Optical Sample Detection
4.3 Absorption coefficient and refractive index
νfi−νL+iγ⊥/2π
∆νh ph(νL, νfi, ∆νh)E(r, t). (4.11) Together with equation (4.8), this expression describes all major aspects of the particle-light interaction, including selection rules, natural lineshape, and the influence of particle orientation due to external fields.
4.3 Absorption coefficient and refractive index
Equations (4.9) describe the evolution of a single particle with resonant fre-quencyνfi in a monochromatic light field of frequencyνL. When going over to a macroscopic sample, characterized by an effective absorption coefficient α and refractive indexn, many-particle effects have to be taken into account.
1The according geometries are well-known from Zeeman spectroscopy. While all types of
∆mtransitions are allowed in transversal observation (z⊥z),π-polarized light cannot drive
∆m= 0 transitions longitudinally (zz).
ABSORPTION COEFFICIENT AND REFRACTIVE INDEX 63 4.3.1 Inhomogeneous broadening
In an ensemble of particles, the individual coupling to the light field might vary due to statistical effects or spatial inhomogeneities, as caused for example by nonuniform external fields. The observed dispersion or absorption lineshape then is a convolution of the distributed single-particle lineshapes. These can differ both in amplitude as well as in resonance frequency, thereby giving rise to inhomogeneous broadening.
A prominent example of inhomogeneous broadening originates from the Doppler effect, which shifts any particle’s observed resonance frequency depend-ing on its relative motion with respect to the light source. In a sdepend-ingle-species gas with particle mass mS at temperature T, this motion is characterized by a Maxwell velocity distribution, which translates into a Gaussian distribution of the observed resonant frequenciespG(νfi, ν0, ∆νDoppler). It is centered around the particles’ unshifted resonant frequencyν0 and has a FWHM of
∆νDoppler = ν0 c
8 ln 2kBT mS .
For dilute gases, Doppler broadening usually dominates all other broadening mechanisms.
In the following, the statistical and spatial shift of the individual resonance frequencies νfi will be characterized by a normalized probability distribution function pih(νfi, νih, ∆νih) centered around νih with a FWHM of ∆νih. The light-induced local polarization P(r, t) of the gas can then be approximately expressed in terms of the single-particle dipole moment (4.11) by taking the ensemble average:
P(r, t) =
nS(r)d(r, t)pih(νfi, νih, ∆νih) dνfi
Here, nS(r) is the local number density of the respective species. In the weak-field limit, whereI(r)/Isat 1 and consequently∆νh≈γ⊥/π, the polarization
are the real and imaginary parts of the complex susceptibilityχ=χ+iχ and χ0(r) =nS(r)∆0|∆m|2|µfi|2
ε0 .
If ph is Lorentzian and pih Gaussian, the above convolution is called a Voigt profile. Its typical shape is exemplified in Figure 4.3.
64 OPTICAL SAMPLE DETECTION
-3 -2 -1 0 1 2 3
Susceptibility
(ºL-ºih)/¢ºih
¢ºih
Â00 Â0
Figure 4.3:Absorption and dispersion in a gas. The real and imaginary parts of the complex susceptibility are plotted to scale for∆νih = 1000·∆νh. The horizontal grid line indicates the shifted zero level forχ.
4.3.2 Absorption and dispersion
In order to find a self-consistent description for the propagation of monochro-matic light through an ensemble of two-level particles, the light field (4.3) has to obey the general wave equation
∆− 1 c2
∂2
∂t2
E(r, t) = 1 ε0c2
∂2
∂t2P(r, t).
Solving it in the weak-field limit by applying the slowly varying envelope ap-proximation (∂2zErk ∂zEr) yields
E(x, y, l, t) =E(x, y,0, t)·e−α(νL)l/2ei kn(νL)l (4.13) for the light field at the end of the interaction lengthl, where the abbreviations
α(νL) =kχ¯(l, νL) and n(νL) = 1 +χ¯(l, νL)
2
are identified as the frequency-dependent absorption coefficient α(νL) and re-fractive index n(νL) of the sample. ¯χ stands for the spatial average of the susceptibility over the interaction length:
¯
χ(l, νL) = 1 l
l
0
χ(x, y, z, νL) dz
A spatially homogeneous sample with uniform χ complies with the standard picture for linear absorption, in which the light intensity at the end of the interaction region is derived from the Lambert-Beer law
I(l) =I(0)·e−α(νL)l , (4.14)
ABSORPTION COEFFICIENT AND REFRACTIVE INDEX 65 which immediately follows from (4.13). As the slowly varying envelope approx-imation is applicable to plane waves as well as Gaussian beams and both are in accordance with (4.3), all of the above results are in particular valid for these two important cases.
For practical purposes, further simplifications of equations (4.12) for spa-tially constant χ in a uniform sample are quite useful to directly relate the refractive index to the absorption coefficient. When homogeneous broadening dominates (∆νih∆νh),
λfi=c/νfiis introduced here as the vacuum transition wavelength. In the oppo-site case with ∆νh∆νih, which is much more common, the integral inn(νL) cannot be reduced, and one is left with
α(νL) =nS∆0|∆m|2 λ2fi some arithmetic relief in (4.16b) when letting ∆νh→0:
n(νL)∆ν≈h→01 + c
4π2
1
νL(νfi−νL) α(νfi) dνfi . (4.17) This approximation also demonstrates that pih has to vanish sufficiently fast to provide convergence towards infinity. (4.17) then yields surprisingly reliable results. It should again be stressed that throughout this thesis nS stands for the number density andn for the refractive index in the sample.
4.3.3 Boltzmann statistics
As is obvious from equations (4.15) and (4.16), for instance, the amount of absorption in any sample is also influenced by the actual value of the so far unappreciated dark field population difference ∆0. Even more, its sign deter-mines whether light is absorbed or coherently emitted. The case of inversion (∆0 <0) is in fact the basis for laser operation.
Under normal circumstances, the initial and final states are only populated by thermal excitation, so that
0ii,ff= 1
66 OPTICAL SAMPLE DETECTION
follow a Boltzmann distribution with the partition sum Q(T) =
over all possible quantum numbers F. Note that the above definition makes 0ii,ff the population of each individual magnetic sublevel, in consistence with the two-level approach.∆0 then becomes
∆0(T) = exp[−Ei/kBT]
In quantitative spectroscopy, absorption line strengths are often characterized in terms of the spectral line intensitySif(T), which is related to the frequency-dependent absorption coefficient by
α(νL) =nSSif(T)p(νL, ν0, ∆ν), (4.18) wherep(νL, ν0, ∆ν) is the relevant normalized lineshape function. The spectral line intensity thus is independent of the specific form of the absorption profile.
Equation (4.18) assumes|∆m|2 = 1, as is adequate in the absence of externally imposed quantization axes. A comparison with the above results forα(νL) yields
Sif(T) =∆0(T) λ2fi 8π Afi
in the present model. When it is given for any reference temperature Tref, the spectral line intensity at other values ofT may consequently be calculated from
Sif(T) =Sif(Tref) Q(Tref) Of course, this requires exact knowledge of the energy of the involved states as well as the value of the partition sum at the respective temperatures.