Research Article Self-Focusing of Hermite-Cosh-Gaussian ...downloads.hindawi.com/archive/2014/942750.pdf · Research Article Self-Focusing of Hermite-Cosh-Gaussian Laser Beams in

Research ArticleSelf-Focusing of Hermite-Cosh-Gaussian Laser Beams in Plasmaunder Density Transition

Manzoor Ahmad Wani and Niti Kant

Department of Physics, Lovely Professional University, Phagwara, Punjab 144411, India

Correspondence should be addressed to Manzoor Ahmad Wani; [email protected]

Received 16 June 2014; Accepted 9 September 2014; Published 29 September 2014

Academic Editor: Kim Fook Lee

Copyright © 2014 M. A. Wani and N. Kant. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Self-focusing of Hermite-Cosh-Gaussian (HChG) laser beam in plasma under density transition has been discussed here.The fielddistribution in the medium is expressed in terms of beam-width parameters and decentered parameter. The differential equationsfor the beam-width parameters are established by a parabolic wave equation approach under paraxial approximation. To overcomethe defocusing, localized upward plasma density ramp is considered, so that the laser beam is focused on a small spot size. Plasmadensity ramp plays an important role in reducing the defocusing effect and maintaining the focal spot size up to several Rayleighlengths. To discuss the nature of self-focusing, the behaviour of beam-width parameters with dimensionless distance of propagationfor various values of decentered parameters is examined by numerical estimates.The results are presented graphically and the effectof plasma density ramp and decentered parameter on self-focusing of the beams has been discussed.

1. Introduction

The self-focusing of laser beams in nonlinear optical mediais a fascinating topic which has inspired theoretical andexperimental interest [1–3]. In self-focusing and self-phasemodulation of Cosh-Gaussian laser beam in magnetoplasmausing variational approach, it is found that the decenteredparameter along with absorption coefficient plays a key rolein the nature of self-focusing/defocusing of the beam [4].However in the propagation of a Cos-Gaussian beam in akerr medium, it is found that although the RMS beam widthbroadens, the central parts of the beam give rise to an initialradial compression and a significant redistribution duringpropagation. The partial collapse of central part of the beamappears while the RMS beam width still increases or remainsconstant. It is further observed that the Cos-Gaussian beameventually converts into a Cosh-Gaussian type beam in akerr medium with low and moderate power [5]. In self-focusing of Cosh-Gaussian laser beam in plasma with weakrelativistic and ponderomotive regime, it is observed that alarge value of absorption level weakens the self-focusing effectin the absence of decentered parameter. However, oscillatory

self-focusing takes place for a higher value of decenteredparameter, 𝑏 = 1, and all curves are seen to exhibit sharpself-focusing effect for 𝑏 = 2 [6]. In ponderomotive self-focusing of a short laser pulse under plasma density ramp, thepulse acquires a minimum spot size. As the laser propagatesthrough the density ramp region, it sees a slowly narrowingchannel. In such a case the oscillation amplitude of the spotsize shrinks, while its frequency increases. Therefore, thelaser pulse propagating in a plasma density ramp tends tobecome more focused. If there is no density ramp, the laserpulse is defocused due to the dominance of the diffractioneffect. As the plasma density increases, self-focusing effectbecomes stronger. Similarly as in case of no density ramp,the beam-width parameter does not increase much. Afterseveral Rayleigh lengths, the beam-width parameter attainsa minimum value and maintains it for a long distance.Consequently, the self-focusing effect is enhanced and thelaser pulse is more focused [7].

Nanda et al. [8] while studying the enhanced relativisticself-focusing ofHermite-CoshGaussian laser beam inplasmaunder density transition observed that the proper selectionof decentered parameter and presence of density transition

Hindawi Publishing CorporationAdvances in OpticsVolume 2014, Article ID 942750, 5 pageshttp://dx.doi.org/10.1155/2014/942750

2 Advances in Optics

results stronger self-focusing of laser beam. In self-focusingof Hermite-Cosh Gaussian laser beam in a magnetoplasmawith a ramp density profile, the authors concluded that thepresence of plasma density ramp andmagnetic field enhancesthe self-focusing effect to a greater extent [9]. The properselection of decentered parameter was very much sensitiveto self-focusing [10]. However in studying the self-focusingof Hermite-Gaussian laser beams in plasma under plasmadensity ramp by Kant et al. [11] the authors found that theeffect of plasma density ramp and initial intensity of the laserbeam are important and play a vital role in laser plasmainteraction and hence in strong self-focusing of laser beam.

Recently, a new laser beam, called the Hermite-Cosh-Gaussian (HChG) beam, has been studied extensively andit has been found that such beams can be generated in thelaboratory by the superposition of two decentered Hermite-Gaussian beams as Cosh-Gaussian ones [12]. In this paper, wemainly study the self-focusing of HChG laser beams propa-gating in underdense plasma under plasma density ramp ofthe form 𝑛(𝜉) = 𝑛

0Tan(𝜉/𝑑) by a ponderomotive mechanism.

Analytical formulas for HChG beams are derived and resultsare discussed.

2. Theoretical Considerations

2.1. Field Distribution of HChG Beams. We employed thepropagation of HChG laser beam along 𝑧-axis in the plasmawith the field distribution in the following form:

𝐸 (𝑟, 𝑧)

=𝐸0

2𝑓𝐻𝑚(√2𝑟

𝑟0𝑓) exp(𝑏

2

4)

× {exp[−( 𝑟

𝑟0𝑓+𝑏

2)

2

] + exp[−( 𝑟

𝑟0𝑓−𝑏

2)

2

]} ,

(1)

where 𝑚 is the mode index associated with the HermitePolynomial 𝐻

𝑚, 𝑟0is the waist width of Gaussian amplitude

distribution, 𝑏 is the decentered parameter, 𝑟 is the radialcoordinate, 𝐸

0is the amplitude of Gaussian beams for the

central position at 𝑟 = 𝑧 = 0, 𝐴0(𝑟, 𝑧) is the amplitude

of HChG beams in cylindrical coordinates, and 𝑓 is thedimensionless beam-width parameter, which is a measure ofboth axial intensity and width of the beam.

2.2. Nonlinear Dielectric Constant. Further, we considerpropagation of HChG laser beam in a nonlinear mediumcharacterized by dielectric constant of the form

𝜀 = 𝜀0+ Φ (𝐸𝐸

∗) (2)

with 𝜀0= 1 − 𝜔

2

𝑝/𝜔2 and 𝜔2

𝑝= 4𝜋𝑛(𝜉)𝑒

2/𝑚.

With

𝑛 (𝜉) = 𝑛0Tan( 𝜉

𝑑) , 𝜉 =

𝑧

𝑅𝑑

,

𝜀0= 1 − (

𝜔2

𝑝0

𝜔2)Tan( 𝜉

𝑑) , 𝜔

2

𝑝0=4𝜋𝑛0𝑒2

𝑚,

(3)

where 𝜀0and Φ represent the linear and nonlinear parts of

dielectric constant, respectively, 𝜔𝑝is plasma frequency, 𝑒 is

the electronic charge, 𝑚 is the electron mass, 𝑛0is the equi-

librium electron density, 𝑅𝑑is the diffraction length, 𝜉 is the

normalized propagation distance, and 𝑑 is a dimensionlessadjustable parameter.

Now, in case of collision-less plasma, the nonlinear-ity in the dielectric constant is mainly due to pondero-motive force and the nonlinear part of dielectric constantis given by

Φ(𝐴𝐴∗) ≈ (

1

2) 𝜀2𝐴2

0(4)

with

𝜀2= 2(

𝜔2

𝑝

𝜔2)𝛼; 𝛼 =

𝑒2𝑀

6𝑚2𝜔2𝐾𝑏𝑇0

, (5)

where 𝑀 is the mass of scatterer in the plasma, 𝜔 is thefrequency of laser used, 𝐾

𝑏is the Boltzmann constant, and

𝑇0is the equilibrium plasma temperature.

3. Self-Focusing

The wave equation governing the propagation of laser beammay be written as follows:

∇2𝐸 + (

𝜔2

𝑐2) 𝜀𝐸 + ∇(

𝐸∇𝜀

𝜀) = 0. (6)

The last term of (6) on left-hand side can be neglectedprovided that 𝑘−2∇2(ln 𝜀) ≪ 1, where “𝑘” represents the wavenumber. Thus,

∇2𝐸 + (

𝜔2

𝑐2) 𝜀𝐸 = 0. (7)

This equation is solved by employing Wentzel-Kramers-Brillouin (WKB) approximation. Employing the WKBapproximation, (7) reduces to a parabolic wave equation asfollows:

𝑖𝜔

𝑐(2√1 −

𝜔2

𝑝0

𝜔2Tan( 𝑧

𝑑𝑅𝑑

)

−(𝜔2

𝑝0/𝜔2) 𝑧Sec2 (𝑧/𝑑𝑅

𝑑)

𝑑𝑅𝑑√1 − (𝜔2

𝑝0/𝜔2)Tan (𝑧/𝑑𝑅

𝑑)

)(𝜕𝐴

𝜕𝑧)

−(𝜔2

𝑝0/𝜔2)𝐴Sec2 (𝑧/𝑑𝑅

𝑑)

𝑑𝑅𝑑√1 − (𝜔2


𝑑)

Advances in Optics 3

−(𝜔2

𝑝0/𝜔2)𝐴𝑧Sec2 (𝑧/𝑑𝑅

𝑑)

𝑑2𝑅2𝑑√1 − (𝜔2


𝑑)

× (Tan( 𝑧

𝑑𝑅𝑑

) +(𝜔2

𝑝0/𝜔2) Sec2 (𝑧/𝑑𝑅

𝑑)

4 (1 − (𝜔2𝑝0/𝜔2)Tan (𝑧/𝑑𝑅

𝑑)))

=𝜔2

𝑐2((𝜔2

𝑝0/𝜔2)𝐴𝑧Sec2 (𝑧/𝑑𝑅

𝑑)

𝑑𝑅𝑑

−(𝜔2

𝑝0/𝜔2)2

𝐴𝑧2Sec4 (𝑧/𝑑𝑅

𝑑)

4𝑑2𝑅2𝑑(1 − (𝜔2


𝑑)))𝜕2𝐴

𝜕𝑟2

+1

𝑟

𝜕𝐴

𝜕𝑟+𝜔2

𝑐2Φ(𝐴𝐴

∗) 𝐴.

(8)

To solve (8) we express 𝐴 as

𝐴 = 𝐴0(𝑟, 𝑧) exp [−𝑖𝑘𝑠 (𝑟, 𝑧)] , (9)

where

𝑘 =𝜔

𝑐

√1 −𝜔2

𝑝0

𝜔2Tan( 𝑧

𝑑𝑅𝑑

), (10)

where “𝐴0” and “𝑆” are real functions of 𝑟 and 𝑧 with

𝑆(𝑟, 𝑧) as eikonal of the beam which determines conver-gence/divergence of the beam. Substituting for 𝐴 from (9) in(8) and equating real and imaginary parts on both sides of theresulting equation, one obtains

𝑐2

𝜔2𝐴0

(𝜕2𝐴0

𝜕𝑟2+1

𝑟

𝜕𝐴0

𝜕𝑟) + Φ (𝐴

2

0)

= (2(1 −𝜔2

𝑝0

𝜔2Tan( 𝑧

𝑑𝑅𝑑

)) −𝜔2

𝑝0

𝜔2

𝑧Sec2 (𝑧/𝑑𝑅𝑑)

𝑑𝑅𝑑

)𝜕𝑆

𝜕𝑧

+ (1 −𝜔2

𝑝0

𝜔2Tan( 𝑧

𝑑𝑅𝑑

))(𝜕𝑆

𝜕𝑟)

2

−𝜔2

𝑝0

𝜔2

Sec2 (𝑧/𝑑𝑅𝑑)

𝑑𝑅𝑑

× (𝑆 + 𝑧 −𝜔2

𝑝0

𝜔2

𝑧Sec2 (𝑧/𝑑𝑅𝑑) (𝑆 − 𝑧/2)

2𝑑𝑅𝑑(1 − (𝜔2


𝑑))) ,

(1 −𝜔2

𝑝0

𝜔2

𝑧Sec2 (𝑧/𝑑𝑅𝑑)

2𝑑𝑅𝑑(1 − (𝜔2


𝑑)))𝜕𝐴2

0

𝜕𝑧

+𝜕𝑆

𝜕𝑟

𝜕𝐴2

0

𝜕𝑟+ (

𝜕2𝑆

𝜕𝑟2+1

𝑟

𝜕𝑆

𝜕𝑟)𝐴2

0

−(𝜔2

𝑝0

𝜔2

Sec2 (𝑧/𝑑𝑅𝑑)

𝑑𝑅𝑑√1 − (𝜔2


𝑑)

+𝜔2

𝑝0

𝜔2

𝑧Sec2 (𝑧/𝑑𝑅𝑑)Tan (𝑧/𝑑𝑅

𝑑)

𝑑2𝑅2𝑑√1 − (𝜔2


𝑑)

)𝐴2

0

− (𝜔2

𝑝0

𝜔2)

2

(𝑧Sec4 (𝑧/𝑑𝑅

𝑑)

4𝑑2𝑅2𝑑(1 − (𝜔2


𝑑)))𝐴2

0

= 0.

(11)

The solutions of (11) for a cylindrically symmetric HChGbeam can be written as follows:

𝑆 =𝑟2

2𝛽 (𝑧) + Φ (𝑧) ,

𝐴2

0=

𝐸2

0

4𝑓2𝐻2

𝑚(√2𝑟

𝑟0𝑓) exp(𝑏

2

2)

× {exp[−2( 𝑟

𝑟0𝑓+𝑏

2)

2

] + exp[−2( 𝑟

𝑟0𝑓−𝑏

2)

2

]

+ 2 exp[−( 2𝑟2

𝑟20𝑓2

+𝑏2

2)]}

(12)

with

𝛽 (z) = 1

𝑓 (𝑧)

𝜕𝑓

𝜕𝑧, (13)

where 𝛽(𝑧) is the inverse radius of curvature of wave frontand Φ(𝑧) is the phase shift.

Under the paraxial approximation, we have establishedthe differential equation of the beam-width parameter for the𝑚 = 0mode as follows:

(1 −𝜔2

𝑝0

𝜔2Tan( 𝜉

𝑑))(

6𝑏2

(2 − 𝑏2) 𝑓)

+𝛼𝐸2

0

2(𝑟0𝜔

𝑐)

2

(𝜔2

𝑝0

𝜔2)(1 −

𝜔2

𝑝0

𝜔2Tan( 𝜉

𝑑))

Tan (𝜉/𝑑)𝑓3

= [(𝜔2

𝑝0

𝜔2)Tan( 𝜉

𝑑) + (

𝜔2

𝑝0

𝜔2)𝜉Sec2 (𝜉/𝑑)

2𝑑− 1]

𝜕2𝑓

𝜕𝜉2

− [(𝜔2

𝑝0

𝜔2)𝜉Sec2 (𝜉/𝑑)

2𝑑]1

𝑓(𝜕𝑓

𝜕𝜉)

2

4 Advances in Optics

𝜔p0/𝜔 = 0.04, b = 0

𝜔p0/𝜔 = 0.03, b = 0

𝜔p0/𝜔 = 0.02, b = 0

𝜉

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

f(𝜉)

(a)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

𝜉

0.00 0.05 0.10 0.15 0.20 0.25 0.30

𝜔p0/𝜔 = 0.04, b = 1

𝜔p0/𝜔 = 0.03, b = 1

𝜔p0/𝜔 = 0.02, b = 1

f(𝜉)

(b)

Figure 1: Variation of beam-width parameter with the normalised propagation distance (𝜉) with a plasma density ramp. The normalizedparameters are (a) 𝑑 = 5, 𝜔

𝑝0/𝜔 = 0.02 (blue curve), 𝜔

𝑝0/𝜔 = 0.03 (red curve), and 𝜔

𝑝0/𝜔 = 0.04 (black curve) at decentered parameter 𝑏 = 0

and (b) 𝑑 = 5, 𝜔𝑝0/𝜔 = 0.02 (blue curve), 𝜔

𝑝0/𝜔 = 0.03 (red curve), and 𝜔

𝑝0/𝜔 = 0.04 (black curve) at decentered parameter 𝑏 = 1.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

𝜉

0.00 0.05 0.10 0.15 0.20 0.25 0.30

𝜔p0/𝜔 = 0.02, b = 1

𝜔p0/𝜔 = 0.02, b = 0

f(𝜉)

(a)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

𝜉

0.00 0.05 0.10 0.15 0.20 0.25 0.30

𝜔p0/𝜔 = 0.03, b = 1

𝜔p0/𝜔 = 0.03, b = 0

f(𝜉)

(b)

Figure 2: Variation of beam-width parameter with the normalised propagation distance (𝜉) with a plasma density ramp. The normalizedparameters are (a) 𝑑 = 5 and decentered parameters are 𝑏 = 0 (red curve) and 𝑏 = 1 (black curve) at 𝜔

𝑝0/𝜔 = 0.02 and (b) 𝑑 = 5 and

decentered parameters are 𝑏 = 0 (red curve) and 𝑏 = 1 (black curve) at 𝜔𝑝0/𝜔 = 0.03.

+ (𝜔2

𝑝0

𝜔2)

Sec2 (𝜉/𝑑)2𝑑

× [

[

1 − (𝜔2

𝑝0

𝜔2)

𝜉Sec2 (𝜉/𝑑)2𝑑 (1 − (𝜔2

𝑝0/𝜔2)Tan (𝜉/𝑑))

]

]

.

(14)

Equations (14) is the required expression for beam-widthparameters 𝑓.

4. Results and Discussions

For an initial plane wave front of the beam, we use theboundary conditions 𝑓 = 1 and 𝜕𝑓/𝜕𝜉 = 0 at 𝜉 = 0. To checkthe validity of the above analysis, we conduct computationalsimulations for solving the beam-width parameter equation.The following parameters are chosen for the purpose ofnumerical calculations: 𝜔 = 10

14 rad/s, 𝜔𝑝0

= 5.640 ×

1012 rad/s, 𝑟

0= 5 × 10

−3 cm, 𝑛0= 9.983 × 10

17 cm−3, and𝑇0= 105 K. The variation of the beam-width parameter 𝑓

Advances in Optics 5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

𝜉

0.00 0.05 0.10 0.15 0.20 0.25 0.30

𝜔p0/𝜔 = 0.04, b = 1

𝜔p0/𝜔 = 0.04, b = 0

f(𝜉)

Figure 3: Variation of beam-width parameter with the normalizedpropagation distance (𝜉) with a plasma density ramp. The normal-ized parameters are 𝑑 = 5 and decentered parameters are 𝑏 = 0 (redcurve) and 𝑏 = 1 (black curve) at 𝜔

𝑝0/𝜔 = 0.04.

as a function of the normalized propagation distance in anunderdense plasma with an upward plasma density ramp isshown in Figures 1, 2, and 3.

From the above figures, it is found that the beam givesa self-focusing effect for 𝑏 ≤ 1. It is further observed thatas the plasma density increases, self-focusing becomes muchstronger. Combining the results of this paper with previousstudies on Gaussian beams [13, 14], we see that HChG beamsgive freedom to additional source parameters mode index(𝑚) and decentered parameter (𝑏), changing the nature ofself-focusing/defocusing significantly. To overcome defocus-ing, localized upward plasma density ramp is introducedand it is obvious that by applying the density ramp the self-focusing effect is enhanced and the laser is more focused; thatis, self-focusing becomes much stronger. Hence, the upwardplasma density ramp plays an important role in enhancinglaser focusing.

5. Conclusion

In the present investigation, the authors have studied the self-focusing of Hermite-Cosh-Gaussian (HChG) laser beams byconsidering plasma density ramp in a parabolic mediumunder paraxial approximation. It is observed that as boththe plasma density and the decentered parameter increase,the self-focusing effect becomes stronger. However sharpself-focusing of such beams occurs for 𝑏 ≤ 1. Hence, byintroducing such a density profile, a much stronger self-focusing is observed and it could produce ultrahigh laserirradiance over distances much greater than the Rayleighlength which can be used for various applications.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

References

[1] R. Bingham, J. T. Mendonca, and P. K. Shukla, “Plasma basedcharged-particle accelerators,” Plasma Physics and ControlledFusion, vol. 46, no. 1, p. R1, 2004.

[2] P. Sprangle, E. Esarey, and J. Krall, “Laser driven electronacceleration in vacuum, gases, and plasmas,” Physics of Plasmas,vol. 3, no. 5, pp. 2183–2190, 1996.

[3] D. N. Gupta and H. Suk, “Frequency chirp for resonanceenhanced electron energy during laser acceleration,” Physics ofPlasmas, vol. 13, Article ID 044507, 2006.

[4] T. S. Gill, R. Kaur, and R. Mahajan, “Relativistic self-focusingand self-phase modulation of Cosh-Gaussian laser beam inmagnetoplasma,” Laser and Particle Beams, vol. 29, pp. 183–191,2011.

[5] R. Chen, Y. Ni, and X. Chu, “Propagation of a cos-Gaussianbeam in a Kerr medium,” Optics and Laser Technology, vol. 43,no. 3, pp. 483–487, 2011.

[6] T. S. Gill, R. Mahajan, and R. Kaur, “Self-focusing of cosh-Gaussian laser beam in a plasma with weakly relativistic andponderomotive regime,” Physics of Plasmas, vol. 18, Article ID033110, 2011.

[7] N. Kant, S. Saralch, andH. Singh, “Ponderomotive self-focusingof a short laser pulse under a plasma density ramp,”Nukleonika,vol. 56, no. 2, pp. 149–153, 2011.

[8] V. Nanda and N. Kant, “Enhanced relativistic self-focusing ofHermite-cosh-Gaussian laser beam in plasma under densitytransition,” Physics of Plasmas, vol. 21, no. 4, Article ID 042101,2014.

[9] V.Nanda,N. Kant, andM.A.Wani, “Self-focusing of aHermite-Cosh Gaussian laser beam in a magnetoplasma with rampdensity profile,” Physics of Plasmas, vol. 20, no. 11, Article ID113109, 2013.

[10] V. Nanda, N. Kant, and M. A. Wani, “Sensitiveness of decen-tered parameter for relativistic self-focusing of hermite-cosh-gaussian laser beam in plasma,” IEEE Transactions on PlasmaScience, vol. 41, no. 8, pp. 2251–2256, 2013.

[11] N. Kant, M. A. Wani, and A. Kumar, “Self-focusing of Hermite-Gaussian laser beams in plasma under plasma density ramp,”Optics Communications, vol. 285, no. 21-22, pp. 4483–4487, 2012.

[12] A. Belafhal and M. Ibnchaikh, “Propagation properties ofHermite-cosh-Gaussian laser beams,” Optics Communications,vol. 186, no. 4–6, pp. 269–276, 2000.

[13] M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, Self Focusing ofLaser Beams in Dielectrics, Semiconductors and Plasmas, Tata-McGraw-Hill, New Delhi, India, 1974.

[14] S. Patil, M. Takale, V. Fulari, and M. Dongare, “Propagation ofHermite-cosh-Gaussian laser beams in non-degenerate germa-nium having space charge neutrality,” Journal of Modern Optics,vol. 55, no. 21, pp. 3529–3535, 2008.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of

Documents

Research Article Self-Focusing of Hermite-Cosh-Gaussian ...downloads.hindawi.com/archive/2014/942750.pdf · Research Article Self-Focusing of Hermite-Cosh-Gaussian Laser Beams in