Gaussian to Flat-top

Zemax Knowledge Base gives a design method to convert Gaussian irradiance profile to flat top profile (see reference [1]).
This post (A) gives another analytic form and (B) examines the 1-D situation.

A.
If the input Gaussian profile has the 1/e² radius of W, the output flat-top has the half-width of K, then the Zemax KB article gives:
        S = K [1-exp(-2X²/W²)]^1/2                                              (1)
This is a ray mapping formula to get the output coordinate value S for every input coordinate X.

For Zemax, this formula can be written in another form. Let R_EP be the radius of the entrance pupil, then

X W

=

X/REP W/REP

= XNA1/2

(2)

where X_N is the normalized coordinate and A is the system apodization factor. Eq(1) can therefore be written as
        S = K [1-exp(-2X_N²A)]^1/2                                              (3)
This formula is better suited for Zemax because both X_N and A are the Zemax direct parameters. Using this formula, the macro file for generating merit function can be simplified.

B. How about 1-D?
Very often people need a 1-D flat-top, e.g., flat-top on X-axis and Gaussian on Y-axis. In this case, the irradiance integration is different.

For input Gaussian profile, the contained power in a 1-D variable slit is:

Pi(X) = Ii ∫

X -X

exp(-2x2/Wx2)dx

∫

+∞ -∞

exp(-2y2/Wy2)dy

(4)

where I_i is the peak irradiance, W_x and W_y are the 1/e² beam radii. For output profile, the contained power in the 1-D variable slit is:

Po(S) = Io ∫

S -S

dξ

∫

+∞ -∞

exp(-2η2/Wη2)dη

(when S<=Wξ)

(5)

Po(S) = Io2Wξ

∫

+∞ -∞

exp(-2η2/Wη2)dη

(when S>Wξ)

(5)

where I_o is the peak irradiance, W_ξ is the half-width of the flat-top and W_η is the 1/e² half-width of the Gaussian profile.

First, let P_i(X=∞) = P_o(S=∞), the conservation of total power gives:

Io Ii

=

π1/2WxWy 21/22WξWη

(6)

To get the ray-mapping relation, let P_i(X) = P_o(S) and plug in Eq.(6):
        S = W_ξ erf (2^1/2X/W_x) = W_ξ erf (2^1/2X_NA^1/2).                                              (7)
where erf is error function. So for 2D and 1D flat-top generation, the ray-mapping formula is different.

To write a macro for generating merit function for 1-D flat-top, one can use
       GETSYSTEMDATA 1
       Apodization = VEC1(2)
to read the apodization factor.

To caculate erf function, Zemax does not have it built-in. One can use an approximation from wikipedia [2].

References:
[1] http://kb-en.radiantzemax.com/Knowledgebase/How-to-Design-a-Gaussian-to-Top-Hat-Beam-Shaper
[2] http://en.wikipedia.org/wiki/Error_function