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GreenPrint
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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.
===
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]
===
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.
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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