Abstract

For the first time, to our knowledge, optical diffraction is shown to be a wavelet transform with the electromagnetic wavelets. We show that the optical wavelets proposed by Onural [Opt. Lett. 18 , 846 (1993)] are the Huygens wavelets under a Fresnel approximation, and the electromagnetic wavelets proposed by Kaiser [A Friendly Guide to Wavelets (Birkhauser, Boston, Mass., 1994)] reduce to Hyugens wavelets in the case of a monochromatic field.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born, E. Wolf, Principle of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 3.
  2. Y. Li, H. Szu, Y. Sheng, H. J. Caulfield, “Wavelet processing and optics,” Proc. IEEE. 84, 720–732 (1996).
    [CrossRef]
  3. G. Strang, T. Nguyen, Wavelet and Filter Banks (Wellesley-Cambridge, Wellesley, 1996), Chap. 11.
  4. L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
    [CrossRef] [PubMed]
  5. L. Onural, M. Kocatepe, “Family of scaling chirp functions, diffraction and holography,” IEEE Trans. Signal Process 43, 1568–1578 (1995).
    [CrossRef]
  6. G. Kaiser, “Wavelet electrodynamics,” Phys. Lett. A 168, 28–34 (1992).
    [CrossRef]
  7. G. Kaiser, A Friendly Guide to Wavelets (Birkhauser, Boston, Mass., 1994).
  8. S. L. Hahn, “Hilbert transforms,” in The Transforms and Applications Handbook, A. D. Poulariskas, ed. (CRC and IEEE, Boca Raton, 1995), Chap. 7.
  9. Y. Sheng, D. Roberge, H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
    [CrossRef]
  10. J. Lu, H. Zou, J. F. Greenleaf, “A new approach to obtain limited diffraction beams,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 850–853 (1995).
    [CrossRef]
  11. G. Kaiser, “Remote sensing with electromagnetic and acoustic wavelets: a variational approach,” in Wavelet Applications II, H. Szu ed., Proc. SPIE2491, 399–410 (1995).
    [CrossRef]
  12. L. Gross, “Norm invariance of mass-zero equations under the conformal group,” J. Math. Phys. 5, 687–695 (1964).
    [CrossRef]

1996 (1)

Y. Li, H. Szu, Y. Sheng, H. J. Caulfield, “Wavelet processing and optics,” Proc. IEEE. 84, 720–732 (1996).
[CrossRef]

1995 (2)

L. Onural, M. Kocatepe, “Family of scaling chirp functions, diffraction and holography,” IEEE Trans. Signal Process 43, 1568–1578 (1995).
[CrossRef]

J. Lu, H. Zou, J. F. Greenleaf, “A new approach to obtain limited diffraction beams,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 850–853 (1995).
[CrossRef]

1993 (1)

1992 (2)

G. Kaiser, “Wavelet electrodynamics,” Phys. Lett. A 168, 28–34 (1992).
[CrossRef]

Y. Sheng, D. Roberge, H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
[CrossRef]

1964 (1)

L. Gross, “Norm invariance of mass-zero equations under the conformal group,” J. Math. Phys. 5, 687–695 (1964).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principle of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 3.

Caulfield, H. J.

Y. Li, H. Szu, Y. Sheng, H. J. Caulfield, “Wavelet processing and optics,” Proc. IEEE. 84, 720–732 (1996).
[CrossRef]

Greenleaf, J. F.

J. Lu, H. Zou, J. F. Greenleaf, “A new approach to obtain limited diffraction beams,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 850–853 (1995).
[CrossRef]

Gross, L.

L. Gross, “Norm invariance of mass-zero equations under the conformal group,” J. Math. Phys. 5, 687–695 (1964).
[CrossRef]

Hahn, S. L.

S. L. Hahn, “Hilbert transforms,” in The Transforms and Applications Handbook, A. D. Poulariskas, ed. (CRC and IEEE, Boca Raton, 1995), Chap. 7.

Kaiser, G.

G. Kaiser, “Wavelet electrodynamics,” Phys. Lett. A 168, 28–34 (1992).
[CrossRef]

G. Kaiser, A Friendly Guide to Wavelets (Birkhauser, Boston, Mass., 1994).

G. Kaiser, “Remote sensing with electromagnetic and acoustic wavelets: a variational approach,” in Wavelet Applications II, H. Szu ed., Proc. SPIE2491, 399–410 (1995).
[CrossRef]

Kocatepe, M.

L. Onural, M. Kocatepe, “Family of scaling chirp functions, diffraction and holography,” IEEE Trans. Signal Process 43, 1568–1578 (1995).
[CrossRef]

Li, Y.

Y. Li, H. Szu, Y. Sheng, H. J. Caulfield, “Wavelet processing and optics,” Proc. IEEE. 84, 720–732 (1996).
[CrossRef]

Lu, J.

J. Lu, H. Zou, J. F. Greenleaf, “A new approach to obtain limited diffraction beams,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 850–853 (1995).
[CrossRef]

Nguyen, T.

G. Strang, T. Nguyen, Wavelet and Filter Banks (Wellesley-Cambridge, Wellesley, 1996), Chap. 11.

Onural, L.

L. Onural, M. Kocatepe, “Family of scaling chirp functions, diffraction and holography,” IEEE Trans. Signal Process 43, 1568–1578 (1995).
[CrossRef]

L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
[CrossRef] [PubMed]

Roberge, D.

Y. Sheng, D. Roberge, H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
[CrossRef]

Sheng, Y.

Y. Li, H. Szu, Y. Sheng, H. J. Caulfield, “Wavelet processing and optics,” Proc. IEEE. 84, 720–732 (1996).
[CrossRef]

Y. Sheng, D. Roberge, H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
[CrossRef]

Strang, G.

G. Strang, T. Nguyen, Wavelet and Filter Banks (Wellesley-Cambridge, Wellesley, 1996), Chap. 11.

Szu, H.

Y. Li, H. Szu, Y. Sheng, H. J. Caulfield, “Wavelet processing and optics,” Proc. IEEE. 84, 720–732 (1996).
[CrossRef]

Y. Sheng, D. Roberge, H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principle of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 3.

Zou, H.

J. Lu, H. Zou, J. F. Greenleaf, “A new approach to obtain limited diffraction beams,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 850–853 (1995).
[CrossRef]

IEEE Trans. Signal Process (1)

L. Onural, M. Kocatepe, “Family of scaling chirp functions, diffraction and holography,” IEEE Trans. Signal Process 43, 1568–1578 (1995).
[CrossRef]

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

J. Lu, H. Zou, J. F. Greenleaf, “A new approach to obtain limited diffraction beams,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 850–853 (1995).
[CrossRef]

J. Math. Phys. (1)

L. Gross, “Norm invariance of mass-zero equations under the conformal group,” J. Math. Phys. 5, 687–695 (1964).
[CrossRef]

Opt. Eng. (1)

Y. Sheng, D. Roberge, H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (1)

G. Kaiser, “Wavelet electrodynamics,” Phys. Lett. A 168, 28–34 (1992).
[CrossRef]

Proc. IEEE. (1)

Y. Li, H. Szu, Y. Sheng, H. J. Caulfield, “Wavelet processing and optics,” Proc. IEEE. 84, 720–732 (1996).
[CrossRef]

Other (5)

G. Strang, T. Nguyen, Wavelet and Filter Banks (Wellesley-Cambridge, Wellesley, 1996), Chap. 11.

G. Kaiser, A Friendly Guide to Wavelets (Birkhauser, Boston, Mass., 1994).

S. L. Hahn, “Hilbert transforms,” in The Transforms and Applications Handbook, A. D. Poulariskas, ed. (CRC and IEEE, Boca Raton, 1995), Chap. 7.

M. Born, E. Wolf, Principle of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 3.

G. Kaiser, “Remote sensing with electromagnetic and acoustic wavelets: a variational approach,” in Wavelet Applications II, H. Szu ed., Proc. SPIE2491, 399–410 (1995).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

Two electromagnetic wavelets of different scales and locations in the space–time domain. In the left-hand column each plane represents 3-D space; in the right-hand column are the 1-D plots of the wavelets.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

E x ,   y ,   z = 1 j λ z exp j   2 π λ   z     E ξ ,   η ,   0 × exp j   π λ z [ ( x - ξ ) 2 + ( y - η ) 2 ] d ξ d η ,
ψ z x ,   y = 1 j λ z exp j   2 π λ   z exp j   π λ z ( x 2 + y 2 ) ,
f ˜ x + jy = -   h * τ f x + τ y d τ = 1 π j - d τ τ - j   f x + τ y ,
f ˜ x + jy = - d u   1 | y |   h * u - x y f u ,
H ξ = 1 π j - d τ τ + j exp - j ξ τ = 2 θ ξ exp - ξ ,
θ ξ = 1 1 / 2 0 if   ξ > 0 if   ξ = 0 if   ξ < 0 .
f ˜ x + jy = 1 2 π 4 R 4 d 4 p 2 θ py F p exp - py exp jpx .
F ( p ) = 2 π δ ( p 2 ) f ˆ ( p ) = 2 π   δ ( p 0 + ω ) + δ ( p 0 - ω ) 2 ω   f ˆ ( p ) ,
f ˜ ( x + jy ) = C d 3 p 16 π 3 ω 3   2 ω 2 θ ( py ) exp ( - py ) f ˆ ( p ) exp ( jpx ) .
Ψ x + jy * p = 2 ω 2 θ py exp - py exp jpx ,
W f ( x + jy ) = f ˜ ( x + jy ) = C d 3 p 16 π 3 ω 3   Ψ x + jy * ( p ) f ˆ p .
C d 3 p 16 π 3 ω 3   | f ˆ p | 2 = E d 3 x d s | W f x ,   s | 2 .
f x = E d 3 x d s ψ x + jy x W f x + jy ,
ψ x + jy x = C d 3 p 16 π 3 ω   2 ω 2 θ py exp - py exp jp x - x .
ψ x ,   t = ψ - + ψ + = 1 π 2 3 cs - jct 2 - | x | 2 [ cs - jct 2 + | x | 2 ] 3 ,
ψ - ( x ,   t ) = 1 2 π 2 j | x | 1 ( cs - jct - j | x | ) 3 , ψ + ( x ,   t ) = - 1 2 π 2 j | x | 1 ( cs - jct + j | x | ) 3 .
- t 2 ψ - r ,   t + 2 ψ - r ,   t = - 2 j π 1 - jct 2   δ x , - t 2 ψ + r ,   t + 2 ψ + r ,   t = 2 j π 1 - jct 2   δ x .
f x ,   t = E x exp j ω 0 t ,
f x + τ s = E x exp j ω 0 t + τ s ,
W f x ,   t + js = 1 π j - d τ τ - j   E x exp j τ ω 0 s × exp j ω 0 t .
W f x t + js = E x exp j ω 0 t θ ω 0 s exp - s ω 0 .
-   f x ,   t exp - j ω 0 t d t = E d x d s   - d tW f × ψ x + js x ,   t exp - j ω 0 t .
-   ψ x + js x ,   t exp - j ω 0 t d t = 1 4 π 2 c 3 j exp j ω 0 | x - x | / c - exp - j ω 0 | x - x | / c | x - x | × θ s ω 0 ω 0 2 exp - s ω 0 .
0   ω 0 2 exp - s ω 0 exp j ω 0 t + | x - x | / c d ω 0 = 2 s - jt + j | x - x | / c 3 = 4 π 2 c 3 j | x - x | ψ - ,
0   ω 0 2 exp - s ω 0 exp j ω 0 t - | x - x | / c d ω 0 = 2 s - jt - j | x - x | / c 3 = 4 π 2 c 3 j | x - x | ψ + .
E x = 1 4 π 2 c 3 j E d 3 x d sE x θ s ω 0 ω 0 2 exp - 2 s ω 0 × exp j ω 0 | x - x | / c - exp - j ω 0 | x - x | / c | x - x | .
- d s θ s ω 0 exp - 2 s ω 0 = 1 2 ω 0 ,
E x = 1 4 π c 2 1 j λ 0 R 3 d x E x × exp j 2 π | x - x | / λ 0 - exp - j 2 π | x - x | / λ 0 | x - x | ,

Metrics