Exploring Second Harmonic Generation in Non-Linear Optics

This paper focuses on the process of second harmonic generation in non-linear optics in materials having second-order non-linearity, i.e, χ(2) 6= 0.It includes the theory involved in the process of conversion of ω → 2ω, calculation of efficiency. I conclude with some applications of this process which has practical importance.

Introduction

Second-Harmonic Generation (SHG) is a non-linear process in which a pump signal of frequency ω , when incident on a non-linear material having second order non-linearity (χ(2) 6= 0), generates an output wave of frequency 2ω.

This is only possible if the incident electric field is so high that the higher order terms in the polarization can- not be neglected.

The polarization is not only propor- tional to the first order of the incident field, but also on the second, third and other higher orders and accord- ingly we have second harmonic generation(SHG), third harmonic generation(THG) and so on.

P = 0χ

(1)E + 0χ

(2)E2 + 0χ

(3)E3 + ........ (1)

where P represents the polarization, E being the electric field, 0 being the absolute permittivity and χ(1),χ(2),χ(3) are the first order,second order and third order non-linear susceptibilities respectively.

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Ordinary light cannot produce such non-linear effects since in ordinary light, the strength of the incident field is not so high to induce these effects.We need highly co- herent and focussed laser beam for this ; since in laser, the incident electric field is high enough and hence,the higher orders of the field can no longer be neglected. Only non-centrosymmetric crystals (that do not pos- sess a centre of inversion) exhibit SHG.

Theoretical Framework

The incident electric field is given by :

E = E0 cos(kz − ωt) (2)

where E0 represents the amplitude of the incident field, k being the propagation vector along z axis, z represents in the length of the non-linear crystal, and ω being the in cident frequency.

PNL = 0χ

(2)E20/2 + 0χ

(2)E20 cos 2(kz − ωt)/2 (3)

So,the non-linear polarization has two terms - the 1st term is independent of frequency and is called the Op- tical Rectification Term.The second term is the term corresponding to the frequency 2ω. This shows that even if we start with a frequency ω at the input, it is possible to generate a frequency of 2ω at the output due to the non-linearities of the material.

The total electric field inside the crystal is given by:

E = E1 cos(k1z − ωt) + E2 cos(k2z − 2ωt) (4)

where k1 = ωn1/c and k2 = ωn2/c represents the prop- agation vectors corresponding to frequencies ω and 2ω respectively.

n1 ⇒ refractive index (RI) of the surrounding material

n2 ⇒ refractive index (RI) of the non-linear crystal

Eω ⇒ Electric field corresponding to frequency ω

E2ω ⇒ Electric field corresponding to frequency 2ω

E(ω) = E01 cos(k1z − ωt+ θ1) (5)

E(2ω) = E02 cos(k2z − 2ωt+ θ2) (6)

where E01 and E2 are real quantities.

PNL = 0χ

E(ω) = E1 exp{i(k1z − ωt)}/2 + cc (8)

E(2ω) = E2 exp{i(k2z − 2ωt)}/2 + cc (9)

Putting (8) and (9) in (7),

(2)E21) exp{2i(k1z − ωt)}+ cc (11)

From Maxwell’s equation,

∇2E − µ0(∂2E/∂t2) = µ0(∂2PNL/∂t2) (12)

Putting (8) ,(9) ,(10)and (11) in (12), we get two differential equations corresponding to the fields E1 and E2.

Assumption: The conversion efficiency of ω → 2ω is low so that for a strong incident electric field, E1 ≈ constant.

E2(z) = (iωz/2cn2)χ

(2)E21(sinx/x) exp{−ikz/2} (15)

where x = kz/2

Calculation of Efficiency

Power is given by :

P = n|E(z)|2S/2cµ0 (16)

S ⇒ surface area

Efficiency of the conversion of ω → 2ω is given by :

e = ω2χ(2)

E21(sinx/x)

2/c2n1n2 (17)

The efficiency is maximum for x = 0 =⇒ k = 0 (since z 6= 0).

The maximum value of the sinc function is 1 at k2 = 2k1 =⇒ n1 = n2

This criteria is called the Phase Matching Criteria and the material should be engineered such that this condition is satisfied so that the efficiency of conversion of ω → 2ω is maximum.

Conclusion

SHG is used to study the surface properties of opti- cal materials since at the surface, inversion symmetry is broken and hence, SHG occurs at the surface of the materials. The intensity of the SHG largely depends on the behaviour of the materials at the surface, like the presence of impurities on the surface. The principle of SHG is also used in non-linear optical microscopy to increase the resolution since these nonlinear processes like SHG occurs at the centre of highly focused laser beam, the centre being the region with max- imum intensity. SHG has wide range of applications in biophysics. For example-it is used to determine the orientation of proteins. This concept is an emerging one and has huge appli- cations in modern day world.

Acknowledgements

Gratitude is extended to Prof. Amit Rai for his guidance on this topic and Prof. Sasmita Mishra for the opportunity to delve into the fascinating world of non-linear optics.