What is the Dispersion Relation for Frequency Dependent Permittivity?

[LocationX]

I am given the permittivity to be \epsilon(\omega) = 1+ \frac{\omega_p^2}{\omega_0^2 -\omega^2}

I am asked to sketch k vs \omega using the dispersion relation k^2 =\frac{\omega^2 \epsilon(\omega)}{c^2}

here is what I have:

k=\frac{\omega}{c} \sqrt{\epsilon(\omega)}
k=\frac{\omega}{c} \sqrt{1+ \frac{\omega_p^2}{\omega_0^2 -\omega^2}}
\sqrt{1+x} = 1+\frac{x}{2} +...
k=\frac{\omega}{c} \left( 1+ \frac{1}{2} \left( \frac{\omega_p^2}{\omega_0^2 -\omega^2}}\right) \right)

plotting this, i get something like: http://sites.google.com/site/question1site/"

I'm not sure if I should expand this or not. Or what exactly accounts for the differences between the two (expanded vs not expanded). Also, I do not see how \omega_p will play a role in the graphs. As in, I know that the asymtotes of the graph are the resonate frequency, but I do not know where to place \omega_p.

The second part of the question asks for the plot of \omega vs k

In the link above, I have rotated and inversed the graph to get \omega vs k

I need to show that for k>0, there are two allowed values of \omega. We see this clearly as there are two angular frequencies for when k>0 (since graph is not one to one).

The question then asks to show that at small k and large k, one of the two modes will have a dispersion relation similar to an EM wave in the vacuum, i.e. \omega = v_p k where v_p is weakly dependent on k and that the other mode has a frequency \omega that is (to lowest order) independent of k. I'm not sure what to do here. Am I supposed to see this from the graph? Or should I solve for ω in terms for k to answer the questions above?

 

[LocationX]

any ideas?

 

[gabbagabbahey]

k=\frac{\omega}{c} \sqrt{1+ \frac{\omega_p^2}{\omega_0^2 -\omega^2}}
\sqrt{1+x} = 1+\frac{x}{2} +...
k=\frac{\omega}{c} \left( 1+ \frac{1}{2} \left( \frac{\omega_p^2}{\omega_0^2 -\omega^2}}\right) \right)

plotting this, i get something like: http://sites.google.com/site/question1site/"

Why are you including negative values of \omega in your plot? Are such values actually physical?

I'm not sure if I should expand this or not. Or what exactly accounts for the differences between the two (expanded vs not expanded). Also, I do not see how \omega_p will play a role in the graphs. As in, I know that the asymtotes of the graph are the resonate frequency, but I do not know where to place \omega_p.

I see no reason to Taylor expand this unless you are asked to examine the behavior of this function near a specific point.

\omega_p seems to just be a scaling factor that determines how quickly/slowly k increases/decreases with changing omega.

The second part of the question asks for the plot of \omega vs k

In the link above, I have rotated and inversed the graph to get \omega vs k

Again-- negative omega?

I need to show that for k>0, there are two allowed values of \omega. We see this clearly as there are two angular frequencies for when k>0 (since graph is not one to one).

I'm not sure pointing at the graph will be enough for your prof...you might also want to solve the equation
k=\frac{\omega}{c} \sqrt{1+ \frac{\omega_p^2}{\omega_0^2 -\omega^2}} for \omega...you will of course have to solve a quadratic, which you can show gives two real values by looking at the discriminant of said quadratic.

The question then asks to show that at small k and large k, one of the two modes will have a dispersion relation similar to an EM wave in the vacuum, i.e. \omega = v_p k where v_p is weakly dependent on k and that the other mode has a frequency \omega that is (to lowest order) independent of k. I'm not sure what to do here. Am I supposed to see this from the graph? Or should I solve for ω in terms for k to answer the questions above?

Once you solve for omega, then taylor expand about k=0 to look at the behavior for small k...what do you get for your two solutions in this case?

 

[LocationX]

I'm not sure pointing at the graph will be enough for your prof...you might also want to solve the equation
k=\frac{\omega}{c} \sqrt{1+ \frac{\omega_p^2}{\omega_0^2 -\omega^2}} for \omega...you will of course have to solve a quadratic, which you can show gives two real values by looking at the discriminant of said quadratic.

Instead of a quadratic equation, I get a quartic equation.

\omega^4 + \omega^2 \left( -\omega_0^2 - \omega_p^2 -c^2 k^2 \right) + c^2 k^2 \omega_0^2 =0

If we let omega^2 = omega', we can solve it quadratically:

\omega'=\frac{\omega_0^2 + \omega_p^2 +c^2 k^2 \pm \sqrt{(-\omega_0^2 - \omega_p^2 -c^2 k^2)^2 -4( c^2 k^2 \omega_0^2)}}{2}

thus:

\omega_1=\sqrt{\frac{\omega_0^2 + \omega_p^2 +c^2 k^2 + \sqrt{(-\omega_0^2 - \omega_p^2 -c^2 k^2)^2 -4( c^2 k^2 \omega_0^2)}}{2}}
\omega_2=-\sqrt{\frac{\omega_0^2 + \omega_p^2 +c^2 k^2 + \sqrt{(-\omega_0^2 - \omega_p^2 -c^2 k^2)^2 -4( c^2 k^2 \omega_0^2)}}{2}}
\omega_3=\sqrt{\frac{\omega_0^2 + \omega_p^2 +c^2 k^2 - \sqrt{(-\omega_0^2 - \omega_p^2 -c^2 k^2)^2 -4( c^2 k^2 \omega_0^2)}}{2}}
\omega_4=-\sqrt{\frac{\omega_0^2 + \omega_p^2 +c^2 k^2 - \sqrt{(-\omega_0^2 - \omega_p^2 -c^2 k^2)^2 -4( c^2 k^2 \omega_0^2)}}{2}}I don't see where to go from here? How can one determine the which is real?

 

[gabbagabbahey]

Let's take a look at \omega^2 first...clearly as long as it is positive and real, omega will be real...so let's see when \omega^2 is positive and real:

\omega^2=\frac{\omega_0^2 + \omega_p^2 +c^2 k^2 \pm \sqrt{(-\omega_0^2 - \omega_p^2 -c^2 k^2)^2 -4( c^2 k^2 \omega_0^2)}}{2} <br />

Let's call \kappa \equiv \omega_0^2 + \omega_p^2 +c^2 k^2 to make things easier...clearly as long as \omega_p, \omega_0 and k are assumed to be real, \kappa&gt;0...and looking at omega square again:

\omega^2=\frac{\kappa \pm \sqrt{\kappa^2 -4( c^2 k^2 \omega_0^2)}}{2} <br />

we see that \sqrt{\kappa^2 -4( c^2 k^2 \omega_0^2)}} will always be less than or equal to \kappa (as long as this quantity is real), so omega squared will never be real and negative...that means we only need to find where omega squared is real...When is that?

 

[LocationX]

omega squared is real when \kappa^2 &gt;4( c^2 k^2 \omega_0^2)?

 

[gabbagabbahey]

Now that I think about it a little bit, I think it's safe to just assume \omega \in \Re and \omega \geq 0 on physical grounds...that rules out your \omega_2 and \omega_4 solutions since they are always negative, and leaves you with two solutions at most...if \kappa^2-4c^2k^2\omega_0^2 = 0, then you will only have one solution, but you should be able to show that this never happens for k>0. (just solve the resulting quadratic for k^2 (its a quartic in k)...you'll find that there are no real solutions, so k>0 precludes the possibility of there being only one solution for omega.)

 

[LocationX]

ok, I'm going to try and taylor expand this:

<br /> \omega^2=\frac{\kappa \pm \sqrt{\kappa^2 -4( c^2 k^2 \omega_0^2)}}{2} <br />I am getting that both frequencies do not depend on k (to lowest order):

http://sites.google.com/site/question2site/

I'm not sure where to get a frequency that is weakly dependent on k since both frequencies are independent of k for first order. And what exactly does it mean to have omenga weakly dependent on k?

 

[gabbagabbahey]

Ugghh...you can cleanup your expressions Out[73] and Out[74] quite a bit by noting that

\sqrt{\omega_0^4+2\omega_0^2\omega_p^2+\omega_p^4}=\sqrt{(\omega_0^2+\omega_p^2)^2}=\omega_0^2+\omega_p^2

You should get:

\text{Out[73]}=\sqrt{\omega_0^2+\omega_p^2}+\frac{c^2 \omega_p^2}{2(\omega_0^2+\omega_p^2)^{3/2}}k^2 +\ldots \approx \sqrt{\omega_0^2+\omega_p^2}

Which gives you your constant term...And for Out[74], you need to explicitly assume that all your variables are >0 before calculating the series with mathematica (or you can do it by hand)...you should get:

\frac{\omega_0 c k}{\sqrt{\omega_0^2+\omega_p^2}}+\ldots

which is dependent on k; and since omega_p is usually MUCH bigger than omega_0 for most mediums, your coefficient is typically pretty small, and so that frequency is 'weakly dependent' on k...try doing the series again while assuming the c>0,omega_0>0, omega_p>0 and k>0.

 

[LocationX]

Ugghh...you can cleanup your expressions Out[73] and Out[74] quite a bit by noting that

\sqrt{\omega_0^4+2\omega_0^2\omega_p^2+\omega_p^4}=\sqrt{(\omega_0^2+\omega_p^2)^2}=\omega_0^2+\omega_p^2

Sorry, I'm doing the expansion by hand and see if I can get what you got.

I'm also asked to show that the two modes exchange their characteristic behavior as one crosses from small k to large k. I think I was supposed to do the expansion for both small k and large k. For each case, what can be ignored?

I was thinking for small k, the k^4 term can be ignored, but that means I would have to expand (-\omega_0^2 - \omega_p^2 -c^2 k^2)^2. Is there a better way of doing the approximation? If we ignored k^2 then the whole term would just be a constant.

For large k, I was thinking about ignoring the omega_0 and omega_p terms:

<br /> \omega_1=\sqrt{\frac{c^2 k^2 + \sqrt{(c^2 k^2)^2 -4( c^2 k^2 \omega_0^2)}}{2}}

would this work?

 

[gabbagabbahey]

For large k, just Taylor expand about the point (1/k)=0

Don't ignore any terms until after you have calculated the series...in each case, just keep the lowest order non-zero term in the series.

 

[gabbagabbahey]

For large k, start by factoring out a k:

\omega_{\pm}=k\sqrt{\frac{\omega_0^2 + \omega_p^2 +c^2 k^2 \pm \sqrt{(-\omega_0^2 - \omega_p^2 -c^2 k^2)^2 -4( c^2 k^2 \omega_0^2)}}{2k^2}}=k\sqrt{\frac{\omega_0^2\delta^2 + \omega_p^2\delta^2 +c^2 \pm \sqrt{(-\omega_0^2\delta^2 - \omega_p^2\delta^2 -c^2)^2 -4( c^2 \delta^2 \omega_0^2)}}{2}}

where \delta \equiv \frac{1}{k}...for large k, delta is small...so taylor expand this in powers of delta and keep the smallest order non-zero term...

 

[LocationX]

I tried using mathematica and I can't seem to get the result of <br /> \frac{\omega_0 c k}{\sqrt{\omega_0^2+\omega_p^2}}+\ldots

I get:

http://sites.google.com/site/question2site/

For large k, I get \omega_+ = kc and I was going to use mathematica to get the \delta dependence of \omega_-.

 

[gabbagabbahey]

The way you've written your mathematica code, your telling it to make the assumptions after calculating the series...instead try doing

Assuming[{k>0,c>0,[itex]\omega_p[/itex]>0,[itex]\omega_0[/itex]>0},Series[...]]

That tells mathematica to evaluate the series while making those assumptions

 

[gabbagabbahey]

To do it by hand, for small k, just let

f_{\pm}(k)\equiv \sqrt{\frac{\omega_0^2 + \omega_p^2 +c^2 k^2 \pm \sqrt{(-\omega_0^2 - \omega_p^2 -c^2 k^2)^2 -4( c^2 k^2 \omega_0^2)}}{2}}

Then use the formula for a taylor expansion about k=0:

f(k)=f(0)+\frac{f&#039;(0)}{1!}k+\frac{f&#039;&#039;(0)}{2!}k^2 + \ldots

what are you getting for f_{\pm}&#039;(0) andf_{\pm}&#039;&#039;(0)?

 

[LocationX]

<br /> f_{\pm}&#039;=\frac{1}{2} \left( \frac{\omega_0^2 + \omega_p^2 +c^2 k^2 \pm \sqrt{(-\omega_0^2 - \omega_p^2 -c^2 k^2)^2 -4( c^2 k^2 \omega_0^2)}}{2} \right)^{-\frac{1}{2}} \left( c^2 k \pm \frac{1}{2} \left( \frac{1}{2} \left( \omega_0^2 + \omega_p^2 +c^2 k^2 - 4 c^2 k^2 \omega_0^2 \right)^{-\frac{1}{2}} \left(2c^2k-8c^2k \omega_0^2\right) \right)

<br /> f_{\pm}&#039;&#039;=-\frac{1}{4} \left( \frac{\omega_0^2 + \omega_p^2 +c^2 k^2 \pm \sqrt{(-\omega_0^2 - \omega_p^2 -c^2 k^2)^2 -4( c^2 k^2 \omega_0^2)}}{2} \right)^{-\frac{3}{2}} \left( c^2 \pm \frac{1}{2} \left( \frac{-1}{4} \left( \omega_0^2 + \omega_p^2 +c^2 k^2 - 4 c^2 k^2 \omega_0^2 \right)^{-\frac{3}{2}} \left(2c^2k-8c^2k \omega_0^2\right)\right)\left(2c^2-8c^2 \omega_0^2\right) \right)

I get f'(0)=0
f''_-=0
f&#039;&#039;_+=\frac{-c^2}{4(\omega_0^2+\omega_p^2)^{3/2}}

 

[gabbagabbahey]

Hmmm...now that I think about, I see why mathematica is having a problem...when I use this method by hand I get:

f_{-}&#039;(0)=\infty... (0^{-1/2} is undefined)

Try expanding \sqrt{(-\omega_0^2 - \omega_p^2 -c^2 k^2)^2 -4( c^2 k^2 \omega_0^2)} first and keep terms up to order k^2...that should give you:

\omega_{\pm}(k) \approx \sqrt{\frac{\omega_0^2 + \omega_p^2 +c^2 k^2 \pm \left( \omega_0^2 + \omega_p^2+c^2k^2 -\frac{2c^2k^2\omega_0^2}{\omega_0^2 + \omega_p^2}\right) }{2}}

...that should make thing easier for you!

 

[LocationX]

<br /> \sqrt{(-\omega_0^2 - \omega_p^2 -c^2 k^2)^2 -4( c^2 k^2 \omega_0^2)}
<br /> = (\omega_0^2 + \omega_p^2 +c^2 k^2) \sqrt{1 -\frac{4( c^2 k^2 \omega_0^2)}{(\omega_0^2 + \omega_p^2 +c^2 k^2)}}
<br /> \approx (\omega_0^2 + \omega_p^2 +c^2 k^2) \left( 1 -\frac{2( c^2 k^2 \omega_0^2)}{(\omega_0^2 + \omega_p^2 +c^2 k^2)} \right)

I don't see where \omega_0^2 + \omega_p^2 appears in the denominator

 

[gabbagabbahey]

What you've done is expand it to first order in powers of \frac{4( c^2 k^2 \omega_0^2)}{(\omega_0^2 + \omega_p^2 +c^2 k^2)} not to second order in powers of k...instead use taylor expansion:

g(k)=\sqrt{(-\omega_0^2 - \omega_p^2 -c^2 k^2)^2 -4( c^2 k^2 \omega_0^2)} \approx g(0)+g&#039;(0)k+\frac{g&#039;&#039;(0)}{2!} k^2

 

[LocationX]

g(0)=\omega_0^2 + \omega_p^2
g&#039;(0)k=0
\frac{g&#039;&#039;(0)}{2!} k^2 =c^2k^2 \frac{\omega_p^2-\omega_0^2}{\omega_0^2 + \omega_p^2}

Used mathematica for second derivative:
http://sites.google.com/site/question2site/

not sure if this matches up

 

[gabbagabbahey]

Your first two are correct, but I get:

\frac{g&#039;&#039;(0)}{2!} k^2 =c^2k^2 -\frac{2c^2k^2\omega_0^2}{\omega_0^2 + \omega_p^2}

Mathematica agrees with me, so I would check your math.

Edit- our two expressions are equivalent

 

[LocationX]

<br /> \omega_{\pm}=k\sqrt{\frac{\omega_0^2\delta ^2 + \omega_p^2\delta^2 +c^2 \pm \sqrt{(-\omega_0^2\delta^2 - \omega_p^2\delta^2 -c^2)^2 -4( c^2 \delta^2 \omega_0^2)}}{2}}

Here's what I get:
<br /> \omega_+ \approx kc

\sqrt{(-\omega_0^2\delta^2 - \omega_p^2\delta^2 -c^2)^2 -4( c^2 \delta^2 \omega_0^2)} \approx c^2+\delta^2(\omega_p^2 - \omega_0^2)

<br /> \omega_{-}\approx k\sqrt{\frac{\omega_0^2\delta ^2 + \omega_p^2\delta^2 +c^2 -(c^2+\delta^2(\omega_p^2 - \omega_0^2))}{2}}

\omega_{-} \approx k \delta \omega_0 = \omega_0

In summary:

small k:

\omega_+ \approx \sqrt{\omega_0^2 + \omega_p^2}

\omega_- \approx \frac{\omega_0 c k}{\sqrt{\omega_0^2 + \omega_p^2}}

large k:

\omega_+ \approx kc

\omega_- \approx \omega_0

When the question asks to to show that the two modes exchange their characteristic behavior as one crosses from small k to large k, is it enough to say that:

\omega_+ at small k is independent of k, and for large k, there is a k dependence, and vice versa for \omega_- ?

 

[gabbagabbahey]

Looks good to me, but I would maybe say "At small k, the \omega_{-} mode is a constant and \omega_{+} is linearly dependent on k; while for large k, these modes switch and the \omega_{+} mode is a constant and \omega_{-} is linearly dependent on k."

 

[LocationX]

Thanks for the check. The last part of the question asks to show that at a given value of k, the higher frequency mode has a phase velocity v_p &gt; c and the lower frequency mode vp &lt; c. Lastly, the question asks to show for the high and low frequency modes that the group velocity is always vg &lt; c. For this part, should this be done for frequencies of small k and large k as well? And what about the frequencies that do not depend on k? Or should the original (non-expanded) form of omega be used?

 

[LocationX]

I can answer the question above for:
<br /> \omega_- \approx \frac{\omega_0 c k}{\sqrt{\omega_0^2 + \omega_p^2}}

and

<br /> <br /> \omega_+ \approx kc

however, for the constant omegas, I am having trouble. Proving the above becomes hard specifically for small k high frequency, and large k low frequency, since the values do not depend on k.

As an example, here is what I tried for \omega_- \approx \omega_0
Since this is a constant, dividing that by k to get the phase velocity doesn't tell us if it's greater or less than c. What I tried was to get higher order terms:

The expansion of \sqrt{(-\omega_0^2\delta^2 - \omega_p^2\delta^2 -c^2)^2 -4( c^2 \delta^2 \omega_0^2)} gives:

g(0)=c^2
g&#039;(0)k=0
g&#039;&#039;(0)k^2/2=\delta^2 (\omega_p^2-\omega_0^2)
g&#039;&#039;&#039;(0)=0
g^{(4)} k^4/4! = \frac{2 \omega_0^2 \omega_p^2 \delta^4}{c^2}<br /> <br /> \omega_{-}\approx k\sqrt{\frac{\omega_0^2\delta ^2 + \omega_p^2\delta^2 +c^2 -\left(c^2+\delta^2(\omega_p^2 - \omega_0^2)+\frac{2 \omega_0^2 \omega_p^2 \delta^4}{c^2} \right)}{2}}

=\omega_0 \sqrt{1+\left( \frac{\omega_p}{ck} \right) ^2}

\approx \omega_0+\frac{\omega_0 \omega_p^2}{2k^2}

I did not see how the extra term gave any help to see whether vp was greater or less than c.

My final idea was to just say this:

for large k, \omega_- is a constant. The phase velocity is \frac{\omega_0}{k} so that if k increases large enough, it will reach a value when \frac{\omega_0}{k}&lt;c thus, proving vp &lt; c for loq frequencies. Similarly for \omega_+, it's a constant divided by k, so that if k is small enough, the phase velocity will be greater than c.

I am not sure if there is any other way to prove this. Are there any other options?

 

[gabbagabbahey]

I think the higher frequency mode only has a phase velocity greater than c for small k; for that mode v_p \approx \frac{\sqrt{\omega_0^2 + \omega_p^2}}{k} which can be greater than c for small enough k...For large k, I think it's safe to just assume \frac{\omega_0}{k}&lt;c

For the group velocity, I would take the derivative of the general formula for \omega_{\pm} and just make a hand waving type argument that it is always less than c.