Oblique Incidence (2-323)

Total internal reflection occurs when the incident wave is entirely reflected, producing no transmitted wave at all.

a) Using the wavefront model qualitatively explain why total internal reflection can occur. 

b) Now use the boundary condition you found at the surface to explain why total internal reflection can occur. 

c) A plane wave with wavelength of 1 cm is incident at 45 degrees on a surface whose index of refraction can change. What is the maximum wavelength of the transmitted plane wave that you can create by varying the index of refraction? Explain. What is the minimum wavelength of the transmitted plane wave that you can create. Explain. 

d) Using the qualitative wavefront again, is it possible to have no reflected wave (perfect transmission)? Explain why or why not?

Wavefronts are equal phase surfaces that are perpendicular to the direction of propagation. When the wavefront of the incoming ray touches the surface of separation between 1 and 2 mediums all points between A and B (of the surface) will begin to oscillate and radiate independently in time t. Hence

$L_1=v_1*t$ and $L_2=v_2 t$ and since $t$ is the same $L_1/v_1 =L_2/v_2$

or $(AB*sin⁡ θ_1)/v_1 =(AB*sin⁡ θ_2 )/v_2$

$sin⁡ θ_1 /sin⁡ θ_2  =v_1/v_2 =n_2/n_1$ (Snell)

a)

Suppose that the incident ray is coming from 2 towards 1 (from the medium having a less tilt angle). Then since the refracted angle is bigger (the refracted ray is more tilted) at some incoming incident angle, the wave fronts of the refracted wave (red in figure above) will become perpendicular to the segment AB (to the interface). If one increases further the incident angle the refracted wave fronts (that were just perpendicular to the surface –segment AB) will re-enter the incoming medium 1 so that there will be total reflection.

b)

the boundary condition at the surface is exactly the Snell law found above.

$sin⁡ θ_1 /(sin ⁡θ_2)=n_2/n_1$ or $sin⁡ θ_1=sin⁡ θ_2 *n_2/n_1$ with $n_2/n_1 >1$

Since $n_2/n_1 >1$ there will be a maximum possible value of $sin⁡ θ_2$ before $sin⁡ θ_1=1$

$1= sin⁡ θ_2 (maximum)*n_2/n_1$

If $θ_2>θ_2$ (maximum) then the incoming wave (from medium 2) will be totally reflected.

c)

In medium 1 there are incident (I ) and reflected (R )waves. In medium 2 there is the transmitted (T) wave .

All are waves which means they can be written as (for E components)

$(overrightarrow{E_I} )=E_I exp⁡ i((K_I )*r-ωt)$

$ (overrightarrow{E_T} )=E_T*exp⁡ i((K_T )*r -ωt)$

and $(overrightarrow{E_R} )=E_R*exp⁡ i((K_R )*r-ωt)$

where $E_I$,$E_R$ and $E_T$ are just (complex) numbers (the waves amplitudes).

At the boundary the boundary conditions tell that incoming+reflected wave need to match transmitted wave. That is

$(…)*exp⁡ [i((K_I )*r-ωt) ]+(…)*exp⁡(i((K_R )*r-ωt) ]=(…)*exp⁡[i((K_T )*r-ωt)]$

Since the time is the same for all 3 waves (when the wavefront touches the surface) the $exp⁡(-iωt)$ simplifies from above. What remains need to be true for all points on the surface between A to B (so that for all possible $r ⃗$ between A and B) regardless the content in parenthesis. Therefore

$(overrightarrow{K_I})*overrightarrow{r}=(overrightarrow{K_R})*overrightarrow{r}=(overrightarrow{K_T} )*(overrightarrow{r})$ (at $z=0$)

Since the figure lies in the $(x z)$ plane then from above if follows that

$K_{I_x}=K_(R_x )=K_(T_x )$

or $K_I*sin⁡ θ_I=K_R*sin⁡ θ_R=K_T*sin⁡ θ_T$

$2π/λ_I *sin⁡ θ_I =2π/λ_T *sin⁡ λ_T$ or $λ_T/λ_I =n_1/n_2$

or $λ_T=1/n_{21} *λ_I$ with $n_{21}=n_2/n_{14}$

Since $n_2>n_1$ (medium 2 is more optical denser than medium 1) it means that $n_{21}>1$. If we take $n_1=1$ and vary $n_2=[1…∞]$ it means $n_{21}=[0…∞]$ and from red equation above the minimum transmitted wavelength is $=0$ $λ_T=0$ for $n_{21}=∞$, and the maximum wavelength of the transmitted wave is $λ_T=λ_I=1 cm$ for $n_{21}=1$.

d)

The qualitative wavefront model explaining at the beginning says that all points between A and B that are touched by the incoming wavefront begin to oscillate one after the other. (The wave fronts of each points that oscillate compose with the others to give the outgoing wave fronts). Since the radiation of an atom can have any possible direction is space (that is, if it were the same medium the wavefront of an oscillating atom from AB segment would be homogeneous spherical) then one cannot explain using this simple wavefront model why the reflected ray vanishes at a certain incident angle.

To explain this phenomenon we need to take into account also the polarization of the incoming and reflected wave. Which means that vectors E of the incoming wave that are parallel to the surface plane are not reflected (E need to strike the surface to be reflected, and all E parallel to the surface do not strike the surface).