- #1
snickersnee
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[Wasn't sure if each problem needed a separate post. Please feel free to edit if needed.]
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/cm2. 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 w2b. 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?
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/cm2. 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 w2b. 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?