# Angle determination of Sum-Frequency Generation in BBO

• Eoraptor
In summary, the individual is seeking help with determining the correct angles for a third-order autocorrelator using two BBO crystals in their setup. They have a laser and are trying to calculate the angles needed for SHG and SFG using equations for non-collinear and o-e-o configurations. They have tried using MATLAB and for loops to solve the equations, but have not had success yet. They are looking for guidance on how to properly embed the angles and solve the system of equations.
Eoraptor
Hi,

1. Homework Statement

I have difficulties to get the correct angles for a third-order autocorrelator and I hope you can help me.
In my setup are two BBO Crystalls, the first for SHG (Second-harmonic Generation) and the second for SFG (Sum-Frequency Generation).

The Laser is a ultrashort-pulse Ti:Sapphire (λ_1 around 800 nm), the BBO Crystals are 50 µm big.

The calculation for the first BBO (SHG) wasn't really easy (I didn't have a "Nonlinear Optics" class), but I needed only to search for one variable, Θ
For my parameters, I got around θ= 28,65° (as turning angle for the crystal)
The complete calculation is here: https://www.overleaf.com/2792476gxqntk

The SFG is Type-II, so non-collinear and I need to determine $$θ_1, θ_2$$ (from the incoming rays) and Θ, the optical axis of the crystal. All I got are two Equations, which is rather difficult (for me) to solve.

## Homework Equations

In general, the equations are:
$$ω_1+ω_2=ω_3$$
and
$$k_1 + k_2 = k_3$$

In the more specific case of SFG in o-e-o Configuration, these are the equations:
$$n_o(ω)\cdot sin(Θ_1) = n(θ+θ_2)\cdot sin(θ_2)$$
$$n_o(ω) \cdot cos(Θ_1) + n(θ+θ_2) \cdot cos(θ_2)=2\cdot n(Θ,2ω)$$

I have found two possible ways to calculate the angles:

The first is similar to the calculation of non-collinear SHG like in this picture from "The fundamentals of photonics"(Saleh&Teich,2007) which I wanted to transfer to SFG.
[CH-21]

Here we have $$θ_1, θ_2$$ and the crystal axis $$θ$$

Another attempt is, that there is $$θ_1, θ_2$$ and $$θ_3$$ as angle for each beam.

## The Attempt at a Solution

For the first way (like in the fundamentals of photonics) I calculated the following:
https://www.overleaf.com/2795897bqfyps

which results in
\begin{align}
n_1&=n_o(\omega) \\
n_2&=n_e=n(\Theta+\Theta_2,2\omega)=\sqrt{\left( \dfrac{\cos^2(\Theta+\Theta_2)}{n_o^2(2\omega)}+\dfrac{\sin^2(\Theta+\Theta_2)}{n_e^2(2\omega)} \right) ^{-1}} \\
n_3&=n_e=n(\Theta,3\omega)=\sqrt{\left( \dfrac{\cos^2(\Theta)}{n_o^2(3\omega)}+\dfrac{\sin^2(\Theta)}{n_e^2(3\omega)} \right) ^{-1}}
\end{align}

For the second way, I got the following results:

\begin{align}
n_1&=n_o(\omega) \\
n_2&=n_e=n(\Theta_2,2\omega)=\sqrt{\left( \dfrac{\cos^2(\Theta_2)}{n_o^2(2\omega)}+\dfrac{\sin^2(\Theta_2)}{n_e^2(2\omega)} \right) ^{-1}} \\
n_3&=n_e=n(\Theta_3,3\omega)=\sqrt{\left( \dfrac{\cos^2(\Theta_3)}{n_o^2(3\omega)}+\dfrac{\sin^2(\Theta_3)}{n_e^2(3\omega)} \right) ^{-1}}
\end{align}

My Question(s):Which is the correct way to embed these angles?

And I always get only two equations but three variables, so the system is undetermined. What are some tricks to solve such a system? I have access to MATLAB and tried my luck with fsolve(…) or made 3 nested for loops, to test between -90° and +90° for each angle if the equations are 0.

But I had no luck so far.

I would really appreciate your help.

Thanks,

Eoraptor

Eoraptor said:
In the more specific case of SFG in o-e-o Configuration, these are the equations:
$$n_o(ω)\sin(Θ_1)=n(θ+θ_2)⋅sin(θ_2)$$
$$n_o(ω)⋅cos(Θ_1)+n(θ+θ_2)⋅cos(θ_2)=2⋅n(Θ,2ω)$$​
These are the equations from the SHG exercise (from the book) and are not the equations for SFG

The SFG equations (my first Version) would be:
$$n_o(ω)\sin(Θ_1)=2n(θ+θ_2,2ω)⋅sin(θ_2)$$
$$n_o(ω)⋅cos(Θ_1)+2n(θ+θ_2,2ω)⋅cos(θ_2)=3⋅n(Θ,3ω)$$​

The equations under "

## The Attempt at a Solution

" should be correct.

## 1. What is Sum-Frequency Generation (SFG) in BBO?

SFG in BBO is a nonlinear optical process in which two input beams with different frequencies are combined in a birefringent crystal called beta-barium borate (BBO). This results in the generation of a new output beam with a frequency equal to the sum of the two input beams.

## 2. How is the angle of SFG in BBO determined?

The angle of SFG in BBO is determined by measuring the intensity of the output beam at different angles of incidence on the crystal surface. The angle at which the intensity is maximum is the angle of SFG. This process is known as angle tuning.

## 3. What factors affect the angle determination of SFG in BBO?

The angle determination of SFG in BBO can be affected by factors such as the polarization state of the input beams, the crystal thickness, and the quality of the crystal surface. The alignment of the input beams and the temperature of the crystal can also impact the angle determination.

## 4. What are the applications of angle determination of SFG in BBO?

The angle determination of SFG in BBO is an important technique in the field of surface science and spectroscopy. It is used to study the structure and composition of surfaces, as well as the interactions between molecules at interfaces. It is also used in the development of new materials and in the study of biological systems.

## 5. How is the angle of SFG in BBO related to the molecular orientation at surfaces?

The angle of SFG in BBO is directly related to the molecular orientation at surfaces. This is because the intensity of the output beam is dependent on the orientation of the molecules with respect to the crystal surface. By measuring the angle of SFG, we can determine the orientation of molecules at the surface, providing valuable information about the surface structure and properties.

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