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

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In summary, the conversation discusses an amplifying medium composed of four-level atoms and a laser pump to induce amplification. The various parameters of the system are provided, including the pump irradiance required to sustain a small signal gain coefficient and the saturation irradiance. The equations for small-signal gain coefficient and effective pump rate density are used to solve for the pump irradiance, with the unknown variable being the phase velocity. The caption of the figure reveals that the phase velocity can be approximated as the energy difference between two levels divided by Planck's constant. The given wavelengths are not needed to solve the problem.
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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 [itex]I_{p}[/itex], 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 [itex]N_{T} = N_{0} + N_{1} + N_{2} + N_{3}[/itex]. The various parameters are:

[itex]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[/itex]
[itex]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}[/itex]

Assuming an ideal four-level laser system determine:

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

Homework Equations


===
The small-signal gain coefficient [itex]γ_{0}[/itex] and the saturation irradiance [itex]I_{S}[/itex] take the form

[itex]γ_{0} = σR_{p2}\tau_{2}[/itex]

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

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

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

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

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

[itex]k = \frac{2\pi}{λ}[/itex]

Where [itex]λ[/itex] is the wavelength
===
The angular frequency [itex]ω[/itex] of a wave can be expressed as

[itex]ω = 2\pi f[/itex]

Where [itex]f[/itex] is the frequency
===
[itex]\pi ≈ 3.14[/itex]
===

The Attempt at a Solution



I start off with the equation for the small-signal gain coefficient [itex]γ_{0}[/itex]

[itex]γ_{0} = σR_{p2}\tau_{2}[/itex] [1]

and plug in the equation for [itex]R_{p2}[/itex] effective pump rate density

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

into [1].

This yields

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

I solve this equation for the pump irradiance [itex]I_{p}[/itex] and get

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

I know that for a closed system

[itex]κ_{3} = κ_{32} + κ_{31} + κ_{30}[/itex]

Looking at the given variables I get

[itex]κ_{3} = κ_{32} + 0 + 0 = κ_{32}[/itex]

Substituting this into [3] yields

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

Simplifying this yields

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

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

[itex]v_{p} = \frac{ω}{k}[/itex]

This however doesn't really help me. So there must be some other way of expressing the phase velocity [itex]v_{p}[/itex] 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.
 
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  • #2
I have just read the caption of the figure and now realize that [itex]v_{p}≈\frac{E_{3} - E_{0}}{h}[/itex]. The only problem now is that I don't know [itex]E_{3}[/itex] or [itex]E_{0}[/itex]. 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?
 
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Related to How Does Pump Irradiance Affect Gain in a Four-Level Laser System?

What is a gain medium?

A gain medium is a material that amplifies light by providing energy to the photons passing through it. This amplification process is essential for the operation of a laser.

What are the most common types of gain media used in lasers?

The most common types of gain media used in lasers are solid-state crystals (such as ruby or Nd:YAG), gases (such as helium-neon or carbon dioxide), and semiconductors (such as gallium arsenide).

How does the gain medium contribute to the production of laser light?

The gain medium is responsible for providing the stimulated emission of photons, which results in the amplification of light. When excited by an external energy source, the gain medium releases photons that are in phase with the incident photons, leading to the formation of a coherent beam of laser light.

What factors affect the performance of a gain medium?

The performance of a gain medium can be affected by factors such as the physical properties of the material (such as its absorption and emission coefficients), the pumping mechanism used to excite the medium, and the temperature and pressure at which it is operated.

Can a gain medium be used in different types of lasers?

Yes, a gain medium can be used in different types of lasers as long as it meets the specific requirements for each type of laser. For example, a ruby crystal can be used as a gain medium in both a pulsed and continuous-wave laser, but the pumping mechanism and other components may differ for each type of laser.

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