Experiment SAS

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Saturated Absorption SpectroscopyExperiment SAS

University of Florida Department of PhysicsPHY4803L Advanced Physics Laboratory

Overview

You will use a tunable diode laser to carry outspectroscopic studies of the rubidium atom.You will measure the Doppler-broadened ab-sorption profiles of the D2 transitions at780 nm and then use the technique of satu-rated absorption spectroscopy to improve theresolution beyond the Doppler limit and mea-sure the nuclear hyperfine splittings, which areless than 1 ppm of the wavelength. A Fabry-Perot optical resonator is used to calibrate thefrequency scale for the measurements.

Saturated absorption experiments werecited in the 1981 Nobel prize in physics and re-lated techniques have been used in laser cool-ing and trapping experiments cited in the 1997Nobel prize as well as Bose-Einstein condensa-tion experiments cited in the 2001 Nobel prize.Although the basic principles are straightfor-ward, you will only be able to unleash the fullpower of saturated absorption spectroscopy bycarefully attending to many details.

References

1. Daryl W. Preston Doppler-free saturatedabsorption: Laser spectroscopy, Amer. J.of Phys. 64, 1432-1436 (1996).

2. K. B. MacAdam, A. Steinbach, and C.Wieman A narrow band tunable diodelaser system with grating feedback, and a

saturated absorption spectrometer for Csand Rb, Amer. J. of Phys. 60, 1098-1111

(1992).

3. John C. Slater, Quantum Theory ofAtomic Structure, Vol. I, (McGraw-Hill,1960)

Theory

The purpose of this section is to outline thebasic features observed in saturated absorp-tion spectroscopy and relate them to simple

atomic and laser physics principles.

Laser interactionstwo-level atom

We begin with the interactions between a laserfield and a sample of stationary atoms havingonly two possible energy levels. Aspects ofthermal motion and multilevel atoms will betreated subsequently.

The difference E = E1 E0 between theexcited state energy E1 and the ground stateenergy E0 is used with Plancks law to deter-mine the photon frequency associated withtransitions between the two states:

E = h0 (1)

This energy-frequency proportionality is whyenergies are often given directly in frequencyunits. For example, MHz is a common energyunit for high precision laser experiments.

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SAS 2 Advanced Physics Laboratory

There are three transition processes involv-ing atoms and laser fields:

Stimulated absorption in which the atomstarts in the ground state, absorbs a pho-ton from the laser field, and then ends upin the excited state.

Stimulated emission in which the atom startsin the excited state, emits a photon withthe same direction, frequency, and polar-ization as those in the laser field, and thenends up in the ground state.

Spontaneous emission in which the atomstarts in the excited state, emits a pho-ton in an arbitrary direction unrelated tothe laser photons, and then ends up in theground state.

Stimulated emission and absorption are as-sociated with external electromagnetic fieldssuch as from a laser or thermal (blackbody)radiation. We consider spontaneous emission

firsta process characterized by a transitionrate or probability per unit time for an atomin the excited state to decay to the groundstate. This transition rate will be denoted and is about 3.6 107/s (or 36 MHz) for therubidium levels studied here.

In the absence of an external field, any ini-tial population of excited state atoms woulddecay exponentially to the ground state witha mean lifetime t = 1/ 28 ns. In the rest

frame of the atom, spontaneous photons areemitted in all directions with an energy spec-trum having a mean E = h0 and a full widthat half maximum (FWHM) E given by theHeisenberg uncertainty principle Et = hor E = h. Expressed in frequency units,the FWHM is called the natural linewidth andgiven the symbol . Thus

=

2(2)

Figure 1: The Lorentzian line shape profile forresonance absorption.

For our rubidium levels, E 2.5 108 eVor 6 MHz.1

The stimulated emission and absorptionprocesses are also described by a transitionratea single rate giving the probability perunit time for a ground state atom to absorba laser photon or for an excited state atom toemit a laser photon. The stimulated transi-tion rate is proportional to the laser intensity

I (SI units of W/m2

) and is only significantlydifferent from zero when the laser frequency is near the resonance frequency 0. This tran-sition rate will be denoted I, where

= 0L(, 0) (3)and

L(, 0) = 11 + 4( 0)2/2 (4)

gives the Lorentzian (or natural resonance)

frequency dependence as shown in Fig. 1.1The natural linewidth normally represents the

sharpest observable energy distributions, but mostattempts to measure it in gases are confounded byDoppler shifts associated with the random thermalmotion of the atoms, which broaden the emission orabsorption spectrum by an order of magnitude ormore. Saturated absorption spectroscopy specificallyovercomes the Doppler-broadening limit by providingfor a two-photon interaction which only occurs foratoms with a lab frame velocity very near zero.

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Saturated Absorption Spectroscopy SAS 3

laserbeam

photodiodevapor cell

Figure 2: Basic arrangement for ordinary laserabsorption spectroscopy.

L(, 0) also describes the spectrum of radia-tion from spontaneous emission and the width is the same for both cases. The maximumtransition rate 0I occurs right on resonance

( = 0) and for the rubidium transitionsstudied here 0 2 106 m2/J.

The value of/0 defines a parameter of theatoms called the saturation intensity Isat

Isat =

0(5)

and is about 1.6 mW/cm2 for our rubidiumtransitions.2 Its significance is that when thelaser intensity is equal to the saturation in-

tensity, excited state atoms are equally likelyto decay by stimulated emission or by sponta-neous emission.

Basic laser absorption spectroscopy

The basic arrangement for ordinary laserabsorption (not saturated absorption) spec-troscopy through a gaseous sample is shownin Fig. 2. A laser beam passes through the

vapor cell and its intensity is measured by aphotodiode detector as the laser frequency is scanned through the natural resonance fre-quency.

When a laser beam propagates through agaseous sample, the two stimulated transi-tion processes change the intensity of the laser

2, , 0, Isat vary somewhat among the vari-ous rubidium D2 transitions studied here. The valuesgiven are only representative.

beam and affect the density of atoms (num-ber per unit volume) in the ground and ex-cited states. Moreover, Doppler shifts associ-

ated with the random thermal motion of theabsorbing atoms must also be taken into ac-count. There is an interplay among these ef-fects which is critical to understanding satu-rated absorption spectroscopy. We begin withthe basic equation describing how the laser in-tensity changes as it propagates through thesample and then continue with the effects ofDoppler shifts and population changes.

Laser absorption

Because of stimulated emission and absorp-tion, the laser intensity I(x) varies as it prop-agates from x to x + dx in the medium.

Exercise 1 Show that

I(x + dx) I(x) = hI(x)(P0 P1)n0dx(6)

Hints: Multiply both sides by the laser beamcross sectional area A and use conservation ofenergy. Keeping in mind thatI(x) is the laserbeam intensity at some position x inside thevapor cell, explain what the left side would rep-resent. With n0 representing the overall den-sity of atoms (number per unit volume) in thevapor cell, what does n0Adx represent? P0 andP1 represent the probabilities that the atomsare in the ground and excited states, respec-tively, or the fraction of atoms in each state.They can be assumed to be given. What thenis the rate at which atoms undergo stimulatedabsorption and stimulated emission? Finally,recall that h is the energy of each photon. Sowhat would the right side represent?

Equation 6 leads to

dI

dx= I (7)

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where the absorption coefficient(fractional ab-sorption per unit length)

= hn0(P0 P1) (8)The previous exercise demonstrates that the

proportionality to P0 P1 arises from thecompetition between stimulated emission andabsorption and it is important to appreciatethe consequences. If there are equal num-bers of atoms in the ground and excited state(P0 P1 = 0), laser photons are as likely tobe emitted by an atom in the excited state

as they are to be absorbed by an atom in theground state and there will be no attenuationof the incident beam. The attenuation maxi-mizes when all atoms are in the ground state(P0P1 = 1) because only upward transitionswould be possible. And the attenuation caneven reverse sign (become an amplification asit does in laser gain media) if there are moreatoms in the excited state (P1 > P0).

In the absence of a laser field, the ratioof the atomic populations in the two energy

states will thermally equilibrate at the Boltz-mann factor P1/P0 = e

E/kT = eh0/kT. Atroom temperature, kT ( 1/40 eV) is muchsmaller than the h0 ( 1.6 eV) energy dif-ference for the levels involved in this exper-iment and nearly all atoms will be in theground state, i.e., P0 P1 = 1. While youwill see shortly how the presence of a stronglaser field can significantly perturb these ther-mal equilibrium probabilities, for now we will

only treat the case where the laser field is weakenough that P0 P1 = 1 remains a good ap-proximation throughout the absorption cell.

Doppler shifts

Atoms in a vapor cell move randomly in all di-rections with each component of velocity hav-ing a distribution of values. Only the com-ponent of velocity parallel to the laser beam

Figure 3: Maxwell-Boltzmann velocity distribu-tion. The density of atoms vs. their velocity com-

ponent in one direction for room temperature ru-bidium atoms.

direction will be important when taking intoaccount Doppler shifts and it is this compo-nent we refer to with the symbol v. The den-sity of atoms dn in the velocity group betweenv and v+dv is given by the Boltzmann velocitydistribution:

dn = n0 m

2kTemv2/2kT

dv (9)

With a standard deviation (proportional tothe width of the distribution) given by:

v =

kT/m (10)

this is just a standard Gaussian distribution

dn = n01

2 vev

2/22vdv (11)

with a mean of zeroindicating the atoms areequally likely to be going in either direction.It is properly normalized so that the integralover all velocities ( ) is n0, the over-all atom density. Note that the distributionsvariance 2v increases linearly with tempera-ture and decreases inversely with atomic mass.

Exercise 2 Determine the width parameterv of the Maxwell-Boltzmann distribution for

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Figure 4: Doppler profile. The absorption coeffi-cient vs. the laser frequency offset from resonance

has a Gaussian lineshape.

room temperature rubidium atoms and com-pare with the plot of Fig. 3.

Atoms moving with a velocity v see the laserbeam Doppler shifted by the amount (v/c).We will take an equivalent, alternate view thatatoms moving with a velocity v have a Dopplershifted resonance frequency

0 = 0

1 +v

c

(12)

in the lab frame. The sign has been chosen tobe correct for a laser beam propagating in thepositive direction so that the resonance fre-quency is blue shifted to higher frequencies ifv is positive and red shifted if v is negative.

The absorption coefficient d from a veloc-ity group dn at a laser frequency is thenobtained from Eq. 8 by substituting dn for n0

(keeping in mind its dependence on v throughEq. 9) and by adjusting the Lorentzian de-pendence of so that it is centered on theDoppler shifted resonance frequency 0 (keep-ing in mind its dependence on v throughEq. 12).

d = h0(P0 P1)L(, 0)dn (13)The absorption coefficient from all atoms

is then found by integrating over all velocity

groups. We treat the weak-laser case first set-ting P0 P1 = 1 so that

d = n0h0

m2kT

L(, 0)emv2/2kTdv(14)

Exercise 3 (a) Show that the Lorentzianfunction L(, 0) at a particular is signifi-cantly different from zero only for atoms withvelocities within a very narrow range around

vprobe = c(/0 1) (15)

(b) Determine the rough size of this rangeassuming = 6 MHz. ( 0 = c/ where = 780 nm.) (c) Show that over thisrange, emv

2/2kT remains relatively constant,i.e., compare the range with v (Eq. 10).Consequently, the integral of Eq. 14 can beaccurately determined as the integral of theLorentzian times the value of the exponen-tial at v = vprobe. The integration of theLorentzian L

(, 0)dv is easier to look up af-

ter using Eq. 12 to convert dv = (c/0)d0.(d) Show that integrating the absorption co-efficient

d over velocity (from to )

then gives a Gaussian dependence on cen-tered around the resonance frequency 0

= 0e(0)2/22 (16)

with the width parameter given by

= 0

kTmc2

(17)

and

0 = n0h0

m

2kT

c

0

2(18)

(e) Evaluate the Doppler-broadened width pa-rameter for room temperature rubidiumatoms and compare your results with the pro-file illustrated in Fig. 4.

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Populations

Now we would like to take into account the

changes to the ground and excited state pop-ulations arising from a laser beam propagat-ing through the cell. The rate equations forthe ground and excited state probabilities orfractions become:

dP0dt

= P1 I(P0 P1)dP1dt

= P1 + I(P0 P1) (19)

where the first term on the right in each equa-tion arises from spontaneous emission and thesecond term arises from stimulated absorptionand emission.

Exercise 4 (a) Show that steady state condi-tions (i.e., where dP0/dt = dP1/dt = 0) leadto probabilities satisfying

P0 P1 = 11 + 2I/

(20)

Hint: You will also need to use P0 + P1 = 1.(b) Show that when this result is used in Eq. 8(for stationary atoms) and the frequency de-pendence of as given by Eqs. 3-4 is also in-cluded, the absorption coefficient again takesthe form of a Lorentzian

=hn00

1 + 2I/IsatL(, 0) (21)

where L

is a standard Lorentzian

L(, 0) = 11 + 4( 0)2/2

(22)

with a power-broadened width parameter

=

1 + 2I/Isat (23)

The approach to the steady state probabili-ties is exponential with a time constant around

[+2I]1, which is less than 28 ns for our ru-bidium transitions. Thus as atoms, mostly inthe ground state, wander into the laser beam

at thermal velocities, they would only travel afew microns before reaching equilibrium prob-abilities.

C.Q. 1 Take into account velocity groups andtheir corresponding Doppler shifts for the caseP0 P1 = 1. The calculation is performedas in Exercise 3. Show that the result afterintegrating over all velocity groups is:

= 0e(0)2/22 (24)

where the width parameter, is the same asbefore (Eq. 17), but compared to the weak-fieldabsorption coefficient (Eq. 18), the strong-fieldcoefficient decreases to

0 =0

1 + 2I/Isat(25)

Laser absorption through a cell

In the weak-field case at any frequency isgiven by Eq. 16 (with Eq. 18) and is indepen-dent of the laser intensity. In this case, Eq. 7is satisfied by Beers law which says that theintensity decays exponentially with distancetraveled through the sample.

I(x) = I0ex (26)

In the general case, is given by Eq. 24(with Eq. 25) and at any frequency is propor-

tional to 1/

1 + 2I/Isat. The general solutionto Eq. 7 for how the laser intensity varies withthe distance x into the cell is then rather morecomplicated than Beers law and is given inan appendix that can be found on the lab website. However, the strong-field I >> Isat be-havior is easily determined by neglecting the1 compared to I/Isat so that Eq. 7 becomes

dI

dx= k

I (27)

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probebeam

pumpbeam

photodiodevapor cell

Figure 5: Basic arrangement for saturated ab-sorption spectroscopy.

where

k = 0

Isat/2e(0)2/22 (28)

Saturated absorptionsimple model

Up to now, we have considered only a singlelaser beam propagating through the cell. Nowwe would like to understand what happenswhen a second laser propagates through thecell in the opposite direction. This is the ba-sic arrangement for saturated absorption spec-troscopy shown in Fig. 5. The laser beamtraveling to the rightthe one we have been

considering up to now and whose absorptionis measuredis now called the probe beam.The second overlapping laser beam propagat-ing in the opposite direction is called the pumpbeam. Both beams will be from the same laserand so they will have the same frequency, evenas that frequency is scanned through the res-onance.

With only a single weak laser propagat-ing through the sample, P0 P = 1 wouldbe a good approximation throughout the cell.With the two beams propagating through thecell, the probe beam will still be kept weakweak enough to neglect its affect on the pop-ulations. However, the pump beam will bemade strongstrong enough to significantlyaffect the populations and thus change themeasured absorption of the probe beam. Tounderstand how this comes about, we willagain have to consider Doppler shifts.

As mentioned, the stimulated emission andabsorption rates are nonzero only when thelaser is near the resonance frequency. Thus,

we will obtain P0P1 from Eq. 20 by giving aLorentzian dependence on the Doppler shiftedresonance frequency:

= 0L(, 0 ) (29)

with the important feature that for the pumpbeam, the resonance frequency for atoms mov-ing with a velocity v is

0 = 0

1 v

c

(30)

This frequency is Doppler shifted in the direc-tion opposite that of the probe beam becausethe pump beam propagates through the vaporcell in the negative direction. That is, the res-onant frequency for an atom moving with asigned velocity v is 0(1 + v/c) for the probebeam and 0(1 v/c) for the pump beam.

Exercise 5 (a) Plot Eq. 20 versus the detun-ing parameter

= 0 (31)

for values ofI/Isat from 0.1 to 1000. (Use =6 MHz.) (b) Use your graphs to determine howthe FWHM of the dips in P0 P1 change withlaser power.

Exercise 5 should have demonstrated thatfor large detunings ( ), P0 P1 = 1 im-plying that in this case the atoms are in theground statethe same as when there is nopump beam. On resonance, i.e., at = 0,P0 P1 = 1/(1 + 2I/Isat), which approacheszero for large values of I. This implies thatatoms in resonance with a strong pump beamwill have equal populations in the ground andexcited states (P0P1 = 0). That is, a strong,

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Figure 6: Hole burning by the pump beam. Thedensity of ground state atoms is plotted vs. theirvelocity and becomes depleted near the velocitythat Doppler shifts the laser frequency into reso-nance with 0.

resonant laser field causes such rapid transi-tions in both directions that the two popula-tions equilibrate. The laser is said to satu-

rate the transition, which is the origin of thename of the technique.According to Eq. 30, the resonance condi-

tion ( = 0) translates to v = vpump where

vpump = c(1 /0) (32)Consequently, for any frequency within theDoppler profile, only atoms near this veloc-ity will be at zero detuning and will have val-ues ofP0 P1 perturbed from the pump off

value of unity. Of course, whether there area significant number of atoms in either levelnear v = vpump depends on the width of the

Boltzmann velocity distribution and the rela-tive values of and 0. But if vpump is withinthe distribution, the atoms near that velocitywill have perturbed populations.

The density of atoms in the ground statedn1 = P1 dn plotted as a function of v will fol-low the Maxwell-Boltzmann distribution ex-cept very near v = vpump where it will dropoff significantly as atoms are promoted to theexcited state. This is called hole burning

as there is a hole (a decrease) in the densityof atoms in the ground state near v = vpump(and a corresponding increase in the density ofatoms in the excited state) as demonstrated inFig. 6.

How does hole burning affect the probebeam absorption? In Exercise 3 you learnedthat the absorption at any frequency arisesonly from those atoms moving with veloc-ities near vprobe = c(/0 1). Also re-call, the probe beam absorption is propor-tional to P0 P1, which we have just seenremains constant ( 1) except for atoms hav-ing nearly the exact opposite velocities nearvpump = c(1 /0). Therefore, when thelaser frequency is far from the natural reso-nance (| 0| ), the probe absorptionarises from atoms moving with a particular ve-locity in one direction while the pump beam isburning a hole for a completely different set of

atoms with the opposite velocity. In this case,the presence of the pump beam will not affectthe probe beam absorption which would followthe standard Doppler-broadened profile.

Only when the laser frequency is very nearthe resonance frequency ( = 0, vprobe =vpump = 0) will the pump beam burn a hole foratoms with velocities near zero which wouldthen be the same atoms involved with theprobe beam absorption at this frequency. The

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Figure 7: Absorption coefficient vs. the laser fre-quency offset from resonance for a two-level atomat values of I/Isat = 0.1, 1, 10, 100, 1000 (fromsmallest dip to largest). It shows a Gaussian pro-file with the saturated absorption dip at = 0.

absorption coefficient would be obtained bytaking P1P0 as given by Eq. 20 (with Eqs. 29and 30) to be a function of the laser frequency

and the velocity v, using it in Eq. 13 (withEqs. 9 and 12), and then integrating over ve-locity. The result is a Doppler-broadened pro-file with whats called a saturated absorptiondip (or Lamb dip) right at = 0. Numeri-cal integration was used to create the profilesshown in Fig. 7 for several values of I/Isat.

Multilevel effects

Real atoms have multiple upper and lowerenergy levels which add complexities to thesimple two-level model presented so far. Forthis experiment, transitions between two lowerlevels and four upper levels can all bereached with our laser and add features calledcrossover resonances and a process called op-tical pumping. Crossover resonance are addi-tional narrow absorption dips arising becauseseveral upper or lower levels are close enough

0

1

2

V-system

0

1

2

-systemE

h2h1 h1

h0

Figure 8: Energy levels for two possible three-level systems. The V-configuration with two up-per levels and the) -system with two lower levels.

in energy that their Doppler-broadened pro-files overlap. Optical pumping occurs whenthe excited level can spontaneously decay tomore than one lower level. It can significantlydeplete certain ground state populations fur-ther enhancing or weakening the saturated ab-

sorption dips.The basics of crossover resonances can be

understood within the three-level atom in ei-ther the - or V-configurations shown in Fig. 8where the arrows represent allowed sponta-neous and stimulated transitions. In this ex-periment, crossover resonances arise from mul-tiple upper levels and so we will illustrate withthe V-system. Having two excited energy lev-els 1 and 2, the resonance frequencies to the

ground state 0 are 1 and 2, which are as-sumed to be spaced less than a Doppler widthapart.

Without the pump beam, each excited statewould absorb with a Doppler-broadened pro-file and the net absorption would be the sum oftwo Gaussian profiles, one centered at 1 andone centered at 2. If the separation |1 2|is small compared to the Doppler width, theywould appear as a single broadened absorption

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Figure 9: Absorption coefficient vs. the laser fre-

quency for a V-type three-level atom. The threeBell shaped curves are with the pump off and givethe Doppler-broadened absorption from the indi-vidual resonances at 1 and 2 and their sum. Thepump on curve shows normal saturated absorp-tion dips at = 1 and = 2 and a crossoverresonance midway between.

profile.

When the pump beam is turned on, two

holes are burned in the ground state velocitydistribution at velocities that put the atoms inresonance with 1 and 2. These two velocitieswould depend on the laser frequency . For ex-ample, at = 1, the probe absorption involv-ing upper state 1 is from v 0 atoms while theprobe absorption involving the higher-energystate 2 arises from some nonzero, negative-velocity atoms. At this frequency, the pumpbeam burns one hole in the ground state for

v 0 atoms due to upper state 1 and anotherhole for some nonzero, positive-velocity atomsdue to upper state 2. As with the two-levelsystem, the hole at v 0 leads to a decreasedabsorption to upper state 1 and produces asaturated absorption dip at = 1. A similarargument predicts a saturated absorption dipat = 2. Thus at = 1 and at = 2there will be saturated absorption dips similarto those occurring in two-level atoms.

A third dip, the crossover resonance, arisesat a frequency midway betweenat 12 =(1 + 2)/2. All three dips are illustrated in

Fig. 9 assuming I/Isat = 10 and assumingthat the excited levels have the same spon-taneous transitions rate and the same stim-ulated rate constant 0.

At the crossover frequency, the pump andprobe beams are resonant with the sametwo opposite velocity groups: v c (2 1)/212. Atoms at one of these two velocitieswill be resonant with one excited state andatoms at the opposite velocity will be resonant

with the other excited state. The pump beamburns a hole in the ground state populationsat both velocities and these holes affect theabsorption of the probe beam, which is simul-taneously also arising from atoms with thesetwo velocities.

C.Q. 2 Explain why the crossover resonancedip disappears as the laser frequency changesby a few linewidths from the mid-frequency

where the dip occurs. Remember, in this regioneach beam will still be interacting with two ve-locity groups: one group for one excited stateand another for the other excited state. For ex-ample, consider the case where the laser goesabove the mid-frequency by a small amount.How would the velocities change for the twogroups of atoms involved in probe beam absorp-tion? How would the velocities change for thetwo groups of atoms involved in pump beam

absorption?

Optical pumping in rubidium occurs whereone excited level can decay to two differentlower levels. It can be modeled in terms of the-type three-level atom, where the two lowerlevels are separated in energy by much morethan a Doppler width.

Assume the laser beam is resonant withatoms in only one of the lower levels, but

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atoms in the excited level spontaneously de-cay to either lower level more or less equally.Then, each time an atom in the resonant

lower level is promoted by the laser to the ex-cited level, some fraction of the time it willdecay to the non-resonant lower level. Oncein the non-resonant lower level, the atom nolonger interacts with the laser field. It be-comes shelved and unable to contribute tothe absorption. Analysis of the level popula-tions then requires a model where atoms out-side the laser beam interaction volume ran-domly diffuse back into it thereby replenishing

the resonant lower level populations. Depend-ing on the laser intensity and beam geometry,the lower level populations can be significantlyaltered by optical pumping.

As with the population variations arisingfrom stimulated emission and absorption, op-tical pumping also depends on the laser fre-quency and atom velocities. Optical pumpingdue to the pump beam can drastically depleteresonant ground state atoms and can signif-

icantly increase the size of the saturated ab-sorption dips. Optical pumping due to theprobe beam can also affect saturated absorp-tion measurements, particularly if the probelaser intensity is high. In this case, the probebeam shelves the atoms involved in the ab-sorption at all frequencies, not just at the sat-urated absorption dips.

Surprisingly, optical pumping does not sig-nificantly change the predicted form for theabsorption signals. Largely, its effect is tochange the widths (s), the saturation inten-sities (Isats) and the strengths (s) appearingin the formulae.

Energy levels in rubidium

One common application of saturated absorp-tion spectroscopy is in measuring the hyper-fine splittings of atomic spectral lines. They

are so small that Doppler broadening normallymakes it impossible to resolve them. Youwill use saturated absorption spectroscopy to

study the hyperfine splittings in the rubidiumatom and thus will need to know a little aboutits energy level structure.

In principle, quantum mechanical calcula-tions can accurately predict the energy levelsand electronic wavefunctions of multielectronatoms. In practice, the calculations are diffi-cult and this section will only present enoughof the results to appreciate the basic structureof rubidiums energy levels. You should con-

sult the references for a more in-depth treat-ment of the topic.

The crudest treatment of the energy lev-els in multielectron atoms is called the centralfield approximation (CFA). In this approxima-tion the nuclear and electron magnetic mo-ments are ignored and the atomic electronsare assumed to interact, not with each other,but with an effective radial electric field aris-ing from the average charge distribution from

the nucleus and all the other electrons in theatom.

Solving for the energy levels in the CFAleads to an atomic configuration in which eachelectron is described by the following quantumnumbers:

1. The principal quantum number n (al-lowed integer values greater than zero)characterizes the radial dependence of thewavefunction.

2. The orbital angular momentum quantumnumber (allowed values from 0 to n 1) characterizes the angular dependenceof the wavefunction and the magnitudeof the orbital angular momentum of anindividual electron.3

3Angular momentum operators such as satisfy aneigenvalue equation of the form 2 = ( + 1)h2.

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3. The magnetic quantum number m (al-lowed values from to ) further char-acterizes the angular dependence of the

wavefunction and the projection of on achosen quantization axis.4

4. The electron spin quantum number s(only allowed value equal to 1/2) charac-terizes the magnitude of the intrinsic orspin angular momentum s of an individ-ual electron.

5. The spin projection quantum number ms(allowed values

1/2) characterizes the

projection of s on a chosen quantizationaxis.

The rubidium atom (Rb) has atomic num-ber 37. In its lowest (ground state) config-uration it has one electron outside an inertgas (argon) core and is described with the no-tation 1s22s22p63s23p63d104s24p65s. The inte-gers 1 through 5 above specify principal quan-tum numbers n. The letters s, p, and d specifyorbital angular momentum quantum numbers as 0, 1, and 2, respectively. The superscriptsindicate the number of electrons with thosevalues of n and .

The Rb ground state configuration is saidto have filled shells to the 4p orbitals and asingle valence (or optical) electron in a 5s or-bital. The next higher energy configurationhas the 5s valence electron promoted to a 5porbital with no change to the description ofthe remaining 36 electrons.

Fine structure levels

Within a configuration, there can be severalfine structure energy levels differing in the en-ergy associated with the coulomb and spin-orbit interactions. The coulomb interaction is

4Angular momentum projection operators such asz satisfy an eigenvalue equation of the form z =mh.

associated with the normal electrostatic po-tential energy kq1q2/r12 between each pair ofelectrons and between each electron and the

nucleus. (Most, but not all of the coulomb in-teraction energy is included in the configura-tion energy.) The spin-orbit interaction is as-sociated with the orientation energy B ofthe magnetic dipole moment of each electronin the internal magnetic field B of the atom.The form and strength of these two interac-tions in rubidium are such that the energylevels are most accurately described in the L-S or Russell-Saunders coupling scheme. L-S

coupling introduces new angular momentumquantum numbers L, S, and J as describednext.

1. L is the quantum number describing themagnitude of the total orbital angularmomentum L, which is the sum of theorbital angular momentum of each elec-tron:

L =

i (33)

2. S is the quantum number describing themagnitude of the total electronic spin an-gular momentum S, which is the sum ofthe spin angular momentum of each elec-tron:

S =

si (34)

3. J is the quantum number describing themagnitude of the total electronic angularmomentum J, which is the sum of the to-

tal orbital and total spin angular momen-tum:

J = L + S (35)

The values for L and S and J are specifiedin a notation (2S+1)LJ invented by early spec-troscopists. The letters S, P, and D (as withthe letters s, p, and d for individual electrons)are used for L and correspond to L = 0, 1,and 2, respectively. The value of (2S + 1) is

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called the multiplicity and is thus 1 for S = 0and called a singlet, 2 for S = 1/2 (doublet),3 for S = 1 (triplet), etc. The value of J (with

allowed values from |L S| to L + S) is anno-tated as a subscript to the value of L.5

The sum of i or si over all electrons inany filled orbital is always zero. Thus for Rbconfigurations with only one valence electron,there is only one allowed value for L and S:just the value of i and si for that electron.In its ground state (5s) configuration, Rb isdescribed by L = 0 and S = 1/2. The onlypossible value for J is then 1/2 and the fine

structure state would be labeled 2S1/2. Itsnext higher (5p) configuration is described byL = 1 and S = 1/2. In this configurationthere are two allowed values of J: 1/2 and 3/2and these two fine structure states are labeled2P1/2 and

2P3/2.

Hyperfine levels

Within each fine structure level there can be

an even finer set ofhyperfine levels differing inthe orientation energy (again, a B typeenergy) associated with the nuclear magneticmoment in the magnetic field of the atom.The nuclear magnetic moment is much smallerthan the electron magnetic moment and this iswhy the hyperfine splittings are so small. Thenuclear magnetic moment is proportional tothe spin angular momentum I of the nucleus,whose magnitude is described by the quantum

5The terms singlet, doublet, triplet etc. are associ-ated with the number of allowed values of J typicallypossible with a given L and S (ifL S). Historically,the terms arose from the number of closely-spacedspectral lines typically (but not always) observed inthe decay of these levels. For example, the sodiumdoublet at 589.0 and 589.6 nm occurs in the decay ofthe 2P1/2 and

2P3/2 fine structure levels to the2S1/2

ground state. While the 2P1/2 and2P3/2 are truly a

doublet of closely-spaced energy levels, the 2S1/2 statehas only one allowed value ofJ.

number I. Allowed values for I depend on nu-clear structure and vary with the isotope.

The hyperfine energy levels depend on the

the total angular momentum F of the atom:the sum of the total electron angular momen-tum J and the nuclear spin angular momen-tum I:

F = J + I (36)

The magnitude of F is characterized by thequantum number F with allowed values from|J I| to J + I. Each state with a differentvalue ofF will have a slightly different energydue to the interaction of the nuclear magnetic

moment and the internal field of the atom.There is no special notation for the labelingof hyperfine states and F is usually labeledexplicitly in energy level diagrams.

There are two naturally occurring isotopesof Rb: 72% abundant 87Rb with I = 3/2and 28% abundant 85Rb with I = 5/2. Forboth isotopes, this leads to two hyperfine lev-els within the 2S1/2 and

2P1/2 fine structurelevels (F = I1/2 and F = I+ 1/2) and fourhyperfine levels within the

2

P3/2 fine structurelevel (F = I 3/2, I 1/2, I+ 1/2, I+ 3/2).

The energies of the hyperfine levels can beexpressed (relative to a mean energy EJ forthe fine structure state) in terms of two hyper-fine constants A and B by the Casimir formula

EF = EJ + A

2(37)

+ B3( + 1)/4 I(I + 1)J(J + 1)

2I(2I

1)J(2J

1)

where = F(F+ 1) J(J+ 1) I(I+1). (Ifeither I = 1/2 or J = 1/2, the term containingB must be omitted.)

Figure 10 shows the configuration-finestructure-hyperfine energy structure of the Rbatom. The scaling is grossly inaccurate. Theenergy difference between the 5s and 5p con-figurations is around 1.6 eV, the difference be-tween the 2P1/2 and

2P3/2 fine structure levels

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Iso fss A B85Rb 2S1/2 1011.91

2P1/2 120.72 2P3/2 25.01 25.88

87Rb 2S1/2 3417.34 2P1/2 406.20 2P3/2 84.80 12.52

Table 1: Hyperfine constants A and B (in MHz)for the lowest three fine structure states in thetwo naturally-occurring Rb isotopes

is around 0.03 eV, and the differences betweenhyperfine levels are less than 0.00003 eV.

Transitions

The 2S1/2 to2P1/2 transitions are all around

795 nm while the 2S1/2 to2P3/2 transitions are

all around 780 nm. We will only discuss the780 nm transitions that can be reached with

5s

5p

2S1/2

2P1/2

2P3/2

configurationenergy

coulomb andspin-orbit

andhyperfine F=

12

34

32

3

2

nl2S+1LJ

energy

794.8 nm

780.0 nm

Figure 10: Energy structure of the lowest levelsin 85Rb, (I = 5/2). Energy increases toward thetop, but the levels are not shown to scale; theseparation between the 5s and 5p configurationenergies is about 50 times bigger than the spacingbetween the 2P1/2 and

2P3/2 fine structure levels,and it is about 105 times bigger than the spacingwithin any group of hyperfine levels.

the laser used in this experiment. Dipole tran-sitions follow the selection rule F = 0, 1.Thus, in each isotope, the allowed transitions

from the 2S1/2 to 2P3/2 fall into two groups ofthree, with each group labeled by the F of the2S1/2 state. Because the hyperfine splitting be-tween the two 2S1/2 levels is large compared tothe hyperfine splittings among the four 2P3/2levels, the groups will be well separated fromeach other. Within each group, the three pos-sible transitions can be labeled by the F ofthe 2P3/2 state. These three transitions willbe more closely spaced in energy.

C.Q. 3 The energy level diagrams of Fig. 11are based on the information in Table 1 withEq. 37. Use the same information to cre-ate a table and scaled stick spectrum for alltwenty-four 780 nm resonances (including thecrossover resonances) occurring in both iso-topes. The horizontal axis should be at least

25 cm long and labeled in MHz. For the fig-ure, draw a one- or two-centimeter verticalline along the horizontal frequency axis at theposition of each resonance. Label the isotopesand the F of the 2S1/2 level for each of thefour groups and within each group label thevalue of F in the 2P3/2 level for each of thethree normal resonances and the two values ofF for each of the three crossover resonances.Also label their transition energies (in MHz)

relative to the overall 385 THz average transi-tion frequency. Hint: you should get the lowesttwo resonances in 87Rba normal resonancebetween the F = 2 and F = 1 levels 2793.1MHz below the average and a crossover reso-nance between the F = 2 and F = 1, 2 levels2714.5 MHz below the average. The highestenergy resonance is again in 87Rb; a normalresonance from F = 1 to F = 2 4198.7 MHzabove the average.

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Figure 11: Energy levels for 85Rb and 87Rb (not to scale). Note that the hyperfine splittings areabout an order of magnitude larger in the ground 2S1/2 levels compared to the excited

2P3/2 levels.

Apparatus

Laser Safety

The diode laser used in this experiment hassufficient intensity to permanently damageyour eye. In addition, its wavelength (around780 nm) is nearly invisible. Safety gogglesmust be worn when working on this ex-

periment. You should also follow standardlaser safety procedures as well.

1. Remove all reflective objects from yourperson (e.g., watches, shiny jewelry).

2. Make sure that you block all stray reflec-tions coming from your experiment.

3. Never be at eye level with the laserbeam (e.g., by leaning down).

4. Take care when changing optics in yourexperiment so as not to inadvertentlyplace a highly reflective object (glass, mir-ror, post) into the beam.

The layout of the saturated absorption spec-trometer is shown in Fig. 12. Two weak beamsare reflected off the front and rear surfaces of a

thick beam splitter (about 4% each). One (theprobe beam) is directed through the rubidiumcell from left to right and then onto a photo-diode detector. The other is directed througha Fabry-Perot interferometer and onto a sec-ond photodiode detector for calibrating thefrequency scan.

The strong beam is transmitted through thethick beam splitter. The two lenses after thebeam splitter should be left out at first andonly used if needed to expand the laser beam.This is the pump beam passing through thecell from right to left, overlapping the probebeam and propagating in the opposite direc-tion. (The intersection angle is exaggerated inthe figure.)

You should not adjust the components fromthe thick beam splitter back to the laser.

The diode laser

A schematic of our diode laser is shown inFig. 13. The American Journal of Physics ar-ticle on which its construction is based as wellas manufacturer literature on the laser diodeused in the laser can be found in the auxiliary

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laser

TBS

PD2Fabry-Perot

Rb cell

pump beam

probe beam

A1

APP PBS /4 L2L1

/4

M

M

PD1NBS

M

M

M

FPM FPM

Figure 12: Optical layout of the saturated absorption spectrometer. Key: M mirror; A aperture; APPanamorphic prism pair; PBS polarizing beam splitter; TBS thick beam splitter; /4 quarter waveplate; L lens; NBS non-polarizing beam splitter; FPM Fabry-Perot mirror PD photodiode detector.

material.The frequency of the laser depends on three

parameters:

The laser temperature The laser current The position and orientation of the grat-

ing

The laser temperature and current set thefrequency range over which the diode laserwill operate (coarse tuning). Within thisrange, the laser frequency can be continuouslyscanned using the position and orientation ofthe grating (fine tuning). Fine tuning of thelaser frequency is accomplished by a piezoelec-

tric transducer (PZT) located in the gratingmount. The PZT expands in response to avoltage (0-150 V) from the PZT controller (de-scribed later). The voltage can be adjustedmanually or under computer-control.

As the laser cavity length varies, becausethe number of waves in the cavity will stayconstant (for a while), the wavelength andtherefore the laser frequency will vary as well.But if the cavity length change goes too far,

the number of half-wavelengths inside the cav-ity will jump up or down (by one or two) sothat the laser frequency will jump back intothe range set by the temperature and current.These jumps are called mode hops. Continu-ous frequency tuning over the four hyperfinegroups (

8GHz) without a mode hop can

only be achieved if the settings for the tem-perature and current are carefully optimized.

The beam profile (intensity pattern in across-section of the laser beam) from the laserhead is approximately a 1.5 3 mm ellipticalspot. An anamorphic prism pair is used totransform this to a circular pattern. In addi-tion, lenses may be used to change the beamdiameter.

Beam Paths

Several beams are created from the singlebeam coming from the laser. The polarizingbeam splitter and the quarter-wave plate helpprevent the pump beam from feeding back intothe laser cavity thereby affecting the laser out-put.

The thick beam splitter creates the three

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ODVHUGLRGH

YHUWWLOW

FRDUVHIUHT

JUDWLQJ

OHQV

EHDP

HQFORVXUH

3=7

IRFXV

7

7

EDVHSODWH

Figure 13: Schematic of our diode laser. Ex-cept for the thermistors (T), the temperature con-trol elements and electrical connections are notshown.

beams used in the experiment. Two beamsone reflected off the front surface and one re-flected off the back surfacecreate the probebeam and the beam sent through the Fabry-Perot interferometer. Each of these beamscontains about 4% of the laser output. About92% passes through this beam splitter to be-come the pump beam.

You should also check for additional beams,

particularly from the polarizing and non-polarizing beam splitters, and use beam blocksfor any that are directed toward areas wherepeople have access.

Photodetectors

Two photodetectors (PDs) are used to moni-tor the laser beam intensities. A bias voltagefrom a battery ( 22 V) is applied to the PD

L

Figure 14: The optical resonator or Fabry-Perotinterferometer consists of two partially transmit-ting curved mirrors facing each other.

which then acts as a current source. Thecurrent is proportional to the laser power im-pinging on the detector and is converted toa voltage (for measurement) either by send-ing it through a low noise current amplifier orby letting it flow through a resistor to ground.With the latter method, the voltage developedacross the resistor subtracts from the batteryvoltage and the resistance should be adjusted

to keep it below one volt.

Fabry-Perot Interferometer

Once the laser temperature has stabilized anda laser current has been set, a voltage rampwill be applied to the PZT causing the laserfrequency to smoothly sweep over several res-onance frequencies of the rubidium atoms inthe cell. The laser frequency during the sweep

is monitored using a confocal Fabry-Perot in-terferometer. The Fabry-Perot consists of twopartially transmitting, identically-curved mir-rors separated by their radius of curvature asshown in Fig. 14.

With the laser well aligned going into theFabry-Perot cavity, Fig. 15 shows how muchis transmitted through the cavity as a func-tion of the laser frequency. The Fabry-Perotonly transmits the laser beam at a set of dis-

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FSR

FWHM

Figure 15: The transmitted intensity for a wellaligned Fabry-Perot interferometer shows sharppeaks equally spaced in frequency.

crete frequencies separated by the free spectralrange (FSR):

FSR =c

4L(38)

where L is the distance between mirrors. Con-sequently, as the frequency of the laser beamgoing into the cavity is swept, the transmission(as monitored by a photodetector on the otherside of the cavity) will show a series of peaksseparated by the FSR. These peaks (or fringes)will be monitored simultaneously with absorp-tion through the rubidium and will serve asfrequency markers for how the laser frequencyis changing during the sweep.

Rb vapor cell

The Rb vapor cell is a glass cell filled with nat-ural rubidium having two isotopes: 85Rb and87Rb. The vapor pressure of rubidium insidethe cell is determined by the cell temperatureand is about 4105 Pa at room temperature.

Low-noise current amplifier

The low noise current amplifier can be used

to amplify and filter the photodetector signal.This signal will change at a rate which de-pends on the rate at which you scan the laserfrequency. For a very slow scan, the interest-ing parts of the signal will change very slowlyand you can use the low pass filter function ofthe amplifier to reduce the higher frequencynoise like 60Hz pick-up. The low pass filtershould not change your signal if the cutoff fre-quency is set properly. Another possibility isto scan rather fast and use the bandpass filterfunction to reduce the low and high frequencynoise. You should play with these options abit and try to find the optimum setting foryour scan rate and detector noise.

Alignment procedure

Most students have difficulties aligning theposition and propagation direction of a laserbeam simultaneouslyusually due to a mix-

ture of inexperience and impatience. In thefollowing we describe the standard alignmentprocedure. As usual, standard procedures arenot applicable to all cases but they should giveyou an idea how do do it.

Task: Align a laser beam to propagatealong a particular axis in the apparatus, e.g.,the axis of our Fabry-Perot resonator. Thisalignment procedure is called beam walkingand requires two mirrors in adjustable mirrormounts.

Coarse alignment:1. Pick two reference points along the

target axis. These points shouldbe reasonably far apart from eachother.

2. Use the first alignment mirror toalign the laser on the first referencepoint.

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a

b

mirror a

mirror b

laser beam

Figure 16: Procedure to align the laser beam

(dashed line) along a prescribed line (solid lineab). Pick two reference points a and b and usethe first mirror to align the laser to point a andthe second mirror to align the laser to point b.Iterate until laser beam is perfectly aligned.

3. Use the second alignment mirror toalign the laser on the second refer-ence point.

4. Iterate points 2 and 3 until the laser

beam is aligned as well as possiblewith respect to the optical axis. Youmight overshoot in step 2 or step 3occasionally to accelerate the proce-dure.

Fine alignment or optimizing the signal:

1. Display the signal you wish to op-timize. For example, the PD signal

after the Fabry-Perot resonator.2. Optimize the signal with the first

mirror in the horizontal and verticaldirections.

3. Optimize the signal with the secondmirror in the horizontal and verticaldirections.

4. Use the first mirror to de-optimizethe signal very slightly in the hori-

zontal direction. Remember the di-rection you moved the mirror.

5. Use the second mirror to re-optimizethe signal again.

6. Compare this signal with the signalyou had before step 4. If the sig-nal improved repeat step 4 movingin the same direction. If the signalgot worse, repeat step 4 moving inthe opposite direction.

7. Iterate until you cant further im-prove the signal.

8. Do the same for the vertical direc-tion until you cant improve the sig-nal anymore.

9. Iterate steps 4 to 8 until you cantimprove the signal furthermore.

This procedure requires some patience and cannot be done in five minutes.

The Data Acquisition Program

Read the reference material for detailed in-formation on our PCI-MIO-E4 multifunctiondata acquisition (DAQ) board located insidethe computer. Here we will only present whatis needed for this experiment. The func-tionality used here is the DAQ boards abil-ity to measure analog voltages using analog-to-digital converters (ADCs) and to produceanalog voltages using digital-to-analog con-

verters (DACs). The input and output volt-ages from the ADCs and DACs are connectedto and from the experiment through the BNC-2090 interface box.

The DAQ board will be programmed to gen-erate a (discrete) voltage sweep which is sentto the PZT controller to sweep the laser fre-quency. While the voltage sweep is occurring,the DAQ board is programmed to simultane-ously read any photodetectors monitoring the

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probe, pump, and/or the Fabry-Perot laserbeam intensities.

There are two, 12-bit DACs (DAC0 and

DAC1) on our DAQ board and both are usedto generate the sweep. They are set to run inunipolar mode so that when they are sent inte-gers N from 0 to 4095, they produce propor-tional voltages vout from 0 to some referencevoltage vref.

vout = vrefN

4095(39)

The voltages appear at the DAC0OUT and

DAC1OUT outputs on the DAQ interface box.The reference voltage vref for either DAC canbe chosen to be an on-board 10 V source orany value from 0 to 11 V as applied to theEXTREF input on the DAQ interface box.

Our data acquisition program runs DAC1 inthe 10 V internal reference mode to create afixed voltage at DAC1OUT, which is then con-nected to the EXTREF input and becomes thevoltage reference vref for DAC0. The DAC0

programming integers are a repeating triangu-lar ramp starting N=0, 1, 2 ..., Nmax and thengoing backward from Nmax back to 0 (whereNmax is chosen by the user). This program-ming thereby generates a triangular voltagewaveform from zero to vmax and then back tozero, where

vmax = vrefNmax4095

(40)

The properties of the triangle ramp arespecified on the programs front panel settingsfor the DAC1 voltage (vref) (labeled ramp ref(V)) and the value ofNmax (labeled ramp size).The speed of the sweep is determined by thefront panel setting of the sampling rate control.It is specified in samples per second and givesthe rate at which the DAC0 output changesduring a sweep. Thus, for example, with theramp size set for 2048, the DAC1OUT set for

4 V, and the Sweep rate set for 10000/s, theramp will take about 204.8 ms to go from 0to 2 V (2048

4V/4096) and another 204.8 ms

to come back down to zero, after which it re-peats.

The triangle ramp should be connected toEXT INPUT on the MDT691 Piezo Driver.This module amplifies the ramp by a factor of15 and adds an offset controlled by an OUT-PUT ADJ knob on the front of the module toproduce the actual voltage ramp at its OUT-PUT. This output drives the PZT in the laserhead which in turn changes the laser cavity

length and thus the laser frequency. To changethe starting point of the ramp sent to the PZTin the laser head, you should simply offset itusing the OUTPUT ADJ knob. Alternatively,the front panel control ramp start can be setat a non-zero value to have the DAC0 pro-gramming integers begin at that value. Forexample, setting the ramp start to 1024 andthe ramp size to 2048 makes the programmingintegers sent to DAC0 to run from 1024 to

3072 (and back down again). Then, for exam-ple, ifvref were set for 8 V, the ramp would gofrom 2 V to 6 V.

Voltages from the photodetector outputsshould be connected to any of the inputs tothe analog-to-digital converter (ADC) on theinterface box. These inputs are connected toa multiplexer, which is a fast, computer con-trolled switch to select which input is passedon to be measured by the ADC. Before be-

ing measured by the ADC, the signal is ampli-fied by a programmable gain amplifier (PGA).The gain of this amplifier and the configura-tion (unipolar or bipolar) is determined by theRange control on the front panel. You shouldgenerally use the most sensitive range possiblefor the signal you wish to measure so that the12-bit precision of the ADC is most fully uti-lized. When measuring more than one analoginput, it is also preferable if they all use the

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same range so that the PGA does not have tochange settings as it reads each input.

Measurements

Turning the Diode Laser On and Off

1. Ask the instructor to show you how toturn on the laser electronics. The LFI-3526 temperature controller will maintainthe aluminum baseplate at a temperatureabout 10 degrees below room temperatureusing a thermoelectric cooler. The home-made interface box (with a white panelon the front and a power switch on theback) contains a second temperature con-trol stage which heats the diode mount-ing block about 2 degrees with a resis-tive heater. Check that the fan under thelaser head comes on and is suitably po-sitioned to blow on the laser head heatsink. Watch out that you dont skin yourfingers moving the fan as the blade is un-shielded. The fan sits on a thin foam base

to minimize vibrations.

2. Do not change the temperature controllerset point, but learn how to read it andwrite its present value in your lab note-book. The setting is the voltage acrossa 10 k thermistor mounted to the laserheads aluminum baseplate and used forfeedback to stabilize the temperature. Itshould be set to produce the correct tem-perature to allow the laser to operate ata wavelength of 780 nm. The laser fre-quency depends strongly on temperatureso wait for the temperature to stabilizebefore making other adjustments.

3. Check that the laser beam paths are allclear of reflective objects.

4. Turn on the LFI-4505 laser diode currentcontroller. When you push the OUTPUT

ON button, the laser current comes onand the diode laser starts lasing. Thelaser current needed is typically some-

where in the range from 60-90 mA. Fornow, look at the laser output beam withan infrared viewing card and watch thelaser beam come on as you increase thelaser current past the 40 mA or so thresh-old current needed for lasing. Watch thebeam intensify as you bring the currentup to around 70 mA.

5. Later, when you need to turn the laseroff, first turn down the laser current tozero, then hit the OUTPUT ON buttonagain, then push the power button. Af-ter the laser is powered off, the rest ofthe laser electronics can turned off by theswitch on the outlet box. The computerand detector electronics should be pow-ered from another outlet box and shouldbe turned off separately. Also rememberto turn off the battery-powered photodi-ode detectors.

Tuning the laser

6. Align the beam through the center of theF-P and onto its detector. Align theprobe beam through the center of the Rbcell and onto its detector. Align the pumpbeam to be collinear with the probe beambut propagating in the opposite direction.Turn on the detectors and the PAR cur-

rent preamp.7. Focus the video camera on the Rb cell.

Block the probe beam.

8. Wait for the temperature to stabilize andthen adjust the laser current in the rangefrom 60-100 mA while monitoring the ru-bidium cell with the video camera. Whenthe current is set properly, the moni-tor should show strong fluorescence all

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(a) (b)

Figure 17: Single-beam Rb absorption spectra. (a) Shows a laser frequency sweep of about 8 GHz

without a mode hop over the appropriate region of interest for the Rb D2 hyperfine levels near 780nm. (b) Shows a sweep with a mode hop around 0.2 s.

along the pump beam path inside the cell.There can be up to four distinct reso-nances. They are all very sharp and thelaser frequency may drift so you may needto repeatedly scan the laser current backand forth over a small range to continueobserving the fluorescence. Ask the in-

structor for help if no current in this rangeshows strong Rb fluorescence.

9. With the strong pump fluorescence ob-served on the monitor, block the pumpbeam and unblock the probe beam to seethe path of the probe beam and its weakerfluorescence on the monitor. The reso-nances may be observed at two laser cur-rents separated by twenty milliamps ormore. If so, set it for the current closest

to 70 mA.

10. Start the Analog IO data acquisition pro-gram. Set the ramp ref (V) control to 3 V.Set the ramp size to 2000 channels. Leavethe ramp start control at zero and the sam-pling rate at 20,000 samples per second.This makes about a 1.5 V sweepup andback downrun at about 5/s. The 1.5 Vramp would normally be sufficient to pro-

duce a laser frequency sweep of 6-9 GHzor more. However, the laser typicallymode hops during the ramp.

11. In this and the next few steps, you willlook for evidence of mode hops in yourabsorption spectrum and/or in the F-P

markers and you will adjust the temper-ature, current, and PZT offset to get along sweep that covers all four Rb reso-nances without a mode hop. All adjust-ments should be performed while runningthe sweep described above and monitor-ing the computer display of the absorp-tion spectrum and F-P markers, as wellas the video display of the fluorescence.At this point, the back and forth sweep

should produce a blinking fluorescence onthe camera video. Turn off all preampfiltering. You may need to adjust thegain settings on the DAQ boards ADCso that the measured signals are not satu-rating the ADC. Try to adjust the currentpreamplifier gain so that the absorptionspectrum has just under 0.5 V baseline(the signal level away from the peaks) andthen be sure to adjust the ADC gain for

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0-0.5V signals. Also adjust the F-P detec-tor resistor so that the F-P peaks are alsojust under a 0.5 V and adjust the ADC

gain for 0-0.5 V signals.

12. A good absorption spectrum with F-Pmarkers (for only the half-sweep up ofthe up and down triangular waveform) isshown in Fig. 17(a). Fig. 17(b) showswhat these two signals look like with asingle mode hop about halfway into thesweep. Notice the break in the F-P mark-ers and how the section of the absorption

spectrum repeats after the 0.2 s mark asthe laser frequency continues scanning af-ter the abrupt change in frequency causedby the mode hop. Even without any Rbabsorption, the lasers intensity may notbe very constant over the entire sweep.If this is the case, the photodetector cur-rent between absorption peaks from oneside of the sweep to the other will showa significant slope. This is partly becausethe beam from the laser changes direction

a small amount as the piezo scans and thebeam may then move partially on and offthe detector by a small amount. However,if the detected laser intensity varies morethan about 10%, the beam may be hittingvery near the edge of the detector or oneof the mirrors or beam splitters therebymaking the effect bigger than it could be.Fix this problem if it occurs by better cen-tering the beam on the detectors, mirrors,

or splitters.

13. Adjust the PZT offset knob to get thelongest possible sweep without a modehop while also trying to cover the fre-quency range for the four absorptionpeaks. Adjust the laser current 1 or 2 mAone way or the other and then go back tothe PZT knob to see if the sweep is im-proving in length or frequency coverage.

Repeat until you have a feel for how thisworks and have roughly found the bestpossible sweep.

14. If no combination of laser current andPZT offset gives a frequency sweep thatshows all four Rb hyperfine groups, thelaser temperature will need to be ad-justed. Save the best spectrum you cur-rently have for later comparisons. Thetemperature should be changed in smallsteps corresponding to about 0.05-0.10 kon the 10 k thermistor used to control it.

Start by increasing the setting by 0.05 k(lowering the temperature). After chang-ing it, go back to the previous step. Ifthe frequency sweep improved, but is stillnot long enough, change the tempera-ture controller by another 0.05 k in thesame direction. Otherwise, change it inthe opposite direction. Continue iterat-ing these two steps until you have the de-sired sweep. Ask the instructor for help ifthis step is taking more than 60 minutes

or if the temperature controller has beenchanged by 0.25 k without getting theneeded sweep. Keep in mind, however,that on some days it seems the best thelaser can do is a sweep that only coversthree out of the four peaks. In this case,the four peaks can usually be observedthree at a time in two separate sweepswith slightly different settings on the PZTor laser current controller.

Preliminary Measurements

15. Measure the width (FWHM) and spac-ing between the F-P fringe peaks and de-termine the finesse, which is the ratio ofthe spacing to the FWHM. If the finesseis below three, you may want to adjustthe beam steering to better align it tothe Fabry-Perot. Note the spacing be-

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tween the fringes. Measure the F-P mir-ror spacing and calculate the free spectralrange. Measure the distance (in channels)

between the F-P frequency markers atvarious places within the sweep and notethe nonlinearity of the sweep. The non-linearity will complicate the analysis be-cause the frequency spacing between datapoints will not be constant.

16. Change the sweep to 4000 points withvref = 1.5 V to produce about the samesize sweep and explain how the F-P peaks

are affected. make sure all absorptionspectra have the F-P frequency markersalong with them. They will be needed toconvert the spacing between Rb absorp-tion features (such as peak positions andwidths) to frequency units.

There are several ways the F-P markers canbe used to perform the conversion. The ab-sorption spectra are normally plotted vs. timeor channel number. In one method, the lo-

cation of the F-P markers in time or chan-nels would be determined using cursors. Thensince the spacing between frequency markersis constant and is given by the FSR (lets callit ), the frequency at consecutive markerswould be taken as 0, 0 + , 0 + 2,..., The frequency 0 is about 3.85 1014 Hzand not really determined with our appara-tus. However, we will only be interested infrequency differences so this is not a prob-

lem. Then the frequency at the markers canbe fit to a quadratic or cubic polynomial in thetime or channel number where the F-P peakoccurred and the polynomial coefficients canthen be used to convert the time or channelnumber of spectral features to frequency units.

A second, slightly simpler technique wouldbe to count whole marker spacings betweenfeatures and interpolate the location of fea-tures as fractions of the spacing relative to the

markers located near them. The whole andfractional parts multiplied by the FSR wouldthen give the frequency spacing between the

features.

Doppler broadening and Laser Absorp-

tion

17. Obtain a spectrum of the four absorptiongroups (with the F-P frequency markers)and determine the spacing of the peaksin frequency. Use this information withyour stick spectrum to identify the four

groups. Does the frequency increase ordecrease as the PZT ramp voltage goesup?

18. For each group, measure the Doppler-broadened FWHM and compare with pre-dictions. Do you see any effects in theFWHM due to the fact that each groupincludes three transitions at different fre-quencies?

19. Measure the absorption as a function ofincident laser intensity for at least oneof the four Doppler-broadened groups.That is, use one graph cursor to mea-sure the probe detector voltage (and con-vert to a current using the preamp gain)at maximum absorption. This readingwould correspond to the transmitted in-tensity at resonance. Set the other cur-sor toeyeball a background reading at

the same placewhat the voltage read-ing would be if the absorption peak werenot there. This reading, again convertedto a photocurrent) would correspond tothe incident intensity. So that you canget the highest laser power, move the PDto the other side of the Rb cell so it willmeasure the strong pump beam ratherthan the weak probe beam. Replace thebeam splitter with a mirror so that the

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full pump beam intensity will be avail-able to pass through the cell. Use neutraldensity filters to lower the intensity of the

beam.

Be sure to keep track of the gain setting onthe current preamp so that you will be ableto convert the PD voltages to photocurrents.The photocurrent is proportional to the powerincident on the detector and for the Thor-labs DET110 detector, the responsivity (con-version factor from incident laser power to cur-rent) is around 0.5 A/W at 780 nm. You can

also use the Industrial Fiber Optics Photome-ter at a few of the laser intensities for compari-son purposes. It reads directly in watts, but iscalibrated for a HeNe laser. The reading needsto be scaled down by a factor of about 0.8 forthe 780 nm light used here. To get intensity,you will also have to estimate the laser beamdiameter and determine how well you are hit-ting the detector. If the beam is smaller thanthe detector, the area to use is the beam area.

If the beam is larger than the detector, thenthe detector area (13 mm2 for the DET110)should be used. Keep in mind that the beamprobably has a roughly circular intensity pro-file with the central area fairly uniform andcontaining around 90% of the power. The di-ameter of this central area is probably around1/2 to 1/3 that of the beam as measured onthe IR viewing card.

Make sure to get to low enough laser inten-

sities that you see the fractional transmissionbecome constant (Beers law region). Plot thedata in such a way that you can determinewhere the Beers law region ends and the frac-tional transmission starts to increase. Thishappens near Isat, so see if you can use yourdata to get a rough estimate of this quantity.

Plot your data in such a way that it showsthe prediction for the absorption or transmis-sion for the strong-field case. There are several

ways to do this. Explain your plot and how itdemonstrates the prediction.

C.Q. 4 Use the measured weak-field trans-mission fraction and Beers law (Eq. 26) withthe cell length to determine 0. Use this 0with Eq. 18 to determine 0 and then use Eq. 5to determine Isat. Compare this Isat with therough value determined by the graphwhereBeers law starts to be violated and with theapproximate value of 1.6 mW/cm2 given in thetext.

Rb spectra

20. Remove the mirror and replace it with thebeam splitter. Move the detector back tothe other side of the Rb cell so it measuresthe probe beam. Realign the probe beamthrough the splitter and onto the PD.Realign the pump beam to be collinearwith the probe beam. Observe the sat-urated absorption dips from the probe

beam. Readjust the pump beam overlapto maximize the depth of the dips. Notehow the dips disappear when you blockthe pump beam. Measure the saturatedabsorption dips for each of the four hyper-fine groups and explain your results. Useshort sweeps that cover only one group ata time. Experiment with the sweep speed,the number of points in the sweep and thepreamp gain and filter settings to see how

they affect your measurements.

21. Measure the FWHM of one or more dipsas a function of the pump and probebeam intensities and explain your results.How do you get the narrowest resonances?Why cant you get a FWHM as low asthe predicted natural resonance widthof 6 MHz? Hints: Is the laser trulymonochromatic?

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22. If the displayed spectrum is just aboutperfectly stable from sweep to sweep, youmight try turning on the programs aver-

aging feature to improve the quality of thespectra. However, if the spectrum movesaround on the displayeven a littleaveraging will blur the spectrum and de-crease its quality.

23. Measure the relative splittings withineach group and use these results to iden-tify the direct and crossover resonances.

CHECKPOINT: You should have obtainedthe no pump spectrum (Doppler-broadenedwithout saturated absorption dips) and thesaturated absorption spectrum (with thepump beam and showing the saturated ab-sorption dips). Both spectra should cover atleast three of the four possible groups of hyper-fine transitions (with the same lower 2S1/2 hy-perfine level) and they should include the over-layed Fabry-Perot peaks. The Fabry-Perotmirror spacing should be measured and used

to determine the free spectral range, whichshould be used with the overlayed fringesto determine (in MHz) the following spec-tral features: (a) The separation between theDoppler-broadened peaks from the same iso-tope of the no pump spectrum. Comparewith expectations based on the energy level di-agrams in Fig. 11. (b) The FWHM of the eachDoppler-broadened peak. Compare with thepredictions based on Eq. 17. (c) The FWHM

of the narrowest saturated absorption dip.

April 7, 2010

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Experiment SAS