--- title: Spectroscopic Techniques and Frequency Combs module: Lasers and Spectroscopy moduleNumber: 8 lessonNumber: 2 order: 802 summary: > A tunable laser turns spectroscopy from photographing a spectrum into interrogating a single transition, but at room temperature the Doppler width buries the natural linewidth under a thousandfold-broader Gaussian. Saturated absorption and two-photon spectroscopy defeat the first-order Doppler shift by selecting the zero-velocity class or cancelling the shift between counter-propagating photons, recovering natural-width features. Laser-induced fluorescence pushes sensitivity to single atoms, and the optical frequency comb converts an optical frequency into a countable radio-frequency beat, giving absolute frequency measurement across the visible spectrum. topics: [Lasers and Spectroscopy] sources: - book: Demtröder ref: "Ch. 9 — Doppler-Limited Spectroscopy; Ch. 10 — Nonlinear and Doppler-Free Spectroscopy; §10.5 Optical Frequency Combs" - book: Foot ref: "Ch. 8 — Doppler-Free Laser Spectroscopy; §8.3 Saturated Absorption, §8.4 Two-Photon Spectroscopy" - book: Loudon ref: "The Quantum Theory of Light, 3rd ed. — line shapes and coherence" - book: NIST ASD ref: "reference wavelengths and A-coefficients — https://www.nist.gov/pml/atomic-spectra-database" draft: false --- Spectroscopy measures the frequencies of atomic transitions. With a tunable single-mode laser the measurement becomes active: sweep the laser across a transition and record how much light the atoms absorb or how brightly they fluoresce, and the lineshape is read out directly rather than dispersed onto a photographic plate. The precision then runs into a wall built by thermal motion. The natural width of an allowed optical transition is of order $10\ \mathrm{MHz}$, but at room temperature the [Doppler broadening](/atomic-physics/radiative-transitions-and-line-shapes/lifetimes-and-line-shapes) of the same line is a thousand times larger, so the quantity that carries the physics — the line center to a fraction of the natural width — is invisible under a Gaussian smear. This lesson develops the techniques that beat the Doppler width and the instrument that measures the surviving frequencies absolutely. Saturated-absorption spectroscopy selects the single velocity class at rest along the beam; two-photon spectroscopy cancels the first-order Doppler shift for every atom at once. Both recover natural-width features from a Doppler-broadened sample. The optical frequency comb then ties any optical frequency to a countable radio-frequency beat against an atomic clock, the advance that made [optical clocks](/atomic-physics/modern-atomic-physics/optical-clocks-precision) possible. ## Absorption and emission spectroscopy Two complementary measurements read a transition. In **absorption**, light of known frequency passes through the sample and the transmitted intensity is recorded against frequency. Over a path length $\ell$ the transmission follows Beer's law, $$ I(\nu) = I_0\,e^{-\kappa(\nu)\ell}, \qquad \kappa(\nu) = \left(N_1 - \frac{g_1}{g_2}N_2\right)\sigma(\nu), $$ with $\kappa$ the absorption coefficient built from the same cross section $\sigma(\nu)$ that appears in [gain](/atomic-physics/lasers-and-spectroscopy/laser-principles), now with the equilibrium sign that makes it a loss. The measured quantity is the lineshape $\sigma(\nu)$ and, through its integral, the absolute transition strength. In **emission**, the sample is excited — thermally, by discharge, or by a laser — and the spontaneously emitted light is dispersed and recorded. The intensity of a line is proportional to the upper-level population and the Einstein $A$-coefficient, $I_{21} \propto N_2 A_{21} h\nu_{21}$, so emission reports which upper levels are populated and how fast they decay. Absorption and emission see the same transition frequencies but weight them differently: absorption favors transitions from populated lower levels, emission favors fast transitions from populated upper levels. Both, taken with a broadband source or a scanned laser, are **Doppler-limited** — the recorded line is the Gaussian convolution of the natural profile with the thermal velocity distribution. ## The Doppler limit An atom moving with velocity component $v_z$ along the laser beam sees the laser frequency $\nu$ shifted into resonance with its rest-frame transition frequency $\nu_0$ when $$ \nu\left(1 - \frac{v_z}{c}\right) = \nu_0 \quad\Longleftrightarrow\quad v_z = c\,\frac{\nu - \nu_0}{\nu_0}. $$ Each velocity class absorbs at a different laser frequency. The one-dimensional velocity distribution is Maxwellian, $$ P(v_z)\,\d v_z = \sqrt{\frac{m}{2\pi k_{\mathrm B}T}}\, \exp\!\left(-\frac{m v_z^2}{2 k_{\mathrm B}T}\right)\d v_z, $$ so mapping velocity to frequency gives a Gaussian absorption profile of full width at half maximum $$ \Delta\nu_{\mathrm D} = \frac{\nu_0}{c}\sqrt{\frac{8 k_{\mathrm B}T\ln 2}{m}} = \nu_0\sqrt{\frac{8 k_{\mathrm B}T\ln 2}{m c^2}}. $$ The scale is set by the thermal speed against $c$. For the sodium $D$ lines ($\nu_0 \approx 5.1\times 10^{14}\ \mathrm{Hz}$, $m = 23\ \mathrm{u}$) at $T = 500\ \mathrm{K}$, $\Delta\nu_{\mathrm D} \approx 1.7\ \mathrm{GHz}$, against a natural width $\Gamma/2\pi \approx 10\ \mathrm{MHz}$. The Doppler width exceeds the natural width by more than two orders of magnitude, and it hides the fine and hyperfine structure that lies within it. Cooling narrows the Gaussian only as $\sqrt{T}$, so reaching the natural width thermally would demand impractical temperatures; the Doppler-free methods sidestep the broadening instead of reducing the motion. ## Saturated-absorption spectroscopy The trick is to make two counter-propagating beams interrogate the _same_ atoms and arrange that they can only agree at line center. A strong **pump** beam and a weak **probe** beam, derived from the same laser so they share a frequency $\nu$, pass through the vapor cell in opposite directions along the same axis. The transmitted probe is recorded. $$ % caption: A saturated-absorption setup: a beamsplitter derives a strong pump and a % weak probe from one laser; they counter-propagate through the cell and the probe % is detected. Only atoms at rest along the beam interact with both. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} % cell \fill[black!8] (2.5,-0.5) rectangle (6.5,0.5); \draw[black] (2.5,-0.5) rectangle (6.5,0.5); \node[black, anchor=south] at (4.5,0.55) {vapor cell}; % laser + beamsplitter \node[draw, minimum width=9mm, minimum height=6mm] (las) at (-0.6,0) {laser}; \draw[black!70, very thick, ->] (las.east) -- (8.2,0); \node[black!70, anchor=south] at (7.4,0.05) {pump}; \draw[fill=black!8] (1.2,-0.3) -- (1.8,0.3) -- (1.8,-0.3) -- cycle; \node[anchor=north east, font=\scriptsize] at (1.25,-0.4) {splitter}; % probe path (down from splitter, back through cell) \draw[acc, thick, ->] (1.5,-0.3) -- (1.5,-1.3) -- (8.6,-1.3) -- (8.6,-0.25) -- (2.3,-0.25); \node[acc, anchor=north, font=\scriptsize] at (5,-1.3) {probe (counter-propagating)}; % detector \node[draw, minimum width=9mm, minimum height=6mm] (det) at (0.4,-0.25) {detector}; \draw[acc, ->] (2.3,-0.25) -- (det.east); \end{tikzpicture} $$ Follow the pump first. At a detuning $\nu - \nu_0$ the pump saturates — bleaches — the single velocity class that it Doppler-shifts into resonance, namely $v_z = +c(\nu-\nu_0)/\nu_0$. Those atoms spend a large fraction of their time in the upper level, so the population difference $\Delta N$ for that class is driven toward zero: the pump burns a **hole** in the velocity distribution of ground-state atoms. $$ % caption: The pump saturates one velocity class, burning a hole in the ground-state % velocity distribution. On line center that class is v_z at rest, and the % counter-propagating probe addresses the very same atoms. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (-3.6,0) -- (3.8,0) node[right] {velocity along beam}; \draw[->, black] (-3.6,0) -- (-3.6,3.0) node[above] {ground population}; % maxwellian \draw[black, very thick] plot[smooth, domain=-3.3:3.5, samples=60] ({\x}, {2.6*exp(-\x*\x/3.0)}); % hole burned at v = +1.1 \draw[acc, very thick] (1.1,{2.6*exp(-1.1*1.1/3.0)}) -- (1.1,{2.6*exp(-1.1*1.1/3.0)-1.1}); \fill[white] (1.1,{2.6*exp(-1.1*1.1/3.0)-0.55}) circle (2pt); \node[acc, anchor=south] at (1.1,{2.6*exp(-1.1*1.1/3.0)+0.05}) {hole}; \node[black, anchor=north, font=\scriptsize] at (1.1,0) {saturated class}; \draw[black, dashed] (0,0) -- (0,2.6); \node[black, anchor=south, font=\scriptsize] at (0,2.6) {at rest}; \end{tikzpicture} $$ The probe, travelling the other way, resonates with atoms of the _opposite_ sign, $v_z = -c(\nu-\nu_0)/\nu_0$. For any detuning $\nu \neq \nu_0$ the probe and pump talk to different atoms, and the probe sees the ordinary Doppler-broadened absorption. At exactly $\nu = \nu_0$ both beams address the same class, the one with $v_z = 0$ — and the pump has already bleached it. The probe therefore finds reduced absorption in a narrow window at line center: a **Lamb dip** whose width is set by the natural linewidth and the power broadening, not by the Doppler width. > **Definition (Lamb dip).** The narrow decrease in absorption at line center in a > counter-propagating pump–probe experiment, arising because both beams saturate the > single zero-velocity class there. Its width approaches the homogeneous (natural > plus power-broadened) width, recovering sub-Doppler resolution from a > Doppler-broadened sample. The depth and width of the dip are set by the pump's **saturation parameter** $s = I/I_{\text{sat}}$, the ratio of the pump intensity to the saturation intensity of the transition. The saturated absorption coefficient of the pumped class scales as $\kappa_0/(1 + s)$, so a strong pump ($s \gg 1$) bleaches the class almost completely and maximizes the dip, at the cost of **power broadening** the dip to a width $\Delta\nu_{\text{nat}}\sqrt{1 + s}$. Practical saturated-absorption spectroscopy trades dip contrast against dip width by choosing $s$ of order unity, keeping the recovered width within a small multiple of the natural width while retaining a detectable signal. The line center located this way is the frequency reference that locks lasers to atomic transitions — a saturated-absorption cell of rubidium or iodine is the standard optical frequency anchor in the laboratory. When several transitions share a lower level and lie within one Doppler width, the pump for one and the probe for another can address the same class at the midpoint frequency, producing **crossover resonances** halfway between each pair of true lines. These extra dips are diagnostic, not artifacts: their positions fix the true line separations, and they are often stronger than the real dips. ## Two-photon Doppler-free spectroscopy A second route cancels the Doppler shift for every atom at once. An atom absorbs two photons from two counter-propagating beams of the same frequency $\nu$, reaching a level with $E_f - E_i = 2h\nu$. In the atom's frame the two beams are Doppler-shifted oppositely, $$ \nu\left(1 + \frac{v_z}{c}\right) + \nu\left(1 - \frac{v_z}{c}\right) = 2\nu, $$ so the sum is $2\nu$ independent of $v_z$: the first-order Doppler shift cancels term by term for _every_ velocity class. The entire thermal ensemble contributes to a single narrow resonance at $2\nu = (E_f - E_i)/h$, rather than being spread across a Gaussian. $$ % caption: Two-photon absorption from two counter-propagating beams of equal % frequency. The two Doppler shifts are equal and opposite, so their sum is % velocity-independent and every atom contributes to one narrow line. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} % atom \fill[black] (0,0) circle (0.45); \draw[black] (0,0) circle (0.45); \node[black!70] at (0,0) {atom}; % velocity \draw[black, ->, thick] (0,0.7) -- (1.1,0.7); \node[black, anchor=south, font=\scriptsize] at (0.55,0.7) {velocity}; % beam 1 from left \draw[acc, very thick, ->] (-4,0) -- (-0.6,0); \node[black, anchor=south] at (-2.3,0.05) {beam 1}; \node[black, anchor=north, font=\scriptsize] at (-2.3,-0.05) {shift up}; % beam 2 from right \draw[acc, very thick, ->] (4,0) -- (0.6,0); \node[black, anchor=south] at (2.3,0.05) {beam 2}; \node[black, anchor=north, font=\scriptsize] at (2.3,-0.05) {shift down}; \end{tikzpicture} $$ The two-photon transition connects states of the same parity ($\Delta\ell = 0, 2$), which are inaccessible to a single dipole photon, so the method reaches transitions a one-photon experiment cannot. The residual broadening is the natural width plus the **second-order Doppler shift** from relativistic time dilation, $$ \frac{\Delta\nu}{\nu_0} = -\frac{v^2}{2c^2}, $$ which does not cancel because it is even in velocity. This term is the systematic that limits confined-atom and two-photon spectroscopy, and removing it is one of the motivations for [laser cooling](/atomic-physics/modern-atomic-physics/laser-cooling-doppler). The canonical application is the hydrogen $1S\to 2S$ transition, whose $2S$ level is metastable and whose narrow two-photon line is one of the most precisely measured frequencies in physics, a benchmark for the Rydberg constant and the [Lamb shift](/atomic-physics/qed-corrections-and-hyperfine-structure/lamb-shift-qed). ## Laser-induced fluorescence Absorption spectroscopy measures a small dip on a large transmitted signal; sensitivity is limited by the noise on that background. Laser-induced fluorescence inverts the geometry: excite the atoms with the laser and detect the spontaneously emitted photons at right angles, against a dark background. Every excited atom re-emits, and with the laser tuned to resonance an atom can be cycled through excitation and emission millions of times per second, scattering enough photons to be seen individually. - **Background-free detection.** The scattered fluorescence is collected away from the laser axis, so the signal is photons counted against darkness rather than a small change on a bright beam. Single-atom and single-ion detection rests on this. - **State selectivity.** Tuning the laser to a chosen transition excites only atoms in the corresponding lower level, so the fluorescence intensity reports the population of that specific state — the readout used in [optical clocks](/atomic-physics/modern-atomic-physics/optical-clocks-precision) and cold-atom experiments. Combined with a Doppler-free excitation geometry, fluorescence detection yields natural-width lines with the sensitivity to work at vanishingly low atom numbers. ## The optical frequency comb Measuring an optical frequency absolutely means comparing it to the cesium standard, which ticks at $9.19\ \mathrm{GHz}$. An optical frequency is near $5\times 10^{14}\ \mathrm{Hz}$, five orders of magnitude higher, and no electronic counter reaches it. The optical frequency comb bridges the gap in a single device: it is a gear that meshes the radio-frequency and optical domains. A **mode-locked** laser emits a train of identical ultrashort pulses separated by the cavity round-trip time $T = 1/f_{\text{rep}}$. The Fourier transform of a train of pulses spaced by $T$ is a comb of discrete frequencies spaced by $f_{\text{rep}}$. Because the pulse envelope travels at the group velocity while the carrier travels at the phase velocity, the carrier slips relative to the envelope by a fixed phase $\Delta\varphi$ each round trip, and this offsets the whole comb from the harmonics of $f_{\text{rep}}$ by the **carrier-envelope offset frequency** $$ f_{\text{ceo}} = \frac{\Delta\varphi}{2\pi}\,f_{\text{rep}}. $$ The frequency of the $n$-th comb tooth is therefore $$ \nu_n = n\,f_{\text{rep}} + f_{\text{ceo}}, \qquad n \sim 10^6, $$ with $n$ a large integer and both $f_{\text{rep}}$ and $f_{\text{ceo}}$ in the radio-frequency range where they can be counted against an atomic clock.[^demtroeder-comb] $$ % caption: The comb in the frequency domain: teeth spaced by the repetition rate, % the whole set offset from zero by the carrier-envelope frequency. Two counters at % radio frequency fix every optical tooth. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (10,0) node[right] {frequency}; % teeth \foreach \x in {1.6,2.4,3.2,4.0,4.8,5.6,6.4,7.2,8.0,8.8} { \draw[acc, very thick] (\x,0) -- (\x,2.2); } % offset from origin \draw[black, thick, <->] (0,0.5) -- (1.6,0.5); \node[black, anchor=south, font=\scriptsize] at (0.8,0.5) {$f_c$}; % spacing \draw[black, <->] (4.0,2.45) -- (4.8,2.45); \node[black, anchor=south, font=\scriptsize] at (4.4,2.45) {rep. rate}; \node[acc, anchor=south] at (7.2,2.25) {teeth}; \end{tikzpicture} $$ ### Self-referencing The offset $f_{\text{ceo}}$ is measured by an $f$–$2f$ interferometer, which requires the comb to span at least one octave (a factor of two in frequency), achieved by broadening the spectrum in a nonlinear fiber. Take a tooth $\nu_n = n f_{\text{rep}} + f_{\text{ceo}}$ from the red end and frequency-double it in a nonlinear crystal to $2\nu_n = 2n f_{\text{rep}} + 2 f_{\text{ceo}}$. Compare it with the existing tooth $\nu_{2n} = 2n f_{\text{rep}} + f_{\text{ceo}}$ at the blue end. Their beat note is $$ 2\nu_n - \nu_{2n} = f_{\text{ceo}}, $$ a radio-frequency signal read directly off a photodetector. With $f_{\text{rep}}$ and $f_{\text{ceo}}$ both counted and both locked to the clock, every optical tooth is known absolutely. ### Absolute frequency measurement An unknown laser at $\nu_L$ is measured by overlapping it with the comb on a photodetector and recording the beat $f_b$ against the nearest tooth $\nu_n$: $$ \nu_L = n\,f_{\text{rep}} + f_{\text{ceo}} + f_b. $$ Three radio-frequency quantities — $f_{\text{rep}}$, $f_{\text{ceo}}$, $f_b$ — and one integer $n$ (fixed to within one tooth by a coarse wavemeter) determine an optical frequency to the accuracy of the reference clock. The gear ratio is the tooth index: with $f_{\text{rep}} = 250\ \mathrm{MHz}$ and an optical frequency near $5\times 10^{14}\ \mathrm{Hz}$, the beat is against tooth $n \approx 2\times 10^6$, so a fractional error $\delta f_{\text{rep}}/f_{\text{rep}}$ in the microwave rate maps onto the same fractional error in the optical frequency. Counting the two radio-frequency signals to one part in $10^{15}$ therefore fixes the optical frequency to the same fractional precision, and because the comb spectrum is phase-coherent across its full octave, the ratio $n$ is exact rather than approximate: no accumulation of error occurs between the microwave and optical ends. The comb also runs in reverse: locked to an optical clock transition, it transfers that transition's stability down to the radio-frequency $f_{\text{rep}}$, delivering an optical clock's precision as a countable microwave output. $$ % caption: The f-2f self-referencing scheme. An octave-spanning comb has its red % end doubled and beat against its blue end; the beat note is the offset frequency, % closing the link between the comb and the clock. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (10,0) node[right] {frequency}; \draw[black, dashed] (5,-0.2) -- (5,2.6); \node[black, anchor=south, font=\scriptsize] at (5,2.6) {one octave}; % red-end tooth n \draw[acc, very thick] (1.6,0) -- (1.6,1.6); \node[black, anchor=south, font=\scriptsize] at (1.6,1.6) {tooth n}; % doubling arrow \draw[black, thick, ->] (1.6,1.1) .. controls (4,2.6) and (6.4,2.6) .. (8.4,1.7); \node[black, anchor=south east, font=\scriptsize] at (3.4,2.45) {double}; % blue-end tooth 2n and doubled tooth \draw[acc, very thick] (8.4,0) -- (8.4,1.6); \node[black, anchor=south, font=\scriptsize] at (8.4,1.9) {tooth 2n}; \draw[acc, very thick, densely dashed] (8.7,0) -- (8.7,1.2); \draw[black, thick, <->] (8.4,0.6) -- (8.7,0.6); \node[black, anchor=west, font=\scriptsize] at (8.85,0.6) {beat}; \end{tikzpicture} $$ ## What the techniques deliver The three ideas stack. Doppler-free excitation — saturated absorption or two-photon — recovers a natural-width line from a warm sample; fluorescence detection reads it out at single-atom sensitivity; the comb measures its center frequency absolutely against the clock. Together they take an atomic transition from a Gaussian blur on a plate to a frequency known to fifteen significant figures. The next lesson turns from generating and measuring spectra to reading them: how the [NIST Atomic Spectra Database](/atomic-physics/lasers-and-spectroscopy/line-catalog-nist-asd) organizes real level and transition data, and how a measured line is matched to a tabulated one. [^demtroeder-comb]: Demtröder, _Atoms, Molecules and Photons_, 2nd ed., §10.5 — the frequency comb $\nu_n = n f_{\text{rep}} + f_{\text{ceo}}$ from a mode-locked pulse train, the carrier-envelope offset from the group–phase velocity mismatch, and $f$–$2f$ self-referencing on an octave-spanning spectrum. . The 2005 Nobel Prize (Hall and Hänsch) recognized the technique.