- Email: [email protected]
Second harmonic generation: Effects of the multiple reflections of the fundamental and the second harmonic waves on the Maker fringes
Optics Communications 279 (2007) 183–195 www.elsevier.com/locate/optcom
Second harmonic generation: Effects of the multiple reflections of the fundamental and the second harmonic waves on the Maker fringes Gildas Tellier *, Christian Boisrobert IREENA, Universite´ de Nantes, BP 92208, 2 rue de la Houssiniere, 44322 Nantes Cedex 03, France Received 26 April 2007; received in revised form 19 June 2007; accepted 19 June 2007
Abstract The Maker fringes technique is commonly used for the determination of nonlinear optical coefficients. In this article, we present a new formulation of Maker fringes in parallel-surface samples, using boundary conditions taking into account the anisotropy of the crystal, the refractive-index dispersion, and the reflections of the fundamental and the second harmonic waves inside the material. Complete expressions for the generated second harmonic intensity are given for birefringent crystals for the case of no pump depletion. A comparison between theory and experimental results is made, showing the accuracy of our theoretical expressions. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Second harmonic generation; Maker fringes; Multiple reflections; Anisotropy
1. Introduction Since 1970, the determination of the nonlinear optical coefficients of the tensor v(2) associated with second harmonic generation (SHG) has been performed almost exclusively by the Maker fringe technique [2], as described in detail by Jerphagnon and Kurtz [1]. In this theoretical development, the boundary conditions did not take into account the reflections of the second harmonic issued from the second interface on the first interface of the nonlinear material. Moreover, the theory presented did not consider the general case of anisotropic media. Bloembergen and Pershan gave an expression for the second harmonic fields generated by an isotropic transparent nonlinear parallel slab [3], but the reflections of the second harmonic and fundamental waves in the media were ignored in this expression. Taking into account the anisotropy of the nonlinear media, a theoretical expression has been developed by Okamoto et al. [4] in the case of slabs with C1m space symmetry, but once again, multiple reflections were neglected. An expression using complete boundary conditions, including multiple reflections of both the fundamental and the second harmonic waves, has been developed in the case of SHG setups using non-parallel slabs [5], but this approach does not apply to the Maker Fringes Technique. Referring to studies carried out on the third harmonic generation [6,7], the first expression taking into account multiple reflections in the media and developed for the Maker Fringes was obtained by Herman and Hayden [8]. This expression has been established for uniaxial anisotropic materials and uses complete boundary conditions for the second harmonic wave to solve the problem. Unfortunately, the multiple reflections of the fundamental waves are not considered in this approach. Moreover, the expression has been developed for uniaxial materials but in one direction only. *
Corresponding author. Tel.: +33 68 877 1707. E-mail address: [email protected] (G. Tellier).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.06.048
184
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
In this paper, we report on the development of a theoretical expression of the Maker fringes, using complete boundary conditions with multiple reflection of both the fundamental and the second harmonic waves, in the general case of anisotropic biaxial materials. We here intend to provide a complete general framework for the second harmonic generation analysis using the Maker fringes technique. 2. Theoretical developments 2.1. Description of the problem In this section, we present a general approach of the boundary-value problem for SHG. The configuration is shown in Fig. 1. The origin is defined at the center of the nonlinear layer of thickness L. We consider the case of no pump depletion. ! !! The propagating waves are taken plane, monochromatic, and can thus be written as E eið k : r xtÞ . The fundamental and the second harmonic waves can be polarized in the plane of incidence (polarization p) or perpendicular to the plane of incidence (polarization s). In the following development we will only consider the polarization configuration p–p. The results obtained for the other configurations will be given in Appendixes. The fundamental wave is incident on the parallel slab at an angle h. We firstly describe the wave propagation conditions in these three regions: Region I (air): This media is supposed to be isotropic with an index equal to 1. The waves propagating in this region are – The incident fundamental and reflected fundamental waves (x): !r ! L! ! ! L! ! ! E1i þ Er1i ¼ E1i eið q1 :ð r þ2 z ÞxtÞ e^1a þ Er1i eið q1 :ð r þ2 z ÞxtÞ e^r1a
ð1Þ
– the reflected second harmonic wave (2x), from the nonlinear media: !r ! L! ! E2a ¼ Reið q2 :ð r þ2 z Þ2xtÞ e^r2a
ð2Þ
With the following wave vectors: 2p ! q1 ¼ ðsinðhÞ; 0; cosðhÞÞ k !r 2p q1 ¼ ðsinðhÞ; 0; cosðhÞÞ k !r 4p q2 ¼ ðsinðhÞ; 0; cosðhÞÞ k
ð3Þ
Since the second harmonic waves and the fundamental waves are polarized in the plane of incidence (configuration p–p), the polarization unit vectors are e^1a ¼ ðcosðhÞ; 0; sinðhÞÞ
ð4Þ
e^r1a ¼ e^r2a ¼ ð cosðhÞ; 0; sinðhÞÞ
Region II (nonlinear material): This media is considered anisotropic biaxial and its dielectric permittivities can be represented by the tensor: 0 2 1 0 nix 0 B C r ðixÞ ¼ ri ¼ @ 0 n2iy 0 A ð5Þ 0
0
n2iz z
k 1s
k 2s
III θ 2s II
k2
I
q1
θ2 θ
x
θ 1s
k 2r q 2r
output medium (region III) z=L/2
k1 θ1 θ
k 1r non–linear material (region II) z=L/2
q 1r air (region I)
Fig. 1. Geometry of the problem of second harmonic generation in a nonlinear parallel slab.
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
185
with i = 1, 2 and where (nix, niy, niz) are the principal refraction indices at the fundamental (i = 1) and the second harmonic (i = 2) frequencies. In Region II, the refractive indices are depending on the propagation direction and can be written as follows: n1(h1) for the fundamental (pulsation x) and n2(h2) for the second harmonic (pulsation 2x). These refraction indices obey to the Snell laws and, using the index ellipsoid, the angle-dependant indices can be written as 2 1 cos ðhi Þ sin2 ðhi Þ 2 ni ðhi Þ ¼ þ ; i ¼ 1; 2 ð6Þ n2ix n2iz The fundamental wave propagating in this media is the sum of two waves propagating in two opposite directions. These waves correspond to the multiple reflections of the fundamental at the input face of the sample and the output face of the sample: !r ! !! ! E1 ¼ E1 eið k 1 : r xtÞ^e1 þ Er1 eið k 1 : r xtÞ e^r1 ð7Þ The second harmonic wave can be decomposed in three different waves: the second harmonic bound wave generated by the fundamental wave and two free waves propagating, as the fundamental wave, in two opposite directions. These two free waves correspond to the multiple reflections of the second harmonic at the input face of the sample and the output face of the sample. The total second harmonic wave can thus be written as !r ! !! ! ! ð8Þ E2 ¼ eb þ E2^e2 eið k 2 : r 2xtÞ þ Er2^er2 eið k 2 : r 2xtÞ with the following wave vectors: ! 2pn1 ðh1 Þ k1 ¼ ðsinðh1 Þ; 0; cosðh1 ÞÞ k !r 2pn1 ðh1 Þ ðsinðh1 Þ; 0; cosðh1 ÞÞ k1 ¼ k ! 4pn2 ðh2 Þ ðsinðh2 Þ; 0; cosðh2 ÞÞ k2 ¼ k !r 4pn2 ðh2 Þ ðsinðh2 Þ; 0; cosðh2 ÞÞ k2 ¼ k
ð9Þ
! ! Since the nonlinear media is anisotropic, the electric field E is not collinear to the displacement vector D , thus is not ! perpendicular to the wave vector k . The electric field deviates from the displacement vector by the walk-off angle ci with i = 1, 2 (respectively, for x and 2x), as shown in Fig. 2. When the waves are polarized in the plane of incidence (configuration p–p), the polarization unit vectors are e^1 ¼ ðcosðh1 c1 Þ; 0; sinðh1 c1 ÞÞ e^r1 ¼ ð cosðh1 c1 Þ; 0; sinðh1 c1 ÞÞ e^2 ¼ ðcosðh2 c2 Þ; 0; sinðh2 c2 ÞÞ
ð10Þ
e^r2 ¼ ð cosðh2 c2 Þ; 0; sinðh2 c2 ÞÞ z
ki θi
θi
x
γi ei
γi Di
Fig. 2. Definition of the walk-off angle.
186
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
Region III (output medium): In most cases, the output medium is the same as the input medium, but in order to differentiate the contributions of the waves reflected on the input and the output faces, this output medium will be considered as an isotropic media with a refraction index of n1s at x and n2s at 2x. In this region, the fundamental propagating wave is ! ! L! ! E1t ¼ E1t eið k 1s :ð r 2 z ÞxtÞ e^1s
ð11Þ
The second harmonic wave transmitted and propagating in the output medium is ! ! L! ! E2s ¼ T eið k 2s :ð r 2 z Þ2xtÞ e^2s
ð12Þ
with the following wave vectors: ! 2n1s p ðsinðh1s Þ; 0; cosðh1s ÞÞ; k 1s ¼ k
! 4n2s p ðsinðh2s Þ; 0; cosðh2s ÞÞ k 2s ¼ k
ð13Þ
In this region, the polarization unit vectors are e^1s ¼ ðcosðh1s Þ; 0; sinðh1s ÞÞ e^2s ¼ ðcosðh2s Þ; 0; sinðh2s ÞÞ
ð14Þ
2.2. Expression of the walk-off angle c In an anisotropic media, the displacement vector can be written as follows: 0
n2ix ! B 0 D ¼ 0 @ 0
0 n2iy
1 0 ! 0 C AE
0
n2iz
ð15Þ
with i = 1, 2 depending on the pulsation. Using Maxwell’s equations, Snell’s law and the index ellipsoid, we obtain the following expressions of the walk-off angle ci as a function of the incident angle h:
2 ni ðhi Þ cosðci Þ cosðhi Þ nix 2 ni ðhi Þ sinðhi ci Þ ¼ cosðci Þ sinðhi Þ niz nix niz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðci Þ ¼ 2 ni ðhi Þ n2ix þ n2iz ni ðhi Þ cosðhi ci Þ ¼
ð16Þ
niz sinðhÞ sinðci Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðniz nix Þ þ ðn2iz n2ix Þ sinðhÞ2 2.3. Expression of the magnitude of the second harmonic transmitted field as a function of the bound wave magnitude At the region I–region II interface (z ¼ L2), the boundary conditions for the electric and magnetic fields lead to the equations: L cosðhÞR ¼ e þ cosðh2 c2 ÞðE2 ei/2 Er2 ei/2 Þ ð17Þ 2 bx L R ¼ cb þ n2 ðh2 Þ cosðc2 ÞðE2 ei/2 þ Er2 ei/2 Þ ð18Þ 2 by The continuity of the same fields at the region II–region III interface (z ¼ L2) requires
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
187
L cosðh2s ÞT ¼ e þ cosðh2 c2 ÞðE2 ei/2 Er2 ei/2 Þ ð19Þ 2 bx L n2s T ¼ cb þ n2 ðh2 Þ cosðc2 ÞðE2 ei/2 þ Er2 ei/2 Þ ð20Þ 2 by where /2 = 2pLn2(h2)cos(h2)/k and b L2 ; b L2 are the magnitudes of the magnetic the bound wave at z = L/2 and z = L/2. After solving this system for E2 and Er2 , the magnitude of the second harmonic field can be expressed as a function of the bound wave magnitude: " !# 1 2i/2 L L þ þ 2i/2 u v e T ¼ ua v e þ 2n2 ðh2 Þ cosðh2 c2 Þ cosðc2 Þ e þ cosðhÞcb ð21Þ D a 2 bx 2 by with the following coefficient: 2i/2 þ 2i/2 uþ D ¼ u s ua e a us e
u a ¼ cosðh2 c2 Þ n2 ðh2 Þ cosðc2 Þ cosðhÞ u s ¼ n2s cosðh2 c2 Þ n2 ðh2 Þ cosðc2 Þ cosðh2s Þ L L þ þ cosðhÞcb a ¼e 2 bx 2 by L L a ¼ n2s e cosðh2s Þcb 2 bx 2 by L L v ¼ cosðh2 c2 Þcb n2 ðh2 Þ cosðc2 Þe 2 by 2 bx
ð22Þ
2.4. Expression of the magnitudes of the fundamental waves propagating in the nonlinear medium In order to express the magnitudes of the propagating fundamental waves in the nonlinear medium, the boundary conditions are written, as in the previous section, but this time at the x pulsation. At the region I–region II interface (z ¼ L2), the continuity of the electric and magnetic fields requires cosðhÞðE1i Er1i Þ ¼ cosðh1 c1 ÞðE1 ei/1 Er1 ei/1 Þ E1i þ Er1i ¼ n1 ðh1 Þ cosðc1 ÞðE1 ei/1 þ Er1 ei/1 Þ
ð23Þ
with /1 = pLn1(h1)cos(h1)/k. Moreover, at the region II–region III interface z ¼ L2 , the continuity of the electric and magnetic fields requires cosðh1s ÞE1t ¼ cosðh1 c1 ÞðE1 ei/1 Er1 ei/1 Þ n1s E1t ¼ n1 ðh1 Þ cosðc1 ÞðE1 ei/1 þ Er1 ei/1 Þ
ð24Þ
Solving these equations and using the Fresnel transmission and reflection coefficients at the input and output faces of the slab, E1 and Er1 can thus be written as functions of the magnitude of the incident fundamental wave E1i and the Fresnel reflection and transmission coefficients: ei/1 E1 ¼ taf 1 2i/ E1i e 1 rfa1 rfs1 e2i/1 ð25Þ ei/1 r E1 ¼ taf 1 rfs1 2i/ E1i e 1 rfa1 rfs1 e2i/1 With the coefficients: taf 1 ¼
2 cosðhÞ ; uþ a1
rfa1 ¼
u a1 ; uþ a1
rfs1 ¼
u s1 uþ s1
u a1 ¼ cosðh1 c1 Þ n1 ðh1 Þ cosðc1 Þ cosðhÞ u s1 ¼ n1s cosðh1 c1 Þ n1 ðh1 Þ cosðc1 Þ cosðh1s Þ
ð26Þ
188
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
2.5. Expression of the bound wave ! ! Since the fundamental wave in region II is the sum of two contra-propagative waves E1 and Er1 , the induced nonlinear polarization is also decomposed as a sum: ! ! ! ! r ! r ! ! ! ! ! ! i2 k 1 r xt i2 k 1 r xt i k þ k r 2xt 1 1 P NL ¼ P NL1 e þ P NL2 e þ P NL3 e ð27Þ with ! ! ! ð28Þ P NL1 ¼ 0 E1 E1 vð2Þ e^1 e^1 ; P NL2 ¼ 0 Er1 Er1 vð2Þ e^r1 e^r1 ; P NL3 ¼ 20 Er1 E1 vð2Þ e^1 e^r1 The bound wave, written as: ! ! ! ! ! ! i k b1 r 2xt i k b2 r 2xt i k b3 r 2xt ! þ ! eb2 e þ ! eb3 e ð29Þ eb ¼ ! eb1 e is a particular solution of the propagation equation: 2 2 2x 2x 1 ! ! ! rotðrotðeb ÞÞ r2 eb ¼ P NL c c 0 with ! ! ! ! ! ! ! k b1 ¼ 2k 1 ; k b2 ¼ 2k r1 ; k b2 ¼ k 1 þ k r1 We obtain the following relations for the bound waves: 2 3 ! ! ! 2 NL1 ! n2 ðh1 Þ ! 41 P NL1 k b1 ð k b1 P Þ5 eb1 ¼ r2 2 2 2 0 ðn1 ðh1 Þ n2 ðh1 Þ Þ n n Þ ð2x c 2x 2z 2 3 ! ! ! 2 NL2 ! Þ ð k P n ðh Þ k 2 1 ! 41 P NL2 b2 b2 5 eb2 ¼ r2 0 ðn1 ðh1 Þ2 n2 ðh1 Þ2 Þ n n Þ2 ð2x c 2x 2z eb3z ¼
ð30Þ
ð31Þ
ð32Þ
n22x P NL2 z n22 ðh2 Þ cos2 ðh2 Þn22z 0
2.6. Expression of the relation between the magnitude of the transmitted second harmonic wave and the magnitude of the incident fundamental wave At the region I–region II interface (z ¼ L2) and at the region II–region III interface (z ¼ L2), the electric and magnetic fields can be written as: L e ¼ eb1x e2i/1 þ eb2x e2i/1 2 bx L b ¼ bb1y e2i/1 þ bb2y e2i/1 þ bb3y 2 by ð33Þ L e ¼ eb1x e2i/1 þ eb2x e2i/1 2 bx L b ¼ bb1y e2i/1 þ bb2y e2i/1 þ bb3y 2 by Using identities and coefficients such as eið2/1 2/2 Þ ¼ 2i sinð2/1 /2 Þei/2 þ e2i/1 v i ¼ cosðh2 c2 Þcbbiy n2 ðh2 Þ cosðc2 Þebix
for i ¼ 1; 2
ð34Þ
v 3 ¼ cosðh2 c2 Þcbb3y in Eq. (21), we obtain T ¼
1 þ i/2 i/2 þ i/2 ½u v 2i sinð2/1 þ /2 Þei/2 uþ u þ uþ a v1 2i sinð2/1 /2 Þe a v2 2i sinð2/1 /2 Þe a v2 2i sinð2/1 þ /2 Þe D a 1 i/2 þ i/2 þ u þ uþ a v3 2i sinð/2 Þe a v3 2i sinð/2 Þe
ð35Þ
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
189
Furthermore, from Eq. (32), we find that:
0 1 ! NL ^ P e n ðh Þ cosðh Þ n ðh Þ cosðh Þ 1 1 1 2 2 2 @ 2 1 A v 1 ¼ 2 2 ! n22x NL ^r 0 ðn1 ðh1 Þ n2 ðh1 Þ Þ e P 1 0 2 ! 1 2 NL ^ e P n ðh Þ n ðh Þ cosðh Þ n ðh Þ cosðh Þ 2 2 1 1 1 1 2 2 2 2 A @ v 2 ¼ ! n22x NL ^r 0 ðn1 ðh1 Þ2 n2 ðh1 Þ2 Þ e2 P 2 ! 1 v ðe^2 þ e^r2 Þ P NL 3 ¼ 3 2n2 ðh2 Þ cosðh2 Þ n2 ðh1 Þ
2
ð36Þ
The Fresnel transmission and reflection coefficients at the input and output faces of the slab, are rfa2 ¼
u a ; uþ a
rfs2 ¼
u s ; uþ s
tfs2 ¼
2n2 ðh2 Þ cosðc2 Þ cosðh2 c2 Þ uþ s
ð37Þ
The uses of these coefficients together with Snell’s law, in Eq. (35), give, for the transmitted second harmonic magnitude: 2
T¼
2
2
i tfs2 n2 ðh1 Þ ðn1 ðh1 Þ n2 ðh2 Þ Þ 2pL 1 0 n2 ðh2 Þ cosðc2 Þ cosðh2 c2 Þ n22x ðn1 ðh1 Þ2 n2 ðh1 Þ2 Þ k ðe2i/2 rfa2 rfs2 e2i/2 Þ
! sinðWÞ i/ ! sinðUÞ i/ ! sinðWÞ i/ ! sinðUÞ i/ e 2 þ rfa2 ðe^r2 P NL e 2 þ rfa2 ðe^r2 :P NL e 2 þ ðe^2 P NL e 2 ðe^2 P NL 1 Þ 1 Þ 2 Þ 2 Þ W U W U 2 2 2 2 ! ! sinð/2 Þ i/ 1 n22x ðn1 ðh1 Þ n2 ðh1 Þ Þ 1 n22x ðn1 ðh1 Þ n2 ðh1 Þ Þ ^r NL sinð/2 Þ i/2 ^ rfa2 ð e ðe^2 þ e^r2 Þ P NL þ e Þ P e e 2 2 3 3 2 2 2 2 2 2 2 2 n2 ðh1 Þ ðn1 ðh1 Þ n2 ðh2 Þ Þ /2 2 n2 ðh1 Þ ðn1 ðh1 Þ n2 ðh2 Þ Þ /2
#
ð38Þ with W = 2/1 /2 and U = 2/1 + /2. 2.7. Expression of the transmitted second harmonic intensity as a function of the incident fundamental wave’s intensity Ix and the incident angle h Using Eqs. (25) and (28), we can write ! e^2 P NL1 ¼ 0 ! e^r2 P NL1 ¼ 0 ! e^2 P NL2 ¼ 0
t2af 1 e2i/1 2
e4i/1 þ ðrfa1 rfs1 Þ e4i/1 2rfa1 rfs1 t2af 1 e2i/1 2
e4i/1 þ ðrfa1 rfs1 Þ e4i/1 2rfa1 rfs1
2 ð2Þ
jE1i j veff1 2 ð2Þ
jE1i j veff1r
2
ðtaf 1 rfs1 Þ e2i/1 e4i/1 þ ðrfa1 rfs1 Þ2 e4i/1 2rfa1 rfs1
! e^r2 P NL2 ¼ 0
ðtaf 1 rfs1 Þ2 e2i/1
ð2Þ
jE1i j2 veff2
ð39Þ
2 ð2Þ
jE1i j veff2r 2 e4i/1 þ ðrfa1 rfs1 Þ e4i/1 2rfa1 rfs1 ! t2af 1 rfs1 2 ð2Þ ðe^2 þ e^r2 Þ P NL3 ¼ 20 jE1i j veff3 2 e4i/1 þ ðrfa1 rfs1 Þ e4i/1 2rfa1 rfs1 with ð2Þ
and
veff1r ¼ e^r2 vð2Þ e^1 e^1
ð2Þ
and
veff2r ¼ e^r2 vð2Þ e^r1 e^r1
veff1 ¼ e^2 vð2Þ e^1 e^1 veff2 ¼ e^2 vð2Þ e^r1 e^r1
ð2Þ ð2Þ
ð40Þ
ð2Þ
veff3 ¼ ðe^2 þ e^r2 Þ vð2Þ e^1 e^r1 The incident fundamental intensity Ix and the second harmonic transmitted intensity I2x are given by 1 1 2 2 I x ¼ c0 jE1i j and I 2x ¼ c0 jT j ð41Þ 2 2 When developing Eq. (41) using Eqs. (38) and (39), we finally obtain the second harmonic intensity as a function of the fundamental intensity and of its incident angle on the slab:
190
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
I 2x
2 t2fs2 2 n2 ðh1 Þ ¼ 2 2 c0 ðn2 ðh2 Þ cosðc2 Þ cosðh2 c2 ÞÞ n2x
!2
!2 2 2 2 ðn1 ðh1 Þ n2 ðh2 Þ Þ 2pL 1 2 2 2 k ðn1 ðh1 Þ n2 ðh1 Þ Þ ð1 þ ðrfa2 rfs2 Þ 2rfa2 rfs2 cosð4/2 ÞÞ
I 2x t4af 1 2
4
3
2
ð1 þ 4ðrfa1 rfs1 Þ þ ðrfa1 rfs1 Þ 4 cosð4/1 Þðrfa1 rfs1 þ ðrfa1 rfs1 Þ Þ þ 2ðrfa1 rfs1 Þ cosð8/1 ÞÞ
½aðhÞ2 þ bðhÞ2 þ cðhÞ2 þ dðhÞ2 þ eðhÞ2 þ f ðhÞ2 þ 2 cosð4/1 ÞðaðhÞdðhÞ þ bðhÞcðhÞÞ 2 cosð2ð/1 þ /2 ÞÞðaðhÞeðhÞ þ cðhÞf ðhÞÞ 2 cosð2ð/1 /2 ÞÞðbðhÞf ðhÞ þ dðhÞeðhÞÞ 2 cosð2/1 ÞðaðhÞf ðhÞ þ bðhÞeðhÞ þ cðhÞeðhÞ þ dðhÞf ðhÞÞ þ 2 cosð2/2 ÞðaðhÞbðhÞ þ cðhÞdðhÞ þ eðhÞf ðhÞÞ þ 2aðhÞcðhÞ cosð2UÞ þ 2bðhÞdðhÞ cosð2WÞ
ð42Þ
with the following coefficients: sinðWÞ W ð2Þ sinðUÞ bðhÞ ¼ rfa2 veff1r U ð2Þ sinðWÞ 2 cðhÞ ¼ rfa2 rfs1 veff2r W sinðUÞ ð2Þ dðhÞ ¼ r2fs1 veff2 U 2 2 n22x ðn1 ðh1 Þ n2 ðh1 Þ Þ ð2Þ sinð/2 Þ eðhÞ ¼ rfa2 rfs1 veff3 2 2 2 /2 n2 ðh1 Þ ðn1 ðh1 Þ n2 ðh2 Þ Þ ð2Þ
aðhÞ ¼ veff1
f ðhÞ ¼ rfs1
n22x
2
2
2
2
ðn1 ðh1 Þ n2 ðh1 Þ Þ 2
n2 ðh1 Þ ðn1 ðh1 Þ n2 ðh2 Þ Þ
ð2Þ
veff3
ð43Þ
sinð/2 Þ /2
This relation is only correct for the polarization case of p–p generation. The relations for the other cases (s–s, s–p, p–s) are given in Appendix section and can be obtained by similar calculations that will not be developed in this article. 3. Comparison between theory and experimental results In order to validate our theoretical results we compared these theoretical results to experimental measurements obtained with a second harmonic generation experimental setup developed in our laboratory (Fig. 3). The fundamental wave is generated by a Yag Laser with a 1064 nm wavelength. A second harmonic wave is generated with LiNbO3 parallel slabs. An example is given on Fig. 4. In this figure, experimental points measured on an approximatively 500 lm thick Y-cut sample. The polarization is s for the fundamental wave and s for the second harmonic wave (x-axis in the theoretical development), and was taken along the extraordinary axis of the crystal. The continuous line was calculated with the classical theory [1] using the following parameters: nex = 2.1563, ne2x = 2.2346, L = 497.5 lm, and with the second-order tensor’s coefficients for LiNbO3 congruent melt: ð2Þ
v22 ¼ 2:1 pm V1 ;
ð2Þ
v31 ¼ 4:35 pm V1 ;
ð2Þ
v33 ¼ 27:2 pm V1
ð44Þ
The indices used here were those given by Shoji et al. [9] and the second-order coefficients by Nikogosyan [10]. Fig. 4 shows that the experimental points oscillate around the theoretical curve. These high frequency oscillations are due to the multiple reflections inside the nonlinear material. The theoretical model used for the fit on this figure does not include these reflections, this is why we have developed our theoretical model, for a better fit of the Maker fringes obtained experimentally. In our theoretical expression the Maker fringes are modulated by two Fabry–Perot functions F Px and F P2x, resonating at the fundamental frequency and at the sum frequency. These two functions, cause of the high frequency oscillations observed on the experimental results, are
Fig. 3. Second harmonic generation experimental setup.
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
0.5
Calulated intensity Experimental points
0.45
Measured amplitude (V)
191
0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
–40
–20
0
20
40
Angle of incidence (º) Fig. 4. Simulated Maker fringes (continuous line) for an Y-cut LiNbO3 sample and experimental points obtained with our second harmonic generation experimental setup.
1
F Px ¼ F P 2x
2
2
3
2
ð1 þ ðrfa1 rfs1 Þ ð4 þ ðrfa1 rfs1 Þ Þ 4 cosð4/1 Þðrfa1 rfs1 þ ðrfa1 rfs1 Þ Þ þ 2ðrfa1 rfs1 Þ cosð8/1 ÞÞ 1 ¼ 2 ð1 þ ðrfa2 rfs2 Þ 2rfa1 rfs1 cosð4/2 ÞÞ
ð45Þ
In Fig. 5, we see a comparison between experimental measures obtained on an approximatively 500 lm thick Y-cut LiNbO3 sample in the case of p p polarizations. The chosen polarization was along the extraordinary axis when h = 0°. The theoretical curve was calculated with a thickness of L = 497.912 lm and with the following indices [9]: n1y ¼ n1z ¼ nox ¼ 2:2322; n1x ¼ nex ¼ 2:1563 n2y ¼ n2z ¼ no2x ¼ 2:3231; n2x ¼ ne2x ¼ 2:2346
ð46Þ
In this case, the effective coefficients of the second-order tensor are given by ð2Þ
ð2Þ
ð2Þ
ð2Þ
veff1 ¼ 2v22 sinðh2 c2 Þ sinðh1 c1 Þ cosðh1 c1 Þ v22 cosðh2 c2 Þ sin2 ðh1 c1 Þ þ v22 cosðh2 c2 Þ cos2 ðh1 c1 Þ ð2Þ
ð2Þ
ð2Þ
ð2Þ
veff1r ¼ 2v22 sinðh2 c2 Þ sinðh1 c1 Þ cosðh1 c1 Þ þ v22 cosðh2 c2 Þ sin2 ðh1 c1 Þ v22 cosðh2 c2 Þ cos2 ðh1 c1 Þ ð2Þ
ð2Þ
ð2Þ
ð2Þ
veff2 ¼ 2v22 sinðh2 c2 Þ sinðh1 c1 Þ cosðh1 c1 Þ v22 cosðh2 c2 Þ sin2 ðh1 c1 Þ þ v22 cosðh2 c2 Þ cos2 ðh1 c1 Þ ð2Þ ð2Þ veff2r ¼ 2v22 ð2Þ veff3 ¼ 0
sinðh2 c2 Þ sinðh1 c1 Þ cosðh1 c1 Þ þ
ð2Þ v22
2
cosðh2 c2 Þ sin ðh1 c1 Þ
0.4
ð47Þ
2
cosðh2 c2 Þ cos ðh1 c1 Þ
Experimental curve Theoretical curve
0.35
Measured amplitude (V)
ð2Þ v22
0.3 0.25 0.2 0.15 0.1 0.05 0
–40
–30
–20
–10
0
10
Angle of incidence (o )
20
30
40
Fig. 5. Second harmonic intensity generated by an Y-cut LiNbO3 sample measured experimentally (continuous curve) and calculated theoretically (dotted curve).
192
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
In Fig. 5 we can see that the experimental and theoretical curves have the same oscillations, even at the high frequency level. The set of Figs. 5 and 6 represents the same curves but, respectively, for angles of incidence ranging from 45° to 45° and 15° to 15°. These figures confirm the fact that the high frequency oscillations due to the multiple reflections of both the fundamental and the second harmonic waves have the same frequency in the experimental and in the theoretical case, but the magnitude of these high frequency oscillations is different. Actually, the difference between the theoretical oscillations and the experimental oscillations increases when the angle of incidence increases. Measurements made with higher angular resolution (0.001° against 0.1°) are shown in Fig. 7 and in Fig. 8. These two figures, respectively, plotted for a p–p case and for a s–s case, prove that the frequency and aspect of the oscillations are the same. It is clear on these two figures that the only difference between the theoretical curve calculated with our expression and the experimental results are the magnitude of these oscillations. This difference is thought to be due to the slight variation of the sample’s thickness. The differences between theory and experimental results come from the fact that the diameter of our incident beam on the sample is 3 mm. The second harmonic intensity detected is thus an average of the generated second harmonic intensity on the illuminated surface of the sample. Within the 3 mm diameter illuminated surface their are some irregularities in the sample thickness. We introduced the average of the sample thickness in the theoretical expression, but the value of the second harmonic field which is generated varies from one point to another of the illuminated surface, causing a difference between the expected theoretical results and the experimental results. As shown in Eq. (42), the oscillations depend on 2/1, 2/2, 4/1, 4/2, . . . These coefficients are importantly thickness dependant. For example, when the angle of incidence is null, a phase inversion of the coefficient 2/2 is obtained for a 100 nm thickness variation, and for 4/2, a variation of 60 nm is sufficient. Moreover, these thickness variations decrease when the angle of incidence increases. It is clear that a small variation of the thickness of the studied sample can lead to a significative change on the intensity of the generated second harmonic wave.
0.4
Experimental curve Theoretical curve
Measured amplitude (V)
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
–15
–10
–5
0
Angle of incidence (o)
5
10
15
Fig. 6. Second harmonic intensity generated by an Y-cut LiNbO3 sample measured experimentally (continuous curve) and calculated theoretically (dotted curve).
0.4
Experimental curve Theoretical curve
Measured amplitude (V)
0.35 0.3
0.25 0.2
0.15 0.1
0.05 –7
–6
–5
–4
–3
–2
Angle of incidence (º) Fig. 7. Second harmonic intensity generated by an Y-cut LiNbO3 sample measured experimentally (continuous curve) and calculated theoretically (dotted curve).
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
0.35
193
Experimental curve Theoretical curve
Measured amplitude (V)
0.3 0.25 0.2 0.15 0.1 0.05
–4
–2
0
2
4
Angle of incidence (º) Fig. 8. Second harmonic intensity generated by an Y-cut LiNbO3 sample measured experimentally (continuous curve) and calculated theoretically (dotted curve).
4. Conclusion In this paper we have presented new expressions in the analysis of data obtained from Maker fringes SHG experiments. They include the effects of reflections of the second harmonic wave and of the fundamental wave in the nonlinear material and are valid for any incidence. These reflections, currently neglected, can significantly change the second harmonic spectrum obtained with a Maker Fringe experimental setup. This is the reason why we have developed this new theoretical expression in the general case of anisotropic biaxial materials. From the results obtained on LiNbO3 samples, we show that our theoretical approach gives accurate results for second harmonic generation. More precisely on the high frequency oscillations due to the multiple reflections of the second harmonic and of the fundamental waves. We show that the high frequency oscillations due to the multiple reflections of both the fundamental and the second harmonic waves have the same frequency in the experimental and in the theoretical case, the same aspect, and that the differences of magnitude of these high frequency oscillations are due to slight variations of the samples thickness. Moreover, for the first time to our knowledge, we have presented an accurate expression of the Maker fringes concerning anisotropic materials. Appendix A. Case of a s polarized fundamental wave and a s polarized transmitted second harmonic wave In the case of a s polarized fundamental wave and of a s polarized transmitted second harmonic wave, the intensity of the transmitted second harmonic wave is given by the relation: 2 t2fs2 2I 2x 2pL I 2x ¼ c0 ðn2y cosðh2 ÞÞ2 ð1 þ ðrfa2 rfs2 Þ2 2rfa2 rfs2 cosð4/2 ÞÞ k
t4af 1 2
4
2
3
2
ð1 þ 4ðrfa1 rfs1 Þ þ ðrfa1 rfs1 Þ 4 cosð4/1 Þðrfa1 rfs1 þ ðrfa1 rfs1 Þ Þ þ 2ðrfa1 rfs1 Þ cosð8/1 ÞÞ 2
2
2
½aðhÞ þ bðhÞ
2
2
þ cðhÞ þ dðhÞ þ eðhÞ þ f ðhÞ þ 2 cosð4/1 ÞðaðhÞdðhÞ þ bðhÞcðhÞÞ 2 cosð2ð/1 þ /2 ÞÞðaðhÞeðhÞ þ cðhÞf ðhÞÞ 2 cosð2ð/1 /2 ÞÞðbðhÞf ðhÞ þ dðhÞeðhÞÞ 2 cosð2/1 ÞðaðhÞf ðhÞ þ bðhÞeðhÞ þ cðhÞeðhÞ þ dðhÞf ðhÞÞ þ 2 cosð2/2 ÞðaðhÞbðhÞ þ cðhÞdðhÞ þ eðhÞf ðhÞÞ þ 2aðhÞcðhÞ cosð2UÞ þ 2bðhÞdðhÞ cosð2WÞ with the following coefficients: sinðWÞ W ð2Þ sinðUÞ bðhÞ ¼ rfa2 veff1 U 2 ð2Þ sinðWÞ cðhÞ ¼ rfa2 ðrfs1 Þ veff2 W sinðUÞ 2 ð2Þ dðhÞ ¼ ðrfs1 Þ veff2 U ð2Þ
aðhÞ ¼ veff1
ðA:1Þ
194
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
sinð/2 Þ /2 sinð/ ð2Þ 2Þ f ðhÞ ¼ rfs1 veff3 /2 ð2Þ
eðhÞ ¼ rfa2 rfs1 veff3
ðA:2Þ
where ð2Þ
veff1 ¼ e^2 vð2Þ e^1 e^1 ;
ð2Þ
veff2 ¼ e^2 vð2Þ e^r1 e^r1 ;
ð2Þ
veff3 ¼ e^2 vð2Þ e^1 e^r1
In the s–s case, the Fresnel transmission and reflection coefficients are given by 2 cosðhÞ 2n2y cosðh2 Þ ; tfs2 ¼ taf 1 ¼ n1y cosðh1 Þ þ cosðhÞ n2y cosðh2 Þ þ n2s cosðh2s Þ n1y cosðh1 Þ cosðhÞ n2y cosðh2 Þ cosðhÞ ; rfa2 ¼ rfa1 ¼ n1y cosðh1 Þ þ cosðhÞ n2y cosðh2 Þ þ cosðhÞ n1y cosðh1 Þ n1s cosðh1s Þ n2y cosðh2 Þ n2s cosðh2s Þ ; rfs2 ¼ rfs1 ¼ n1y cosðh1 Þ þ n1s cosðh1s Þ n2y cosðh2 Þ þ n2s cosðh2s Þ
ðA:3Þ
ðA:4Þ
Appendix B. Case of a s polarized fundamental wave and a p polarized transmitted second harmonic wave In the case of a s polarized fundamental wave and of a p polarized transmitted second harmonic wave, the intensity of the transmitted second harmonic wave is given by the same relation as in the p–p case, with the following substitutions: ðB:1Þ e^1 ¼ e^r1 ¼ ð0; 1; 0Þ and n1(h1) = n1y, which gives c1 ¼ 0
ðB:2Þ
These coefficients are also changed: 2 cosðhÞ n1y cosðh1 Þ þ cosðhÞ n1y cosðh1 Þ cosðhÞ rfa1 ¼ n1y cosðh1 Þ þ cosðhÞ n1y cosðh1 Þ n1s cosðh1s Þ rfs1 ¼ n1y cosðh1 Þ þ n1s cosðh1s Þ taf 1 ¼
ðB:3Þ
All the other coefficients are unchanged and similar to the p–p case. Appendix C. Case of a p polarized fundamental wave and a s polarized transmitted second harmonic wave In this case, the intensity of the transmitted second harmonic wave is given by the same relation as in the s–s case, with the following substitutions: e^1 ¼ ðcosðh1 c1 Þ; 0; sinðh1 c1 ÞÞe^r1 ð cosðh1 c1 Þ; 0; sinðh1 c1 ÞÞ
ðC:1Þ
and n1y = n1(h1), therefore: 2 cosðhÞ cosðh1 c1 Þ þ n1 ðh1 Þ cosðc1 Þ cosðhÞ cosðh1 c1 Þ n1 ðh1 Þ cosðc1 Þ cosðhÞ rfa1 ¼ cosðh1 c1 Þ þ n1 ðh1 Þ cosðc1 Þ cosðhÞ n1s cosðh1 c1 Þ n1 ðh1 Þ cosðc1 Þ cosðh1s Þ rfs1 ¼ n1s cosðh1 c1 Þ þ n1 ðh1 Þ cosðc1 Þ cosðh1s Þ
taf 1 ¼
All the other coefficients are unchanged and similar to the s–s case. References [1] J. Jerphagnon, S.K. Kurtz, Journal of Applied Physics 41 (4) (1970) 1667. [2] P.D. Maker, R.W. Terhune, M. Nisenoff, C.M. Savage, Physical Review Letters 8 (1) (1962) 21.
ðC:2Þ
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195 [3] [4] [5] [6] [7] [8] [9] [10]
N. Bloembergen, P.S. Pershan, Physical Review 128 (2) (1962) 606. N. Okamoto, Y. Hirano, O. Sugihara, Journal of the Optical Society of America B 9 (1992) 2083. R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, R. Ito, Japanese Journal of Applied Physics 27 (1988) L1134. D. Neher, A. Wolf, C. Bubeck, G. Wegner, Chemical Physics Letters 163 (1989) 116. F. Krausz, E. Wintner, Physical Review B 39 (1989) 3701. W.N. Herman, L.M. Hayden, Journal of the Optical Society of America B 12 (1995) 416. I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, R. Ito, Journal of Optical Society of America B 14 (1997) 2268. D.N. Nikogosyan, Properties of Optical and Laser Related Materials; A Handbook, John Wiley, 1997.
195
Second harmonic generation: Effects of the multiple reflections of the fundamental and the second harmonic waves on the Maker fringes Gildas Tellier *, Christian Boisrobert IREENA, Universite´ de Nantes, BP 92208, 2 rue de la Houssiniere, 44322 Nantes Cedex 03, France Received 26 April 2007; received in revised form 19 June 2007; accepted 19 June 2007
Abstract The Maker fringes technique is commonly used for the determination of nonlinear optical coefficients. In this article, we present a new formulation of Maker fringes in parallel-surface samples, using boundary conditions taking into account the anisotropy of the crystal, the refractive-index dispersion, and the reflections of the fundamental and the second harmonic waves inside the material. Complete expressions for the generated second harmonic intensity are given for birefringent crystals for the case of no pump depletion. A comparison between theory and experimental results is made, showing the accuracy of our theoretical expressions. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Second harmonic generation; Maker fringes; Multiple reflections; Anisotropy
1. Introduction Since 1970, the determination of the nonlinear optical coefficients of the tensor v(2) associated with second harmonic generation (SHG) has been performed almost exclusively by the Maker fringe technique [2], as described in detail by Jerphagnon and Kurtz [1]. In this theoretical development, the boundary conditions did not take into account the reflections of the second harmonic issued from the second interface on the first interface of the nonlinear material. Moreover, the theory presented did not consider the general case of anisotropic media. Bloembergen and Pershan gave an expression for the second harmonic fields generated by an isotropic transparent nonlinear parallel slab [3], but the reflections of the second harmonic and fundamental waves in the media were ignored in this expression. Taking into account the anisotropy of the nonlinear media, a theoretical expression has been developed by Okamoto et al. [4] in the case of slabs with C1m space symmetry, but once again, multiple reflections were neglected. An expression using complete boundary conditions, including multiple reflections of both the fundamental and the second harmonic waves, has been developed in the case of SHG setups using non-parallel slabs [5], but this approach does not apply to the Maker Fringes Technique. Referring to studies carried out on the third harmonic generation [6,7], the first expression taking into account multiple reflections in the media and developed for the Maker Fringes was obtained by Herman and Hayden [8]. This expression has been established for uniaxial anisotropic materials and uses complete boundary conditions for the second harmonic wave to solve the problem. Unfortunately, the multiple reflections of the fundamental waves are not considered in this approach. Moreover, the expression has been developed for uniaxial materials but in one direction only. *
Corresponding author. Tel.: +33 68 877 1707. E-mail address: [email protected] (G. Tellier).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.06.048
184
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
In this paper, we report on the development of a theoretical expression of the Maker fringes, using complete boundary conditions with multiple reflection of both the fundamental and the second harmonic waves, in the general case of anisotropic biaxial materials. We here intend to provide a complete general framework for the second harmonic generation analysis using the Maker fringes technique. 2. Theoretical developments 2.1. Description of the problem In this section, we present a general approach of the boundary-value problem for SHG. The configuration is shown in Fig. 1. The origin is defined at the center of the nonlinear layer of thickness L. We consider the case of no pump depletion. ! !! The propagating waves are taken plane, monochromatic, and can thus be written as E eið k : r xtÞ . The fundamental and the second harmonic waves can be polarized in the plane of incidence (polarization p) or perpendicular to the plane of incidence (polarization s). In the following development we will only consider the polarization configuration p–p. The results obtained for the other configurations will be given in Appendixes. The fundamental wave is incident on the parallel slab at an angle h. We firstly describe the wave propagation conditions in these three regions: Region I (air): This media is supposed to be isotropic with an index equal to 1. The waves propagating in this region are – The incident fundamental and reflected fundamental waves (x): !r ! L! ! ! L! ! ! E1i þ Er1i ¼ E1i eið q1 :ð r þ2 z ÞxtÞ e^1a þ Er1i eið q1 :ð r þ2 z ÞxtÞ e^r1a
ð1Þ
– the reflected second harmonic wave (2x), from the nonlinear media: !r ! L! ! E2a ¼ Reið q2 :ð r þ2 z Þ2xtÞ e^r2a
ð2Þ
With the following wave vectors: 2p ! q1 ¼ ðsinðhÞ; 0; cosðhÞÞ k !r 2p q1 ¼ ðsinðhÞ; 0; cosðhÞÞ k !r 4p q2 ¼ ðsinðhÞ; 0; cosðhÞÞ k
ð3Þ
Since the second harmonic waves and the fundamental waves are polarized in the plane of incidence (configuration p–p), the polarization unit vectors are e^1a ¼ ðcosðhÞ; 0; sinðhÞÞ
ð4Þ
e^r1a ¼ e^r2a ¼ ð cosðhÞ; 0; sinðhÞÞ
Region II (nonlinear material): This media is considered anisotropic biaxial and its dielectric permittivities can be represented by the tensor: 0 2 1 0 nix 0 B C r ðixÞ ¼ ri ¼ @ 0 n2iy 0 A ð5Þ 0
0
n2iz z
k 1s
k 2s
III θ 2s II
k2
I
q1
θ2 θ
x
θ 1s
k 2r q 2r
output medium (region III) z=L/2
k1 θ1 θ
k 1r non–linear material (region II) z=L/2
q 1r air (region I)
Fig. 1. Geometry of the problem of second harmonic generation in a nonlinear parallel slab.
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
185
with i = 1, 2 and where (nix, niy, niz) are the principal refraction indices at the fundamental (i = 1) and the second harmonic (i = 2) frequencies. In Region II, the refractive indices are depending on the propagation direction and can be written as follows: n1(h1) for the fundamental (pulsation x) and n2(h2) for the second harmonic (pulsation 2x). These refraction indices obey to the Snell laws and, using the index ellipsoid, the angle-dependant indices can be written as 2 1 cos ðhi Þ sin2 ðhi Þ 2 ni ðhi Þ ¼ þ ; i ¼ 1; 2 ð6Þ n2ix n2iz The fundamental wave propagating in this media is the sum of two waves propagating in two opposite directions. These waves correspond to the multiple reflections of the fundamental at the input face of the sample and the output face of the sample: !r ! !! ! E1 ¼ E1 eið k 1 : r xtÞ^e1 þ Er1 eið k 1 : r xtÞ e^r1 ð7Þ The second harmonic wave can be decomposed in three different waves: the second harmonic bound wave generated by the fundamental wave and two free waves propagating, as the fundamental wave, in two opposite directions. These two free waves correspond to the multiple reflections of the second harmonic at the input face of the sample and the output face of the sample. The total second harmonic wave can thus be written as !r ! !! ! ! ð8Þ E2 ¼ eb þ E2^e2 eið k 2 : r 2xtÞ þ Er2^er2 eið k 2 : r 2xtÞ with the following wave vectors: ! 2pn1 ðh1 Þ k1 ¼ ðsinðh1 Þ; 0; cosðh1 ÞÞ k !r 2pn1 ðh1 Þ ðsinðh1 Þ; 0; cosðh1 ÞÞ k1 ¼ k ! 4pn2 ðh2 Þ ðsinðh2 Þ; 0; cosðh2 ÞÞ k2 ¼ k !r 4pn2 ðh2 Þ ðsinðh2 Þ; 0; cosðh2 ÞÞ k2 ¼ k
ð9Þ
! ! Since the nonlinear media is anisotropic, the electric field E is not collinear to the displacement vector D , thus is not ! perpendicular to the wave vector k . The electric field deviates from the displacement vector by the walk-off angle ci with i = 1, 2 (respectively, for x and 2x), as shown in Fig. 2. When the waves are polarized in the plane of incidence (configuration p–p), the polarization unit vectors are e^1 ¼ ðcosðh1 c1 Þ; 0; sinðh1 c1 ÞÞ e^r1 ¼ ð cosðh1 c1 Þ; 0; sinðh1 c1 ÞÞ e^2 ¼ ðcosðh2 c2 Þ; 0; sinðh2 c2 ÞÞ
ð10Þ
e^r2 ¼ ð cosðh2 c2 Þ; 0; sinðh2 c2 ÞÞ z
ki θi
θi
x
γi ei
γi Di
Fig. 2. Definition of the walk-off angle.
186
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
Region III (output medium): In most cases, the output medium is the same as the input medium, but in order to differentiate the contributions of the waves reflected on the input and the output faces, this output medium will be considered as an isotropic media with a refraction index of n1s at x and n2s at 2x. In this region, the fundamental propagating wave is ! ! L! ! E1t ¼ E1t eið k 1s :ð r 2 z ÞxtÞ e^1s
ð11Þ
The second harmonic wave transmitted and propagating in the output medium is ! ! L! ! E2s ¼ T eið k 2s :ð r 2 z Þ2xtÞ e^2s
ð12Þ
with the following wave vectors: ! 2n1s p ðsinðh1s Þ; 0; cosðh1s ÞÞ; k 1s ¼ k
! 4n2s p ðsinðh2s Þ; 0; cosðh2s ÞÞ k 2s ¼ k
ð13Þ
In this region, the polarization unit vectors are e^1s ¼ ðcosðh1s Þ; 0; sinðh1s ÞÞ e^2s ¼ ðcosðh2s Þ; 0; sinðh2s ÞÞ
ð14Þ
2.2. Expression of the walk-off angle c In an anisotropic media, the displacement vector can be written as follows: 0
n2ix ! B 0 D ¼ 0 @ 0
0 n2iy
1 0 ! 0 C AE
0
n2iz
ð15Þ
with i = 1, 2 depending on the pulsation. Using Maxwell’s equations, Snell’s law and the index ellipsoid, we obtain the following expressions of the walk-off angle ci as a function of the incident angle h:
2 ni ðhi Þ cosðci Þ cosðhi Þ nix 2 ni ðhi Þ sinðhi ci Þ ¼ cosðci Þ sinðhi Þ niz nix niz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðci Þ ¼ 2 ni ðhi Þ n2ix þ n2iz ni ðhi Þ cosðhi ci Þ ¼
ð16Þ
niz sinðhÞ sinðci Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðniz nix Þ þ ðn2iz n2ix Þ sinðhÞ2 2.3. Expression of the magnitude of the second harmonic transmitted field as a function of the bound wave magnitude At the region I–region II interface (z ¼ L2), the boundary conditions for the electric and magnetic fields lead to the equations: L cosðhÞR ¼ e þ cosðh2 c2 ÞðE2 ei/2 Er2 ei/2 Þ ð17Þ 2 bx L R ¼ cb þ n2 ðh2 Þ cosðc2 ÞðE2 ei/2 þ Er2 ei/2 Þ ð18Þ 2 by The continuity of the same fields at the region II–region III interface (z ¼ L2) requires
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
187
L cosðh2s ÞT ¼ e þ cosðh2 c2 ÞðE2 ei/2 Er2 ei/2 Þ ð19Þ 2 bx L n2s T ¼ cb þ n2 ðh2 Þ cosðc2 ÞðE2 ei/2 þ Er2 ei/2 Þ ð20Þ 2 by where /2 = 2pLn2(h2)cos(h2)/k and b L2 ; b L2 are the magnitudes of the magnetic the bound wave at z = L/2 and z = L/2. After solving this system for E2 and Er2 , the magnitude of the second harmonic field can be expressed as a function of the bound wave magnitude: " !# 1 2i/2 L L þ þ 2i/2 u v e T ¼ ua v e þ 2n2 ðh2 Þ cosðh2 c2 Þ cosðc2 Þ e þ cosðhÞcb ð21Þ D a 2 bx 2 by with the following coefficient: 2i/2 þ 2i/2 uþ D ¼ u s ua e a us e
u a ¼ cosðh2 c2 Þ n2 ðh2 Þ cosðc2 Þ cosðhÞ u s ¼ n2s cosðh2 c2 Þ n2 ðh2 Þ cosðc2 Þ cosðh2s Þ L L þ þ cosðhÞcb a ¼e 2 bx 2 by L L a ¼ n2s e cosðh2s Þcb 2 bx 2 by L L v ¼ cosðh2 c2 Þcb n2 ðh2 Þ cosðc2 Þe 2 by 2 bx
ð22Þ
2.4. Expression of the magnitudes of the fundamental waves propagating in the nonlinear medium In order to express the magnitudes of the propagating fundamental waves in the nonlinear medium, the boundary conditions are written, as in the previous section, but this time at the x pulsation. At the region I–region II interface (z ¼ L2), the continuity of the electric and magnetic fields requires cosðhÞðE1i Er1i Þ ¼ cosðh1 c1 ÞðE1 ei/1 Er1 ei/1 Þ E1i þ Er1i ¼ n1 ðh1 Þ cosðc1 ÞðE1 ei/1 þ Er1 ei/1 Þ
ð23Þ
with /1 = pLn1(h1)cos(h1)/k. Moreover, at the region II–region III interface z ¼ L2 , the continuity of the electric and magnetic fields requires cosðh1s ÞE1t ¼ cosðh1 c1 ÞðE1 ei/1 Er1 ei/1 Þ n1s E1t ¼ n1 ðh1 Þ cosðc1 ÞðE1 ei/1 þ Er1 ei/1 Þ
ð24Þ
Solving these equations and using the Fresnel transmission and reflection coefficients at the input and output faces of the slab, E1 and Er1 can thus be written as functions of the magnitude of the incident fundamental wave E1i and the Fresnel reflection and transmission coefficients: ei/1 E1 ¼ taf 1 2i/ E1i e 1 rfa1 rfs1 e2i/1 ð25Þ ei/1 r E1 ¼ taf 1 rfs1 2i/ E1i e 1 rfa1 rfs1 e2i/1 With the coefficients: taf 1 ¼
2 cosðhÞ ; uþ a1
rfa1 ¼
u a1 ; uþ a1
rfs1 ¼
u s1 uþ s1
u a1 ¼ cosðh1 c1 Þ n1 ðh1 Þ cosðc1 Þ cosðhÞ u s1 ¼ n1s cosðh1 c1 Þ n1 ðh1 Þ cosðc1 Þ cosðh1s Þ
ð26Þ
188
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
2.5. Expression of the bound wave ! ! Since the fundamental wave in region II is the sum of two contra-propagative waves E1 and Er1 , the induced nonlinear polarization is also decomposed as a sum: ! ! ! ! r ! r ! ! ! ! ! ! i2 k 1 r xt i2 k 1 r xt i k þ k r 2xt 1 1 P NL ¼ P NL1 e þ P NL2 e þ P NL3 e ð27Þ with ! ! ! ð28Þ P NL1 ¼ 0 E1 E1 vð2Þ e^1 e^1 ; P NL2 ¼ 0 Er1 Er1 vð2Þ e^r1 e^r1 ; P NL3 ¼ 20 Er1 E1 vð2Þ e^1 e^r1 The bound wave, written as: ! ! ! ! ! ! i k b1 r 2xt i k b2 r 2xt i k b3 r 2xt ! þ ! eb2 e þ ! eb3 e ð29Þ eb ¼ ! eb1 e is a particular solution of the propagation equation: 2 2 2x 2x 1 ! ! ! rotðrotðeb ÞÞ r2 eb ¼ P NL c c 0 with ! ! ! ! ! ! ! k b1 ¼ 2k 1 ; k b2 ¼ 2k r1 ; k b2 ¼ k 1 þ k r1 We obtain the following relations for the bound waves: 2 3 ! ! ! 2 NL1 ! n2 ðh1 Þ ! 41 P NL1 k b1 ð k b1 P Þ5 eb1 ¼ r2 2 2 2 0 ðn1 ðh1 Þ n2 ðh1 Þ Þ n n Þ ð2x c 2x 2z 2 3 ! ! ! 2 NL2 ! Þ ð k P n ðh Þ k 2 1 ! 41 P NL2 b2 b2 5 eb2 ¼ r2 0 ðn1 ðh1 Þ2 n2 ðh1 Þ2 Þ n n Þ2 ð2x c 2x 2z eb3z ¼
ð30Þ
ð31Þ
ð32Þ
n22x P NL2 z n22 ðh2 Þ cos2 ðh2 Þn22z 0
2.6. Expression of the relation between the magnitude of the transmitted second harmonic wave and the magnitude of the incident fundamental wave At the region I–region II interface (z ¼ L2) and at the region II–region III interface (z ¼ L2), the electric and magnetic fields can be written as: L e ¼ eb1x e2i/1 þ eb2x e2i/1 2 bx L b ¼ bb1y e2i/1 þ bb2y e2i/1 þ bb3y 2 by ð33Þ L e ¼ eb1x e2i/1 þ eb2x e2i/1 2 bx L b ¼ bb1y e2i/1 þ bb2y e2i/1 þ bb3y 2 by Using identities and coefficients such as eið2/1 2/2 Þ ¼ 2i sinð2/1 /2 Þei/2 þ e2i/1 v i ¼ cosðh2 c2 Þcbbiy n2 ðh2 Þ cosðc2 Þebix
for i ¼ 1; 2
ð34Þ
v 3 ¼ cosðh2 c2 Þcbb3y in Eq. (21), we obtain T ¼
1 þ i/2 i/2 þ i/2 ½u v 2i sinð2/1 þ /2 Þei/2 uþ u þ uþ a v1 2i sinð2/1 /2 Þe a v2 2i sinð2/1 /2 Þe a v2 2i sinð2/1 þ /2 Þe D a 1 i/2 þ i/2 þ u þ uþ a v3 2i sinð/2 Þe a v3 2i sinð/2 Þe
ð35Þ
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
189
Furthermore, from Eq. (32), we find that:
0 1 ! NL ^ P e n ðh Þ cosðh Þ n ðh Þ cosðh Þ 1 1 1 2 2 2 @ 2 1 A v 1 ¼ 2 2 ! n22x NL ^r 0 ðn1 ðh1 Þ n2 ðh1 Þ Þ e P 1 0 2 ! 1 2 NL ^ e P n ðh Þ n ðh Þ cosðh Þ n ðh Þ cosðh Þ 2 2 1 1 1 1 2 2 2 2 A @ v 2 ¼ ! n22x NL ^r 0 ðn1 ðh1 Þ2 n2 ðh1 Þ2 Þ e2 P 2 ! 1 v ðe^2 þ e^r2 Þ P NL 3 ¼ 3 2n2 ðh2 Þ cosðh2 Þ n2 ðh1 Þ
2
ð36Þ
The Fresnel transmission and reflection coefficients at the input and output faces of the slab, are rfa2 ¼
u a ; uþ a
rfs2 ¼
u s ; uþ s
tfs2 ¼
2n2 ðh2 Þ cosðc2 Þ cosðh2 c2 Þ uþ s
ð37Þ
The uses of these coefficients together with Snell’s law, in Eq. (35), give, for the transmitted second harmonic magnitude: 2
T¼
2
2
i tfs2 n2 ðh1 Þ ðn1 ðh1 Þ n2 ðh2 Þ Þ 2pL 1 0 n2 ðh2 Þ cosðc2 Þ cosðh2 c2 Þ n22x ðn1 ðh1 Þ2 n2 ðh1 Þ2 Þ k ðe2i/2 rfa2 rfs2 e2i/2 Þ
! sinðWÞ i/ ! sinðUÞ i/ ! sinðWÞ i/ ! sinðUÞ i/ e 2 þ rfa2 ðe^r2 P NL e 2 þ rfa2 ðe^r2 :P NL e 2 þ ðe^2 P NL e 2 ðe^2 P NL 1 Þ 1 Þ 2 Þ 2 Þ W U W U 2 2 2 2 ! ! sinð/2 Þ i/ 1 n22x ðn1 ðh1 Þ n2 ðh1 Þ Þ 1 n22x ðn1 ðh1 Þ n2 ðh1 Þ Þ ^r NL sinð/2 Þ i/2 ^ rfa2 ð e ðe^2 þ e^r2 Þ P NL þ e Þ P e e 2 2 3 3 2 2 2 2 2 2 2 2 n2 ðh1 Þ ðn1 ðh1 Þ n2 ðh2 Þ Þ /2 2 n2 ðh1 Þ ðn1 ðh1 Þ n2 ðh2 Þ Þ /2
#
ð38Þ with W = 2/1 /2 and U = 2/1 + /2. 2.7. Expression of the transmitted second harmonic intensity as a function of the incident fundamental wave’s intensity Ix and the incident angle h Using Eqs. (25) and (28), we can write ! e^2 P NL1 ¼ 0 ! e^r2 P NL1 ¼ 0 ! e^2 P NL2 ¼ 0
t2af 1 e2i/1 2
e4i/1 þ ðrfa1 rfs1 Þ e4i/1 2rfa1 rfs1 t2af 1 e2i/1 2
e4i/1 þ ðrfa1 rfs1 Þ e4i/1 2rfa1 rfs1
2 ð2Þ
jE1i j veff1 2 ð2Þ
jE1i j veff1r
2
ðtaf 1 rfs1 Þ e2i/1 e4i/1 þ ðrfa1 rfs1 Þ2 e4i/1 2rfa1 rfs1
! e^r2 P NL2 ¼ 0
ðtaf 1 rfs1 Þ2 e2i/1
ð2Þ
jE1i j2 veff2
ð39Þ
2 ð2Þ
jE1i j veff2r 2 e4i/1 þ ðrfa1 rfs1 Þ e4i/1 2rfa1 rfs1 ! t2af 1 rfs1 2 ð2Þ ðe^2 þ e^r2 Þ P NL3 ¼ 20 jE1i j veff3 2 e4i/1 þ ðrfa1 rfs1 Þ e4i/1 2rfa1 rfs1 with ð2Þ
and
veff1r ¼ e^r2 vð2Þ e^1 e^1
ð2Þ
and
veff2r ¼ e^r2 vð2Þ e^r1 e^r1
veff1 ¼ e^2 vð2Þ e^1 e^1 veff2 ¼ e^2 vð2Þ e^r1 e^r1
ð2Þ ð2Þ
ð40Þ
ð2Þ
veff3 ¼ ðe^2 þ e^r2 Þ vð2Þ e^1 e^r1 The incident fundamental intensity Ix and the second harmonic transmitted intensity I2x are given by 1 1 2 2 I x ¼ c0 jE1i j and I 2x ¼ c0 jT j ð41Þ 2 2 When developing Eq. (41) using Eqs. (38) and (39), we finally obtain the second harmonic intensity as a function of the fundamental intensity and of its incident angle on the slab:
190
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
I 2x
2 t2fs2 2 n2 ðh1 Þ ¼ 2 2 c0 ðn2 ðh2 Þ cosðc2 Þ cosðh2 c2 ÞÞ n2x
!2
!2 2 2 2 ðn1 ðh1 Þ n2 ðh2 Þ Þ 2pL 1 2 2 2 k ðn1 ðh1 Þ n2 ðh1 Þ Þ ð1 þ ðrfa2 rfs2 Þ 2rfa2 rfs2 cosð4/2 ÞÞ
I 2x t4af 1 2
4
3
2
ð1 þ 4ðrfa1 rfs1 Þ þ ðrfa1 rfs1 Þ 4 cosð4/1 Þðrfa1 rfs1 þ ðrfa1 rfs1 Þ Þ þ 2ðrfa1 rfs1 Þ cosð8/1 ÞÞ
½aðhÞ2 þ bðhÞ2 þ cðhÞ2 þ dðhÞ2 þ eðhÞ2 þ f ðhÞ2 þ 2 cosð4/1 ÞðaðhÞdðhÞ þ bðhÞcðhÞÞ 2 cosð2ð/1 þ /2 ÞÞðaðhÞeðhÞ þ cðhÞf ðhÞÞ 2 cosð2ð/1 /2 ÞÞðbðhÞf ðhÞ þ dðhÞeðhÞÞ 2 cosð2/1 ÞðaðhÞf ðhÞ þ bðhÞeðhÞ þ cðhÞeðhÞ þ dðhÞf ðhÞÞ þ 2 cosð2/2 ÞðaðhÞbðhÞ þ cðhÞdðhÞ þ eðhÞf ðhÞÞ þ 2aðhÞcðhÞ cosð2UÞ þ 2bðhÞdðhÞ cosð2WÞ
ð42Þ
with the following coefficients: sinðWÞ W ð2Þ sinðUÞ bðhÞ ¼ rfa2 veff1r U ð2Þ sinðWÞ 2 cðhÞ ¼ rfa2 rfs1 veff2r W sinðUÞ ð2Þ dðhÞ ¼ r2fs1 veff2 U 2 2 n22x ðn1 ðh1 Þ n2 ðh1 Þ Þ ð2Þ sinð/2 Þ eðhÞ ¼ rfa2 rfs1 veff3 2 2 2 /2 n2 ðh1 Þ ðn1 ðh1 Þ n2 ðh2 Þ Þ ð2Þ
aðhÞ ¼ veff1
f ðhÞ ¼ rfs1
n22x
2
2
2
2
ðn1 ðh1 Þ n2 ðh1 Þ Þ 2
n2 ðh1 Þ ðn1 ðh1 Þ n2 ðh2 Þ Þ
ð2Þ
veff3
ð43Þ
sinð/2 Þ /2
This relation is only correct for the polarization case of p–p generation. The relations for the other cases (s–s, s–p, p–s) are given in Appendix section and can be obtained by similar calculations that will not be developed in this article. 3. Comparison between theory and experimental results In order to validate our theoretical results we compared these theoretical results to experimental measurements obtained with a second harmonic generation experimental setup developed in our laboratory (Fig. 3). The fundamental wave is generated by a Yag Laser with a 1064 nm wavelength. A second harmonic wave is generated with LiNbO3 parallel slabs. An example is given on Fig. 4. In this figure, experimental points measured on an approximatively 500 lm thick Y-cut sample. The polarization is s for the fundamental wave and s for the second harmonic wave (x-axis in the theoretical development), and was taken along the extraordinary axis of the crystal. The continuous line was calculated with the classical theory [1] using the following parameters: nex = 2.1563, ne2x = 2.2346, L = 497.5 lm, and with the second-order tensor’s coefficients for LiNbO3 congruent melt: ð2Þ
v22 ¼ 2:1 pm V1 ;
ð2Þ
v31 ¼ 4:35 pm V1 ;
ð2Þ
v33 ¼ 27:2 pm V1
ð44Þ
The indices used here were those given by Shoji et al. [9] and the second-order coefficients by Nikogosyan [10]. Fig. 4 shows that the experimental points oscillate around the theoretical curve. These high frequency oscillations are due to the multiple reflections inside the nonlinear material. The theoretical model used for the fit on this figure does not include these reflections, this is why we have developed our theoretical model, for a better fit of the Maker fringes obtained experimentally. In our theoretical expression the Maker fringes are modulated by two Fabry–Perot functions F Px and F P2x, resonating at the fundamental frequency and at the sum frequency. These two functions, cause of the high frequency oscillations observed on the experimental results, are
Fig. 3. Second harmonic generation experimental setup.
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
0.5
Calulated intensity Experimental points
0.45
Measured amplitude (V)
191
0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
–40
–20
0
20
40
Angle of incidence (º) Fig. 4. Simulated Maker fringes (continuous line) for an Y-cut LiNbO3 sample and experimental points obtained with our second harmonic generation experimental setup.
1
F Px ¼ F P 2x
2
2
3
2
ð1 þ ðrfa1 rfs1 Þ ð4 þ ðrfa1 rfs1 Þ Þ 4 cosð4/1 Þðrfa1 rfs1 þ ðrfa1 rfs1 Þ Þ þ 2ðrfa1 rfs1 Þ cosð8/1 ÞÞ 1 ¼ 2 ð1 þ ðrfa2 rfs2 Þ 2rfa1 rfs1 cosð4/2 ÞÞ
ð45Þ
In Fig. 5, we see a comparison between experimental measures obtained on an approximatively 500 lm thick Y-cut LiNbO3 sample in the case of p p polarizations. The chosen polarization was along the extraordinary axis when h = 0°. The theoretical curve was calculated with a thickness of L = 497.912 lm and with the following indices [9]: n1y ¼ n1z ¼ nox ¼ 2:2322; n1x ¼ nex ¼ 2:1563 n2y ¼ n2z ¼ no2x ¼ 2:3231; n2x ¼ ne2x ¼ 2:2346
ð46Þ
In this case, the effective coefficients of the second-order tensor are given by ð2Þ
ð2Þ
ð2Þ
ð2Þ
veff1 ¼ 2v22 sinðh2 c2 Þ sinðh1 c1 Þ cosðh1 c1 Þ v22 cosðh2 c2 Þ sin2 ðh1 c1 Þ þ v22 cosðh2 c2 Þ cos2 ðh1 c1 Þ ð2Þ
ð2Þ
ð2Þ
ð2Þ
veff1r ¼ 2v22 sinðh2 c2 Þ sinðh1 c1 Þ cosðh1 c1 Þ þ v22 cosðh2 c2 Þ sin2 ðh1 c1 Þ v22 cosðh2 c2 Þ cos2 ðh1 c1 Þ ð2Þ
ð2Þ
ð2Þ
ð2Þ
veff2 ¼ 2v22 sinðh2 c2 Þ sinðh1 c1 Þ cosðh1 c1 Þ v22 cosðh2 c2 Þ sin2 ðh1 c1 Þ þ v22 cosðh2 c2 Þ cos2 ðh1 c1 Þ ð2Þ ð2Þ veff2r ¼ 2v22 ð2Þ veff3 ¼ 0
sinðh2 c2 Þ sinðh1 c1 Þ cosðh1 c1 Þ þ
ð2Þ v22
2
cosðh2 c2 Þ sin ðh1 c1 Þ
0.4
ð47Þ
2
cosðh2 c2 Þ cos ðh1 c1 Þ
Experimental curve Theoretical curve
0.35
Measured amplitude (V)
ð2Þ v22
0.3 0.25 0.2 0.15 0.1 0.05 0
–40
–30
–20
–10
0
10
Angle of incidence (o )
20
30
40
Fig. 5. Second harmonic intensity generated by an Y-cut LiNbO3 sample measured experimentally (continuous curve) and calculated theoretically (dotted curve).
192
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
In Fig. 5 we can see that the experimental and theoretical curves have the same oscillations, even at the high frequency level. The set of Figs. 5 and 6 represents the same curves but, respectively, for angles of incidence ranging from 45° to 45° and 15° to 15°. These figures confirm the fact that the high frequency oscillations due to the multiple reflections of both the fundamental and the second harmonic waves have the same frequency in the experimental and in the theoretical case, but the magnitude of these high frequency oscillations is different. Actually, the difference between the theoretical oscillations and the experimental oscillations increases when the angle of incidence increases. Measurements made with higher angular resolution (0.001° against 0.1°) are shown in Fig. 7 and in Fig. 8. These two figures, respectively, plotted for a p–p case and for a s–s case, prove that the frequency and aspect of the oscillations are the same. It is clear on these two figures that the only difference between the theoretical curve calculated with our expression and the experimental results are the magnitude of these oscillations. This difference is thought to be due to the slight variation of the sample’s thickness. The differences between theory and experimental results come from the fact that the diameter of our incident beam on the sample is 3 mm. The second harmonic intensity detected is thus an average of the generated second harmonic intensity on the illuminated surface of the sample. Within the 3 mm diameter illuminated surface their are some irregularities in the sample thickness. We introduced the average of the sample thickness in the theoretical expression, but the value of the second harmonic field which is generated varies from one point to another of the illuminated surface, causing a difference between the expected theoretical results and the experimental results. As shown in Eq. (42), the oscillations depend on 2/1, 2/2, 4/1, 4/2, . . . These coefficients are importantly thickness dependant. For example, when the angle of incidence is null, a phase inversion of the coefficient 2/2 is obtained for a 100 nm thickness variation, and for 4/2, a variation of 60 nm is sufficient. Moreover, these thickness variations decrease when the angle of incidence increases. It is clear that a small variation of the thickness of the studied sample can lead to a significative change on the intensity of the generated second harmonic wave.
0.4
Experimental curve Theoretical curve
Measured amplitude (V)
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
–15
–10
–5
0
Angle of incidence (o)
5
10
15
Fig. 6. Second harmonic intensity generated by an Y-cut LiNbO3 sample measured experimentally (continuous curve) and calculated theoretically (dotted curve).
0.4
Experimental curve Theoretical curve
Measured amplitude (V)
0.35 0.3
0.25 0.2
0.15 0.1
0.05 –7
–6
–5
–4
–3
–2
Angle of incidence (º) Fig. 7. Second harmonic intensity generated by an Y-cut LiNbO3 sample measured experimentally (continuous curve) and calculated theoretically (dotted curve).
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
0.35
193
Experimental curve Theoretical curve
Measured amplitude (V)
0.3 0.25 0.2 0.15 0.1 0.05
–4
–2
0
2
4
Angle of incidence (º) Fig. 8. Second harmonic intensity generated by an Y-cut LiNbO3 sample measured experimentally (continuous curve) and calculated theoretically (dotted curve).
4. Conclusion In this paper we have presented new expressions in the analysis of data obtained from Maker fringes SHG experiments. They include the effects of reflections of the second harmonic wave and of the fundamental wave in the nonlinear material and are valid for any incidence. These reflections, currently neglected, can significantly change the second harmonic spectrum obtained with a Maker Fringe experimental setup. This is the reason why we have developed this new theoretical expression in the general case of anisotropic biaxial materials. From the results obtained on LiNbO3 samples, we show that our theoretical approach gives accurate results for second harmonic generation. More precisely on the high frequency oscillations due to the multiple reflections of the second harmonic and of the fundamental waves. We show that the high frequency oscillations due to the multiple reflections of both the fundamental and the second harmonic waves have the same frequency in the experimental and in the theoretical case, the same aspect, and that the differences of magnitude of these high frequency oscillations are due to slight variations of the samples thickness. Moreover, for the first time to our knowledge, we have presented an accurate expression of the Maker fringes concerning anisotropic materials. Appendix A. Case of a s polarized fundamental wave and a s polarized transmitted second harmonic wave In the case of a s polarized fundamental wave and of a s polarized transmitted second harmonic wave, the intensity of the transmitted second harmonic wave is given by the relation: 2 t2fs2 2I 2x 2pL I 2x ¼ c0 ðn2y cosðh2 ÞÞ2 ð1 þ ðrfa2 rfs2 Þ2 2rfa2 rfs2 cosð4/2 ÞÞ k
t4af 1 2
4
2
3
2
ð1 þ 4ðrfa1 rfs1 Þ þ ðrfa1 rfs1 Þ 4 cosð4/1 Þðrfa1 rfs1 þ ðrfa1 rfs1 Þ Þ þ 2ðrfa1 rfs1 Þ cosð8/1 ÞÞ 2
2
2
½aðhÞ þ bðhÞ
2
2
þ cðhÞ þ dðhÞ þ eðhÞ þ f ðhÞ þ 2 cosð4/1 ÞðaðhÞdðhÞ þ bðhÞcðhÞÞ 2 cosð2ð/1 þ /2 ÞÞðaðhÞeðhÞ þ cðhÞf ðhÞÞ 2 cosð2ð/1 /2 ÞÞðbðhÞf ðhÞ þ dðhÞeðhÞÞ 2 cosð2/1 ÞðaðhÞf ðhÞ þ bðhÞeðhÞ þ cðhÞeðhÞ þ dðhÞf ðhÞÞ þ 2 cosð2/2 ÞðaðhÞbðhÞ þ cðhÞdðhÞ þ eðhÞf ðhÞÞ þ 2aðhÞcðhÞ cosð2UÞ þ 2bðhÞdðhÞ cosð2WÞ with the following coefficients: sinðWÞ W ð2Þ sinðUÞ bðhÞ ¼ rfa2 veff1 U 2 ð2Þ sinðWÞ cðhÞ ¼ rfa2 ðrfs1 Þ veff2 W sinðUÞ 2 ð2Þ dðhÞ ¼ ðrfs1 Þ veff2 U ð2Þ
aðhÞ ¼ veff1
ðA:1Þ
194
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195
sinð/2 Þ /2 sinð/ ð2Þ 2Þ f ðhÞ ¼ rfs1 veff3 /2 ð2Þ
eðhÞ ¼ rfa2 rfs1 veff3
ðA:2Þ
where ð2Þ
veff1 ¼ e^2 vð2Þ e^1 e^1 ;
ð2Þ
veff2 ¼ e^2 vð2Þ e^r1 e^r1 ;
ð2Þ
veff3 ¼ e^2 vð2Þ e^1 e^r1
In the s–s case, the Fresnel transmission and reflection coefficients are given by 2 cosðhÞ 2n2y cosðh2 Þ ; tfs2 ¼ taf 1 ¼ n1y cosðh1 Þ þ cosðhÞ n2y cosðh2 Þ þ n2s cosðh2s Þ n1y cosðh1 Þ cosðhÞ n2y cosðh2 Þ cosðhÞ ; rfa2 ¼ rfa1 ¼ n1y cosðh1 Þ þ cosðhÞ n2y cosðh2 Þ þ cosðhÞ n1y cosðh1 Þ n1s cosðh1s Þ n2y cosðh2 Þ n2s cosðh2s Þ ; rfs2 ¼ rfs1 ¼ n1y cosðh1 Þ þ n1s cosðh1s Þ n2y cosðh2 Þ þ n2s cosðh2s Þ
ðA:3Þ
ðA:4Þ
Appendix B. Case of a s polarized fundamental wave and a p polarized transmitted second harmonic wave In the case of a s polarized fundamental wave and of a p polarized transmitted second harmonic wave, the intensity of the transmitted second harmonic wave is given by the same relation as in the p–p case, with the following substitutions: ðB:1Þ e^1 ¼ e^r1 ¼ ð0; 1; 0Þ and n1(h1) = n1y, which gives c1 ¼ 0
ðB:2Þ
These coefficients are also changed: 2 cosðhÞ n1y cosðh1 Þ þ cosðhÞ n1y cosðh1 Þ cosðhÞ rfa1 ¼ n1y cosðh1 Þ þ cosðhÞ n1y cosðh1 Þ n1s cosðh1s Þ rfs1 ¼ n1y cosðh1 Þ þ n1s cosðh1s Þ taf 1 ¼
ðB:3Þ
All the other coefficients are unchanged and similar to the p–p case. Appendix C. Case of a p polarized fundamental wave and a s polarized transmitted second harmonic wave In this case, the intensity of the transmitted second harmonic wave is given by the same relation as in the s–s case, with the following substitutions: e^1 ¼ ðcosðh1 c1 Þ; 0; sinðh1 c1 ÞÞe^r1 ð cosðh1 c1 Þ; 0; sinðh1 c1 ÞÞ
ðC:1Þ
and n1y = n1(h1), therefore: 2 cosðhÞ cosðh1 c1 Þ þ n1 ðh1 Þ cosðc1 Þ cosðhÞ cosðh1 c1 Þ n1 ðh1 Þ cosðc1 Þ cosðhÞ rfa1 ¼ cosðh1 c1 Þ þ n1 ðh1 Þ cosðc1 Þ cosðhÞ n1s cosðh1 c1 Þ n1 ðh1 Þ cosðc1 Þ cosðh1s Þ rfs1 ¼ n1s cosðh1 c1 Þ þ n1 ðh1 Þ cosðc1 Þ cosðh1s Þ
taf 1 ¼
All the other coefficients are unchanged and similar to the s–s case. References [1] J. Jerphagnon, S.K. Kurtz, Journal of Applied Physics 41 (4) (1970) 1667. [2] P.D. Maker, R.W. Terhune, M. Nisenoff, C.M. Savage, Physical Review Letters 8 (1) (1962) 21.
ðC:2Þ
G. Tellier, C. Boisrobert / Optics Communications 279 (2007) 183–195 [3] [4] [5] [6] [7] [8] [9] [10]
N. Bloembergen, P.S. Pershan, Physical Review 128 (2) (1962) 606. N. Okamoto, Y. Hirano, O. Sugihara, Journal of the Optical Society of America B 9 (1992) 2083. R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, R. Ito, Japanese Journal of Applied Physics 27 (1988) L1134. D. Neher, A. Wolf, C. Bubeck, G. Wegner, Chemical Physics Letters 163 (1989) 116. F. Krausz, E. Wintner, Physical Review B 39 (1989) 3701. W.N. Herman, L.M. Hayden, Journal of the Optical Society of America B 12 (1995) 416. I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, R. Ito, Journal of Optical Society of America B 14 (1997) 2268. D.N. Nikogosyan, Properties of Optical and Laser Related Materials; A Handbook, John Wiley, 1997.
195