\u003ctitle\u003eNumerical study and optimization of third harmonics generation in two-sectioned periodically poled...

Numerical study and optimization of third harmonicsgeneration in two-sectioned periodically poled LiTaO3

Oleg A. Louchev1 and Satoshi Wada2

1Megaopto Co. Ltd., RIKEN Cooperation Center W414, 2-1 Hirosawa, Wako, Saitama 351-0106,

Japan 2Solid-State Laser for Astronomical Observation Research Unit, RIKEN, 2-1 Hirosawa, Wako,

Saitama 351-0198,Japan

[email protected]

ABSTRACT

The feasibility of the cascaded second and third harmonic generation in two-sectioned periodically poled lithium tantalate crystal is analyzed. Simulation using computational non-linear optical model rigorously coupled with the thermal model suggests that 20-30 % efficiency can be achieved for 3 W power 2.2 ns pulsed 1.064 �m laser operating at frequency 6.8x103 Hz if the crystal is composed with optimized section lengths for: (i) 8.0 �m periodic first-order SHG structure and (ii) 6.6 �m periodic third-order THG structure. Significant inhibition of THG efficiency can be due to the heat release of SH and TH along the crystal, associated thermal dephasing and lenzing which can be effectively inhibited by decreasing the crystal cross-section dimensions to the practical minimum of 200x200 �m.

Keywords: Harmonic generation and mixing; Photothermal effects; Nonlinear optics, materials.

1. INTRODUCTION

The development of quasi-phase matched non-linear optical devices with periodic domain polarization originally suggested in a classical paper1 has shown a significant progress over the last decade and remains the focus of study of many groups using a variety of materials.2-23 Stoichiometric and near-stoichiometric Mg-doped lithium tantalate (SLT) is a promising optical material possessing decreased coercive field required for ferroelectric polarization inversion by the nonstoichiometric defect control durring crystal growth, an excellent transparency within the range of �280-5500 nm, high second order susceptibility and related effective non-linearity deff�10 pm/V,12 and the best heat conductance, �8.5 W/m K,12 known for the dielectric materials24 used for non-linear optical devices. Modeling13,14 suggests that this set of properties is very attractive for various non-linear applications operating at high intensities and powers. The development of reliable fabrication technology for periodically poled (PP) microscale structures25,26 allows the development quasi-phase matched non-linear generation at room temperatures extending the operation of infrared lasers into visible and UV regimes. For instance, recent achievements in this field include high power second harmonics generation (SHG) for pulsed,15,16,20 continuous wave (CW),18,22 optical parametric oscillator operation using PP Mg-doped SLT21 and UV generation in PP LT waveguide.23 In this Communication we consider third harmonics generation (THG) required by many possible UV applications. In particular, we develop a computational model for simulation of the cascaded second and third harmonics generation in PP Mg-doped SLT. The main goal of our study is to estimate possible efficiency and to provide practical guidelines for the experimental implementation of a THG device by using a two-sectioned periodically poled structure for high-power non-linear conversion of nanosecond infrared 1.064 �m laser radiation into 355 nm radiation.

Nonlinear Optics and Applications III, edited by Mario Bertolotti, Proc. of SPIE Vol. 7354, 73540Y · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.820585

Proc. of SPIE Vol. 7354 73540Y-1

2. COMPUTATIONAL MODEL

We consider third harmonics (TH) generation by a two-period structure fabricated inside Mg-doped SLT crystal. This structure is assumed to have two periodically poled sections separated by a small intermediate zone with a constant polarization. The first section with the QPM period 11 2 k��� � �8 �m irradiated by the pulsed fundamental harmonic (FH) beam is assumed to produce second harmonic generation (SHG). After passing through the intermediate zone the obtained SH beam mixed with the remaining FH beam is assumed to undergo the sum-frequency generation within the second section with third-order QPM period 22 6 k��� � �6.6 �m, resulting in high intensity third harmonic (TH) output beam ( is the phase mismatch).

ik� Our model and computational technique used here extends the computational model for SHG in PP crystals used previously13,14 and includes additionally the formalism for THG generation in the second section. In particular, we consider here a nanosecond pulse input FH intensity distribution described by:

� �

� �

� ��

�

� �

� ��� 2

2

20

22

0 expexp),,(��t

ryxItyxI , (1)

where 2

02/3

00 2 rQI ��� is the peak pulse intensity, Q0 is the FH pulse energy, r0 is the laser beam radius and 2� is the

pulse duration time (related to the experimentally used pulse width by �� 3.32ln4 ���t ). For the case of the nanosecond pulses considered here for any starting z=0 point at the crystal front face the electric fields along the z direction, i. e. � �E t z A t z i t k zj j j( , ) ( , ) exp ( )� � j� , the wave amplitudes, , and laser intensities,

, can be described in a one-dimensional slowly varying amplitude (SVA) approximation neglecting the time derivatives in the corresponding coupled amplitude equations. We take into account the optical loss associated with the linear FH, SH, TH absorption (�1 , �2 and �3) and non-linear two-photon SH ( ) and TH absorption ( ). Hence, at a particular moment t and x-y coordinate defining the value of the local input laser beam intensity, and input FH amplitude,

),( ztAj

22I

I t z A t zj j( , ) ( , )� 2

� 23I�

� 2/111 ),,(2 cntyxI�� �A at z=0, the FH-SH-TH energy conversion mechanism along the first and

intermediate PPSLT sections is described by slowly varying amplitude (SVA) approximation:27

111

*122

1

211 5.0)exp()(8 AzkiAAzd

cki

dzdA ���

���� , (2)

� 2*22

*2

212

2

222 5.0)exp()(4 AAAkziAzd

cki

dzdA ����

���� , (3)

, (4) 03 �A and along the second section of the PP SLT structure by:

112*232

1

211 5.0)exp()(8 AzkiAAzd

cki

dzdA ���

���� , (5)

� 2*22

*22

*132

2

222 5.0)exp()(8 AAAzkiAAzd

cki

dzdA

����

����� , (6)

� 3*33

*32212

3

233 5.0)exp()(

8 AAAzkiAAzdcki

dzdA

����

���� , (7)

Proc. of SPIE Vol. 7354 73540Y-2

where A2=0 at z=0, 12 2�� � , 13 3�� � , cnk jjj /�� is the wave vector, 211 2 kkk ��� and 3212 kkkk ���� are the wave vector mismatches for SHG and THG, respectively , and nj(T) is the refractive index as a function of temperature T.

��� 2/* nc� �� 2/nc�� *

The square-wave periodicity of the effective non-linear coupling coefficient along two PP sections is taken into account by:

, (8) � �

� ����

���

�

����

���

LzLzLsignd

Lzzsigndzd

eff

eff

2222

111

,)/)(2cos(

0,)/2cos()(

�

�

where deff1 and deff2 are the effective non-linearities for SHG and THG sections, correspondingly. Within the intermediate zone SHG does not take place due to dephasing and d(z)=0, i.e. A1 and A2 change only due to absorption in Eqs. (2) and (3). The local beam intensity is associated with the wave amplitude by:

2),(

2),( tzA

cntzI j

jj �

� . (9)

The values of refractive indices depend on the local temperature nj(T), changing the values of the wave vector cnk jjj /�� , and most importantly of the wave vector mismatch k� entering Eqs. (2) - (7). To take this effect into account the above SVA approximation is coupled with the temperature distribution inside the crystal (i=cr) and the temperature controlled metal holder (i=h) which is described by 3-D equations:

),(2 tqTktTC iiiiii x�!�""# , (10)

where #i, Ci, ki are the density, specific heat and heat conductance, respectively, and is the heat generation intensity.

),( tqi x

The external temperature of the crystal-metal holder assembly is assumed to be experimentally controlled T=Ts. At the front (z=0) and rear (z=L) of the crystal-holder assembly we assume the heat exchange condition, i.e.

, where h�10 W/m2 K is the estimate of the free convection heat exchange coefficient to air at T0=20 oC. For the metal holder , whereas the heat generation in the PP crystal resulting from linear and non-linear absorption is defined by the derivative of the total intensity I=I1+I2 along the direction of laser beam propagation as:

)( 0TThTk iii ��!�

0),( �tqh x

$��

3

1),(),( tI

dzdtq icr xx . (11)

However, for the nanosecond pulse laser a quasi-state approximation may be used with the distribution of heat generation over one pulse duration as:

),(),( * tqtq crcr xx %�

, (12) & dtIIIIItq

tcr '

�

����� 2333

222211

* ),( �����x (where �t=6� is the period used for the time integration of one laser pulse and % is the laser pulse frequency. This approximation requires recalculation of the distribution defined by Eq. (12) whenever the temperature change over the considered period of time is sufficiently high (i.e. causing significant modification in Ij distributions). Moreover, this approximation may be used for modes when the characteristic time constant of the temperature field stabilization inside the irradiated zone, , is much larger than %-1. crcrcrst kCr /2

0 #� �

Proc. of SPIE Vol. 7354 73540Y-3

3. RESULTS AND DISCUSSION

In our simulation we consider non-linear SHG-THG process for TQPM=33 oC in PP crystal composed of three

sections: (i) with the SHG QPM period 11 2 k��� � =8.01 �m, (ii) without PP structure (constant polarization section) and (iii) with third order THG period 22 2x3 k��� � =6.62 �m. We consider here the case of )=1.064 �m for a fundamental wave length with 2.2 ns laser pulse duration and with repetition rate %=6.8x103 Hz. Due to the absence of absorption data for LiTaO3 we use in all our simulation the set of absorption parameters for a similar material (LiNbO3):24,28 �1= 0.002 cm-1, �2= 0.025 cm-1, �3= 0.025 cm-1 , �=0.1x10-11 m/W and �= 3.5 x10-11 m/W.

0.8

0.6

0.4

0.2

0.0

outp

ut e

ffic

ienc

y

50x10-3 3020100time (s)

*�

�

3�

Fig. 1 Stabilization of the output efficiency for 1st (�), 2nd (2�) and 3rd (3�) harmonics as a function of time for the optimized crystal.

The above model allows one to define reasonable parameters which can be used to obtain effective THG. In Fig. 1 we show the behavior of the output efficiencies for an optimized crystal 0.5x0.5x2.8 mm structure with (i) 0.75 mm for the SHG section, (ii) 0.05 mm for the intermediate section and (iii) 2 mm for the THG section. Fig. 1 shows that for this set of parameters initially THG efficiency achieves the level of 22 % and remains at the level of 20 % even after the final stabilization. The results given in Fig. 2 a, b and c show that for this set of PP crystal parameters the onset of T-field (c) does not lead to a significant modification in the SH (2�) and TH (3�) intensity distributions along the crystal, and the resulting THG efficiency at the crystal output. Fig. 2 a shows that for this crystal structure the SH intensity is depleted at the crystal output providing saturation in TH intensity ( at the pulse maximum). The final temperature increase does not exceed 3 K at the crystal end and does not lead to a significant reduction of TH output. In Fig. 3 we also present the pulse fluence distribution along the radius for (a) t=0 s and (b) t=0.05 s, correspondingly. These figures show that both initially and after the final stabilization TH beam profile is close to the Gaussian-like distribution. In Fig. 4 a, b and c we give the final results of our simulation study which shows the dependence of THG output efficiency versus total crystal length (a and b) and crystal width (c) (with fixed 0.75 mm for SHG section) for the initial temperature TQPM=33 o C and after the onset of final temperature distribution. Fig. 4 a shows that the optimal crystal length for 3 W input energy is about 3 mm giving final output efficiency �20 % (with �25 % for the initial temperature).

Proc. of SPIE Vol. 7354 73540Y-4

1.0

0.8

0.6

0.4

0.2

0.0

norm

aliz

ed i

nten

sitie

s

0.250.200.150.100.050.00z (cm)

(a)

�

*�

3�

+

1.0

0.8

0.6

0.4

0.2

0.0

norm

aliz

ed i

nten

sitie

s

0.250.200.150.100.050.00

z (cm)

(b)�

2�

3�

+

4442403836343230

T (

o C

)

0.250.200.150.100.050.00z (cm)

final TQPM

(c)

Fig. 2 Distribution for (a) initial (t=0 s) and (b) final (t=0.05 s) normalized intensities along

the optimized crystal for FH (�), SH (2�) and TH (3�), and (c) corresponding final temperature distribution.

6

5

4

3

2

1

0

puls

e flu

ence

dis

tribu

tion

( J/ c

m2 )

20015010050radius (�m)

(a)

� - input

�,

*�

3�

6

5

4

3

2

1

0

puls

e flu

ence

dis

tribu

tion

( J/ c

m2 )

20015010050radius (�m)

(b)�,�,input

�

2�

3�

Fig. 3 Optimized crystal laser pulse fluence radial distributions for FH (�), SH (2�) and TH (3�): (a) t=0 s and (b) t=0.05 s.

Proc. of SPIE Vol. 7354 73540Y-5

Fig. 4 a and all previous figures are calculated for deff1= deff2=10 pm/V used for both SHG and THG sections. However, the value of for the THG section can significantly differ from that of SHG section. This follows from the definition for from the classical anharmonic oscillator approximation for sum-frequency generation:

)2(-�effd)2(-

)()()(

)/(),,(

2121

23

2121)2(

��������-

DDDamqN ee

��� , (13)

where , N is the atomic density, qe is electron charge, me is the mass of electron, a is parameter that characterizes the strength of the non-linearity, � is the dipole damping rate and �0 is the resonance frequency. �0� 1016 rad/s.

����� iii iD 2)( 220 ���

Eq. (13) suggests that for the input laser frequency 11 /2 )�� c� �0.19x1016 rad/s the value of for THG in the third section can be significantly higher than deff=10 pm/V. That is, estimating from the absorption edge )0�280 nm the value of �0� 0.67x1016 rad/s one finds that for SH conversion �1=0.19x1016 the value of , whereas for THG section .

)2(-

1)� � 60

21 6.0()2( �� DD

60111 2.0)()2()3( ���� �DDD

0.4

0.3

0.2

0.1

0.0

THG

out

put e

ffic

ienc

y

654321crystal length (mm)

(a) initial final

0.4

0.3

0.2

0.1

0.0

THG

out

put e

ffic

ienc

y

3.02.52.01.51.0crystal length (mm)

initial final

(b)

0.4

0.3

0.2

0.1

0.0

THG

out

put e

ffic

ienc

y

1.00.80.60.40.2

crystal width (mm)

initial final

(c)

Fig. 4 THG output efficiency for the initial temperature (33 oC) and after onset of final temperature distibution.

Proc. of SPIE Vol. 7354 73540Y-6

Fig. 4 b shows the efficiency of the resulting THG using deff�10 pm/V for the 0.075 mm long SHG section and deff�30 pm/V for the THG section as a function of the total crystal length. This figure suggests that 25 % THG efficiency can be obtained for the reduced crystal length of 1.7 mm (0. 9 mm THG section). This is due to significant decrease of UV absorption loss for shorter crystal length. Finally, Fig. 4 c shows that the effect of absorption, high temperature increase and THG efficiency loss can be effectively compensated by decreasing cross-section dimensions of PP crystal. That is, this figure shows THG efficiency as a function of crystal width 0.2 – 1 mm for 1.8 mm long crystal which is shown in Fig. 4 b to provide the maximal initial efficiency 30 %. Fig. 4 c suggests that maximal THG efficiency of �30 % can be maintained even after the final thermal stabilization if the crystal width is reduced to the practical limit of 200x200 �m (with the beam radius 100 �m). This leads to the reduction of the temperature increase over TQPM=33 oC, inhibition of thermal dephasing and associated THG efficiency loss.

4. SUMMARY

In this work we have analyzed a possibility of cascaded SHG-THG process in two-sectioned periodically poled Mg-doped lithium tantalate crystal. Our model and performed simulations suggest that 25-30 % THG efficiency can be achieved for 3 W power 2.2 ns pulsed 1.064 �m laser operating at frequency 6.8x103 Hz by using the optimized crystal dimensions for (i) 0. 8.0 �m period SHG structure and (ii) 6.6 �m period third-order THG section. Our simulation show that the significant inhibition of THG can be due to the heat release and thermal dephasing which can be, nevertheless, effectively inhibited by decreasing crystal cross-section dimensions to the practical minimum of 200x200 �m.

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