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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 17 — Aug. 15, 2011
  • pp: 16448–16454
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Generation of Airy solitary-like wave beams by acceleration control in inhomogeneous media

Sabino Chávez-Cerda, Ulises Ruiz, Victor Arrizón, and Héctor M. Moya-Cessa  »View Author Affiliations


Optics Express, Vol. 19, Issue 17, pp. 16448-16454 (2011)
http://dx.doi.org/10.1364/OE.19.016448


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Abstract

We investigate the propagation of Airy beams in linear gradient index inhomogeneous media. We demonstrate that by controlling the gradient strength of the medium it is possible to reduce to zero their acceleration. We show that the resulting Airy wave beam propagates in straight line due to the balance between two opposite effects, one due to the inhomogeneous medium and the other to the diffraction of the beam, in a similar way as a solitary wave in a nonlinear inhomogeneous medium. Going even further we were able to invert the sign of the acceleration of the beam.

© 2011 OSA

1. Introduction

Since their proposal [1

1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]

,2

2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef] [PubMed]

], finite energy optical Airy beams in free space have been a cause of disconcert due to their uncommon accelerating and non-diffracting propagation characteristics, becoming in this way a subject of many investigations. They share several odd properties with the also controversial non-diffracting Bessel beams [3

3. J. Durnin, J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]

6

6. S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).

] and the Airy quantum wave packets of infinite extent introduced by Berry and Balazs as free particle solutions of the Schrödinger equation [7

7. M. V. Berry and N. L. Balazs, “Nonspreading wave-packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]

].

On propagation finite energy two-dimensional Airy beams continuously bend following parabolic trajectories on the plane of propagation extending towards the non-oscillatory region of the beam [1

1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]

,2

2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef] [PubMed]

]. This was a striking behavior for a two dimensional beam in free space that was different from the non-diffracting transverse parabolic beams previously reported in three dimensions [8

8. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004). [CrossRef] [PubMed]

,9

9. C. López-Mariscal, M. Bandres, J. Gutiérrez-Vega, and S. Chávez-Cerda, “Observation of parabolic nondiffracting optical fields,” Opt. Express 13(7), 2364–2369 (2005). [CrossRef] [PubMed]

]. If observed along their parabolic trajectories, infinite extent Airy beams do not change their amplitude shape. It is for this reason they have also been considered as non-diffracting beams.

Although in the optical waveguides literature they are not referred to as Airy beams, it is well known that the propagation modes in inhomogeneous media with a linear gradient refractive index have a profile determined by the Airy function [10

10. D. Marcuse, “TE modes of graded index slab waveguides,” IEEE J. Quantum Electron. 9(10), 1000–1006 (1973). [CrossRef]

,11

11. C.-L. Chen, Foundations of guided-wave optics (Wiley, New Jersey 2006), Ch 3.

]. In these media the stationary Airy solution propagates invariant parallel to the propagation axis.

Propagation of Airy beams has recently been studied in plasmons, nonlinear photorefractive materials and photonic crystals [12

12. D. N. Christodoulides and T. H. Coskun, “Diffraction-free planar beams in unbiased photorefractive media,” Opt. Lett. 21(18), 1460–1462 (1996). [CrossRef] [PubMed]

16

16. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. 36(7), 1164–1166 (2011). [CrossRef] [PubMed]

]. In the latter case, it was shown that the nonlinear Airy beam in the photonic crystal can invert the direction of the acceleration by carefully adjusting the phase mismatch of the nonlinear processes involved.

In the present work we demonstrate, theoretically and experimentally, that it is possible to control the transverse acceleration of Airy beams when propagating in inhomogeneous media. By modifying the strength of the gradient index, the acceleration can be reduced to the point of creating a beam that propagates similar to a solitary wave with Airy profile. We show that such propagating behavior is the result of two opposite effects one due to the beam and the other to the medium. Even further, we also show the possibility of inverting the acceleration of the Airy beam.

2. Propagation in inhomogeneous media

I. Fermat's ray theory

The equation of light rays propagating in a medium with refractive index n = n(x,y,z) obtained from Fermat's variational principle is [17

17. M. Born and E. Wolf, Principles of Optics, Seventh Ed., (Cambridge University Press, Cambridge 1999), Ch. 3.

],
dds(ndrds)=n,
(1)
where r is the position vector of a point on a ray function of the length of arc s of the ray. For a material with a linear gradient index (GRIN) n 2(x) = n 0 2±n 1 x, with n 1<< n 0, and paraxial rays this equation yields

d2xdz2n12n02=0,
(2)

whose solutions are rays that, launched parallel to the propagation axis z, follow parabolas whose branches extend in the direction of the gradient. This is, if the gradient is negative the rays bend towards the negative x-axis and vice versa, see Figs. 1a
Fig. 1 Top row: Typical behavior of a rays propagating in linear GRIN media. The red dotted line shows qualitatively the gradient of the refractive index and the blue line the corresponding ray trajectory for: a) negative gradient and b) positive gradient. Bottom row: Free space propagation of the Airy mode at the exit of the corresponding inhomogeneous media (see Airy equation at the bottom of the figure) with negative gradient, c), and positive gradient d). The horizontal coordinate corresponds to the transverse coordinate and the vertical one to the evolution coordinate.
) and 1b). Seen in another way, the rays bend in the direction of the increasing of the refractive index (red dotted line). For each case, the corresponding Airy differential equation has also the same sign of the gradient. This is relevant since the Airy differential equation with the positive sign, u” + xu = 0, is rarely quoted in the literature and its solution is a mirror reflection of the solution of the Airy equation with the negative sign. By looking at the corresponding Airy mode, we notice that if it was to propagate in free space it would move in the opposite direction as that of the rays, see Fig. 1c) and 1d) where the modulus squared of the amplitude is plotted. Then, from this it can be deduced that the gradient index of the medium presents an opposite effect to the propagating Airy beam that can be associated to the opposite force described by Berry and Balazs [7

7. M. V. Berry and N. L. Balazs, “Nonspreading wave-packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]

]. This can create a balance such that can make the beam to travel parallel to the longitudinal axis in a similar way as a solitary wave as will be shown below.

II. Airy beam propagation in linear gradient index media

iuz=122ux2+k1xu.
(3)

In order to solve this equation we use an operational approach by defining the operator p = -i∂/∂x. Performing the following transformation [18

18. There exist several forms for producing a solution of this equation, for instance by using the Zassenhaus formula [19], or by simplifying the differential equation via a transformation (see for instance [20] for a quadratic term). The operators to disentangle the exponential of the sum of two operators can also be given in different orderings of the exponentials involved. Of course, they are all equivalent.

-20

20. H. Moya-Cessa and M. Fernández Guasti, “Coherent states for the time dependent harmonic oscillator: the step function,” Phys. Lett. A 311(1), 1–5 (2003). [CrossRef]

]
u(x,z)=exp(i3k1p3)v(x,z),
(4)
and considering a particular case of the Hadamard lemma [21

21. W. H. Louissel, Quantum Statistical Properties of Radiation (Wiley-Interscience, New York 1990), Ch. 3.

,22

22. H. Moya-Cessa and F. Soto-Eguibar, Differential equations: an operational approach, (Rinton Press, New Jersey 2011), Ch. 2.

], eχABe-χA=B+χ[A,B]+(χ2/2!)[A,[A,B]]+…, we simplify the problem of the linear GRIN paraxial wave equation to find the solution of
ivz=k1x2v
(5)
that can immediately be integrated to give

v(x,z)=exp(ik1xz2)v(x,0).
(6)

By considering an Airy initial condition u(x,0) = Ai(kxx) for the paraxial wave Eq. (3), it is transformed by substitution in Eq. (4) as

v(x,0)=exp(i3k1p3)Ai(kxx).
(7)

By writing the Airy function in its integral form [23

23. G. N. Watson, A treatise on the theory of Bessel functions, Ch. VI, Cambridge University Press, Cambridge 1944).

]
Ai(s)=1(2π)2exp(it3/3+st)dt
(8)
we obtain from (7)
v(x,z)=1(2π)2exp(ik1xz2)exp[i(t3/3+kx3t3/3k1+kxxt)]dt
(9)
and using again (5) we obtain the propagated function
u(x,z)=eiΦ(x,z)Ai[kxx+kxz22(k1kx32)],
(10)
where Φ(x,z)=(k1kx32)[xz+z36(k1kx3)].

3. Experiment

We have implemented experimentally the control of acceleration of an Airy beam propagating in a linear GRIN medium. The medium is a solution of sugar in water, where the strength of the gradient refractive index is controlled by the concentration of sugar. It was previously shown that this solution approximately represents a linear GRIN medium [24

24. W. M. Strouse, “Bouncing light beams,” Am. J. Phys. 40(6), 913–914 (1972). [CrossRef]

,25

25. D. Ambrosini, A. Ponticiello, G. S. Spagnolo, R. Borghi, and F. Gori, “Bouncing light beams and the Hamiltonian analogy,” Eur. J. Phys. 18(4), 284–289 (1997). [CrossRef]

]. The Airy beam was generated displaying a cubic phase in a Spatial Light Modulator (SLM) [23

23. G. N. Watson, A treatise on the theory of Bessel functions, Ch. VI, Cambridge University Press, Cambridge 1944).

], which is illuminated with a Gaussian beam. This cubic phase is given by
Ψ(x)=exp[i(2π/α)x3]
(11)
where α is a constant that characterizes the experimental spatial scale. The cubic phase and the Airy beam are connected by the relation
{Ψ(x)}=Ai(qk),
(12)
where α denotes the Fourier transform, Ai represents the Airy beam, k is the conjugate variable (spatial frequency) corresponding to the spatial coordinate x, and q=[ α/(6π)]1/3. The phase optical field Ψ(x) is generated with a phase SLM [26

26. SLM 512, Boulder nonlinear systems.

], and the Fourier transform of the phase field Ψ(x) [in Eq. (12)] is implemented with a lens of positive power. The experimental setup, depicted in Fig. 3
Fig. 3 Components of the experimental setup: Laser, beam expander (BE), SLM, Fourier transforming lens (L), and solution container (SC).
, shows a beam expander (BE) that conditions the beam of a solid state laser (verdi V8, 532 nm) to illuminate the total active area of the SLM (≅1cm2). The illumination on the SLM is oblique, avoiding the necessity of a beam splitter. The lens (L) that produces the Fourier transform of the cubic phase has a focal distance f = 50cm. The dimensions of the solution container (SC) were 60 cm (length), 2.5 cm (height) and 5 cm (width). The back focal plane of the lens, where the Airy beam is obtained, coincides with the front end of the SC. The parameter α=1.287x1010 μm3, of the generated phase modulation Ψ(x), was chosen to achieve an Airy beam deflection of approximately 2 mm in a propagation distance of 600mm, when no sugar is added to the water. We employed 3 different concentrations of sugar: C1 (with no added sugar), C2 (with 6.5 gr of sugar) and C3 (with 13 gr of sugar). To obtain the concentrations C2 and C3, the water with the added sugar is gently stirred for approximately 15 seconds. After that, the solution is allowed to stabilize before capturing images of the propagated beams.

Solitary-like propagation and switching of acceleration of Airy beams has been observed in nonlinear crystals [12

12. D. N. Christodoulides and T. H. Coskun, “Diffraction-free planar beams in unbiased photorefractive media,” Opt. Lett. 21(18), 1460–1462 (1996). [CrossRef] [PubMed]

16

16. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. 36(7), 1164–1166 (2011). [CrossRef] [PubMed]

]. Such manipulation requires the fabrication of the crystals, whose refractive index is modified by complex nonlinear processes. At some point these crystals can be seen as diffused waveguides with a GRIN index as the one described in our experiments.

4. Conclusions

We have demonstrated for the first time the solitary-like long-distance wave propagation of Airy beams in inhomogeneous media. Using Fermat's principle for rays and wave propagation we provided the physics of this behavior as the result of the balance between the free space acceleration of Airy beams in one direction and the effect of the gradient index that induces a deflection in the opposite direction. The Airy beam solutions in GRIN media was fully described theoretically using an operational method. The solution also shows the possibility of inversion of the acceleration by manipulating the gradient index strength that was demonstrated experimentally.

Acknowledgements

References and links

1.

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]

2.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef] [PubMed]

3.

J. Durnin, J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]

4.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987). [CrossRef]

5.

D. DeBeer, S. R. Hartmann, and R. Friedberg, “Comment on “Diffraction-free beams”,” Phys. Rev. Lett. 59(22), 2611 (1987). [CrossRef] [PubMed]

J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Durnin, Miceli, and Eberly reply,” Phys. Rev. Lett. 59(22), 2612 (1987).

6.

S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).

7.

M. V. Berry and N. L. Balazs, “Nonspreading wave-packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]

8.

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004). [CrossRef] [PubMed]

9.

C. López-Mariscal, M. Bandres, J. Gutiérrez-Vega, and S. Chávez-Cerda, “Observation of parabolic nondiffracting optical fields,” Opt. Express 13(7), 2364–2369 (2005). [CrossRef] [PubMed]

10.

D. Marcuse, “TE modes of graded index slab waveguides,” IEEE J. Quantum Electron. 9(10), 1000–1006 (1973). [CrossRef]

11.

C.-L. Chen, Foundations of guided-wave optics (Wiley, New Jersey 2006), Ch 3.

12.

D. N. Christodoulides and T. H. Coskun, “Diffraction-free planar beams in unbiased photorefractive media,” Opt. Lett. 21(18), 1460–1462 (1996). [CrossRef] [PubMed]

13.

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010). [CrossRef] [PubMed]

14.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009). [CrossRef]

15.

I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett. 35(10), 1581–1583 (2010). Ye, Zhuoyi; Liu, Sheng; Lou, Cibo; Zhang, Peng; Hu, Yi; Song, Daohong; Zhao, Jianlin; Chen, Zhigang, Quantum Electronics and Laser Science Conference (QELS) 2011 paper: JTuI32, OSA Technical Digest (CD).

16.

W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. 36(7), 1164–1166 (2011). [CrossRef] [PubMed]

17.

M. Born and E. Wolf, Principles of Optics, Seventh Ed., (Cambridge University Press, Cambridge 1999), Ch. 3.

18.

There exist several forms for producing a solution of this equation, for instance by using the Zassenhaus formula [19], or by simplifying the differential equation via a transformation (see for instance [20] for a quadratic term). The operators to disentangle the exponential of the sum of two operators can also be given in different orderings of the exponentials involved. Of course, they are all equivalent.

19.

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8(4), 962–982 (1967). [CrossRef]

20.

H. Moya-Cessa and M. Fernández Guasti, “Coherent states for the time dependent harmonic oscillator: the step function,” Phys. Lett. A 311(1), 1–5 (2003). [CrossRef]

21.

W. H. Louissel, Quantum Statistical Properties of Radiation (Wiley-Interscience, New York 1990), Ch. 3.

22.

H. Moya-Cessa and F. Soto-Eguibar, Differential equations: an operational approach, (Rinton Press, New Jersey 2011), Ch. 2.

23.

G. N. Watson, A treatise on the theory of Bessel functions, Ch. VI, Cambridge University Press, Cambridge 1944).

24.

W. M. Strouse, “Bouncing light beams,” Am. J. Phys. 40(6), 913–914 (1972). [CrossRef]

25.

D. Ambrosini, A. Ponticiello, G. S. Spagnolo, R. Borghi, and F. Gori, “Bouncing light beams and the Hamiltonian analogy,” Eur. J. Phys. 18(4), 284–289 (1997). [CrossRef]

26.

SLM 512, Boulder nonlinear systems.

OCIS Codes
(000.3860) General : Mathematical methods in physics
(080.1510) Geometric optics : Propagation methods
(080.5692) Geometric optics : Ray trajectories in inhomogeneous media
(070.7345) Fourier optics and signal processing : Wave propagation
(260.2710) Physical optics : Inhomogeneous optical media

ToC Category:
Physical Optics

History
Original Manuscript: June 15, 2011
Revised Manuscript: July 28, 2011
Manuscript Accepted: July 28, 2011
Published: August 11, 2011

Citation
Sabino Chávez-Cerda, Ulises Ruiz, Victor Arrizón, and Héctor M. Moya-Cessa, "Generation of Airy solitary-like wave beams by acceleration control in inhomogeneous media," Opt. Express 19, 16448-16454 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16448


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References

  1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]
  2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef] [PubMed]
  3. J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]
  4. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987). [CrossRef]
  5. D. DeBeer, S. R. Hartmann, and R. Friedberg, “Comment on “Diffraction-free beams”,” Phys. Rev. Lett. 59(22), 2611 (1987).J. Durnin, J. J. Miceli, and J. H. Eberly, “Durnin, Miceli, and Eberly reply,” Phys. Rev. Lett. 59(22), 2612 (1987). [CrossRef] [PubMed]
  6. S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).
  7. M. V. Berry and N. L. Balazs, “Nonspreading wave-packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]
  8. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004). [CrossRef] [PubMed]
  9. C. López-Mariscal, M. Bandres, J. Gutiérrez-Vega, and S. Chávez-Cerda, “Observation of parabolic nondiffracting optical fields,” Opt. Express 13(7), 2364–2369 (2005). [CrossRef] [PubMed]
  10. D. Marcuse, “TE modes of graded index slab waveguides,” IEEE J. Quantum Electron. 9(10), 1000–1006 (1973). [CrossRef]
  11. C.-L. Chen, Foundations of guided-wave optics (Wiley, New Jersey 2006), Ch 3.
  12. D. N. Christodoulides and T. H. Coskun, “Diffraction-free planar beams in unbiased photorefractive media,” Opt. Lett. 21(18), 1460–1462 (1996). [CrossRef] [PubMed]
  13. S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010). [CrossRef] [PubMed]
  14. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009). [CrossRef]
  15. I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett. 35(10), 1581–1583 (2010). Ye, Zhuoyi; Liu, Sheng; Lou, Cibo; Zhang, Peng; Hu, Yi; Song, Daohong; Zhao, Jianlin; Chen, Zhigang, Quantum Electronics and Laser Science Conference (QELS) 2011 paper: JTuI32, OSA Technical Digest (CD).
  16. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. 36(7), 1164–1166 (2011). [CrossRef] [PubMed]
  17. M. Born and E. Wolf, Principles of Optics, Seventh Ed., (Cambridge University Press, Cambridge 1999), Ch. 3.
  18. There exist several forms for producing a solution of this equation, for instance by using the Zassenhaus formula [19], or by simplifying the differential equation via a transformation (see for instance [20] for a quadratic term). The operators to disentangle the exponential of the sum of two operators can also be given in different orderings of the exponentials involved. Of course, they are all equivalent.
  19. R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8(4), 962–982 (1967). [CrossRef]
  20. H. Moya-Cessa and M. Fernández Guasti, “Coherent states for the time dependent harmonic oscillator: the step function,” Phys. Lett. A 311(1), 1–5 (2003). [CrossRef]
  21. W. H. Louissel, Quantum Statistical Properties of Radiation (Wiley-Interscience, New York 1990), Ch. 3.
  22. H. Moya-Cessa and F. Soto-Eguibar, Differential equations: an operational approach, (Rinton Press, New Jersey 2011), Ch. 2.
  23. G. N. Watson, A treatise on the theory of Bessel functions, Ch. VI, Cambridge University Press, Cambridge 1944).
  24. W. M. Strouse, “Bouncing light beams,” Am. J. Phys. 40(6), 913–914 (1972). [CrossRef]
  25. D. Ambrosini, A. Ponticiello, G. S. Spagnolo, R. Borghi, and F. Gori, “Bouncing light beams and the Hamiltonian analogy,” Eur. J. Phys. 18(4), 284–289 (1997). [CrossRef]
  26. SLM 512, Boulder nonlinear systems.

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