LM.3 · Radiation attenuation, laser gain and cavity modes

GRE · GRE Subject Test · GRE 物理 · 知识点 44

44.1

Radiation attenuation 衰减, laser gain and cavity modes

A photon beam has no definite range, yet a heavy charged particle does; shielding calculations fail if the two are swapped.

Prerequisites: 7, 17, 21.

Distinguish charged-particle stopping and range from exponential photon attenuation
Compute attenuated intensity and detected counts with stated coefficients
Explain stimulated emission, population inversion 粒子数反转 and optical cavity resonance conditions

词汇训练

English	中文	拼音
population inversion/ˌpɒpjʊˈleɪʃn ɪnˈvɜːʃn/	粒子数反转	lì zi shù fǎn zhuǎn
attenuation/əˌtenjuːˈeɪʃn/	衰减	shuāi jiǎn

44.2

Choose the interaction model

Heavy charged particles lose energy continuously through many small collisions, so they have an approximate range: under a constant-loss model a particle with initial energy E0 and loss rate dE/dx stops after E0/(dE/dx). Electrons straggle more and radiate. Photons instead interact in single events, so a narrow beam attenuates exponentially as I=I0 e^(−μx) with no definite maximum depth; the half-value layer 半值层 is ln2/μ.

词汇训练

English	中文	拼音
half-value layer/hɑːf ˈvæljuː ˈleɪə/	半值层	bàn zhí céng

44.3

Attenuate and count

Apply the exponential only to the stated narrow-beam geometry: μ=0.2 /cm gives a half-value layer of 3.47 cm and I/I0=e^(−2)≈0.135 after 10 cm. Multiply by detector efficiency for recorded counts: with μx=ln4 and ε=0.25 the recorded fraction is 0.25×1/4=1/16. Build-up from scattered photons makes broad-beam shielding transmit more than the narrow-beam exponential; state the geometry.

44.4

Invert the population

Stimulated emission produces a photon matching the stimulating photon in frequency, direction and phase, giving coherent amplification. It competes with absorption; net gain needs population inversion N2/g2>N1/g1 (N2>N1 for equal degeneracies). Positive-temperature equilibrium has N2/g2<N1/g1 by the Boltzmann factor, so a two-level system in equilibrium cannot lase continuously; practical lasers pump a third level or a metastable state.

44.5

Resonate in the cavity

A linear cavity of length L supports standing modes with L=mλ/2, i.e. frequencies ν_m=mc/(2L) with spacing c/(2L): for L=30 cm the spacing is 500 MHz. Gain must exceed the round-trip losses for oscillation. Interferometers such as Michelson use the same coherence: fringe counts track optical path changes, as in the gas-cell measurement.

44.6

Worked method

A narrow beam has attenuation $I=I_0e^{-\mu x}$ and half-value layer $x_{1/2}=\ln2/\mu$.

$$x_{1/2}=(\ln2)/(0.20\,\mathrm{cm^{-1}})=3.47\,\mathrm{cm}.$$

Three half-value layers transmit 1/8. This model excludes scattered build-up into the detector. For a linear vacuum laser cavity, adjacent longitudinal modes are separated by $c/(2L)$. Net stimulated gain requires $N_2/g_2>N_1/g_1$; the simpler $N_2>N_1$ assumes equal degeneracies.

Radiation attenuation, laser gain and cavity modes: GRE original diagram — Radiation attenuation, laser gain and cavity modes: original GRE teaching diagram.

44.7

Check conditions and vocabulary

Assigning photons a definite range or charged particles a single exponential law, and forgetting that equilibrium two-level populations cannot invert. Check which interaction model the beam species requires.

half-value layer: The material thickness that halves an exponentially attenuated beam, equal to ln2/μ.

population inversion: A non-equilibrium state with more population in the upper laser level than the lower, enabling net stimulated emission.