# How can I make dynamically changing functions?

I am trying to do transfer matrix calculations (optics), where I have a layered sample and want to calculate how light propagates through it, depending on some parameters:

The charm of the method is that you can do a matrix multiplication for every subsequent layer, no matter how many layers you have; you just need a transition matrix T between every layer and a propagation matrix P for passing through the layer, which can be easily defined. An exemplary transfer matrix of a single layer with air on both sides would look like this:
M[some properties] := T[air properties,layer properties].P[layer properties].T[layer properties,air properties]

What I want to do now is a script, where you can give a number of layers such as n=3 and the transfer matrix will be automatically generated (of course you have to give a list of parameters as well). I thought of doing this recursively with a for-loop, such as:

For[i=1, i<n+1, i++, M[..] := M[..].P[properties i].T[properties i, properties i+1]];


but this does not seem to work, probably because I am declaring a function M and then overwrite it (M is initialized as a unity matrix before this function).

The thing is, for a higher number of layers the transition matrix becomes a bit unhandy, and even more so if you want to edit it manually. I was hoping to generate this function dynamically from the input of n and a list of material properties and then store it in a file, to be able to transfer it to a different script later.

Is there a way to do this? Where did I go wrong? I know that := is used to delay the calculation of the value (effectively making the thing left of it a function) and that does not seem suited for recursive use... but I also tried different variations and nothing worked. Also, the input parameters of the function in the brackets M[...] will probably be a problem, because it will be more parameters for each layer. Can I bridge this? I know that I can input anything into a function but I need to name it, because I need the values at specific points of the function.

Somehow I can't believe that Mathematica does not support the dynamic generation of functions depending on input, it seems like such an obvious thing to me. I tried to search for solutions but the name of the Dynamic[] function interferes with my search and I couldn't think of a better word to describe the type of function that I want.

EDIT: so here is some real code:

P[K_, e_, Q_, d_] := ...
T[K_, ea_, eb_, Q_] := ...

M[K_,Q_,e1_,d1_,e2_] := T[K,e0,e1[K],Q].P[K,e1[K],Q,d1].T[K,e1[K],e2[K],Q];

M[K_,Q_,e1_,d1_,e2_,d2_,e3_] := T[K,e0,e1[K],Q].P[K,e1[K],Q,d1].T[K,e1[K],e2[K],Q].P[K,e2[K],Q,d2].T[K,e2[K],e3[K],Q];

M[K_,Q_,e1_,d1_,e2_,d2_,e3_,d3_,e4_] := T[K,e0,e1[K],Q].P[K,e1[K],Q,d1].T[K,e1[K],e2[K],Q].P[K,e2[K],Q,d2].T[K,e2[K],e3[K],Q].P[K,e3[K],Q,d3].T[K,e3[K],e4[K],Q];


these three examples are:

1. air - layer 1 - substrate
2. air - layer 1 - layer 2 - substrate
3. air - layer 1 - layer 2 - layer 3 - substrate

EDIT 2:
I tried it this way now:

P[K_, e_, Q_, d_] := ...;
T[K_, ea_, eb_, Q_] := ...;

M[K_, Q_, e_, d_, m_] :=  T[K, 1, e[[1, K]], Q].For[i = 1, i < m + 1, i++, P[K, e[i, K], Q, d[i]].T[K, e[i, K], e[i + 1, K], Q].];

e = {e1, e2, e3, e4};
d = {d1, d2, d3};
n = 3;

M[Kvalue, Qvalue, e, d, n]



So if only this Matrix multiplication would work with a loop, I would be fine :/ but unfortunately, it doesn't seem to work this way.

Any help would be greatly appreciated!
Best,
Julian

• Why couldn't you use Piecewise[]? May 27, 2020 at 11:48
• I don't really see how I could use Piecewise[] to get the desired result. Would you test for n and then return a different function for each n? If so, I would have to put in all options manually as far as I can see. Also, can I put a function definition with arguments in there? I didn't find an example of that in the documentation... May 27, 2020 at 13:03
• Could you give us a complete example for, say, two layers? at the moment, the question is a bit fuzzy and, although I understand your physical problem in general, I do not know exactly what you need to do in Mathematica so I can't help much. May 27, 2020 at 14:52
• You are trying to modify a function so it generates different matrices based on the layer correct? Couldn't you just write a second function that calls M with the different parameters for each layer? Also, are you aware of Function? May 27, 2020 at 15:02
• @MarcoB I edited the post to include real code (even thought I changed some things, it should represent it adequately). May 27, 2020 at 17:48

You don't need to modify functions, you just need to define the e[i, K] and d[i, K] and call this:

M[K_, Q_, n_] :=
Fold[Dot, T[K, e0, e[1, K], Q],
P[K, e[#, K], Q, d[#, K]].T[K, e[#, K], e[# + 1, K], Q] & /@ Range[n]]

M[K, Q, 1]
(* returns T[K, e0, e[1, K], Q].P[K, e[1, K], Q, d[1, K]].T[K, e[1, K], e[2, K], Q] *)

M[K, Q, 2]
(* returns T[K, e0, e[1, K], Q].P[K, e[1, K], Q, d[1, K]].T[K, e[1, K], e[2, K],
Q].P[K, e[2, K], Q, d[2, K]].T[K, e[2, K], e[3, K], Q] *)

M[K, Q, 3]
(* returns T[K, e0, e[1, K], Q].P[K, e[1, K], Q, d[1, K]].T[K, e[1, K], e[2, K],
Q].P[K, e[2, K], Q, d[2, K]].T[K, e[2, K], e[3, K], Q].P[K, e[3, K],
Q, d[3, K]].T[K, e[3, K], e[4, K], Q] *)

• I didn't know you could use Dot like this... that looks very elegant and exactly what I wanted to do! I will try it out tomorrow and let you know if this works like I want it to. Thanks! May 27, 2020 at 20:12
• Did this work for you and is my answer acceptable? May 31, 2020 at 21:16
• Sorry, as I simplified the problem a bit for this question I needed some time to implement it properly. It does exactly what I want it to, thanks! Jun 3, 2020 at 21:32