Dear David,

Thank you very much for your explanation and the attached files. They helped me a lot to design what I wanted.

Best regards,

Astghik

Let’s take it a step at a time.

First, a few words about the direction cosines: l, m, n. If we think about the direction as a vector in 3-space, (l, m, n) are the cosines of the included angles between our direction vector and the three Cartesian coordinate axes: (x, y, z) respectively. Since we are only concerned with direction, we can consider the magnitude of the vector to be 1. Then the direction cosines are also the projections of our unit magnitude direction vector on the coordinate axes. This means that the direction cosines (l,m,n) are in fact the cartesian coordinates of the unit vector representing the given direction. So the square magnitude of the vector is l^2 + m^2 + n^2 which we have said is 1. So is l^2 + m^2 + n^2 = 1 is an identity.  Notice that while we describe our direction in terms of (l, m, n), the identity can be used to express any of the three components in terms of the other two. So we really need only two components to define a direction. We could think of the two components as a polar angle with respect to the z coordinate axis and an azimuthal angle.

The formula OpticStudio uses for intensity is

   I = I0 Exp[-Gx l^2 - Gy m^2], where Gx and Gy are different parameters.

The formula does not include the direction cosine with respect to the z-axis, which is n. However, we could use the identity given earlier to express this in terms of n and either l or m. So this is not necessarily radially symmetric about z. For that, the intensity needs to be a function of n alone, since that is the cosine of the angle with respect to z.

But look at what happens when Gx = Gy. Then let Gx = Gy = G and rewrite the expression for intensity:

   I = I0 Exp[-G l^2 - G m^2] = I0 Exp[-G (l^2 + m^2)]

We rewrite the identity is l^2 + m^2 + n^2 = 1 as l^2 + m^2 = 1 - n^2 and substitute:

   I = I0 Exp[-G (1-n^2)]

Now the intensity is only a function of the angle subtended between the given direction and the z axis.

We can also see that the intensity falls to 1/e^2 when G (1-n^2) = 2, where n^2 = Cos^2[theta] with theta being the angle between the given direction and the z axis.

Also note that since Sin^2 + Cos^2 =1:

   1 – n^2 = 1 – Cos^2 = Sin^2

So the intensity falls to 1/e^2 when G Sin^2[theta] = 2. For small theta, Sin(theta) is approximately theta so the intensity falls to 1/e^2 when G theta^2 = 2. ( theta in radians )