Abstract

We study the properties of pulsed solutions to the scalar and vector wave equations composed of plane-waves with equal longitudinal spatial frequency. This condition guarantees that, at all times, the field profile is invariant in the longitudinal direction. Particular emphasis is placed on solutions with rotational symmetry. For these solutions, the wave concentrates strongly near the axis at a given time. We provide closed-form expressions for some of these fields, and show that their wavefronts are approximately spherical. Solutions carrying orbital and spin angular momenta are also considered.

© 2016 Optical Society of America

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Corrections

5 December 2016: Corrections were made to the body text; Eqs. (8), (21), and (22); and the reference list.


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References

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  1. A. E. Siegman, Lasers (Univ. Science Books, 1986).
  2. M. A. Alonso and N. J. Moore, “Basis exoansions for monochromatic field propagation in free space,” in Mathematical Optics: Classical, Quantum, and Computational Methods, V. Lakshminarayanan, ed. (CRC Press, 2012), chap. 4, pp. 97–141.
  3. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987).
    [Crossref]
  4. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000).
    [Crossref] [PubMed]
  5. C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Opt. Soc. Am. A 18(10), 2594–2600 (2001).
    [Crossref] [PubMed]
  6. C. J. R. Sheppard, “Generalized Bessel pulse beams,” J. Opt. Soc. Am. A 19(11), 2218–2222 (2002).
    [Crossref] [PubMed]
  7. J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” in Progress in Optics vol. 54 E. Wofl, ed. (Elsevier, 2009), pp. 2–85.
  8. J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferrelec. Freq. Control 39(1), 19–31 (1992).
    [Crossref]
  9. J. Salo, A. T. Friberg, and M. M. Salomaa, “Orthogonal X waves,” J. Phys. A 34, 7079–7082 (2001).
  10. J. W. Goodman, Introduction to Fourier Optics, (Roberts & Co., 2005), pp. 31–62.
  11. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1980), pp. 556–592.
  12. E. G. Williams, Fourier acoustics: sound radiation and nearfield acoustical holography (Academic Press, 1999), pp. 1–13.
  13. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 1965), p. 112.
  14. M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. Math. Gen. 27(11), L391–L398 (1994).
    [Crossref]
  15. C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57(4), 2971–2979 (1998).
    [Crossref]
  16. M. A. Alonso, “The effect of orbital angular momentum and helicity in the uncertainty-type relations between focal spot size and angular spread,” J. Opt. 13(6), 064016 (2011).
    [Crossref]
  17. J. A. Jensen, “Simulation of advanced ultrasound systems using Field II,” in Biomedical Imaging: Nano to Macro, 2004. IEEE International Symposium on (2004), pp. 636–639 Vol. 631.
    [Crossref]
  18. J. A. Jensen, “Field: a program for simulating ultrasound systems,” in 10th Nordibaltic Conference on Biomedical Imaging(1996), pp. 351–3553.
  19. H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24(25), 28659–28668 (2016).
    [Crossref]

2016 (1)

2011 (1)

M. A. Alonso, “The effect of orbital angular momentum and helicity in the uncertainty-type relations between focal spot size and angular spread,” J. Opt. 13(6), 064016 (2011).
[Crossref]

2002 (1)

2001 (2)

J. Salo, A. T. Friberg, and M. M. Salomaa, “Orthogonal X waves,” J. Phys. A 34, 7079–7082 (2001).

C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Opt. Soc. Am. A 18(10), 2594–2600 (2001).
[Crossref] [PubMed]

2000 (1)

1998 (1)

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57(4), 2971–2979 (1998).
[Crossref]

1994 (1)

M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. Math. Gen. 27(11), L391–L398 (1994).
[Crossref]

1992 (1)

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferrelec. Freq. Control 39(1), 19–31 (1992).
[Crossref]

1987 (1)

Abouraddy, A. F.

Alonso, M. A.

M. A. Alonso, “The effect of orbital angular momentum and helicity in the uncertainty-type relations between focal spot size and angular spread,” J. Opt. 13(6), 064016 (2011).
[Crossref]

Berry, M. V.

M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. Math. Gen. 27(11), L391–L398 (1994).
[Crossref]

Chávez-Cerda, S.

Durnin, J.

Friberg, A. T.

J. Salo, A. T. Friberg, and M. M. Salomaa, “Orthogonal X waves,” J. Phys. A 34, 7079–7082 (2001).

Greenleaf, J. F.

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferrelec. Freq. Control 39(1), 19–31 (1992).
[Crossref]

Gutiérrez-Vega, J. C.

Iturbe-Castillo, M. D.

Jensen, J. A.

J. A. Jensen, “Field: a program for simulating ultrasound systems,” in 10th Nordibaltic Conference on Biomedical Imaging(1996), pp. 351–3553.

Kondakci, H. E.

Lu, J.-Y.

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferrelec. Freq. Control 39(1), 19–31 (1992).
[Crossref]

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57(4), 2971–2979 (1998).
[Crossref]

Salo, J.

J. Salo, A. T. Friberg, and M. M. Salomaa, “Orthogonal X waves,” J. Phys. A 34, 7079–7082 (2001).

Salomaa, M. M.

J. Salo, A. T. Friberg, and M. M. Salomaa, “Orthogonal X waves,” J. Phys. A 34, 7079–7082 (2001).

Sheppard, C. J. R.

IEEE Trans. Ultrason. Ferrelec. Freq. Control (1)

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferrelec. Freq. Control 39(1), 19–31 (1992).
[Crossref]

J. Opt. (1)

M. A. Alonso, “The effect of orbital angular momentum and helicity in the uncertainty-type relations between focal spot size and angular spread,” J. Opt. 13(6), 064016 (2011).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Phys. A (1)

J. Salo, A. T. Friberg, and M. M. Salomaa, “Orthogonal X waves,” J. Phys. A 34, 7079–7082 (2001).

J. Phys. Math. Gen. (1)

M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. Math. Gen. 27(11), L391–L398 (1994).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57(4), 2971–2979 (1998).
[Crossref]

Other (9)

J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” in Progress in Optics vol. 54 E. Wofl, ed. (Elsevier, 2009), pp. 2–85.

J. W. Goodman, Introduction to Fourier Optics, (Roberts & Co., 2005), pp. 31–62.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1980), pp. 556–592.

E. G. Williams, Fourier acoustics: sound radiation and nearfield acoustical holography (Academic Press, 1999), pp. 1–13.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, 1965), p. 112.

J. A. Jensen, “Simulation of advanced ultrasound systems using Field II,” in Biomedical Imaging: Nano to Macro, 2004. IEEE International Symposium on (2004), pp. 636–639 Vol. 631.
[Crossref]

J. A. Jensen, “Field: a program for simulating ultrasound systems,” in 10th Nordibaltic Conference on Biomedical Imaging(1996), pp. 351–3553.

A. E. Siegman, Lasers (Univ. Science Books, 1986).

M. A. Alonso and N. J. Moore, “Basis exoansions for monochromatic field propagation in free space,” in Mathematical Optics: Classical, Quantum, and Computational Methods, V. Lakshminarayanan, ed. (CRC Press, 2012), chap. 4, pp. 97–141.

Supplementary Material (3)

NameDescription
» Visualization 1: MOV (670 KB)      Visualization 1
» Visualization 2: MOV (654 KB)      Visualization 2
» Visualization 3: MOV (313 KB)      Visualization 3

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Figures (4)

Fig. 1
Fig. 1 Hyperbola in the 2D Fourier plane defining the iso-phase condition. For forward-propagating fields and when the analytic signal representation is used only the top (blue) branch is used.
Fig. 2
Fig. 2 Excitation signal at the initial plane, for k L c 0 q = 0.001 (a), 1 (b), and 4 (c).
Fig. 3
Fig. 3 Real part of the pulses for (left) two dimensions given in Eq. (16), (middle) three dimensions with unit vorticity, and (right) the three-dimensional pulses in Eq. (21), at five different times. For the first two columns, x [−20/kL, 20/kL] (vertical) and z [0,100/kL] (horizontal), while for the third, x [−20/KL, 20/KL] and z [-50/KL,50/KL].
Fig. 4
Fig. 4 5Mhz medical ultrasound linear array simulation of the acoustic pressure field resulting from the longitudinal iso-phase excitation with additional Gaussian apodization in time and space. Three time points are shown, before and after the convergence of the broadband components at time steps proportional to 2π/ω L . Image size (vertical or transverse dimension x horizontal or axial dimension): 1.55mm x 3.75mm.

Equations (22)

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p ˜ ( x , y , 0 ; ω ) = + + P ˜ ( k x , k y , 0 ; ω ) e i ( k x x + , k y y ) d k x , d k y
P ˜ ( k x , k y , 0 ; ω ) = 1 4 π 2 p ˜ ( x , y , 0 ; ω ) e i ( k x x + , k y y ) d x d y ,
P ˜ ( k x , k y , z ; ω ) = P ˜ ( k x , k y , 0 ; ω ) e i k z z
k z = k 2 k x 2 k y 2 .
p ( x , y , z ; t ) = + + p ˜ ( x , y , z ; t ) e | i ω t d ω = + + + + + + P ˜ ( k x , k y , 0 ; ω ) e i ( k x x + k y y + k z z ω t ) d k x d k y d ω .
k x 2 + k y 2 = ( ω c ) 2 .
k z 2 = ( ω c ) 2 k x 2 k y 2 = k L 2 ,
( ω c ) 2 k x 2 = k L 2 .
k L 2 = ( ω L c ) 2 .
k x 2 = 1 c 0 2 ( ω 2 ω L 2 ) .
P ˜ ( k x , 0 ; ω ) = A ( k x ) δ [ k x 2 1 c 0 2 ( ω 2 ω L 2 ) ] , ω 0 ,
p ( x , z ; t ) = e i k L z k = + ω = 0 + A ( k x ) δ [ k x 2 1 c 0 2 ( ω 2 ω L 2 ) ] e i ( k x x = ω t ) d ω d k x .
p ( x , z ; t ) = e i k L z A ( k x ) k x 2 + k L 2 exp [ i ( k x x c 0 t k x 2 + k L 2 ) ] d k x .
p ( x , z ; t ) = c 0 e i k L z A ( k L sin h η ) exp [ i k L ( x sin h η c 0 t cos h η ) ] d η .
A ( k L sin h η ) = exp [ k L c 0 q cos h η ] ,
p ( x , z , t ) = c 0 2 K 0 [ k L x 2 c 0 2 ( t i q ) 2 ] exp ( i k L z ) ,
x 2 + ( z z 0 ) 2 c 0 2 ( t 2 q 2 ) ,
p ( ρ , z , t ) = c 0 exp [ k L ρ 2 c 0 2 ( t i q ) 2 ] k L ρ 2 c 0 2 ( t i q ) 2 exp ( i k L z ) ,
C ^ ± p = 1 2 [ ( v ± ) ± i v ± × c 0 t v ± c 0 2 2 t 2 ] p .
C ^ A p = ( z × ) p , C ^ R p = × ( z × ) p .
p 2 ( ρ , z , t ) = k L p ( ρ , z , t ) F ( k L K L ) d k L = p ( ρ , z , t ) | k L = K L f ( z + i ρ 2 c 0 2 ( t i q ) 2 ) ,
p 2 ( ρ , z , t ) = k L p ( ρ , z , t ) 1 2 π δ exp [ ( k L K L ) 2 2 δ 2 ] d k L = p ( ρ , z , t ) | k L = K L exp [ δ 2 2 ( z + i ρ 2 c 0 2 ( t i q ) 2 ) 2 ] ,

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