Optics Communications 283 (2010) 417–426

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Optics Communications

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Mathieu–Gauss beams: A thermal consideration

H. Nadgaran a,*, M. Servatkhah a, M. Sabaeian b

a Physics Department, Shiraz University, Shiraz 71454, Iranb Physics Department, Shahid Chamran University, Ahvaz, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 February 2009Received in revised form 8 October 2009Accepted 19 October 2009

0030-4018/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.optcom.2009.10.040

* Corresponding author. Tel./fax: +98 7112284594.E-mail address: nadgaran@susc.ac.ir (H. Nadgaran

The generation of Helmholtz–Gauss beam families such as Mathieu–Gauss (MG) and Bessel–Gauss (BG)beams can be dramatically suffered from heat-induced thermal effects. This work is devoted to the gen-eration of MG beams under severe induced thermal loads. The output MG beams of a solid-state laserwere assumed to pass through an ABCD system matrix. The simulated results which were achieved bycharacterization of the transfer matrix of the thermally affected inhomogeneous medium report system-atic variations of the intensity profile of MG beams from pump power of 1–5 W. It is shown that for highvalues of pump power, the thermal effects are so influential and the beam intensity profile is so changedthat the identification of MG beams is not easily possible at first hand. The results can facilitate this iden-tification and can be utilized for deeper understanding of thermal effects phenomena.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Mathieu beams are considered as an ideal non-diffracting solu-tion to the two dimensional Helmholtz wave equation in ellipticalcylindrical coordinates [1–3]. These non-diffracting solutions areusually apodized by a radial Gaussian to give Mathieu–Gauss(MG) beams. Unlike Mathieu beams, their apodized version, MGbeams, can carry finite amount of energy and are nearly non-dif-fracting because of acquiring low divergence over a long range ofpropagation. MG beams have been realized experimentally byoptical set-ups [4] and their propagation have well been studiedin free space and even through ABCD optical systems [1,4–6]. MGbeams have attracted many scientific and industrial researchersin working with photonic lattices [7], optical tweezers [8,9], local-ized X waves [10] and optical scanners [11]. MG beams can also beconsidered as to have potential applications in wireless communi-cations, laser machining, and surgery [12–15].

Thermal effects which play an important and crucial role in la-ser’s beam quality discussions have been the origin of many re-searches in which high thermal load density of laser gainmedium is inevitable. Induced heat loads which lead to thermo-optical and thermo-mechanical effects cause beam profile degra-dation. Since recognition and discrimination of thermally affectedlaser beams from their pure, and standard non-thermal counter-parts is very important in practice, it is necessary to discuss thepropagation of various special form of laser beams in inhomoge-neous thermal lens media.

ll rights reserved.

).

In our recent paper, we have reported a thermal model whichinvestigates the thermal effects on the generation of anotherGaussian apodized solution of Helmholtz equation known as theBessel–Gauss (BG) beams [16]. In that work which was based onthe Hakola’s design [17] parameters, we have shown the extentof the heat induced effects on the output beam profile.

Recently Kartashov et al. have also been determining the impor-tance of thermal effects in investigating the stability of vortex sol-itons in thermal nonlinear media [18].

The main aim of this work is to present a thermal modeladdressing thermally-affected MG beams. The model simulatesthe beam profile of zero-order MG beams and compare it withthe non-thermal beam profile. The results not only supports thejustification of the thermal effects importance, but also demon-strates that these effects are so pronounced, that MG beam identi-fication at some high pump powers is indeed difficult.

2. Mathieu–Gauss (MG) beams

To give an insight to the Gaussian apodized Mathieu beamsknown as MG beams a complex field amplitude of Helmholtz–Gauss beam A1ð~r1;j1Þ at the input plane of z ¼ z1 of a radial sym-metrical ABCD system, together with its time dependence ofexpð�ixtÞ is defined as [6,19,20]:

A1ð~r1;j1Þ ¼ expð�r21=x

21ÞUð~r1;j1Þ ð1Þ

where~r1 ¼ ðx1; y1Þ shows the transverse coordinates, x1 and q1 sat-isfy x1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2iq1=k

pwhere x1 is the waist of the Gaussian envelope

and q1 shows the input parameter of the Gaussian beam travelingaxially through the ABCD system, and j1 ¼ kt is the transverse wave

Fig. 1. (a) Crystal temperature distribution versus r in the plane of z = 0 and (b)versus Z at r = 0 for 1, 2, 3, 4 and 5 W pump power from bottom to top.

418 H. Nadgaran et al. / Optics Communications 283 (2010) 417–426

number. Uð~r1;j1Þ is the solution of the two dimensional Helmholtzequation of:

ðr2t þ k2

t Þ U ¼ 0 ð2Þ

where

r2t ¼

@2

@x2 þ@2

@y2

In order to find U, we use the separation of variables techniqueby letting:

U ¼ RðnÞUðgÞ ð3Þ

where the elliptical radial coordinate nð0 6 n <1Þ and ellipticalangular coordinate gð0 6 g < 2pÞ are related to Cartesian coordi-nates by:

x ¼ f cosh ncos gy ¼ f sinh nsin g ð4Þ

where it is assumed that the foci are located at ð�f ; oÞ. Therfore thefolowing two equations coming out of the seperation of variablestechnique must satisfy:

d2Udg2 þ ða� 2q cos 2gÞU ¼ 0

d2R

dn2 � ða� 2q cosh 2nÞR ¼ 0 ð5Þ

where q ¼ f 2k2t =4 carries information about the transverse spatial

frequency, kt , and a is the separation constant.U which is known as even-order MG beams can then be written

as [3]:

MGemðn;g; z ¼ 0Þ ¼ Ne

m exp � r2

x20

� �Jemðn; qÞcemðg; qÞ ð6Þ

where m ¼ 0;2;4; . . . for even beams, and Nem is the normalization

factor. Jeemð�Þ and cee

mð�Þ are radial and angular Mathieu functionswhich are the solutions of Eq. (5), respectively.The productJee

mð�Þceemð�Þ can be expanded in Bessel functions, that is [3]:

Jee2nðn; qÞcee

2nðg; qÞ ¼ C2n

X1j¼0

A2j½ð�1Þj2pJ2jðktrÞ � cosð2jhÞ� ð7Þ

Therefore MG beams can be written as an expansion of BGbeams:

MGe2n ¼ Ne

2n2pC2n

X1j¼0

ð�1ÞjA2j � exp � r2

x20

� �J2jðktrÞ cosð2jhÞ ð8Þ

with

Ne2n ¼

2 expðr=2Þffiffiffiffipp

x0ce2nð0; qÞce2nðp=2; qÞ� A0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP1

j¼0ð1þ dj;0ÞI2jðrÞA2

2j

s266664

377775 ð9Þ

where r ¼ k2t x2

0=4, Imð�Þ is the mth-order modified Bessel functionof the first kind and di;j is the Kronecker delta.

2.1. MG beams propagating through an ABCD optical system

We have so far introduced even-order MG beams as an expan-sion of normalized BG beams.

We have adopted the method of Guizar-Sicairos et al. [6] inusing Huygens diffraction integral, and use their relation for thetransverse distribution of a mth-order Mathieu–Gauss beam atoutput plane z ¼ z2 of an ABCD system as:

A2ð~r2;j2Þ ¼ expj1j2B

i2k

� �expðik1ÞAþ B=q1

� expikr2

2

2q2

!Uð~r2;j2Þ ð10Þ

where 1 is the optical path length from the input to the outputplane measured along the optical axis, q2 and j2 are the complexparameter and complex transverse wave number at the outputplane, respectively. These parameters obey the following transfor-mation laws:

q2 ¼Aq1 þ BCq1 þ D

ð11Þ

j2 ¼j1q1

Aq1 þ Bð12Þ

We notice that q complex parameter is defined as:1=q ¼ 1=Rþ 2i=kx2 where R and k are the radius of wavefront cur-vature and wave vector of Gaussian apodization, respectively.

By using Eqs. (8) and (10) the evolution of MG beams throughan ABCD system at plane z ¼ z2 is written as:

MGeOUT

2n ð~r2; j2Þ ¼ Ne2n2pC2n

X1j¼0

ð�1ÞjA2j � expj1j2B

i2k

� �expðik1ÞAþ B=q1

� expikr2

2

2q2

!J2jðj2r2Þ cosð2jhÞ ð13Þ

Fig. 2. (a) Crystal refractive index versus r in the plane of z = 0 and (b) versus Z atr = 0 for 1, 2, 3, 4 and 5 W pump power from bottom to top.

H. Nadgaran et al. / Optics Communications 283 (2010) 417–426 419

3. Thermal model

Eq. (13) shows a non-thermal even-order MG beam as the out-put of a system characterized by ABCD matrix. In order to consider

Fig. 3. Absorption coefficient of Nd:YA

the thermal effects in generation of MG beams, we assume that MGbeams are produced by a solid-state laser pumped by a Gaussian(GP) pump profile or a super-Gaussian (SGP) pump profile. In thisrespect, it is assumed that the homogeneous gain medium is con-verted to an inhomogeneous graded index (GRIN) one by heat-in-duced refractive index variations. MG beams which arepropagating in this inhomogeneous medium have then heavilybeen influenced by thermal effects. In order to characterize thisGRIN medium, we allow that the medium refractive index wouldchange along the propagation direction of the MG beam as wellas along crystal radius, r. Characterization of ABCD transfer matrixwould then require to solve the heat equation [20] of:

@2T@r2 þ

1r@T@rþ @

2T@z2 ¼ �

Sðr; zÞK

ð14Þ

where K is the thermal conductivity for the gain medium, and Sðr; zÞis the heat source density. The proper boundary conditions for edgeand faces of the crystal are [16]:

� K@Tðr; zÞ@r

jr¼a ¼ h½Tðr ¼ a; zÞ � Tr�

K@Tðr; zÞ@z

jz¼0 ¼ h½Tðr;0Þ � Tr �

� K@Tðr; zÞ@z

jz¼l ¼ h½Tðr; lÞ � Tr� ð15Þ

where Tr is the ambient temperature, i.e. the coolant temperature, his the heat transfer coefficient of the medium and a is the radius ofthe crystal. Typical experimental conditions when a rod radius offew millimeters (l = 5 mm in this work) is exposed to a pump beamof radius of tens of microns (xp ¼ 100 lm in this work) are fulfilledby these boundary conditions [18]. In other words, we assume afixed temperature on the rod cylindrical boundary, that is lightlaunched along z, raises the medium temperature but the contactof the boundary of the medium with a coolant system would fixits temperature [18].

We can write the source term as [16]:

Sðr; zÞ ¼ Q GPðSGPÞQðrÞ expð�azÞ ð16Þ

Where QGPðSGPÞ are constants and QðrÞ stands for a Gaussian profile(GP, expð�2r2=x2

pÞ) or a super-Gaussian profile (SGP, expð�2r4=x4pÞ).

xp is the pump beam spot size.The exact solution of Eq. (14) has been comprehensively given

by Usievic et al. [21]. The temperature can be written as:

G crystal versus wavelength [28].

Fig. 4. Non-thermal and thermally-affected MG beam intensity profile at Z2 = 2 cm. (a) under pump power of 1 W. (b)–(e) under pump powers of 2, 3, 4, and 5 W,respectively. Solid plots represent the non-thermal beam, dotted plots show the MG beam under super-Gaussian pump, and dashed plots depict the MG beam under Gaussainpump. Dark pictures is the transverse pattern on XY plane together with their 3D plots.

420 H. Nadgaran et al. / Optics Communications 283 (2010) 417–426

Fig. 4 (continued)

H. Nadgaran et al. / Optics Communications 283 (2010) 417–426 421

Tðr; zÞ ¼X1i¼1

tiðzÞJ0ðmirÞ þ Tr ð17Þ

where J0ðmirÞ’s are the zero-order Bessel functions with mi’s as thepositive roots of the equation hJ0ðmiaÞ � kJ1ðmiaÞ ¼ 0 and tiðzÞ as:

tiðzÞ ¼ tþi expðmizÞ þ t�i expð�mizÞ þ Ci expð�czÞ ð18Þ

where

Ci ¼

R2QGðSGÞQðrÞJ0ðmirÞrdr

a2Kðm2i � a2Þ½J2

0ðmiaÞ þ J21ðmiaÞ�

ð19Þ

and t�i ’s are given as

t�i ¼ðKaþ hÞðKmi � hÞe�mi l � ðKa� hÞðKmi � hÞe�al

ðKmi � hÞ2 expð�milÞ � ðKmi þ hÞ2 expðmilÞ� Ci ð20Þ

Fig. 1 shows the variation of temperature versus r, and z.We can now calculate the change of refractive index using tem-

perature expression [22] as:

Dnðr; zÞ ¼ @n@T½Tðr; zÞ � Tr � þ ðn0 � 1Þezz ð21Þ

The first term results from thermal dispersion and the secondterm is caused by axial strain which in plane stress approximationis given by:

ezz ¼ ð1þ mpÞaT ½Tðr; zÞ � Tr� ð22Þ

Where mp and aT are the Poisson ratio and thermal expansion coef-ficient, respectively.

Chen et al. [23] have used Paraxial Ray Approximation (PRA) toreport the following relation for the refractive index of the GRINmedium:

nðr; zÞ ¼ n0 1� r2c2ðzÞ2

� �ð23Þ

where:

cðzÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12n0

@n@Tþ ðn0 � 1Þð1þ mÞaT

� �X1i¼1

tiðzÞm2i

sð24Þ

Eq. (21) clearly shows that increasing temperature in materialleads to a refractive index growth.

Fig. 5. The same as Fig. 4, but at Z2 = 5 cm.

422 H. Nadgaran et al. / Optics Communications 283 (2010) 417–426

Fig. 2 depicts the refractive index variation of the crystal laseragainst r, and z.

As Eq. (22) shows the main responsible parameter for refractiveindex variation is cðzÞ. The ABCD matrix can be written as:

Fig. 5 (continued)

H. Nadgaran et al. / Optics Communications 283 (2010) 417–426 423

A B

C D

� �¼

1 L

0 1

� �� ½GRIN� ð25Þ

where L is the cavity free length. The GRIN matrix in Eq. (23) repre-sents thermal effects imposed on the gain medium and its detailedcalculation has been given by Sabaeian et al. [16].

In summary, by constructing the ABCD transfer matrix which in-cludes the thermal effects through GRIN matrix, we can now sim-ulate the MG beams suffered by thermal effects.

Before presenting the results of this work, it is worth mentionsome considerations that have been made:

(1) It is assumed that the transverse profile of heat source doesnot change with z, whereas its amplitude is exponentiallydecreasing due to loss. This assumption bears a high levelof justification by stating that the variation of transverseprofile can show its manifestation in the variation of the

pump beam spot size via [24] xpðzÞ ¼ xp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ðz=z0Þ2

qwhere

z0 ¼ npx20=kp is the Rayleigh range. Calculation of ð z

z0Þ2 with

our case as x0 ¼ 100 lm, n0 ¼ 1:816, kp ¼ 808 nm and

z ¼ l ¼ 5 mm will give us the value of ð zz0Þ2 ¼ 0:00501 which

is negligible and therefore xpðzÞ ¼ xp. So, as pump beamspot size is not changing during propagation along the crys-tal, we can safely ignore the transverse profile variation ofthe pump source.

(2) Reshaping of pump beam upon propagation is also neglectedhere, since it is assumed that the crystal warming up profilealmost obeys the pump intensity profile and because of thatthe lasing rate equations are not coupled with the heat equa-tion to account for pump beam reshaping [25–27]. Thereforethis non-coupling of heat-lasing equations is utilized herewithout loss of generality.

(3) As in many other relevant thermal works, it is assumed thatthe laser beam itself, here MG beam, does not heat the lasercrystal, since crystal heating is mainly due to quantum defi-ciency parameter of the crystal, whereas the generated MGbeam must have a large absorption coefficient at the laserwavelength of 1064 nm to act as another major heat source.Fig. 3 shows the absorption coefficient for the Nd ion in YAGbetween the UV and mid-infrared [28] confirming thatNd:YAG absorption coefficient is almost zero at our laser

Fig. 6. The same as Fig. 4, but at Z2 = 40 cm (far field).

424 H. Nadgaran et al. / Optics Communications 283 (2010) 417–426

Fig. 6 (continued)

H. Nadgaran et al. / Optics Communications 283 (2010) 417–426 425

wavelength (1064 nm) and therefore our assumption thatthe laser MG beam cannot be treated as a major heat loadis valid.

4. Results and discussion

Using the models of non-thermal and thermally-affected MGbeams presented so far, as stated earlier we assume an endpumped solid-state Nd:YAG laser [17] crystal of length l = 5 mmand radius of a = 1.5 mm delivers the output MG beams. The laserwas assumed to be longitudinally pumped by a diode laser ofkp = 808 nm and pump waist of xp = 100 lm. The laser beam wave-length and waist are kl = 1064 nm and xl = 300 lm, respectively.The transverse wave vector was taken as j = 22144 cm�1. [16,17].

Using the above parameters, the thermal and non-thermal MGbeams of zero order were simulated.

The non-thermal beam profile is also considered as the outputbeam under very small amount of heat load, that is the GRIN ma-

trix be equal to 1 l0 1

� �in this case, and the ABCD matrix repre-

senting the behavior of the laser cavity becomes essentially1 z2

0 1

� �with z2 ¼ Lþ l.

Fig. 4 shows the MG beam intensity profile at angle of h ¼ 90�.Fig. 4a shows the thermally-affected MG beam for pump power

of 1 W under GP (dashed) and under SGP (dotted) together with itscorresponding intensity pattern on XY plane and its 3D plot atZ2 = 2 cm. In all Figures the non-thermal MG beam (solid black)is also plotted for quick comparison with the thermal ones. The fo-cal length of thermal lens for results of Fig. 4a is about 28 cm.

Fig. 4b–e shows the same MG beam under 2, 3, 4, and 5 Wpump powers, respectively. Their corresponding focal lengths ofthe induced thermal lens are 14, 9, 7, and 5 cm, respectively show-ing stronger lens by pump power increase as expected. From Fig. 4it is clearly evident that gradual increase of pump power leads toconsiderable alteration of the intensity profile of MG beam show-ing its less resemblance to a non-thermal MG beam profile. Thismeans that the induced heat load has severely affected the MGbeam generation. We can hardly see any difference between MGoutputs under GP and MG outputs under SGP pump profiles.

426 H. Nadgaran et al. / Optics Communications 283 (2010) 417–426

Fig. 5a–e shows the same non-thermal and thermal MG beamsfor the same pump powers of Fig. 4, but at Z2 ¼ 5 cm. From Fig. 5we can see even more changes to the MG beam profile as the pumppower increases. Here a systematic comparison between the non-thermal MG and thermal MG beam under different pump powersreveals the extent to which the output MG beam is suffered fromthe thermal effects, so that at pump power of 5 W, we end up witha two discernible intensity peaks in expense of gradual diminishingof the central peak. From Fig. 5 we can also see that the differencebetween the two MG beam outputs under GP and SGP pump pro-files is begun to emerge.

Fig. 6 shows the thermally-affected MG beams at far field ofZ2 ¼ 40 cm. From this Figure, one can interestingly see the gradualappearance of the middle intensity peak compared to non-thermalMG beam plot. Here from Fig. 6 we can again see the difference be-tween the output MG beam under GP and SGP pump profiles.

In summary, a systematic comparison between the non-thermalMG beam profile and its various thermal correspondents shows avery much influence that the thermal effects can have on the gen-eration of MG beams. We should emphasize here that at some highpump powers, the effects of the induced heat is so dramatic thatone can no longer easily identify the output beam from a trueMG beam.

5. Conclusion

Based on our previous work [16] on the effects of induced heatload on the generation of Bessel–Gauss (BG) beams, we have inves-tigated the thermal effects on the generation of Mathieu–Gauss(MG) beams in this work. The thermal model of Reference [16]was used to characterize the inhomogeneous thermally affectedgain medium of a solid-state laser. The output MG beams of this la-ser which have experienced GRIN (Graded Index) medium weresimulated and compared with the non-thermal ones. Two differentpump profiles of Gaussian and super-Gaussian with differentpump powers of up to 5 W were used. A complete comparison be-tween the non-thermal and thermally-affected MG beams not onlyreveals the vital role of thermal effects on the generation of specialtypes of laser beams, MG beams in this work, but also shows howthe intensity profiles of MG beams are suffering from the heat in-duced load. It is shown that under high pump powers leading tostronger thermal lens, the identification of the thermally-affected

MG beams is difficult from their non-thermal one at first hand.The simulations and comparisons performed in this work not onlycan make this identification with ease, but also facilitates a betterunderstanding of thermal phenomena in the generation of specialtypes of laser beams and also in getting a tool for solid-state laserdesign, optimization, and beam quality monitoring.

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