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?