How Does Pump Irradiance Affect Gain in a Four-Level Laser System?

[GreenPrint]

I'm not sure if this question should be posted in the introductory physics section or the advanced physics section.

Consider an amplifying medium, composed of homogeneous broadening four-level atoms as show in figure 26.5, page 557 of textbook.

http://img689.imageshack.us/img689/6108/lt2f.png

Amplification is to occur on the 2-to-1 transition. The medium is pumped by a laser of intensity $I_{p}$, which is resonant with the 3-to-0 transition. The spontaneous decay processes are indicated on the diagram. The total number of gain atoms is $N_{T} = N_{0} + N_{1} + N_{2} + N_{3}$. The various parameters are:

$k_{32} = 10^{8} \frac{1}{s}; k_{21} = 10^{3} \frac{1}{s}; k_{10} = 10^{8} \frac{1}{s}; k_{30} = k_{31} = k_{20} = 0$
$σ_{p} = 4x10^{-19} cm^{2}; σ = 2.5x10^{-18} cm^{2}; λ_{30} = 300 nm; λ_{21} = 600 nm; N_{τ} = 2.0x10^{26} m^{-3}$

Assuming an ideal four-level laser system determine:

a) The pump irradiance required to sustain a small signal gain coefficient of $\frac{0.01}{cm}$
b) The saturation innradiance.

Homework Equations

===
The small-signal gain coefficient $γ_{0}$ and the saturation irradiance $I_{S}$ take the form

$γ_{0} = σR_{p2}\tau_{2}$

$I_{S} = \frac{hv^{'}}{σ\tau_{2}}$
===
$R_{p2}$ is a effective pump rate density

$R_{p2} = \frac{κ_{32}}{κ_{3}}(\frac{σ_{p}I_{p}}{hv_{p}}N_{T})$
===
In a closed system

$κ_{3} = κ_{32} + κ_{31} + κ_{30}$
===
The lifetime $\tau$ of an energy level is defined to be the inverse of the total decay rate from the level so that

$\tau_{n} = \frac{1}{κ_{n}}, n \epsilon Z$
===
Planck's constant $h\ =\ 6.62606876(52)\ \times\ 10^{-34}\ J\ s$
===
The phase velocity $v_{p}$ of a wave can be expressed as

$v_{p} = \frac{ω_{p}}{k_{p}} ≈ \frac{ω}{k}$
===
$k$ is the propagation constant of a wave that can be expressed as

$k = \frac{2\pi}{λ}$

Where $λ$ is the wavelength
===
The angular frequency $ω$ of a wave can be expressed as

$ω = 2\pi f$

Where $f$ is the frequency
===
$\pi ≈ 3.14$
===

The Attempt at a Solution

I start off with the equation for the small-signal gain coefficient $γ_{0}$

$γ_{0} = σR_{p2}\tau_{2}$ [1]

and plug in the equation for $R_{p2}$ effective pump rate density

$R_{p2} = \frac{κ_{32}}{κ_{3}}(\frac{σ_{p}I_{p}}{hv_{p}}N_{T})$ [2]

into [1].

This yields

$γ_{0} = σ\frac{κ_{32}}{κ_{3}}(\frac{σ_{p}I_{p}}{hv_{p}}N_{T})\tau_{2}$

I solve this equation for the pump irradiance $I_{p}$ and get

$I_{p} = \frac{γ_{0}κ_{3}hv_{p}κ_{2}}{σκ_{32}σ_{p}N_{T}}$ [3]

I know that for a closed system

$κ_{3} = κ_{32} + κ_{31} + κ_{30}$

Looking at the given variables I get

$κ_{3} = κ_{32} + 0 + 0 = κ_{32}$

Substituting this into [3] yields

$I_{p} = \frac{γ_{0}κ_{32}hv_{p}κ_{2}}{σκ_{32}σ_{p}N_{T}}$ [3]

Simplifying this yields

$I_{p} = \frac{γ_{0}hv_{p}κ_{2}}{σσ_{p}N_{T}}$ [4]

At this point it looks like I'm very close to solving this problem as all but one variable the phase velocity $v_{p}$ is given. As mentioned in the relevant equations

$v_{p} = \frac{ω}{k}$

This however doesn't really help me. So there must be some other way of expressing the phase velocity $v_{p}$ that I'm not aware of. Once I figure this out I should be able to solve this problem easily. My book doesn't have any examples in this section and I can't seem to find similar questions on the internet, hence I'm stuck and not really sure how to proceed.

Thanks for any help.

 

[GreenPrint]

I have just read the caption of the figure and now realize that $v_{p}≈\frac{E_{3} - E_{0}}{h}$. The only problem now is that I don't know $E_{3}$ or $E_{0}$. Looks like I might be able to solve this.

I was able to solve the problem. My only concern is that I didn't use the given wavelengths in the problem. Are they needed to solve the problem?