Resonator 1D

Introduction

A resonator is a device that can store wave energy. The waves have an amplitude and frequency. If you try to input a wave of arbitrary frequency, then it will be almost totally reflected. If part of the wave is not reflected, its amplitude will quickly be attenuated to zero. The waves can be electromagnetic with frequency all the way from radio frequency to x-ray frequency. Or they can be acoustic (sound or vibrational) waves of any frequency. Here we will discuss waves like those of visible light which are usually considered to have an index of refraction. Resonators for visible light are part of a laser device. An example would be a solid state laser diode which is a staple of our modern technology. Here we have aa transparent material (called a slab since this is a one dimension resonator) partially filling the resonator. Our variables will be the material's index of refraction and its thickness.

Here we have a dielectric slab of thickness, `t_s`, of material centered between perfectly reflecting mirrors at both ends of a linear 1D resonator of width `L` (see Canvas 0).

The Maxwell differential equation for the electric field, `vecE` is: (See Appendix):

`-1/n^2(del^2E(x))/(delx^2)=(k_r)^2E(x)`

where `n` is the index of refraction of the slab and `k_r` is the effective wave vector of the resonator. Obviously `E(x)`, when doubly differentiated, is a self repeating function. The only example of a function like this is the exponential. And since `n` is real, the real versions of the exponential are the sine and cosine. The cosine will be symmetric about the center of the resonator and the sine will be antisymmetric.

The Constants of the Resonator's Maxwell Equation

From the above discussion we may conclude the following: The expressions for `E(x)` in the resonator are: Inside the slab:

`E_s(x)=Scos(k_sx)\ \ \ symmetric`
`E_s(x)=Ssin(k_sx)\ \ \ antisymmetric`

where `k_s=(2*pi)/lambda_s` and `lambda_s` is the wavelength in the slab. Inside the vacuum:

`E_v(x)=Vcos(k_vx) \ \ \ symmetric`
`E_v(x)=Vsin(k_vx) \ \ \ antisymmetric`

where `k_v=(2*pi)/lambda_v` and `lambda_v` is the wavelength in the vacuum. Also we have the boundary condition:

`Vcos(k_vL/2)=0 \ \ \ symmetric `
`Vsin(k_vL/2)=0 \ \ \ antisymmetric`

where `L` is the full length of the resonator. We must now find the relation between `k_v` and `k_s`. We can put our proposed solutions into the Maxwell wave equation and that involves taking their second derivative:

`-(d^2E_s)/(dx^2)=k_s^2E_s` `-(d^2E_v)/(dx^2)=k_v^2E_v`

Repeating the wave equation we have:

`-1/n^2(d^2E(x))/(dx^2)=k_r^2E(x)`

Then

`k_s^2/n^2-k_r^2=0` `k_v^-k_r^2=0`

which just results in the following.

`k_s=nk_r`
`k_v=k_r`

Solving for the Constants

We are interested in solutions for `k_r` where there are a given number of half cycles, `m`, of our sinusoids over the entire width `L`, of the resonator.

`(n_s)*k_rt_s+n_vk_r(L-t_s)=(2m+1)pi \ \ \ symmetric`
`(n_s)*k_rt_s+n_vk_r(L-t_s)=(2m)pi \ \ \ antisymmetric`

where `n_s` is the index of refractiion of the slab, `n_v` is the index ot vacuum and `t_s` is the thickness of the slab. Solving for k_r we get:

`kr=(2m+1)pi/((n_s-1)t_s+L) \ \ \ symmetric`
`kr=(2m)pi/((n_s-1)t_s+L) \ \ \ antisymmetric`

`m` is an integer and is called the mode number. For the Maxwell equation when we have discontinuous values of `n_s` it is impossible to satisty both continuous `E` value and continuous `(dE)/dx` values at the slab boundaries. Therefore we must resort to the use of a graded `n(x)` value. Here that will be called a "swaged pulse" (see Canvas 1 black plot). The swaged pulse has condinuous `n(x)` and continuous `(dn)/dx` and results in continuous `E(x)` and continuous `(dE)/dx`.

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Appendix 1 (Maxwell Wave Equation)

Maxwell Equations

Maxwell introduced the displacement field `vec D` where `vec D=epsilon vec E` and `epsilon` is called the dielcctric constant. The value of `vec D` and `epsilon` depend on the density of dipoles in the medium where the electric field exists. The medium's dipoles tend to cancel out any electric field and therefore the medium is called a dielectric. Strangely, even vacuum behaves like it has a significant amount of polarizability so the vacuum dielectric constant is called `epsilon_0` and has value `8.85x10^-12` Farads per meter. The product `1/sqrt(mu_0epsilon_0)=c` where `c` is the speed of light in vacuum.

The Wave Equations

If there are no "real" currents, then the magnetic intensity `vec H` can be written:

`grad xx vec H=(delD)/(delt)\ \ \ \ (1)`

Also, if there are no magnetic materials in proximity:

`grad xx vec E=-(delB)/(delt) \ \ \ (2)`

where `vec B=mu_0vecH` and `mu_0` is the magnetic permeability of vacuum. Takine the curl of both sides of equation 2 we get:

`grad xx grad xx vec E=-mu_0(del)/(delt)(gradxxvecH) \ \ \ (3)`

Using equation 1 in equation 3 we have:

`grad xx grad xx vec E=-mu_0(del^2)/(delt^2)(D)=-mu_0epsilon(del^2)/(delt^2)(vecE) \ \ \ (4)`

The only vector variable in equation 4 is `vec E`. Equation 4 is called the wave equation for the `vecE`.

Further Simplification of the Wave Equation

An identity for `grad xx grad xx vec E` is:

`grad xx grad xx vec E=-grad * gradE\ \ \ \ (5)`

Then:

`grad * gradE=mu_0epsilon(del^2)/(delt^2)(vecE) \ \ \ (6)`

Since `epsilon` may be a function of `x` we will multiply both sides of the equation by `epsilon_0/epsilon` and get:

`epsilon_0/epsilon grad * gradE=mu_0epsilon_0(del^2)/(delt^2)(vecE) \ \ \ (7)`

A common name for `epsilon/epsilon_0` is the square of the index of refraction, `n^2`, and we already gave the name for `1/(mu_0epsilon_0)` as `c^2` so we use these symbols in equation 7:

`1/n^2 grad * gradE=1/c^2(del^2)/(delt^2)(vecE) \ \ \ (8)`

Also, since we are working with waves we can expect that `vecE` will have periodic time variation at frequency `omega` for example `vec E(t)=E_0cos omegat` so a further simplification of equation 8 becomes:

`1/n^2 grad * gradE=-(omega^2)/c^2(vecE) \ \ \ (9)`

In a resonator with perfect nodes at each end, only certain values of `omega` called the eigenfrequencies will be present. Since `omega/c` has units of a wave vector we will call this quantity `k_r=omega/c` so equation 9 becomes:

`1/(n(x)^2) grad * gradE=-k_r^2(vecE) \ \ \ (10)`

To find the modes of a resonator, we will solve equation 10 for `k_r^2` and `E(x)`.

Appendix 2: (Solving the finite difference matrix for eigenvectors)

Here we start with the finite difference equation as well a valid eigenvalue of that equation and iteratively solve for the eigenvector associated with that eigenvalue.

`a_i=m[i][i"+/"-1]` is an off diagonal element and `d_i=m[i][i]` where `m` is the tridiagonal finite difference equation matrix.

`((d_1-lambda,a_1,0,0),(a_2,d_2-lambda,a_2,0),(0,a_3,d_3-lambda,a_3),(0,0,a_4,d_4-lambda))((v_1),(v_2),(v_3),(v_4))=((0),(0),(0),(0))`

`(((d_1-lambda)v_1+a_1v_2),(a_2v_1+(d_2-lambda)v_2+a_2v_3), (a_3v_2+(d_3-lambda)v_3+a_3v_4),(a_4v_3+(d_4-lambda)v_4))=((0),(0),(0),(0))`

Let `v_1=b` where `b` is the left hand bouldary value. In order to have non-zero eigenvector elements, b cannot be zero but it can be made very small and the rest of the elements will be scaled to it. Then: First row:

`v_2=-((d_1-lambda)v_1)/a_1`

Second row:

`v_3=-(a_(2)v_2+(d_(i-1)-lambda)v_1)/a_2`

For all but the first and last rows:

`v_i=-(a_(i-1)v_(i-2)+(d_(i-1)-lambda)v_(i-1))/a_(i-1)`

Last or Nth row:

`v_N=-(d_N-lambda)*v_(N-1)/a_(N-1)`

So the algorithm is as follows:

`v_1=1;`
`v_2=-((d_1-lambda)v_1)/a_1;`
`"for"(i=3;i"<="N;i"++"")`
`v_i=-(a_(i-1)*v_(i-2)+d_(i-1)*v_(i-1))/a_(i-1)`
`v[N]=-(a_N-1)v_(N-1)")/a_N;`

Modifications for `a_(n,n-1) != a_(m,n+1)`

`((d_11-lambda,a_12,0,0),(a_21,d_22-lambda,a_23,0),(0,a_32,d_33-lambda,a_34),(0,0,a_43,d_4-lambda))((v_1),(v_2),(v_3),(v_4))=((0),(0),(0),(0))`

`(((d_11-lambda)v_1+a_12v_2),(a_21v_1+(d_22-lambda)v_2+a_23v_3), (a_32v_2+(d_33-lambda)v_3+a_34v_4),(a41v_3+(d_44-lambda)v_4))=((0),(0),(0),(0))`