Can Intense Laser Pumping Induce Lasing in a Two-Level System?

[SSJVegetto]

Homework Statement

Assume a two-energy-level system with energetic separation between the two levels of 10000 cm^-1 (as you can easily see, λ = 1 μm equals 1/λ = 10000 cm^-1, and E ∝ 1/λ, i.e. the unit [cm^-1] defines an energy).
a) What is the thermal population in the upper level?
b) If you pump the system with a very intense laser, how much will the
population of the upper level increase? Explain the result!
c) Under these conditions, can you achieve lasing? Explain the answer!

Homework Equations

$$k\ =\ 1.3806503(24)\ \times\ 10^{-23}\ J\ K^{-1}$$
$$h\ =\ 6.62606876(52)\ \times\ 10^{-34}\ J\ s$$
$$\frac{N_{2}}{N_{1}}=\frac{g_{2}}{g_{1}}.e^{-(E_{2}-E_{1})/kT}$$

The Attempt at a Solution

a. I'm assuming the degeneracy can be taken to be 1 for both? And now i use the equation given above to solve this problem and i get $$\frac{N_{2}}{N_{1}}=0.61$$ where i used for the energy difference $$E_{2}-E_{1}=\Delta E=\frac{hc}{\Delta\lambda}$$. Where i used $$\Delta\lambda=10000 cm^{-1}$$ I am wondering if i got the energy right.

b. I really need some help with this one i just don't know how to answer this one. c. I don't know how to answer this i think when you can achieve population inversion the laser should be able to start lasing. But whether to tell if it can achieve that yes or no i don't know how to answer that and would appreciate some help.

Bob

 

[Redbelly98]

Welcome to Physics Forums

Homework Equations

$$k\ =\ 1.3806503(24)\ \times\ 10^{-23}\ J\ K^{-1}$$
$$h\ =\ 6.62606876(52)\ \times\ 10^{-34}\ J\ s$$
$$\frac{N_{2}}{N_{1}}=\frac{g_{2}}{g_{1}}.e^{-(E_{2}-E_{1})/kT}$$

The Attempt at a Solution

a. I'm assuming the degeneracy can be taken to be 1 for both?

Yes, it's strictly a two-state system, one upper state and one lower state.

And now i use the equation given above to solve this problem and i get $$\frac{N_{2}}{N_{1}}=0.61$$ where i used for the energy difference $$E_{2}-E_{1}=\Delta E=\frac{hc}{\Delta\lambda}$$. Where i used $$\Delta\lambda=10000 cm^{-1}$$ I am wondering if i got the energy right.

Uh, not quite. Note that

ΔE = hc/λ, and it's 1/λ (not "Δλ") = 10000 cm^-1, or λ=1 μm​

Also, what are you using for the temperature?

b. I really need some help with this one i just don't know how to answer this one.

A very intense laser has the same effect as making the temperature very high.

c. I don't know how to answer this i think when you can achieve population inversion the laser should be able to start lasing. But whether to tell if it can achieve that yes or no i don't know how to answer that and would appreciate some help.

Bob

Look at the equation for N2/N1, and imagine that T is very large. Will that make N2/N1 greater than 1, i.e. N2>N1?

 

[SSJVegetto]

Welcome to Physics Forums Yes, it's strictly a two-state system, one upper state and one lower state.

Ok and when do i know i have to use degeneracy? When will we be talking about degenerate energy levels? And thank you! :D

Uh, not quite. Note that

ΔE = hc/λ, and it's 1/λ (not "Δλ") = 10000 cm^-1, or λ=1 μm​

Also, what are you using for the temperature?

I used a temperature of 300 K which is room temperature i assume? Also what you are saying i don't really get? The energy difference between two states is equal to the wavelength of the photon that is radiated when going from the upper level to the lower level?

A very intense laser has the same effect as making the temperature very high.

What is called very hot? What temperature do i use to determine how much of the upper level population is increased?

I do see that it never reaches N2/N1 = 1 and thus population inversion is not achieved and the absorption process remains dominant? So lasing is not possible. But how do i calculate how much of the upper level population is increased?

Look at the equation for N2/N1, and imagine that T is very large. Will that make N2/N1 greater than 1, i.e. N2>N1?

Ok by trying to look at the equation more qualitatively i see that this only becomes 1 for an infinite temperature where $$e^{0}=1$$. Thus lasing will not be possible when pumping this.

But when does it start lasing using this equation i mean there is never a system that reaches population inversion? Is it therefore that we need a three or four level system with a fast decaying upper level to the upper laser level?

 

[Redbelly98]

Ok and when do i know i have to use degeneracy? When will we be talking about degenerate energy levels? And thank you! :D

You're welcome.

Use degeneracy when there is some information in the problem statement that indicates you should do so. When that happens is up to your professor.

I used a temperature of 300 K which is room temperature i assume?

Yes.

Also what you are saying i don't really get? The energy difference between two states is equal to the wavelength of the photon that is radiated when going from the upper level to the lower level?

No, the energy difference is not equal to the wavelength! Use the E=hc/λ formula instead, with λ=1 micrometer = 10^-6 m. (Your answer of N2/N1=0.61 is incorrect.)

I am wondering if i got the energy right.

You haven't said what energy you got, so I can't answer that.

What is called very hot? What temperature do i use to determine how much of the upper level population is increased?

I do see that it never reaches N2/N1 = 1 and thus population inversion is not achieved and the absorption process remains dominant? So lasing is not possible.

Correct.

But how do i calculate how much of the upper level population is increased?

It does strike me as an odd question to ask. Perhaps an answer like "the population increases to almost _____", or something along those lines, will be sufficient.

Ok by trying to look at the equation more qualitatively i see that this only becomes 1 for an infinite temperature where $$e^{0}=1$$. Thus lasing will not be possible when pumping this.

Correct.

But when does it start lasing using this equation i mean there is never a system that reaches population inversion? Is it therefore that we need a three or four level system with a fast decaying upper level to the upper laser level?

Yes, that's right.

 

[SSJVegetto]

Thank you for your answers! :D It helps me a lot!

I'm trying to calculate the value now but i don't seem to quite get it. You say i need to calculate it using E = hc/lambda right? That should be equal to the energy difference then right or how should i see this? I said previously that the energy difference is equal to the wavelength, I'm sorry for that for using sloppy words, i meant that the energy difference thus is a function of the wavelength and thus E = hc/lambda. If you ask me you now are assuming that the first energy level the lower population level has an energy equal to zero?

i use:
c = 3.10⁸
h = 6.62606896.10^-34
kb = 1.3806503.10^-23
T = 300
$$\lambda = 1 \mu m$$

Now if i use these and fill these in in this equation:
$$\frac{N_{2}}{N_{1}}=e^{-\frac{hc}{\lambda}.\frac{1}{k_{b}.T}}$$

i get something like:
1.436.10^-21

Is this correct?

 

[Redbelly98]

i get something like:
1.436.10^-21

Is this correct?

Looks good!