cos 2 θ .
I 0 ( x , y , z ) = z π - π / 2 π / 2 d θ z 2 + ρ 2 cos 2 θ = 2 z π 0 π / 2 d θ z 2 + ρ 2 cos 2 θ
I 0 ( x , y , z ) = 1 r             for z > 0 ,
I 0 ( x , y , z ) = - 1 r             for z < 0.
I ( x , y , z , t ) a ( t ) I 0 ( x , y , z ) = a ( t ) r [ Θ ( z ) - Θ ( - z ) ] .
V ˜ ( x , y , z , t ) a ( t + r / c ) - a ( t ) r Θ ( - z ) + a ( t ) - a ( t - r / c ) r × Θ ( z ) + a ( t ) 2 f
V ˜ ( 0 , 0 , 0 , t ) = 1 c d a ( t ) d t + a ( t ) 2 f .
a ( t ) = exp ( - i ω t ) .
V ˜ ( x , y , z , t ) = exp ( - i ω t ) [ exp ( - i k r ) r Θ ( - z ) - exp ( i k r ) r Θ ( z ) + z π - π / 2 π / 2 exp ( i k ρ sin θ ) z 2 + ρ 2 cos 2 θ d θ + z + 2 f π - π / 2 π / 2 exp ( i k ρ sin θ ) ( z + 2 f ) 2 + ρ 2 cos 2 θ d θ ] ,
V ˜ ( 0 , 0 , 0 , t ) = exp ( - i ω t ) ( - i k + 1 2 f ) ,

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