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dnp33
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This is a problem from Boyd Nonlinear Optics chptr 1 problem 2.
Numerical estimate of nonlinear optical quantities. A laser beam of frequency ω carrying 1 W of power is focused to a spot size of 30μm diameter in a crystal having a refractive index of n =2 and a second order susceptibility of [itex]\chi^{(2)}=4\times 10^{-11} m/V[/itex]. Calculate numerically the amplitude P(2ω) of the component of the nonlinear polarization oscillating at frequency 2ω.
[tex]P(2\omega)=\epsilon_0^{(2)}E^2[/tex]
[tex]I=\frac{cn\epsilon_0}{2}E_0^2[/tex]
I solved for [itex]E_0[/itex] assuming a uniform distribution across the spot, with I=P/A and got
[tex] E_0=\sqrt{\frac{2P}{Acn\epsilon_0}}[/tex]
and put that into the equation I gave for [itex]P(2\omega)[/itex]. The value I got was [itex]1.89\times 10^{-11}[/itex], which is almost exactly 4 times the value given in the text of [itex]4.7\times 10^{-11}[/itex]
I feel like it's possible that the value given in the text accidentally uses the diameter of the spot to calculate the area, which would give them a factor of 1/4 that I don't have, but I also thought that maybe my problem lies in my assumption that the spot is uniform. Maybe I actually need to integrate numerically assuming a gaussian beam profile-which would kind of make sense considering the problem title (numerical estimate of nonlinear optical quantities).
Homework Statement
Numerical estimate of nonlinear optical quantities. A laser beam of frequency ω carrying 1 W of power is focused to a spot size of 30μm diameter in a crystal having a refractive index of n =2 and a second order susceptibility of [itex]\chi^{(2)}=4\times 10^{-11} m/V[/itex]. Calculate numerically the amplitude P(2ω) of the component of the nonlinear polarization oscillating at frequency 2ω.
Homework Equations
[tex]P(2\omega)=\epsilon_0^{(2)}E^2[/tex]
[tex]I=\frac{cn\epsilon_0}{2}E_0^2[/tex]
The Attempt at a Solution
I solved for [itex]E_0[/itex] assuming a uniform distribution across the spot, with I=P/A and got
[tex] E_0=\sqrt{\frac{2P}{Acn\epsilon_0}}[/tex]
and put that into the equation I gave for [itex]P(2\omega)[/itex]. The value I got was [itex]1.89\times 10^{-11}[/itex], which is almost exactly 4 times the value given in the text of [itex]4.7\times 10^{-11}[/itex]
I feel like it's possible that the value given in the text accidentally uses the diameter of the spot to calculate the area, which would give them a factor of 1/4 that I don't have, but I also thought that maybe my problem lies in my assumption that the spot is uniform. Maybe I actually need to integrate numerically assuming a gaussian beam profile-which would kind of make sense considering the problem title (numerical estimate of nonlinear optical quantities).