How to calculate the solid angle for a pixel in angle space

  • 11 April 2022
  • 2 replies
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The definition of a solid angle can be found in the Wiki.

https://en.wikipedia.org/wiki/Solid_angle

From the Wiki, you may know a solid angle can be calculated from the following formula.

(Eq.1)

And the relationship between (θx,θy) and (θ,φ) is as shown in the following equation.

The variables in (Eq 1) are (θ,φ). We will convert them to (θx,θy) by the way of variable transformation.

https://math.libretexts.org/Courses/Monroe_Community_College/MTH_212_Calculus_III/Chapter_14%3A_Multiple_Integration/14.7%3A_Change_of_Variables_in_Multiple_Integrals_(Jacobians)

 

Finally we can get the result as below.

(Eq. 2)

(Eq. 2) can be used to calculate the solid angle from a specific pixel coordinate.

However, there is no analytical solution for this equation. We know that a numerical solution can be calculated.

In fact, our algorithm is not this method. When OpticStudio calculate the radiant intensity, a iteration method is used.

Theoretically, the result is the same. One of our users has verified this.


2 replies

@Yihua Hsiao If a Detector Rectangle has angles both spanning from -90 to +90 degrees and has only 1 pixel (1x1), how does equation 2 yield result of 2*pi?

@Ethan 

Regards,

Jiang

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Hi @JJiang  Thanks for your question. I also heard from Ethon about this question. I will reply you later.

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