Solving rate equations by using ode45.

Hi Guys

Iam currently working on two-photon confocal microscopy. But iam having a hard time solving the absorption rate equations for both one-and two-photon absorption of a gaussian beam on a x,z-grid for specified times after excitation. It be should relatively easy using ode45, but i just can't get it to work properly. The problem are without a dout related to my poor matlab experience. I hope someone can help :)

The rate equations are as follows:

dI_OPA/dz=-alpha*(1-R)*I_OPA (also known as Lambert-Beers law)

dn_OPA/dt=(alpha/(hbar*omega_OPA))*I_OPA (Number of excitations per time)

dI_TPA/dz=-beta*(1-R)*I_TPA^2

dn_TPA/dt=(beta/(2*hbar*omega_TPA))*I_TPA^2

I have solved the first equation for dI_OPA/dz yielding the following picture with the following code. _________________________________________________________________

%% Calculation of Gaussian, initial intensity value %%

% Beam radius of approximated Gaussian laser beam in GaN at 400nm.

wz_400=w0_400.*(1+((lambda_400.*zv)./(pi.*w0_400*n400_GaN)).^2);

% Calculation of Gaussian in xz-plane at 400 nm.

Ixz_400=((2.*P_ex)./(pi.*wz_400.^2)).*exp((-2.*(xv-xv(1)).^2)./(wz_400.^2));

[z_OPA I_OPA]=ode45(@IOPA_difeq,zv,Ixz_400); % One photon absorption

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function df = IOPA_difeq(z,I_OPA)

%Constants

alpha_GaN=3.5*10^3.*(1./10^-2); % [1/m] One photon absorption, Muth el at. 1997.

n400_GaN=2.56; % Refractive index at 400 nm excitation.

n_air=1; % Refractive index of air.

R_400=(n_air-n400_GaN)./(n_air+n400_GaN); % Fresnel reflection coefficient at 400 nm.

% Differential equation for depth dependent intensity.

df=-alpha_GaN.*(1-R_400).*I_OPA;

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But how due i code the second function of dn_OPA/dt to be solved with ode45 in which the solution of I_OPA are used as values for a specific set of times (tspan) ?

Best regards SMJ

Answer 1

[Torsten]

Use "dsolve" instead of "ode45". Your complete system has a simple analytical solution.

Best wishes

Torsten.