## Basic diffraction phenomena in time domain

Optics Express, Vol. 18, Issue 11, pp. 11083-11088 (2010)

http://dx.doi.org/10.1364/OE.18.011083

Acrobat PDF (1463 KB)

### Abstract

Using a recently developed technique (SEA TADPOLE) for easily measuring the complete spatiotemporal electric field of light pulses with micrometer spatial and femtosecond temporal resolution, we directly demonstrate the formation of theo-called boundary diffraction wave and Arago’s spot after an aperture, as well as the superluminal propagation of the spot. Our spatiotemporally resolved measurements beautifully confirm the time-domain treatment of diffraction. Also they prove very useful for modern physical optics, especially in micro- and meso-optics, and also significantly aid in the understanding of diffraction phenomena in general.

© 2010 OSA

## 1. Introduction

2. A. Rubinowicz, “Thomas Young and the theory of
diffraction,” Nature **180**(4578),
160–162 (1957). [CrossRef]

4. Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary
diffraction wave theory,” Phys. Rev. E Stat.
Nonlin. Soft Matter Phys. **63**(2), 026601
(2001). [CrossRef] [PubMed]

5. P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, “Crossed-beam spectral interferometry: a
simple, high-spectral-resolution method for completely characterizing complex
ultrashort pulses in real time,” Opt.
Express **14**(24),
11892–11900 (2006)
(and references therein). [CrossRef] [PubMed]

4. Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary
diffraction wave theory,” Phys. Rev. E Stat.
Nonlin. Soft Matter Phys. **63**(2), 026601
(2001). [CrossRef] [PubMed]

6. Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. Kovács, “Experimental investigation of the boundary
wave pulse,” Opt. Commun. **239**(4-6),
243–250 (2004). [CrossRef]

8. M. Vasnetsov, V. Pas’ko, A. Khoroshun, V. Slyusar, and M. Soskin, “Observation of superluminal wave-front
propagation at the shadow area behind an opaque disk,”
Opt. Lett. **32**(13),
1830–1832 (2007). [CrossRef] [PubMed]

9. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant
localized light waves,” Phys. Rev. Lett. **79**(21),
4135–4138 (1997). [CrossRef]

11. H. Valtna-Lukner, P. Bowlan, M. Lõhmus, P. Piksarv, R. Trebino, and P. Saari, “Direct spatiotemporal measurements of
accelerating ultrashort Bessel-type light bullets,”
Opt. Express **17**(17),
14948–14955 (2009). [CrossRef] [PubMed]

## 2. Theoretical description of the boundary wave pulse

4. Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary
diffraction wave theory,” Phys. Rev. E Stat.
Nonlin. Soft Matter Phys. **63**(2), 026601
(2001). [CrossRef] [PubMed]

*a*is given by the wave-function [12]where

*r*and

*z*are, respectively, the radial and axial coordinates of the field point;

*ω*

_{0}and

*k*

_{0}are, respectively, the carrier frequency and wave-number,

*ω*

_{0}=

*c k*

_{0};

*v*is the pulse envelope and

*s*= (

*r*

^{2}+

*z*

^{2}+

*a*

^{2}-2

*ra*cos

*ϕ*)

^{1/2}. The integration over

*ϕ*stems from contour integration along the disk’s boundary and is one-dimensional—this is the known advantage of the BDW theory as compared to common diffraction theories based on 2D surface integrals. The BDW pulse in the case of a circular hole is given also by Eq. (1), but with the opposite sign [4

**63**(2), 026601
(2001). [CrossRef] [PubMed]

## 3. Experimental results in comparison with simulations

13. P. Bowlan, P. Gabolde, and R. Trebino, “Directly measuring the spatio-temporal
electric field of focusing ultrashort pulses,”
Opt. Express **15**(16),
10219–10230 (2007). [CrossRef] [PubMed]

14. P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field
of tightly focused ultrashort pulses with sub-micron spatial
resolution,” Opt. Express **16**(18),
13663–13675 (2008). [CrossRef] [PubMed]

*E*(

*λ*) for that spatial point. Then to measure the spatial dependence of the field, we simply scan the fiber point by point through the space where the unknown light field propagates, so that

*E*(

*λ*) is measured at each position, yielding

*E*(

*λ,x,z*). This field can be Fourier transformed to the time domain to yield

*E*(

*t,x,z*). The plots from our SEA TADPOLE measurements, which are shown below, can be viewed as still images or “snapshots in flight,” since they are spatiotemporal slices of the magnitude of the electric field |

*E*(

*x*,

*z*,

*t*)| of the pulses. While we could also scan along the

*y*-dimension of the beam, our set up had cylindrical symmetry, so we only scanned along the

*x*-dimension with the fiber at

*y*= 0.

*J*

_{0}) radial profile. Moreover, the spot is delayed in time with respect to the main pulse front, and this delay decreases with

*z*, indicating a superluminal propagation speed along the

*z*axis (the GW pulse front propagates at

*c*). This occurs, because, as

*z*(the distance from the screen) increases, the extra distance that the boundary waves must propagate (compared to the GW pulse front) to reach the

*z*axis (

*x*= 0) decreases, so the relative delay of the boundary waves on the axis decreases. As a result, the axial group velocity of the Arago spot—geometrically located at one pole of a luminally expanding spindle torus formed by the boundary diffraction wave pulse—varies from infinity at

*z*= 0 to

*c*for very large values of

*z*. Therefore, the spot of Arago is in fact just a decelerating superluminal Bessel pulse like that recently generated using compound refractive optical elements and also studied with SEA TADPOLE [11

11. H. Valtna-Lukner, P. Bowlan, M. Lõhmus, P. Piksarv, R. Trebino, and P. Saari, “Direct spatiotemporal measurements of
accelerating ultrashort Bessel-type light bullets,”
Opt. Express **17**(17),
14948–14955 (2009). [CrossRef] [PubMed]

*z*-axis) by the expanding spherical-wave constituents, which travel at an angle with respect to the

*z*-axis. It is important to realize that the central interference region of the BDW pulse is not moving at this tilt angle. Its phase (and pulse) fronts are perpendicular to the

*z*-axis and move along this axis. Its Poynting vector, indicating the direction of energy flow, also lies along the

*z*-axis. However, the energy flux is not superluminal. The superluminal pulse’s velocity should not, of course, be confused with the signal velocity. As is well known, Maxwell’s equations, or the wave equation for electromagnetic fields, does not allow superluminal signaling.

*z*= 0mm is the location of the disc. The simulations show the creation of the Arago spot at the moment when the expanding ring torus of the boundary-wave pulse becomes an expanding spindle torus. In the close vicinity after the disc, speeds much greater than

*c*can be seen, where the boundary wave pulse literally jumps out of it. Due to the discrete color scale of the animations, the expanding spindle torus shape of the BDW pulse seems discontinuous during the first few picoseconds of the spot evolution, albeit it is only low in intensity in these particular directions.

*z*= 0mm is bound to the plane-wave pulse moving at velocity

*c*to the right. The fringe pattern in the axial region of the boundary-wave pulse stretches during the propagation as the angle of intersection between the elementary wavelets decreases continuously. Correspondingly, the pulse velocity decreases as the fringe periodicity increases. Since the expansion rate of the spindle-torus is constant (equal to

*c*), the spot on the axis propagates superluminally, decelerating toward the limiting value

*c*. Interestingly, the first seconds of the video also reveal the back-diffracted pulse, which follows from the direct evaluation of Eq. (1). This backward propagating contribution is expected since the spherical waves generated at the boundary of the disk are emitted at all angles in the

*x*-

*z*plane. Of course, the intensity of the backward-moving pulse quickly decreases towards negative values of

*z*and practically ceases to exist within the first millimeter of propagation.

## 4. Conclusions

## Acknowledgements

## References and links

1. | G. A. Maggi, “Sulla propagazione libera e perturbata delle
onde luminose in mezzo isotropo,” Ann. di
Mat IIa ,” |

2. | A. Rubinowicz, “Thomas Young and the theory of
diffraction,” Nature |

3. | g. See, monograph M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1987, 6th ed) and references therein. |

4. | Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary
diffraction wave theory,” Phys. Rev. E Stat.
Nonlin. Soft Matter Phys. |

5. | P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, “Crossed-beam spectral interferometry: a
simple, high-spectral-resolution method for completely characterizing complex
ultrashort pulses in real time,” Opt.
Express |

6. | Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. Kovács, “Experimental investigation of the boundary
wave pulse,” Opt. Commun. |

7. | D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, “Direct measurement of the central fringe
velocity in Young-type experiments,” Phys. Lett.
A |

8. | M. Vasnetsov, V. Pas’ko, A. Khoroshun, V. Slyusar, and M. Soskin, “Observation of superluminal wave-front
propagation at the shadow area behind an opaque disk,”
Opt. Lett. |

9. | P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant
localized light waves,” Phys. Rev. Lett. |

10. | P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of
ultrashort Bessel-X pulses,” Opt. Lett. |

11. | H. Valtna-Lukner, P. Bowlan, M. Lõhmus, P. Piksarv, R. Trebino, and P. Saari, “Direct spatiotemporal measurements of
accelerating ultrashort Bessel-type light bullets,”
Opt. Express |

12. | P. Piksarv, MSc thesis, University of Tartu (2009). |

13. | P. Bowlan, P. Gabolde, and R. Trebino, “Directly measuring the spatio-temporal
electric field of focusing ultrashort pulses,”
Opt. Express |

14. | P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field
of tightly focused ultrashort pulses with sub-micron spatial
resolution,” Opt. Express |

15. | If viewing the plots without magnification in a computer screen the Moiré effect may obscure the actual small period of the intensity oscillations and increase of the period. |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(320.5550) Ultrafast optics : Pulses

(320.7100) Ultrafast optics : Ultrafast measurements

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: March 29, 2010

Revised Manuscript: April 30, 2010

Manuscript Accepted: April 30, 2010

Published: May 11, 2010

**Citation**

Peeter Saari, Pamela Bowlan, Heli Valtna-Lukner, Madis Lõhmus, Peeter Piksarv, and Rick Trebino, "Basic diffraction phenomena
in time domain," Opt. Express **18**, 11083-11088 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-11-11083

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### References

- G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in mezzo isotropo,” Ann. di Mat IIa,” 16, 21–48 (1888).
- A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180(4578), 160–162 (1957). [CrossRef]
- g. See, monograph M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1987, 6th ed) and references therein.
- Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2), 026601 (2001). [CrossRef] [PubMed]
- P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, “Crossed-beam spectral interferometry: a simple, high-spectral-resolution method for completely characterizing complex ultrashort pulses in real time,” Opt. Express 14(24), 11892–11900 (2006) (and references therein). [CrossRef] [PubMed]
- Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239(4-6), 243–250 (2004). [CrossRef]
- D. Chauvat, O. Emile, M. Brunel, and A. Le Floch, “Direct measurement of the central fringe velocity in Young-type experiments,” Phys. Lett. A 295(2-3), 78–80 (2002). [CrossRef]
- M. Vasnetsov, V. Pas’ko, A. Khoroshun, V. Slyusar, and M. Soskin, “Observation of superluminal wave-front propagation at the shadow area behind an opaque disk,” Opt. Lett. 32(13), 1830–1832 (2007). [CrossRef] [PubMed]
- P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997). [CrossRef]
- P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34(15), 2276–2278 (2009). [CrossRef] [PubMed]
- H. Valtna-Lukner, P. Bowlan, M. Lõhmus, P. Piksarv, R. Trebino, and P. Saari, “Direct spatiotemporal measurements of accelerating ultrashort Bessel-type light bullets,” Opt. Express 17(17), 14948–14955 (2009). [CrossRef] [PubMed]
- P. Piksarv, MSc thesis, University of Tartu (2009).
- P. Bowlan, P. Gabolde, and R. Trebino, “Directly measuring the spatio-temporal electric field of focusing ultrashort pulses,” Opt. Express 15(16), 10219–10230 (2007). [CrossRef] [PubMed]
- P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express 16(18), 13663–13675 (2008). [CrossRef] [PubMed]
- If viewing the plots without magnification in a computer screen the Moiré effect may obscure the actual small period of the intensity oscillations and increase of the period.

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