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RJ Jones Optical Sciences OPTI 511L Fall 2008
Experiment: Saturation Spectroscopy _____ ~ 2 lab sessions
In this experiment we explore the use of a single mode tunable diode laser to perform saturation spectroscopy in Cesium vapor.
Objectives:
I To show that a semiconductor laser can be used as the basis of a high-power, single frequency tunable laser system. We will characterize the laser in terms of frequency stability, and calibrate the laser frequency tuning. At the end of the lab we will lock the laser frequency to an atomic transition.
II To study the level structure of a real atom, as opposed to the familiar, idealized 2- level atom. We will set up and perform saturated absorption spectroscopy in a room temperature, low pressure vapor of Cs. This will allow us to observe the Cs Doppler profile, the Lamb dips and cross-over dips. We can also directly observe the effects of power broadening on an atomic transition.
III To understand the complex hyperfine structure of the Cs D 2 transition. This is accomplished using the quantum theory of angular momenta. The insight that we gain is applicable also in other atomic and molecular systems. In particular we will show that the separation between hyperfine states obey Landé's interval rule.
Prelab Questions:
Be sure to include these in your lab notebook.
(1) Diode Laser Setup: Determine the spectral response of the semiconductor cavity (assume no gain/loss in the semiconductor material) - that is: compute the free spectral range and the width of the cavity resonances. Assume that the diode laser cavity length is approximately l = 0.3 mm , that the semiconductor index of refraction is approximately n = 3.6, and that laser facets have a reflectivity of 30%. Compute also the free spectral range of the external cavity, assuming that external cavity length L ≈ 2 cm. How many external cavity modes fall inside the FWHM of a semiconductor cavity mode? Given the homogeneous gain of the semiconductor laser, is it reasonable to assume that only one of these external cavity modes will laser? Give an order-of-magnitude estimate for the expected single frequency tuning range when we scan the length of the external cavity . (2) Cs atomic transition. Calculate the Doppler broadened linewidth of a Cs vapor at room temperature ( m Cs = 2.21× 10 −^25 Kg ). The vapor pressure of Cs in the absorption cell is roughly 10 −^10 atmosphere. Calculate the number density of Cs atoms (use ideal gas law). Given a homogeneous linewidth of5.22 MHz , and the Doppler width calculated above, what fraction of the atoms interact with the laser at the center of the Doppler profile? Assuming a resonant scattering cross section of 3 λ^2 2 π , estimate the resonant absorption coefficient for a low intensity beam. What is the attenuation of a resonant beam passing though the 5 cm long Cs cell? Given the Cs hyperfine structure shown in fig. 6) (see attached discussion), compute the relative position of the Lamb and cross-over dips in the saturated absorption spectrum, for the multiplets starting from the F = 3 and F = 4 ground states respectively. Use these numbers to label the positions of the of the real transitions and cross-over resonances in your saturation absorption experiment.
Hyperfine structure:
- Modify your setup for a saturated absorption experiment (Fig. 2 gives one simple example for the configuration, your TA will show you other possibilities which may be more practical), taking appropriate care to avoid optical feedback to the laser while still maintaining good beam overlap in the vapor cell. Use the function generator to ramp the laser frequency repeatedly across one of the hyperfine multiplets, and display the transmission vs. frequency on an oscilloscope. You will see a number of dips in the absorption - these lines arise when the frequency of the counter-propagating laser beams coincide with various hyperfine transitions in the Cs vapor. Measure the relative (frequency) separations of these features (eventually you can calibrate the scan using the known frequency separations).
- Identify the transitions you are measuring^1. Based on the results from problem (ii) above, assign the observed features to saturation and cross-over lines. Confirm the Landé interval rule (see last page of attached discussion). Are you seeing the multiplet starting from the F = 3 or the F = 4 ground state?
Natural linewidth, power broadening:
- Measure the linewidth of one of the saturated absorption features as a function of laser power if possible. Compare functional dependence to theory for the power- broadened 2-level atom. Are you able to extrapolate to zero power and estimate the natural linewidth? (Ideally, you could float the laser table to minimize laser frequency jitter, but this will not be possible in our experiment.)
(^1) For now, at least record all the information you can in the scan. We will discuss Lande’s Interval rule and the atomic structure of Cs in our next lecture.
Optical pumping:
- You will observe that the Doppler profiles for the multiplets corresponding to F = 3 and F = 4 ground states are centered on the F = 3 → F ′= 2 and F = 4 → F ′= 5 transitions respectively (that is, the Doppler broadened background is not symmetric) Can you explain this in terms of optical pumping between hyperfine states?
Laser frequency locking:
One important use of saturation spectroscopy is to stabilize the frequency of a tunable laser with respect to an atomic transition. The technique is crucial for experiments on laser cooling and trapping. We accomplish frequency locking by deriving an error signal proportional to the instantaneous frequency deviation, and use it to push the laser frequency back towards the desired value. The feedback loop is established using a "lock-box" containing the necessary signal conditioning electronics. The electronics allow us to set the lock point (the center laser frequency) and to adjust the feedback loop gain to achieve maximum laser stability. The TA will help explain the basic principle of operation of the feedback loop.
- Establish the feedback loop, and lock the laser somewhere on the red-detuning (low- frequency) side of the absorption profile. Adjust the lock reference level until the laser is locked ~75 MHz to the red of the F = 4 → F ′ = 5 transition. Adjust the feedback gain until the lock goes into oscillations, then reduce the gain again, until oscillations just disappear. The lock now operates close to optimum conditions. How robust is the lock against external disturbance? What happens if you gently knock the table near the laser box?
- Evaluate the frequency stability of the locked laser. Calibrate the error signal from the saturation setup, i.e. determine the RMS deviation in mV from the set point and calibrate the scan (MHz/volt) using the atomic transitions. Tweak the lock gain to obtain the best compromise between frequency stability and lock robustness.
collimation lens
diffraction grating 1200 grooves/mm
semiconductor laser 1x3μm strip waveguide
output beam
external cavity
Fig. 1): Schematic outline of the external cavity, grating stabilized diode laser. At one specific wavelength the -1 diffraction order from the diffraction grating is fed back into the diode laser, thereby defining an external cavity. The grating is mounted so that both the diffraction angle and the cavity length can be finely tuned, and laser tunability can be achieved.
Saturated absorption spectroscopy
Please note: in this discussion, Planck's constant is used a number of times. Because of computer and font incompatibilities, = shows up as h.
The Doppler width of a room temperature Cs vapor is comparable to the separations between atomic hyperfine levels in the excited state. It is therefore not possible to resolve the excited state hyperfine structure of Cs by measuring the frequency dependent absorption in a Cs vapor. One might speculate if it is possible to reduce the Doppler width of a Cs vapor by cooling the cell. Unfortunately the Doppler width varies slowly with temperature, as T 1 2^ , while the vapor pressure varies very fast, approximately as exp (− η T ), so the absorption vanishes before the Doppler profile is significantly narrowed. In this experiment we investigate the Cs hyperfine spectrum through the use of a spectroscopic technique known as saturated absorption spectroscopy, which is capable of achieving sub-Doppler resolution.
Case of a 2-level atom. The Lamb dip. Consider a 2-level atom interacting with two counterpropagating plane waves, in the configuration of fig. 2. We assume that the plane waves both have polarization εˆ and
frequency ω. The waves propagate in the ± z ˆ directions, and have amplitudes E 1 and E 2 ,
E 1 >> E 2. We refer to the plane waves (1) and (2) as the "pump" and the "probe" respectively. The electric fields are:
G
E 1 = εˆ E 1 e − i^ (^ ω t^ −^ kz^ )^ ,
G
E 2 = εˆ E 2 e − i^ (^ ω t^ + kz^ )
The distribution of atomic velocities along the ˆ z -axis is Maxwell-Boltzmann, i.e. the number density of atoms with velocities in the interval between v and v + dv is
N ( ) vdv = N
m
2 π kB T
e −^ mv
(^2 2) k (^) BT dv ,
where N is the total number density of atoms. In the presence of the pump wave only,
we can find the steady state number densities N g ( v ) and N e ( v ) of atoms in the ground
and excited states (see e. g. "Lasers", Milonni & Eberly, chapter 7):
N e ( ) v =
N ( ) v σ ω −( kv ) Φ
A + 2 σ ω −( kv ) Φ
, N g ( v ) = N ( v ) − Ne ( ) v ,
where
3 λ^2
A^2
4 Δ^2 + A^2
where I 0 = A 2 σ (ω 0 )=ω is the saturation intensity (≈ 1.1 mW cm 2 for the Cs D 2 transition). For I << I 0 the width of the hole is the natural linewidth A (≈ 5.22 MHz for the Cs D 2 transition).
v (^) (m s )
v = Δ^ k^ Ng ( ) v
Fig. 3): Hole-burning in N g ( ) v , the velocity dependent number density of ground
state atoms. In this example T = 300 K , Δ = − 10 A and I ≈ I 0.
Because the probe beam is much less intense than the pump beam, its presence does not significantly affect the number density of ground and excited state atoms. We then find the following expression for the extinction coefficient experienced by the probe at
frequency ω :
a ( ω ) = σ ω +( kv )[ N (^) g ( ) v − Ne ( ) v ] dv −∞
∞ ∫
The transmission through a cell of length l can now be found as e −^ a (^ ω^ ) l^. Note that saturated absorption will result in increased transmission, and thus a more intense output beam in the setup of fig. 2.
A more thorough analysis shows that, for I << I 0 and in the case of an optically thin gas, there is a minimum in the absorption of the probe with a FWHM equal to the homogeneous linewidth A.
Fig. 4 shows the detuning dependent absorption of the probe in the presence of the pump. A sharp dip - the Lamb dip - is evident for Δ = 0, i. e for a laser frequency close
to the atomic transition frequency. We can understand the physical origin of the Lamb
dip as follows: the pump depletes N g ( v ) for a velocity class around vpump = Δ k. At the
same time the probe interacts with a velocity class around vprobe = − Δ k. As long as these velocity classes are distinct, i. e. as long as the pump and probe interacts with different atoms, the presence of the pump does not affect the absorption of the probe. However, when Δ = 0 we have vpump = vprobe , i. e. the pump and the probe interacts with the same velocity class. The depletion of ground state atoms caused by the pump then leads to a reduction in the absorption of the probe. By measuring the position of the Lamb dip we can then determine the atomic frequency with a resolution comparable to the natural linewidth A , even if the Doppler broadened linewidth is many times larger.
a (^) (m −^1 )
Δ A
Fig. 4): Lamb dip in a gas of 2-level atoms with the mass and transition parameters of Cs. Parameters as in fig. 3), also N = 10 10 cm -^.
Case of a 3-level atom. The cross-over dip. We now consider saturated absorption by a sample of atoms with a single ground state g and two excited states e 1 , e 2 , as shown in fig. 5. As the pump/probe frequency is scanned across the absorption profile one observes two independent Lamb
dips, one for ω =ω 1 and one for ω = ω 2. As explained above these Lamb dips occur
whenever the pump/probe frequency is equal to an atomic transition frequency. An additional feature is found in the saturated absorption spectrum of this multilevel
atom. Consider a velocity class around a velocity v such that ω + kv =ω 2 and
ω − kv =ω 1. For these atoms the pump will excite the transition g ↔ e 1 , and deplete
Optical Pumping. Consider a sample of Cs atoms (see level scheme in fig. 6) initially in the F = 4 ground state. If these atoms are excited on the F = 4 ↔ F ′= 5 transition, there will be some population of the F ′= 5 excited state. Because of the selection rule Δ F = 0,±1, atoms in the F ′= 5 excited state can only decay to the F = 4 ground state. Such a transition is referred to as a closed transition, since the total number of atoms in the F = 4 and F ′ = 5 states is obviously conserved.
Consider now a sample of Cs atoms initially in the F = 4 ground state, but let these atoms be excited on the F = 4 ↔ F ′ = 4 transition, so that there will be some population of the F ′= 4 excited state. Atoms in the F ′= 4 excited state can decay both to the F = 4 and F = 3 ground state. An atom that decays to the F = 4 ground state can be excited again; however an atom that decays to the F = 3 ground state stops interacting with the light field, since all transitions beginning from the F = 3 ground state are very far detuned from resonance (see fig. 6 ). Atoms that decay to the F = 3 ground state will not be excited again, and can never be transferred back to the F = 4 ground state. Hence, after a few absorption/spontaneous decay cycles, all atoms in the sample will be transferred to the F = 3 ground state, and the sample will become transparent. This process is referred to as optical pumping.
The Cs atomic structure:
Electron configuration: The only stable, naturally abundant isotope of Cesium is 13355 Cs. The ground state
electronic configuration is
(^1 s^2 2 s^^2 2 p^6 3 s^2 3 p^6 3 d^10 4 s^2 4 p^6 4 d^10 5 s^2 5 p^6 )^6 s
i. e. the atom has a single 6 s electron outside a core composed of completely filled subshells. The first excited state of the Cs atom corresponds to a transfer of the 6 s electron to a 6 p state. It has long been known that when one looks at the 6 s − 6 p transition in the spectrum of light emitted from a Cs discharge, one sees two closely spaced resonance lines, historically named the D 1 and D 2 lines. This splitting of the 6 s − 6 p spectral line is known as fine structure. As demonstrated in this experiment, saturated absorption spectroscopy reveal many more closely spaced resonance lines - this splitting in turn is known as hyperfine structure. The hyperfine levels involved in the D 2 transition are shown in fig. 6). All the splittings between the hyperfine levels have very simple physical interpretations in terms of the interaction between the magnetic moments associated with the electron orbital and spin angular momentum, and the angular momentum of the atomic nucleus. To permit such an interpretation, we briefly review the basic elements of angular momentum theory in quantum mechanics.
Orbital angular momentum, the quantum number L: In quantum mechanics electrons moving in the Coulomb potential created by the nucleus have orbital angular momentum L ˆ. L ˆ is defined as in classical mechanics: L ˆ = ˆ r × p ˆ , but L ˆ and the position ˆ r and the momentum ˆ p are operators. Consider now
the component L ˆ z = y ˆ p ˆ x − x ˆ p ˆ y. The Heisenberg uncertainty relations prevent a
simultaneous measurement of e. g. ˆ x and ˆ px ; hence one cannot simultaneously measure
all three components of L ˆ. More specifically, one can use the commutation relations for r ˆ and ˆ p to derive the commutation relations for the components of L ˆ ; this proves that
the components of L ˆ do not commute and therefore are not simultaneously observable. In other words: an atomic state cannot correspond to well defined values of all three components L ˆ x , L ˆ y and L ˆ z. If we define L ˆ^2 = L ˆ^2 x^ + L ˆ^2 (^) y^ + L ˆ^2 (^) z^ , it can however be shown that
atomic states correspond to well defined values of L ˆ^2 and L ˆ z. This simply means that
only the magnitude of L ˆ and its projection on one axis can be measured simultaneously. The eigenvalues (possible outcomes of a measurement) are
L ˆ^2 : =^2 L L ( + 1 ) L ≥ 0, integer
L^ ˆ z : = m (^) L − L ≤ m (^) L ≤ L , integer (1)
S ˆ^2 : =^2 S S ( + 1 ) = 3 =^2 4 S = 1 2
S ˆ (^) z : = m (^) S m (^) S = ±1 2. (2)
Total electron angular momentum, the quantum number J : Orbital and spin angular momenta of all the electrons in an atom can be added, resulting in a total angular momentum J ˆ = L ˆ + S ˆ. Again we have eigenvalues and quantum numbers
J ˆ^2 : =^2 J J ( + 1 ) L − S ≤ J ≤ L + S , integer or half - integer
J^ ˆ (^) z : = m (^) J − J ≤ m (^) J ≤ J , integer or half − integer (3)
Whenever two angular momenta J ˆ 1 and J ˆ (^) 2 are added to yield a total angular momentum
J ˆ = J ˆ 1 + J ˆ (^) 2 , then the quantum number J takes on all the values J 1 − J 2 ≤ J ≤ J 1 + J 2. If for example J 1 = 4, J 2 = 2 , then the possible values of J are 2,3, 4,5,.
We can now add the angular momenta of all the electrons in an atom. There is some significance to the order in which this is done; we will however ignore this question here. One important result is that the total angular momentum of all the electrons in a filled subshell is zero. This is easy to show by arranging electrons into pairs with quantum numbers ( m (^) L , − m (^) L ) and ( m (^) S ,− m (^) S ); hence (^) ∑ m (^) L + (^) ∑ mS = 0 and J ˆ (^) z = 0. But if the only
allowed eigenvalue of J ˆ (^) z is zero, then J = 0. For an atom such as Cs , consisting of a series of closed shells plus a single unpaired electron with orbital and spin angular quantum numbers L and S = 1 2 , the possible values of the quantum number for the total angular momentum are J = L ± 1 2.
Fine-structure: With an electron in an atomic state one can associate a probability current J that
circulates around the atomic nucleus - this is the quantum mechanical analogy of the classical concept of an orbiting particle. Since an electron is charged, the probability current is equivalent to a circulating electrical current e J. This current in turn gives rise
to a magnetic dipole moment μ L ∝ L ˆ.
Similarly, an electron has an intrinsic magnetic moment μ S =μ B S^ ˆ , where μ B is the
Bohr magneton.
Depending on the relative orientation of the angular momenta L ˆ and S ˆ , there is an interaction energy associated with the magnetic dipoles μ L and μ S. Thus there is a shift Δ E of the energy of the atomic states which is proportional to μ L ⋅ μ S , and therefore
proportional to L ˆ ⋅ S ˆ. But if we remember that
L ˆ ⋅ S ˆ =
J^ ˆ^2 − L ˆ^2 − S ˆ^2
and also remember that the eigenvalues of the operators J ˆ^2 , L ˆ^2 and S ˆ^2 are given by eqs. 1, 2 and 3, then we see that
Δ E ∝ J J ( + 1 ) − L L ( + 1 ) − 3 4. (5)
This allows us to compute the relative values of the fine-structure shift for states with the same quantum number L but different values of the quantum number J.
Consider now the ground state of the Cs atom. Because the unpaired electron is in a 6 s state we have L = 0 , S = 1 2 and J = 1 2. Because there is only one possible value of J there is only one ground state. Also, the value of the energy shift is Δ E = 0 (Note: there are other contributions to the fine-structure in atoms that do lead to a shift of the ground state; these contributions do not produce a splitting of states and are therefore ignored here).
Consider then the first excited state of the Cs atom. The unpaired electron is in the 6 p state and we have L = 1 , S = 1 2 and J = 1 2,3 2. The two states corresponding to
different values of J are shifted by Δ E 1 2 = − 2 κ 6 p and Δ E 3 2 = κ 6 p , where κ 6 p is a
constant that depends in a very complicated way on the electronic structure of the Cs atom. For atomic species (such as Cs) with a valence subshell that is less than half filled,
the constant κ is positive, whereas for a valence subshell that is more than half filled κ
is negative. We now see why the Cs 6 s − 6 p transition is actually split into two lines
with a separation given by the upper state fine-structure splitting.
From eq. 5 we see that the energy splitting of two states with angular momentum quantum numbers J and J − 1 is
δ EJ = Δ EJ − Δ E J − 1 ∝ J J ( + 1 ) − ( J − 1 ) J ∝ J , (6)
i. e. the fine-structure splitting is proportional to the total angular momentum of the upper state. This result is known as Landé's interval rule. For Cs the fine structure splitting of the 6 s − 6 p transition is approximately 42 nm. We are therefore not able to observe fine
structure in this experiment.
Hyperfine structure: We have so far determined the (relative) fine structure splitting of atomic states characterized by different quantum numbers J for the total electron angular momentum. We must now take into account the interaction between the total magnetic moment of the electrons and the nuclear magnetic moment. The nuclear magnetic moment is
