Complex angle in Snell's law; what does it mean
nutanna...@gmail.com
Could any enlightened soul help me with this.
In the Snell's law,
n1* sin(theta1)=n2*sin(theta2),
I have two quick questions,
1. Does one use only the real part of refractive index or full complex
value, if one of the refractive index is complex.
2. If one has to use complex value, what is the physical meaning of
the angles involved i.e. if (4+4i)*sin(pi/2)=(3+3i)*sin(theta2) then
what is the physical interpretation of theta2.
Nutan
Salmon Egg
<62ce060f-fb89-4d70...@f11g2000vbf.googlegroups.com>,
nutanna...@gmail.com wrote:
The usual reason you get a complex angle in a classroom environment is
that you have total internal reflection. When that is the case, it just
means that there is no solution for a beam transmitting with a real
angle. Not knowing the context of your question, I have no clue as to
what the physical context of your question is.
Bill
--
Private Profit; Public Poop! Avoid collateral windfall!
Timo A. Nieminen
You need to use the full complex refractive index.
I don't have a good explanation of what complex angles mean in this case.
Jackson (Classical electrodynamics) covers this, using complex angles; I
don't like this way of doing it.
You can avoid complex angles. Here's how:
Start with the wavevector of the incident wave. If this term is new to
you, it's a vector in the direction of propagation, with magnitude
2*pi/wavelength. A uniform plane wave will vary in space and time as
exp(ik.r - iwt) where k is the wavevector, r is the position vector, k.r
is the scalar product of these two, and w is the angular frequency,
w=2*pi*f. Note that in free space, k = w/c. In a medium of refractive
index n, k = nw/c (even if n is complex).
Let's call the direction parallel to the interface x, and the direction
normal to the interface z, and choose directions so that the wavevector
lies in the xz plane.
The angle of incidence gives you the values of k_x and k_z, the x and z
components of k.
In the two adjacent media, k_x is the same.
Find k_z using Pythagoras' theorem: k^2 = k_x^2 + k_y^2. Note that this
gives 2 possible answers for k_z (+ and -) - in the first medium, one of
these signs will give you the incident wave, and the other one the
reflected wave (and k_z has the same magnitude for both of these waves -
you've proved Euclid's law of reflection in passing!). In the 2nd medium,
choose the sign that gives propagation away from the interface.
In the 2nd medium, k_x is real. If k is real, then real k_z means you have
a propagating wave in the 2nd medium, and imaginary k_z means you have an
evanescant wave in the 2nd medium, with the amplitude falling off
exponentially away from the interface.
If k is complex, then you'll have complex k_z. The complex part of k_z
will result in exponential decay of the wave as it moves into the medium,
in the usual case of an absorbing medium. If you have a medium with gain,
the wave will grow.
You might be interested in past discussions of this in sci.optics
(especially what A. Siegman ("AES") had to say about complex wavevectors).
--
Timo Nieminen
Benj
> Hey Guys,
> I have two quick questions,
>
> 1. Does one use only the real part of refractive index or full complex
> value, if one of the refractive index is complex.
Depends on what you are modeling with complex numbers. Most likely
you'd be using the magnitude of the complex value if you weren't using
the full complex value. Generally in this case complex mathematics is
used to model the solutions to TE and TM plane waves in source-free
space. A complex value usually implies both lossless propagation as
well as losses the given media.
> 2. If one has to use complex value, what is the physical meaning of
> the angles involved i.e. if (4+4i)*sin(pi/2)=(3+3i)*sin(theta2) then
> what is the physical interpretation of theta2.
I suppose one might work up some model using complex angles but what
that would mean isn't clear. I say this because you have the hint in
the identity (i sin Z = sinh iz) where the switch from sin (lossless
propagation) to sinh (lossy decaying propagation) is related to the
complex angle. BUT, generally speaking in Snells law the angle of the
various waves on the media interfaces are just that. ANGLES! You know,
like geometry? Complex angles are often used to designate phase, but
how that applies here would take some interpretation. I think it'd be
best to stick with geometric angles and interpret complex values
according to the plane wave solutions. OK?
Unless you consider math more real than reality. Then make up your own
interpretation!