- #1

snickersnee

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Also \~ and \^ are tilde and hat respectively.

**1a. Homework Statement**

Use perturbation theory to derive the 3rd order nonlinear susceptibility[itex] \chi^{(3)}(3w;w,w,w)[/itex] (problem gives potential energy, etc. but I already know what I have to do, I just need help calculating it)

**1b. Relevant equations**

The equation I need to solve is this: [itex]\ddot{\tilde{x}}^{(3)}+2\gamma \dot{\tilde{x}}^{(3)} + w_0^2 \tilde{x}^{(3)} +2a\~x^{(1)}\~x^{(2)}=0[/itex]

I need to solve it for [itex]\~x^{(3)}[/itex].

The first-order and second-order solutions were given in lecture. I plugged them into the equation, in the 4th term on the left hand side, getting some horrendous expression.

We are looking for solutions of the form (*) [itex]\~x^{(3)}(t)=x^{(3)}(3w)e^{-3iwt}[/itex] (is that the right form of the solution?) Since the differential equation has derivatives with respect to time, I guess I need to differentiate (*) twice, but is it only the exponential that depends on time? What about the x, and omega? Are those constants?

Also, E times its complex conjugate is |E|^2, right?

**1c. The attempt at a solution**

(explained in section 1b)

--------------------------------------

**2a. Homework Statement**

Nonlinear crystal has an EM wave propagating in x direction, linear polarization along [itex]1/\sqrt{2}(\^y+\^z)[/itex] direction, frequency w, intensity 1MW/cm

^{2}. The second order non-linear optical susceptibility tensor for second harmonic generation has only one nonzero component, [itex]\chi^{(2)}_{zzz}(2w,w,w)=10pm/V (10^{-11} m/V)[/itex]

- calculate amplitude and direction of nonlinear polarization at frequency 2w. (use Poynting vector to get E field, be careful with geometry and various factors of 2)

- calculate amplitude and direction of linear polarization P^(1) at frequency w

**2b. Relevant equations**

k vector is in x direction.

Poynting vector is [itex]\vec{S}=\vec{E} \times \vec{H}[/itex]. But I thought there was no magnetic field in optical materials at optical frequencies, so when taking the cross product wouldn't everything just go to 0? I wasn't given any magnetic field info. I guess the intensity is what I'd plug in for S.

I also know the formulas for second-order susceptibility and polarization: [itex]\chi^{(2)}(2w)=-\frac{a(e/m)^2 E^2}{D(2w)D^2(w)},\ \~P^{(2)}=\epsilon_0 \chi^{(2)}\~E^2(t)[/itex]

**2c. The attempt at a solution**

given in section 2b--------------------------------------

**3a. Homework Statement**

Write all elements of d_il matrix (3x6) for lithium niobate (crystal symmetry 3m)

(values given: d_33, d_31=d_15, and d_22.)

**3b. Relevant equations**

(from table 1.5.1, Boyd)

Form of the 2nd order susceptibility tensor. Each element denoted by Cartesian indices

For 3m crystal class: xzx=yzy, xxz=yyz, zxx=zyy,zzz,yyy=-yxx=-xxy=-xyx (mirror plane perpendicular to x^)

To convert d_ijk to d_il:

http://snag.gy/kIrzU.jpg

**3c. The attempt at a solution**

The answer is supposed to be this:

http://snag.gy/WlbCk.jpg

But I don't know how to get that from the Cartesian indices given above. maybe someone could please do the first one as an example, (xzx=yzy) and then I could figure out the rest?

Also, when it says "zxx=zyy,zzz,yyy=-yxx=-xxy=-xyx" is that one big equation? --------------------------------------

**4a. Homework Statement**

Calculate the d_eff value for second harmonic generation in a nonlinear crystal with only one nonlinear coefficient d_33=10pm/V. Input beam at frequency w incident along x-axis, polarized along [itex]1/\sqrt{2}(\^y+\^z)[/itex] and:

- (case a) emission at frequency 2w along x-axis and polarized along ^z direction

- (case b) polarized along ^y direction

- what is the physical meaning of the value of d_eff in case b?

**4b. Relevant equations**

For SHG, [itex]P(2w)=2\epsilon_0 d_{eff}E(w)^2[/itex], and

http://snag.gy/ALUI9.jpg

**4c. The attempt at a solution**

We're told that d_il has only one nonzero component (d_33) so the right hand side of the matrix equation reduces to [itex]2\epsilon_0 d_{33}E_z(w)^2[/itex], right? But it seems that P and E are both unknown, so how can we solve for d_eff?