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In a transparent solid light mainly interacts with Electric dipoles. The electric field wave power of a large array of dipoles that have been activated by the incoming electric field radiation is calculated here. Here we calculate the power produced in the far field Vs angle from the center of the dielectric solid which is assumed to be small compared to the far field distance. We use two assumptions about the dipole array
1.Where the dipoles are uniformly spaced along the x axis one light wavelength apart. This is similar to a crystal where the molecules are spaced uniformly.
2.Where the dipoles are randomly spaced over a limited domain in the `xy` plane. This is similar to an amorphous solid where the molecules do not have uniform spacing.
In each case, the angular width of the dipole emission power becomes narrower as the number of diodes is increased. The angular width is similar to the angular behavior of stimulated emission in a laser. However, in the case of a laser, the angular width is determined by the curvatures of the mirrors which determines the diameter of the excited dipole density.`bb S(bb r)=(3P_o)/(8pir^2)bb hat rsin^2theta` (1)
where `r` is the observation distance from the dipole, `theta` is the angle between the observation direction and the dipole axis, and`P_o=(ck_o^4)/(12piepsilon_o)d_o^2` (2)
where `d_o=qd` where +/-q are the charges at the end of the dipole and d is the spacing between the two charges. and where `k_o=nomega/c=(2pi)/lambda` omega is the radian frequency of the dipole, `epsilon_o` is the dielectric constant of the solid, `c` is the speed of light in vacuum and the index of refraction `n=sqrt(epsilon)`. The electric field is proportional to the square root of `bb S(bb r)` so`E_d=sqrt(S(r)(mu_0/epsilon_o)` (3)
Combining equation 3 with equation 1 we obtain:`E_d=sqrt(mu_o/epsilon_o)sqrt((ck_o^4)/(32pi^2epsilon_or^2)d_o^2)sintheta` (4)
Simplifying the square root term we have`E_d=sqrt(mu_o/epsilon_o)(k_o^2 d_o)/(4pir)sqrt((c)/epsilon_o)sintheta` (5)
We will first sum the fields produced by the array of dipoles when the dipole groups are set one wavelength apart along the x axis and all oscillating at the incident wave frequency `omega`. In order to have a large `bb S` vector in the x direction, the phasing of the dipoles will be:`bb d_o(x,t)=bb hatz exp(iomegat-k_"o"x)` (6)
where `t` is time and `x` is the location of the dipole on the `x` axis. Now we will compute the radiation field at a long distance `bbr` from the center of a linear array of dipoles centered on the x axis. These dipoles have been excited as shown in the previous equation. Let us define an observation point,`(x,y,z)` relative to an individual dipole as`|bb r|^2=(x-x_m)^2+y^2+z^2` (7)
where `x_m` is the `x` position of the mth dipole on the x axis `x_m=[-mlambda,-(m-1)lambda...0...(m-1)lambda,mlambda]`. Since the distance 'x' is very long compared to the length of the array we can let the angle between the `x` axis and `bbr` will be approximated as`sinphi~sqrt((y^2+z^2)/(x^2+y^2+z^2))` (8)
At (x,y,z) the phase delay for the mth dipole radiation is`phi(x,y,z,x_m)=k_o(sqrt((x-x_m)^2+y^2+z^2)))` (9)
Note that if (y,z)=(0,0) then the single term in the phase delay sum is`phi(x,y,z,x_m)=k_o(x-x_m))` (10)
If (y,z)gt(0,0) then the sum over phases of the contributing dipoles becomes`E(r)=E_dsum_mk_o(sqrt((x-x_m)^2+y^2+z^2)))` (11)
which is a much smaller magnitude because of negative values associated with the (y,z) magnitudes. The Appendix shows that we can also obtain an algebraic result for this sum which arises from the standard algebra for the geometric series . We will plot the algebraic result as well as the series sum result. For `x` much greater than any `x_m` we can rewrite `E(r)` as:
`E(r)~E_dsum_mexpjk_o(sqrt(-2"x"x_m+x^2+y^2+z^2)))` (12)
`E(r)~E_dsum_mexpjk_o(rsqrt(1-(2"x"x_m)/r^2))` (13)
`E(r)~E_dsum_mexpjk_o(r(1-("x"x_m)/r^2))` (13a)
`E(r)~exp(jk_"o"r)E_dsum_mk_"o"x_m(x/r))` (13b)
`E(r)~E_dsum_mexpjk_o(x_mcosphi)exp(jk_"o"r)` (14)
`E(r)~E_dexp(jk_"o"r)sum_(i=-(N/2))^(i=(N/2))expj(2pii(cosphi))` (15)
`E(r)~E_dexp(jk_"o"r)sin(Npicosphi)/sin(picosphi)` (16)
For numerical calculation, when `cosphi=1` we have to increment the phase `picosphi` to `picosphi+10^(-8)` to avoid a divide by zero error. So when `cosphi!=1` since the phase of the terms are not then separated by `2pi` we can expect much smaller values for the sum. In other words, the emitted power will be centered on the x axis but be reduced at larger (y,z) values.Canvases 1 and 3 show the incoming electric wave vector as well as the dipoles. On Canvas 1 the dipoles are spaced at 1 wavelength intervals and centered on the x axis. On Canvas 3 the dipoles are randomly placed inside the dielectric slab. It is assumed that there is no permanent dipole moment in the dielectric slab so the dipoles rotate about their center. The dielectric slab is shown as gray and it has a thin antireflection coating (dark gray) at each end. This coating avoids the need to show reflected waves from either end which would greatly complicate the graphics. Then the electric field inside the slab is reduced by the factor `1/sqrt(epsilon)` (where epsilon is the slab's dielectric constant) with respect to the electric field in the vaccuum at each end of the slab. Canvases 2 and 4 show the Power Vs angle that is emitted at a far away observation point (x,y). The angle is relative to the x axis and is with respect to the center of the dipole array and is defined as
since the diode arrays are centered at `(x,y)=(0,0)`. Canvases 2 and 4 show both a linear plot Vs angle and a radial plot where the power maximum at small angles has small red circles and the more minimal power at large angles has larger blue circles. Both the linear and the radial plots are scaled relative to their maximum power rather than absolute power.The first slider controls the dielectric constant `epsilon`. You will notice that the wavelength inside the slab gets shorter as `epsilon` is increased since the index of refraction `n=sqrt(epsilon)` and slab wavelength is `lambda=lambda_("vacuum")/n`. As mentioned before, when `epsilon` is increased, the wave amplitude in the slab is decreased.
The second slider controls the number of uniformly spaced dipoles in Canvas 1. Increaseing the number of dipoles results in a narrower angular distribution of power which can be seen in the Canvas 2 Plots.
The third slider controls the number of randomly positioned dipoles in Canvas 3. Increaseing the number of dipoles results in a narrower angular distribution of power which can be seen in the Canvas 4 Plots.
We have the sum
`sum_(i=1)^(i=n) cos(ipsi)` A1
But this is just the real part of a similar expression:`Re[sum_(i=1)^(i=n) exp(ijpsi)]` A1
where `j=sqrt(-1)`.The geometric series sum for `r^i`:
`sum_(i=0)^nr^i=(1-r^n)/(1-r)` A2
`sum_(i=1)^nr^i=(1-r^n)/(1-r)-1=(r-r^(n+1))/(1-r)=1/r(1-r^(n+1))/(1-r)` A2
Thus our series sum is`Re[1/exp(jpsi)(1-exp((n+1)jpsi))/(1-exp(jpsi))]` A2
The magnitude of our denominator is
`|1-exp(jpsi)|=sqrt((1-cospsi)^2+sin^2psi)`
`=1-2cospsi+cos^2psi+sin^2psi=2-2cospsi=4sin^2(psi/2)`
A2
`|1/exp(jpsi)(1-exp((n+1)jpsi))/(1-exp(jpsi))|=1*sqrt((1-cos(n+1)psi)^2+sin^2(n+1)psi)`
`=1-2cos(n+1)psi+cos^2(n+1)psi+sin^2(n+1)psi=2-2cos(n+1)psi=4sin^2((n+1)psi/2)`
A2
`|N|^2/|D|^2=(4sin^2((n+1)psi/2))/(4sin^2(psi/2))` A2