## A diffractive mechanism of focusing |

Optics Express, Vol. 20, Issue 25, pp. 27253-27262 (2012)

http://dx.doi.org/10.1364/OE.20.027253

Acrobat PDF (1326 KB)

### Abstract

We examine the free time evolution of a rectangular one dimensional Schrödinger wave packet of constant phase during the early stage which in the paraxial wave approximation is identical to the diffraction of a scalar field from a single slit. Our analysis, based on numerics and the Cornu spiral reveals considerable intricate detail behavior in the density and phase of the wave. We also point out a concentration of the intensity that occurs on axis and propose a new measure of width that expresses this concentration.

© 2012 OSA

## 1. Introduction

2. G. Möllenstedt and C. Jönsson, “Elektronenmehrfachinterferenz an regelmäßig hergestellen Feinspalten,” Z. f. Phys. **155**, 427–474 (1959). [CrossRef]

3. C. Jönsson, “Electron diffraction at multiple slits,” Am. J. Phys. **42**, 4–11 (1974). [CrossRef]

4. J. A. Leavitt and F. A. Bills, “Single-slit diffraction pattern of a thermal atomic potassium beam,” Am. J. Phys. **37**, 905–912 (1969). [CrossRef]

5. A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys. **60**, 1067–1073 (1988). [CrossRef]

6. O. Nairz, M. Arndt, and A. Zeilinger, “Quantum interference experiments with large molecules,” Am. J. Phys. **71**, 319–325 (2003). [CrossRef]

7. In a seminal paper, Marcos Moshinsky studied the propagation of a matter wave suddenly released from a shutter. For this reason these functions are sometimes called Moshinsky functions. See M. Moshinsky, “Diffraction in time,” Phys. Rev. **88**, 625–631 (1952). [CrossRef]

25. T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E **64**, 066613 (2001). [CrossRef]

26. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**, 1499–1501 (1987). [CrossRef] [PubMed]

*z*≈

*a*

^{2}/3

*λ*where

*a*and

*λ*are the width of the slit and the wavelength of the incident beam, respectively, and iii) a phase plateau around this focus. To quantitatively describe this focusing effect we introduce a new measure. We conclude in section 3 with a summary and an outlook.

## 2. Near-field pattern

*m*along the

*x*-axis, governed by the Schrödinger equation in the absence of a potential. However, we emphasize that this problem is equivalent to the diffraction of a scalar field from a single slit described in the paraxial approximation [27].

### 2.1. Probability density and phase

*a*according to the Schrödinger equation of a free particle. For this purpose it is convenient to introduce dimensionless variables such as position

*χ*≡

*x*/

*a*and time

*τ*≡

*ht*/(

*ma*

^{2}) where

*h*is Planck’s constant. For the diffraction problem

*τ*translates into the distance

*z*from the plane of the slit by the substitution law

*τ*=

*zλ*/

*a*

^{2}.

*τ*reads which can be expressed in terms of the standard Fresnel integrals [29] in the form giving rise to This expression can be evaluated numerically to give the near-field probability density |

*ψ*|

^{2}and the phase pattern of |

*ψ*| in the

*τ*−

*χ*plane shown in Figs. 1(a) and 1(b), respectively. We observe a series of maxima culminating in a dominant one near

*τ*= 1/3 and

*χ*= 0 where the intensity is 1.8 times the one within the slit. In the neighborhood of this point, the phase is approximately constant in space. On the way towards the slit a structure of ever increasing complexity and fineness emerge.

### 2.2. Cornu spiral essentials

*F*. The complex value

*F*(

*w*) is given either by the real and imaginary parts

*FC*(

*w*) and

*FS*(

*w*) of the point on the spiral depicted in Fig. 1(c) and 1(d) with arc length

*w*, or by the separation |

*F*(

*w*)| of this point from the origin together with the angle with respect to the real axis. In order to provide the background for our analysis of Figs. 1(a) and 1(b) we now briefly review the essential properties of the Cornu spiral.

*π*/4 as indicated in Fig. 1(c) by the blue arrow marked

*n*= 0. At this point we have the identity

*FC*(

*w*) =

*FS*(

*w*).

*w*further and thus continue on the spiral the separation |

*F*(

*w*)| decreases till we have reached again an angle of approximately

*π*/4. Now we have reached a minimum of |

*F*(

*w*)|.

*w*the separation and thereby |

*F*(

*w*)| increases again and we obtain another maximum when the angle of the arrow assumes the value

*π*/4. However, due to the spiral nature of the curve this maximum is now smaller than the first one. This second maximum is marked in Fig. 1(c) by

*n*= 1.

*w*. However, the values of |

*F*| at these extremal points approach a common limit as we spiral into the center.

*F*(

*w*)| occur for points on the spiral with

*FC*=

*FS*which yields with the help of the identity and the definition Eq.(4) of

*FC*and

*FS*the condition Hence, |

*F*| assumes extrema for arguments

*l*= 0 we find the largest maximum, for

*l*= 1 the lowest minimum, for

*l*= 2 again a maximum and so on. Thus the arguments

*w*corresponding to maxima of |

_{n}*F*| read where

*n*= 0, 1, 2, 3.... The two largest maxima determined by

*n*= 0 and

*n*= 1 are depicted in Fig. 1(c).

*F*| are given by with

*m*= 0 and

*m*= 1 providing us with the smallest values.

*F*given by the angle with respect to the horizontal axis. After the first maximum of |

*F*| this angle and thus the phase oscillates around

*π*/4 with a decaying amplitude. The two largest values of the phase are indicated in Fig. 1(c) by the two red arrows marked

*k*= 0 and

*k*= 1.

### 2.3. Application to the single-slit

*ψ*is given by the sum of two Fresnel integrals at different arguments. If we neglect, for the moment, the interference between these contributions, the maxima of |

*ψ*| will be determined by the maxima of the two Fresnel integrals with arguments With the help of the expression Eq. (9) for the location of the maxima of |

*F*(

*w*)|, the individual contributions in Eq. (5) have a maximum at and respectively.

*χ*= 0, that is the behavior along the optical axis. Here the two terms in Eq. (5) are identical and the interference is obviously constructive, generating a series of maxima at

*τ*= 1/(3 + 8

*n*), the largest of these occurring for

*τ*= 1/3. This point corresponds to the focus, where the central peak is narrowest.

*χ*= 0, as shown in Fig 1(d). Thus we expect that their sum gives a reduced amplitude but with the same phase.

*χ*= ±1/2 and

*τ*= 0. For

*τ*> 0, the curves propagate in the region between the edges and intersect at many points. This gives a clear comparison of the intensity pattern with an emerging network of blue and red curves coming from left and right edges respectively. The points of intersection coincide with the bright regions of the pattern.

*χ*= 0 are the maxima at

*τ*= 1/(3 + 8

*n*) already noted. Crossings of these lines for

*χ*≠ 0 are relative maxima of the terms for different

*n*-values. But here we again refer to the graph of the Cornu spiral and note that the two terms in Eq. (5) have the same phase and thus the sum is also a maximum. The probability amplitude Eq. (4) is a maximum at these points.

### 2.4. Focusing effect

*x*

^{2}〉 increases quadratically with time [20

20. K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics **375**, 133–143 (2010). [CrossRef]

*κ*with dimensions of inverse length and the expectation value is performed with respect to the quantum mechanical probability density |

*ψ*|

^{2}. In the limit of vanishing

*κ*our measure reduces to the second moment. Moreover, it increases monotonically as a function of time when the initial wave packet is a Gaussian - a feature that any

*bona fide*definition of a measure of width should have.

*ψ*(

*χ*,

*τ*) given by Eq. (5) into the definition Eq. (14) and performing the average numerically. The relevant length parameter is given by the dimensionless product

*κa*. Here it is convenient to compare 𝒲 with its original value at

*τ*= 0 in the form of a normalized width

*δ*𝒲 ≡ 𝒲(

*κ*,

*τ*)/𝒲(

*κ*, 0). The numerical results are shown in Fig. 3 with a plot of

*δ*𝒲 as a function of

*κa*and

*τ*. The resulting surface clearly shows that for each cut of the form

*κa*= constant there is a minimum at

*τ*≈ 1/3 indicating the focusing point. The plot in Fig. 3 also shows that there is a value of

*κa*for which the measure yields a

*global*minimum. Therefore, the best way to capture the shrinkage of the packet is by choosing the extremal point corresponding to

*κa*≈ 4.5. For

*τ*> 1/3,

*i.e.*after the focus, the measure

*δ*𝒲 increases due to expansion in agreement with Fig. 1(a).

31. M. J. W. Hall, “Incompleteness of trajectory-based interpretations of quantum mechanics,” J. Phys. A: Math. Gen. **37**, 9549–9556 (2004). [CrossRef]

*χ*at the center, separate the two parts and put a constant distribution between them. The four panels of Fig. 4 show the resulting spatio-temporal distributions for four increasing values of the width Δ

*χ*, that is for decreasing sharpness of the rectangular distribution. Indeed, we eventually lose the features characteristic for the rectangular distribution such as the focusing and the fine strucure but it is remarkable how robust they are for small values of Δ

*χ*. We can explain Fig. 4 with more detail by noting that the slope of the soft edges decreases faster than exponentially with an increasing width Δ

*χ*. The upper panels show that the focusing point in the red region is visible after increasing Δ

*χ*from 1/100 (left panel) to 1/10 (right panel), while the intensity peak decreases its value from 1.8 to 1.7. On the other hand, the lower row of Fig. 4 shows that increasing Δ

*χ*from 1 (left panel) to 10 (right panel) destroys the focusing effect as the maximum intensity decreases from 1.4 to 1.

## 3. Conclusions and outlook

*a*

^{2}-dependence of the focusing effect. Unfortunately this method of focusing without a lens but by selecting takes a toll on the integrated intensity.

*λ*the distance

*z*from the slit to the main intensity peak is determined by the relation This condition, together with the one to observe diffraction

*a*/

*λ*≫ 1, leads us to the estimate for the location of the focus. For the G-line of a Mercury vapor lamp with 435.8nm, this estimate gives the condition

*a*> 138.7 nm. Preliminary results of an experiment along these lines have been obtained [33].

34. A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A **82**, 023613 (2010). [CrossRef]

35. T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B **79**, 1013–1019 (2004). [CrossRef]

## Acknowledgments

## References and links

1. | M. Born and E. Wolf, |

2. | G. Möllenstedt and C. Jönsson, “Elektronenmehrfachinterferenz an regelmäßig hergestellen Feinspalten,” Z. f. Phys. |

3. | C. Jönsson, “Electron diffraction at multiple slits,” Am. J. Phys. |

4. | J. A. Leavitt and F. A. Bills, “Single-slit diffraction pattern of a thermal atomic potassium beam,” Am. J. Phys. |

5. | A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys. |

6. | O. Nairz, M. Arndt, and A. Zeilinger, “Quantum interference experiments with large molecules,” Am. J. Phys. |

7. | In a seminal paper, Marcos Moshinsky studied the propagation of a matter wave suddenly released from a shutter. For this reason these functions are sometimes called Moshinsky functions. See M. Moshinsky, “Diffraction in time,” Phys. Rev. |

8. | J. Goldemberg and H. M. Nussenzveig, “On the possibility of the experimental observation of diffraction in time effects,” Rev. Mex. Fis. |

9. | M. Bozic, D. Arsenovic, and L. Vuskovic, “Transverse momentum distribution of atoms in an interferometer,” Z. Naturforsch. |

10. | M. Oberthaler and T. Pfau, “One-, two-and three-dimensional nanostructures with atom lithography,” J. Phys.: Condens. Matter |

11. | W. Schnitzler, N. M. Linke, R. Fickler, J. Meijer, F. Schmidt-Kaler, and K. Singer, “Deterministic ultracold ion source targeting the Heisenberg limit,” Phys. Rev. Lett. |

12. | T. Sleator, T. Pfau, V. Balykin, and J. Mlynek, “Imaging and focusing of an atomic beam with a large period standing light wave,” Appl. Phys. B. |

13. | W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express |

14. | A. Turlapov, A. Tonyushkin, and T. Sleator, “Talbot-Lau effect for atomic de Broglie waves manipulated with light,” Phys. Rev. A |

15. | See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett. |

16. | A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. |

17. | Z. Liao, M. Al-Amri, and M.S. Zubairy, “Quantum lithography beyond the diffraction limit via Rabi oscillations,” Phys. Rev. Lett. |

18. | I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich, “In- and outbound spreading of a free-particle s-wave,” Phys. Rev. Lett. |

19. | M. Andreata and D. Dodonov, “On shrinking and expansion of radial wave packets,” J. Phys. A: Math. Gen. |

20. | K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics |

21. | R. Mack, V. P. Yakovlev, and W. P. Schleich, “Correlations in phase space and the creation of focusing wave packets,” J. Mod. Opt. |

22. | T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A |

23. | L. Novotny, “The history of near-field optics” in |

24. | D. Courjon, |

25. | T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E |

26. | J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

27. | The paraxial approximation is expected to hold as long as the wavelength is much less than the slit width. |

28. | R. P. Feynman and A. R. Hibbs, |

29. | M. Abramowitz and I. A. Stegun, |

30. | R. W. Wood, |

31. | M. J. W. Hall, “Incompleteness of trajectory-based interpretations of quantum mechanics,” J. Phys. A: Math. Gen. |

32. | E. Sadurní, W. B. Case, and W. P. Schleich, in preparation. |

33. | M. Gonçalves (personal communication, 2011). |

34. | A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A |

35. | T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(260.1960) Physical optics : Diffraction theory

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: September 5, 2012

Revised Manuscript: October 22, 2012

Manuscript Accepted: October 24, 2012

Published: November 19, 2012

**Citation**

W. B. Case, E. Sadurni, and W. P. Schleich, "A diffractive mechanism of focusing," Opt. Express **20**, 27253-27262 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-25-27253

Sort: Year | Journal | Reset

### References

- M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, London, 1964).
- G. Möllenstedt and C. Jönsson, “Elektronenmehrfachinterferenz an regelmäßig hergestellen Feinspalten,” Z. f. Phys.155, 427–474 (1959). [CrossRef]
- C. Jönsson, “Electron diffraction at multiple slits,” Am. J. Phys.42, 4–11 (1974). [CrossRef]
- J. A. Leavitt and F. A. Bills, “Single-slit diffraction pattern of a thermal atomic potassium beam,” Am. J. Phys.37, 905–912 (1969). [CrossRef]
- A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys.60, 1067–1073 (1988). [CrossRef]
- O. Nairz, M. Arndt, and A. Zeilinger, “Quantum interference experiments with large molecules,” Am. J. Phys.71, 319–325 (2003). [CrossRef]
- In a seminal paper, Marcos Moshinsky studied the propagation of a matter wave suddenly released from a shutter. For this reason these functions are sometimes called Moshinsky functions. See M. Moshinsky, “Diffraction in time,” Phys. Rev.88, 625–631 (1952). [CrossRef]
- J. Goldemberg and H. M. Nussenzveig, “On the possibility of the experimental observation of diffraction in time effects,” Rev. Mex. Fis.VI.3, 105–115 (1957).
- M. Bozic, D. Arsenovic, and L. Vuskovic, “Transverse momentum distribution of atoms in an interferometer,” Z. Naturforsch.56a, 173–177 (2001).
- M. Oberthaler and T. Pfau, “One-, two-and three-dimensional nanostructures with atom lithography,” J. Phys.: Condens. Matter15, R233–R255 (2003). [CrossRef]
- W. Schnitzler, N. M. Linke, R. Fickler, J. Meijer, F. Schmidt-Kaler, and K. Singer, “Deterministic ultracold ion source targeting the Heisenberg limit,” Phys. Rev. Lett.102, 070501 (2009). [CrossRef] [PubMed]
- T. Sleator, T. Pfau, V. Balykin, and J. Mlynek, “Imaging and focusing of an atomic beam with a large period standing light wave,” Appl. Phys. B.54, 375–379 (1992). [CrossRef]
- W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express17(23), 20966–20974 (2009). [CrossRef] [PubMed]
- A. Turlapov, A. Tonyushkin, and T. Sleator, “Talbot-Lau effect for atomic de Broglie waves manipulated with light,” Phys. Rev. A71, 043612 (2005). [CrossRef]
- See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett.88, 083601 (2002). [CrossRef] [PubMed]
- A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85, 2733–2736 (2000). [CrossRef] [PubMed]
- Z. Liao, M. Al-Amri, and M.S. Zubairy, “Quantum lithography beyond the diffraction limit via Rabi oscillations,” Phys. Rev. Lett.105, 183601 (2010). [CrossRef]
- I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich, “In- and outbound spreading of a free-particle s-wave,” Phys. Rev. Lett.89, 060404 (2002). [CrossRef] [PubMed]
- M. Andreata and D. Dodonov, “On shrinking and expansion of radial wave packets,” J. Phys. A: Math. Gen.36, 7113–7128 (2003). [CrossRef]
- K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics375, 133–143 (2010). [CrossRef]
- R. Mack, V. P. Yakovlev, and W. P. Schleich, “Correlations in phase space and the creation of focusing wave packets,” J. Mod. Opt.57, 1437–1444 (2010). [CrossRef]
- T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A79, 053823 (2009). [CrossRef]
- L. Novotny, “The history of near-field optics” in Progress in Optics vol. 50, E. Wolf, ed. (Elsevier, Amsterdam, 2007) pp. 137–184. [CrossRef]
- D. Courjon, Near-field Microscopy and Near-Field Optics (World Scientific Publishing, Singapore, 2003). [CrossRef]
- T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E64, 066613 (2001). [CrossRef]
- J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987). [CrossRef] [PubMed]
- The paraxial approximation is expected to hold as long as the wavelength is much less than the slit width.
- R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
- R. W. Wood, Physical Optics (Optical Society of America, Washington, 1988).
- M. J. W. Hall, “Incompleteness of trajectory-based interpretations of quantum mechanics,” J. Phys. A: Math. Gen.37, 9549–9556 (2004). [CrossRef]
- E. Sadurní, W. B. Case, and W. P. Schleich, in preparation.
- M. Gonçalves (personal communication, 2011).
- A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A82, 023613 (2010). [CrossRef]
- T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B79, 1013–1019 (2004). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.