Princeton University

Fall Semester Junior Paper

Probe Laser Intensity Profile Within aMultipass Cell for Optical Magnetometry

Author:Horace Zhang

Advisor:Professor Michael Romalis

January 6, 2015

Abstract

Modern magnetometers have diverse applications ranging from mineral detection tobiomedical imaging [5]. We describe the operational principle of a high sensitivity atomicmagnetometer which measures the Faraday rotation angle of a linearly polarized laser beamafter propagation through a multipass optical cell of alkali metal vapor [11]. This rotationangle is proportional to the atomic spin in the measurement direction so rotation signalnoise from spin uncertainty gives insight about the fundamental magnetometer sensitivitylimit. We then describe the intensity profile of the laser within the optical cell. This profileis important in calculation of the number of atoms measured N and quantification of atomicdiffusions effect on rotation signal noise [11]. We numerically calculate intensity integrals inthe expression for N and describe a simplified regime ignoring laser beam overlap within thecell so integration is reduced from three dimensions to a single dimension. In this approxi-mation, we find multipass cell geometries so approximate case calculations agree within 5%of general case calculations, allowing for less computationally expensive N calculations insome cell configurations.

1 Introduction

This paper explains first how magnetic fields are measured by measuring the Faraday ro-tation angle of probe laser light after propagation through alkali metal atom vapor within amultipass optical cell. After we review this atomic physics, we characterize the intensity profileof the laser within the cell. We show that is proportional to the average spin value F inthe direction of the probe laser. This proportionality demonstrates that measurement of ,and thus magnetic field sensitivity, is fundamentally limited by precision in spin measurementsubject to the uncertainty principle. Field sensitivity is the smallest change in field the magne-tometer detects and has units of T/

Hz. This represents the measurement precision after one

second of interrogation which improves as the square root of measurement time [5]. Currentdevelopments in magnetometry have achieved sensitivity in the sub-femtotesla range [11].

We find the static magnetic field Bz in the z direction by measuring the frequency of Larmorspin precession L of the atomic spin vector about the field. The Zeeman effect splits thedegenerate alkali atom ground state into two states with an energy difference proportional toL, proportional to the field strength [5]. To study magnetometer sensitivity, we examine thenoise signal 2(t) of the rotation measurement due to spin uncertainty and other sources ofexperimental noise. We show that the Fourier transform of this noise signal has a Lorenztianline shape peaked at the Larmor frequency L [4] [11]. Thus, examination of the power spectrumof 2(t) gives both a quantification for the quantum noise present and the value of L fromwhich Bz is determined.

The experimental apparatus is described in section two and the rotation angle of the probelaser is measured by two photodetectors. This method of measurement introduces an additionalsource of noise, photon shot noise (PSN), into measurement due to quantization of the pho-tons arriving at the photodetectors [3]. To resolve PSN from noise from uncertainty in spinmeasurement (spin projection noise), we examine the power spectra of the two noise signals.The ratio of the power spectra of spin projection noise to photon shot noise is proportional tothe laser path length within the vapor cell [1] [2]. Thus, an advantage of using multipass cellsis increased probe path length, leading to dominance of the spin projection noise signal whichimposes a fundamental limit on measurement certainty.

The latter part of this paper characterizes the beam intensity profile within the cell andexplores numerical computation of integrals involving this profile. This intensity profile isimportant in calculating the number of atoms participating in measurement and the effect ofdiffusion of atoms in and out of the probe beam on the rotation noise signal [11] [4]. We derive anexpression for the effective number of atoms participating in measurement based upon spatialintensity profile integrals and then quantitatively model the laser intensity distribution. We

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also determine the position of the beam in the cell using the ABCD ray matrix approachto find the beam pattern at each cell cross section. Finally, we numerically calculate theintensity profile integrals to find the effective beam area in the expression for the numberof atoms participating in measurement. These intensity profile integrals can be simplified inthe approximation of ignoring beam overlap. Within this simplified regime, we show thatintegration is reduced from three dimensions to a single dimension and area calculations usingthe simplified calculation agree within 5% of the general method calculation. Thus, we providea regime where a computationally expensive 3D integration is avoided, allowing for quick areacalculations.

1.1 Fundamental Uncertainty Limit

We first show that any spin measurement has a fundamental limit to its precision imposed bythe uncertainty principle. Consider an atom in the spin state Fz = F . Using the spin angularmomentum commutation relationship [Fx, Fy] = iFz, (in this notation the ~ is dropped) andthe generalized uncertainty principle

AB | 12i[A,B]| (1)

where is the standard deviation, the uncertainty in Fx is given by:

(Fx) =

F

2. (2)

We assume that (Fx) = (Fy). Using Na atoms each with uncertainty given by (2) reducesthe uncertainty to the standard quantum limit (SQL)

(Fx) =

F

2Na. (3)

This uncertainty is known as spin projection noise and causes noise in the rotation signal.

2 Experimental Apparatus

Figure 1: Schematic of experimental design to measure rotation angle of linearly polarizedprobe light. Figure is adapted from the experimental apparatus given in figure 1 of [4].

Figure (1) shows a basic schematic of the experimental design. A glass cell contains alkalimetal vapor and is heated to 100C. The cell is surrounded by multilayer magnetic shieldsto prevent interference from external magnetic fields. The cell is also filled with N2 buffer gasto prevent diffusion of atoms to the walls of the cell. The magnetic field Bz in the T range isapplied perpendicular to the probe propagation direction, x. The probe beam passes througha linear polarizer and enters the multipass cell where it makes N passes through the cell andexits through the same hole. Then the beam hits a polarimeter oriented at 45 to the initial

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probe polarization angle. The rotation signal is measured by the signal difference between thetwo photodiodes PD1 and PD2 [2] [4]. The design of the multipass cell is described in detail inthe Gaussian beam optics section of this paper.

3 The Energy Levels of Alkali Metal Atoms

n_ state: m = -1/2 n+ state m = +1/2

5S1/2: l = 0

5P1/2: l =1

E = 0

E =L

m = +1/2 m = -1/2

Figure 2: The D1 optical transition withinone hyperfine state of alkali metal atoms.The degenerate ground state is lifted by theZeeman effect with E = ~L, where L isthe Larmor precession frequency.

n_ m = -1/2 n+ m = +1/2 5S

5P

+ -

Figure 3: Interaction of circularly po-larized light with the n 5S ground state.Conservation of angular momentum requiresthat + light interacts with n atoms andvice versa.

Alkali metal atoms have a single electron in the outer energy shell and energy calculationsare well approximated by consideration of the contribution from this single electron and thenucleus [5]. Measurement of the rotation noise signal involves excitation of this single electronfrom the 5S (angular momentum l = 0 state) to the 5P (l = 1) state by the probe laser. In thissection, we explain the energy structure of the alkali metal atom ground and excited states.

Defining J = L+S, the P state has 2 sub states with j = 1/2 and j = 3/2, as l+s = 1+1/2 =1/2 or 3/2. The 5S to 5P1/2 transition is known as the D1 transition and the 5S to 5P3/2 isknown as the D2 transition. These excitations are the optical transitions and characterizedby an energy difference E = ~0, where 0 is the optical resonance frequency. Figure (2)illustrates the D1 optical transition.

Due to the nonzero nuclear spin of alkali metal atoms, the ground state of these atoms issplit into two hyperfine components with angular momentum F = I 1/2 by the interactionHamilitonian H = AI S, where A is some constant. Here F is the total atomic spin F =L+ S+ I, where I is the nuclear spin and S is the electronic spin. Within each hyperfine state,the degeneracy of the ground state is lifted by the Zeeman effect due to the Bz field. The Bzfield is small enough ( 1T ) so that the Zeeman Hamiltonian HZ = e2m(L + 2S) Bz canbe treated as a perturbation to the unperturbed wavefunction [4]. According to first orderperturbation theory, the correction to the energy is:

E1Z = BgJBzmj (4)

where B is the Bohr magneton, gJ is the Lande g-factor, andmj is the spin quantum number [6].For example, within one hyperfine component, the ground state of the valence electron in the5S state has l = 0, j = s = 1/2, and thus mj = 1/2. By (4), the previously degenerate groundstate has shifted into two energy levels. The magnitude of this energy splitting is quantified by~L, where L is the Larmor precession frequency (see section 5.1) [5]. Figure (2) shows an

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illustration of Zeeman split sub levels and the D1 optical transition within one hyperfine stateof the alkali metal atom. Quantum fluctuations transfer atoms between the n+ and n statesand the next section explains that this fluctuation leads to the rotation signal.

4 Paramagnetic Faraday Rotation

In this section, we explain how rotation of the plane of polarization of linearly polarizedprobe light occurs and show that is proportional to F , the atomic spin in the direction ofprobe propagation. The linearly polarized probe light can be decomposed into two oppositelyrotating circularly polarized waves . Conservation of angular momentum requires that +

light only interact with m = 1/2 atoms and vice versa (figure 3). Due to this interaction,we show that each distinct polarization direction has a different frequency, and thus a differentindex of refraction. This phenomena is known as dispersion, as the phase velocity of eachpolarization direction is frequency dependent. When the probe beam is rewritten as a linearlypolarized wave, this frequency difference leads to a rotation of the plane of polarization by .

The phase velocity v of a wave is defined as

v =

k=c

n(5)

where k is the wavenumber, is the frequency, c is speed of light, and n is the index of refraction,given by

n =

00. (6)

In this model, the relative permeability /0 is taken to unity [9]. Equations (5) and (6)suggest that examination of the the difference in () between the two polarizations providesthe frequency difference.

To derive an expression for electric permittivity as a function of frequency , considerthe equation of motion for a single electron subject to a damping force measured by and aharmonic force

m(~x+ ~x+ 20~x) = e ~E(~x, t) (7)where x denotes the temporal derivative. We try as a solution x = eit. Plugging in andrearranging gives

~x = eit =em

(20 2 i)1 ~E. (8)The dipole moment is then

p = e~x = e2

m(20 2 i)1 ~E = 0e ~E (9)

where the last equality is the definition of the polarization density, with e as the electricsusceptibility. We identify the susceptibility as:

e =e2

0m(20 2 i)1. (10)

From the definition ()/0 = 1 +e with e

0 [2]. The expressione2

0mecan be rewritten as 4pirec

2, where re is the electron radius. Thus,for the D1 transition in alkali metal atoms, we have

()

0= 1 +Nfosc4pirec

2(20 2 i)1. (12)

Although the frequency of the probe laser is detuned from the resonant frequency 0 [4], both and 0 >> so 0 >> . Thus, when compared to , we assume that 0. Usingthis approximation, the above equation is written

()

0= 1 +

2piNfoscrec2

0

0 + i/2(0 )2 + (/2)2 . (13)

We now decompose the linearly polarized probe light into two counter rotating circularlypolarized light waves, . Let n denote the atomic density of atoms in the spin 12 and spin12 states respectively. Optical transitions from the 5S to 5P require that atoms in each stateabsorb a photon from the probe beam. photons carry angular momentum ~ (m = 1),respectively [5]. Atoms with m = 1/2 in the n state may absorb a photon with m = +1 totransition into the m = +1/2 level of the 5P1/2 state. On the other hand, atoms in the groundstate with m = +1/2 may absorb a photon with m = 1 to transition into the m = 1/2 levelof the 5P1/2 state. These interactions between

light and the n states are shown in Figure(3). However, an atom in the n+ state cannot interact with

+ light as there is no excitedstate with an additional +1 angular momentum. From (13), fluctuations of the atom numberbetween the two states cause a difference in () for the two polarization directions, leading toFaraday rotation.

The linearly polarized light wave is written as two circularly polarized light waves in the xyplane as

+ : E1x = cos(k+x t) and E1y = sin(k+x t) (14) : E2x = cos(kx t) and E2y = sin(kx t). (15)

The difference in indices of refraction for is contained in the difference in the wavenumberk. If we write the wavenumber k as

k = + i/2, (16)

(5) relates k to the frequency dependency of the permittivity so

2 2/4 = 2

c2Re[()/0]. (17)

We can ignore the as it is an exponential decay when the wave is written as eikx (ex) andaffects both equally.

Using the Taylor expansion

1 + x 1 +x/2, from (13) and (17) we write the wavenumberfor the two polarization directions

k+ =

c

1 + 2n

2pifoscrec2

0D()

c+ nfoscrecD() (18)

k =

c

1 + 2n+

2pifoscrec2

0D()

c+ n+foscrecD(). (19)

In (18) and (19), N from (13) is replaced by either 2n+ or 2n. Derivation of () assumesunpolarized light, but we assume spherical symmetry so () for light coming uniformly inall directions is equivalent to () for all the light coming from one direction. So choosing a

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polarization direction such as , the total atomic density N is given by either 2n+ or 2n. Forexample, as far as + light is concerned, all atoms are in the m = 1/2 state, and assumingequal distribution of atoms between the states, ntotal = 2n. D() is identified as the dispersionprofile written in terms of frequency . Using = 2pi, it is given by

D() =0

(0 )2 + (/2)2 . (20)

From (14) and (15), the probe field in the x direction is given by

E1x + E2x = cos (x/c t+ nfoscrecD()x) + cos (x/c t+ n+foscrecD()x) . (21)

If we let A x/c t and define the polarization P asP =

n+ nn+ + n

with n n+ + n (22)

then (21) is written

E1x + E2x = cos (A+ nfoscrecD()x) + cos (A+ nfoscrecD()x+ nPfoscrecD()x) . (23)

Using cos(X) + cos(X + Y ) = 2 cos(Y/2) cos(X + Y/2) (23) is:

2 cos(nPfoscrecD()x/2) cos(A+ nfoscrecD()x+ nPfoscrecD()x/2). (24)

Using the same procedure for E1y + E2y with sin(X) sin(X + Y ) = 2 sin(Y/2) cos(X + Y/2)

gives

E1y + E2y = 2 sin(nPfoscrecD()x/2) cos(A+ nfoscrecD()x+ nPfoscrecD()x/2). (25)

The tangent of the rotation angle of the plane of polarization is given by the ratio of the fieldin the y direction to x direction

tan() =E1y + E

2y

E1x + E2x

= tan(nP

foscrec

2D()x

). (26)

Thus, identifying x as the path length l of the probe laser, and using tan(A) = tan(A) wehave for

=1

2nlPfoscrecD( 0) (27)

where the dispersion profile from (20) is rewritten

D( 0) = 0( 0)2 + (/2)2 . (28)

The rotation angle in (27) is proportional to atomic spin as the definition of polariza-tion in (22) is proportional to the x component of the atomic spin polarization vector ~P (Fx, Fy, Fz)t [10] [1]. If nuclear spin is ignored, for a spin 1/2 particle, Fx = P/2. Ac-counting for nuclear spin adds a factor of 2I+ 1 in the denominator of P/2, but proportionalityis unchanged [4]. Thus, (27) shows that the measurement of the rotation angle, and thus themagnetic field, is fundamentally limited by uncertainty in spin measurement (see (3)).

5 Analysis of the Rotation Signal

In this section we derive the Larmor precession frequency L of the spin vector about thestatic Bz field and show that this frequency is proportional to the magnetic field strength. Wethen examine the contribution of spin spin relaxation to the sinusoidal rotation signal oscillatingwith frequency L. After accounting for this contribution, we examine the power spectrum of thenoise 2 of the rotation signal and show it has a Lorenztian shape with total noise given by thearea under its peak centered at L. Finally, we derive an expression relating the rotation angle to the voltage across the photodetectors PD1 and PD2. This expression will be important toderive the expression for effective number of atoms probed.

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5.1 Larmor Precession

In addition to causing an energy shift of the alkali metal atoms, Bz causes Larmor spinprecession of the atomic ensemble about Bz. To derive the frequency of this precession, considera spin 1/2 particle in the ground state subject to a field of magnitude B0 in the z direction.Ignoring nuclear spin, the Hamiltonian is H = B0Fz, where is the gyromagnetic ratio [6].The eigenstates are the eigenvectors of this Hamiltonian

+ =

(10

)with eigenenergy E+ = B0~/2 (29)

=(

01

)with eigenenergy E = B0~/2. (30)

Because the Hamiltonian is time independent, the solution to the Schrodinger equation

i~

t = H (31)

can be written in terms of the stationary states as

(t) =

(aeiB0t/2

beiB0t/2

)(32)

where a and b are determined by initial conditions at (0). If we let a = cos(/2) and b =sin(/2) and take expectation values of the Fx and Fy operators we find

Fx = (t)Fx(t) = ~/2 sin cos(B0t) (33)and

Fy = (t)Fy(t) = ~/2 sin sin(B0t). (34)Equations (33) and (34) show the spin vector F is titled at an angle about the z axis

and precesses at the Larmor frequency given by

L = B0. (35)

Equation (35) shows that the magnitude of the static field is proportional to the Larmor fre-quency. Equations (27), (33), and (34) show that without the effect of spin exchange relaxationdiscussed next, the rotation signal would be sinusoidal in time with frequency L.

5.2 Spin Exchange Relaxation

An additional factor affecting the rotation signal (t) is collisions between alkali metal atoms.In these spin exchange collisions, the combined spin of the atoms is conserved, but the spins ofthe atoms involved may rotate. An example for atoms A and B is given by:

A() +B() A() +B(). (36)

Because the alkali metal ground state is split into two hyperfine states with angular mo-mentum F = I 1/2, spin exchange collisions transfer atoms between the two hyperfine states.These random processes cause the atomic angular momenta vectors to acquire random angleswith respect to each other, and thus the spin vector F decays from angle to its equilibriumposition with no transverse (x,y) spin components [3] [7]. This decay of the transverse com-ponent of the spin vector is analogous to the evolution of the magnetization vector in nuclearmagnetic resonance (NMR), and is characterized by a spin spin relaxation time T2. The decayof the transverse spin component is thus given by the Bloch Equations, which describe the

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time evolution of the magnetization vector with given relaxation times [8]. A solution to thisequation for spin exchange relaxation is given by

F xy(t) = F xy(0)et/T2 . (37)

Thus, the effect of spin spin relaxation adds an exponential decay to the sinusoidal noise signal

(t) = 0 sin(Lt)et/T2 (38)

where 0 is the maximum Faraday rotation angle given by (27) in the regime where 0 >> .From (28), D( 0) 1/( 0) and 0 is [2]

0 =1

2nlrecPfosc/( 0). (39)

5.3 The Power Spectrum of the Rotation Noise Signal

Figure 4: Plot of Faraday rotation signalgiven by (38). This exponentially decayingsinusoid oscillating at L also describes thetime evolution of the magnetization vectorin response to a transverse rf pulse.

Figure 5: The power spectrum of the rotationnoise signal 2 given by the Fourier transformof the response in Figure (4). It has a Lorentzianshape peaked at L and the total noise presentis the area under this plot.

Our investigation is focused on the power spectrum of 2, the noise (fluctuations) ofthis rotation signal. To determine the behavior of this power spectrum, we use the fluctuationdissipation theorem, which states that the power spectrum of fluctuations is proportional to thefrequency response of the system to an external perturbation [1] [12]. Although originally for-mulated for systems in thermal equilibrium, it applies equally well to quantum spin fluctuationsystems like ours [13]. This theorem links the fluctuations of the system about its equilibriumstate to its response to a small driving force. Thus, to obtain the power spectrum for fluctuationsof the rotation angle , we examine its frequency response to a small perturbation.

As explained in section 5.2, the atomic spin vector in our system is analogous to the mag-netization vector in NMR. Thus, to examine the frequency response of F, we examinethe response of this magnetization vector to a small perturbation. In NMR, the static Bz fieldcauses the magnetization vector to align in the z direction. A small perturbation by an rf (radiofrequency) pulse directed in the transverse plane tilts this vector away from the z axis andcauses precession of the vector about the z axis at Larmor frequency L [7]. This precessionwas derived quantum mechanically in section 5.1. We also explained in 5.2 that the magnetiza-tion vector decays to its original orientation with a decay time characterized by T2. Thus, thetime evolution of the magnetization vector in response to the small rf pulse is an exponentiallydecaying sinusoid oscillating at L like the structure of the rotation signal in (38).

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To convert the temporal evolution of the magnetization vector to its frequency response,we take the Fourier transform of the exponentially decaying sinusoidal evolution. The Fouriertransform of a decaying sinusoid has a Lorentzian line shape peaked at L. Thus, by the fluctu-ation dissipation theorem, the power spectrum of fluctuations 2 is also a Lorentzian peakedat L. Figure (4) shows the temporal response of the magnetization vector to a transverse rfpulse. Figure (5) shows the Fourier transform of the response given in figure (4), which is equiv-alent to the power spectrum of the rotation noise signal 2. Figure (5) has units of rad2/Hzto account for the effect of measurement time on rotation sensitivity. It has a Lorentzian lineshape peaked at L, and this frequency is proportional to magnetic field strength. The values ofthe power spectrum within a small frequency range give the noise for that range so integrationof the power spectrum over all frequencies gives the total noise present [4] [18].

5.4 The Output Voltage Signal

As shown in figure (1), after the probe beam exits the multipass cell, it is split by a polarizingbeam splitter (PBS) oriented at 45 to the initial direction of probe beam polarization [2]. Thetwo photodetectors PD1 and PD2 receive an intensity signal proportional to |E|2 of the probebeam. Without the Faraday rotation angle , the signal received by the two detectors wouldbe equal. To examine the signal difference, we write for the two detectors,

I1 = I0 cos2( pi

4) and I2 = I0 sin

2( pi4

) (40)

where I1 + I2 = I0. The voltage output from the polarimeters is proportional to the intensity,and to measure the rotation angle, we take the voltage difference of the two detectors (see figure1)

Vout = V1 V2 = V0 sin(2). (41)For small rotation angles sin and as V0 = V1 + V2, we have

=V1 V2

2(V1 + V2). (42)

6 Effective Number of Atoms Probed

In this section we develop a formula for number of atoms measured and show it is relatedto an effective measurement area and depends on spatial probe laser intensity profile integrals.If we assume that the probe beam is of uniform intensity, the number of atoms participatingin measurement would be N = nlA, where A is the photodetector area, n is atomic density,and l is probe path length . However, laser beams are Gaussian with intensity not uniformlydistributed across each detector so we find an expression for the effective detector area.

To find an expression for this area, consider (42), which relates the rotation angle to thevoltage across each detector. We know that is proportional to the atomic spin F, and forunpolarized atoms the root mean square (rms) spin fluctuations scale as

Fx2 1/

N (see

equation (3)) [4]. So 2 scales as 1/N and by examining the variance of the noise signal, wecan find an expression for N and Aeffective.

We divide each photodetector into a grid so the total voltage is the sum of the voltage ineach grid block i, and assume that for each detector, photons incident on adjacent grids areindependent. From (42) we write

2 =

(

i V1,i

i V2,i)2

4(

i V1,i +

i V2,i)2

=

i(V

21,i + V

22,i)

4(

i(V1,i + V2,i))2

. (43)

9

The cross terms from squaring the numerator vanish after taking the average as voltage acrossadjacent blocks is independent. Using the orientation shown in figure (1), the photodetectorsare in the yz plane and we identify the numerator of (43) as

I(y, z)2dydz and the denominator

as [I(y, z)dydz]2 as the total voltage across each detector is given by the area integral of the

intensity. Using N 1/2, the reciprocal of (43) gives N as

N = nlAeffective = nl[I(y, z)dydz]2I(y, z)2dydz

. (44)

This expression assumes that in the probe direction, the square of the intensity profile is uniformin the yz plane. Relaxing this assumption, we integrate the denominator over the entire volumeof the multipass cell so N is

N = nl2[I(x, y, z)dydz]2I(x, y, z)2dV

(45)

where

Aeff = l[I(x, y, z)dydz]2I(x, y, z)2dV

. (46)

It is these two intensity profile integrals we wish to calculate.

7 Optics of Gaussian Beams

To calculate these integrals, we must develop a general formula for the intensity of the laserbeam within the cell. We first explain the geometry of the multipass cell and then develop thetheory for the intensity distibution. The laser beam is initially circular when it enters the cell.But due to the geometry of the mirrors, the beam becomes generally astigmatic with ellipticallight spots at each cross section as it passes through the cell [14]. We first review the theory ofpropagation of a stigmatic (circular) Gaussian laser beam and then generalize to the astigmaticcase. Then using the ABCD ray matrix approach, we trace the position of the center of thebeam as it passes through the mirrors. Given the position of the beam centers and the generalbeam intensity description, we calculate the beam pattern at each cell cross section and theintensity distribution.

7.1 Geometry of the Multipass Cell

Figure 6: The multipass cell consists of twocylindrical mirrors M1 and M2 separated bydistance d oriented at an angle relative toeach other. Figure is taken from figure (1)of [2]

Figure 7: The entrance of the probe beaminto the multipass cell. It may have an anglefrom the z axis of with a twist angle aboutthe z axis .

Figure (6) shows the multipass cell which consists of two identical cylindrical mirrors M1and M2 with equal focal lengths of 50mm separated by a distance d and oriented at a twist

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angle relative to each other. For the rest of this paper, let z be the axis of propagation of theprobe beam instead of x so the integrals in the numerators of (45) and (46) are in the xy plane.The probe laser enters through a hole in M1 at an angle relative to the z axis and may alsobe twisted about the z axis by an angle (figure (7)). Using the ABCD ray matrix approach,we trace the path of the laser within the cell and find suitable separation distances and twistangles so the beam makes N passes and exits through the same hole. Taking a cross sectionof figure (6) shows that there are N beam spots at each cross section except at the planes ofM1 and M2, which have N/2 beam spots due to complete overlap. We choose the origin of thecoordinate system where the beam ray enters the cell. The initial configuration of the beam rayin figure (7) is specified by the vector (x0, y0, x

0, y0)t, where (x0, y0) are initial positions and

(x0, y0) are the projections of the ray direction unit vector onto the x and y axes [14]. For smallinput angles sin so the beam configuration vector is written

(0, 0, cos, sin)t. (47)

7.2 Intensity Distribution of Stigmatic Gaussian Beam

We consider the case where the electric field E of the beam has a propagation direction (z)and varies in the transverse direction r =

x2 + y2. Beam like solutions of the wave equation

are given in the form E(x, y, x) = (x, y, z)eikz where is localized about the z axis andhas spherical surfaces of constant phase over a small region [15]. We show the beam intensityI = |E|2 exhibits a Gaussian radial dependence, confirming localization. We find that thebeam is characterized by a complex parameter q(z), which can be written in terms of tworeal parameters, the beam waist w(z), and the radius of curvature of the phase front R(z).Physically, the beam waist is where I is 1/e2 of its maximal value. We seek to describe theintensity profile of the beam everywhere using the known parameters, , the laser wavelength,and w0, the initial waist size.

For any component E of an electromagnetic wave, the time independent propagation is givenby the Helmholtz wave equation

2E + k(~r)2E = 0. (48)From equations (5) and (6) and c = 1/

00, we identify k =

()() = 2pi/ as the

wavenumber. We try the beam like solution E(x, y, x) = (x, y, z)eikz and assume that thebeam remains paraxial (see 7.4.1), so varies slowly with z. This allows us to ignore the 2/z2

term so2

x2+2

y2 2ik

z= 0. (49)

For we try the fundamental Gaussian beam solution

(x, y, z) = exp

{i[P (z) +

k

2q(z)r2]}

. (50)

In the above equation P (z) is a phase shift, and q(z) is the complex beam parameter. Both Pand q must be specified as explicit functions of z for complete beam description. Substituting(50) into (49) yields:

2k(dP

dz+

i

q(z)

)(

k2

q2(z) k

2

q2(z)

dq

dz

)r2 = 0. (51)

Because his equation must hold for all values of r, each term is zero and we obtain two differentialequations for q(z) and P (z):

dq

dz= 1 = q(z) = q0 + z (52)

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dP

dz=iq(z)

(53)

where q0 is the integration constant, the value of q at an origin to be specified. For a physicalinterpretation of q, we write it in terms of the two real parameters w(z) and R(z)

1

q=

1

R ipiw2

. (54)

With this definition (50) is rewritten

= exp

{i[P (z) +

kr2

2

(1

R ipiw2

)]}. (55)

We choose the origin z = 0 where there is no radial phase variation in the field . From (55),this choice requires R at z = 0. Equation (54) at the origin gives

1

q0= i

piw20(56)

where w(0) = w0 is the beamwaist. Plugging the value of q0 from (56) into (52) gives q as afunction of z

q(z) =ipiw20

+ z. (57)

From (57) and using the general definition of q in (54), we have expressions for R(z) and w(z)in terms of only the known parameters w0 and

w2(z) = w20

[1 +

(z

piw20

)2](58)

R(z) = z

[1 +

(piw20z

)2]. (59)

Equation (58) emphasizes that w0 is the minimum beam waist which expands along z and (59)gives the radius of curvature for the spherical phase front at z [15].

Equations (54) (58) and (59) provide a complete description for q(z) and it remains todetermine P (z). Using (57) we can solve the differential equation (53) for P (z)

P (z) = i ln(w(z)

w0

) tan1

(z

piw20

). (60)

Inserting this expression for P (z) into (55) and taking the beam intensity, I = ||2 we find

I(x, y, z) 1w2(z)

exp

( 2r2w(z)2

)(61)

This shows that the beam waist is where I is 1/e2 of its maximal value and confirms the Gaussianbeam intensity distribution. Proportionality is sufficient because from (45), the constant ofproportionality cancels when evaluating the effective atom number N .

7.3 The Astigmatic Case

The thin lenses of the multipass cell are cylindrical mirrors which are astigmatic due todifferent focal lengths in the x and y directions (fx 6= fy). Upon reflection by M2, the incominginitially circular beam is transformed into a simple astigmatic beam with elliptical light spots.The different directional focal lengths cause the previously circular beam with equal waists

12

Figure 8: The simple astigmatic Gaussianbeam with elliptical cross sections (green andblue) and unchanging orientation angle alongpropagation. Figure adapted from figure (3)of [16]

Figure 9: The generally astigmatic Gaus-sian beam. The cross sections are still ellipti-cal but change orientation along propagation.Figure adapted from figure (3) of [16]

(wx = wy) to have elliptical cross sections (wx 6= wy). This simple astigmatic beam does notchange orientation angle as it propagates. When this simple astigmatic beam hits M1 after M2reflection, it becomes generally astigmatic with w1(z) 6= w2(z) and changes orientation alongpropagation [17] [16]. The xy subscripts are dropped as beam rotation means the waists areno longer measured relative to the xy axes. Figures (8) and (9) show the simple and generalastigmatic beam.

For the simple astigmatic beam the electric field can be written

= (q1q2)1/2 exp

[ik

2

(x2

q21+y2

q22

)](62)

which satisfies the wave equation (49). Rotation of this beam by a complex angle around thez axis describes the beam with general astigmatism [17] [16] and is given by

= (q1q2)1/2 exp

{ik

2[Q1x

2 +Q2y2 + tan(2)(Q2 Q1)xy]

}(63)

where

Q1 =cos2

q1+

sin2

q2and Q2 =

sin2

q1+

cos2

q2(64)

and the real part of represents the physical beam rotation.From (61) we write the intensity distribution of the general astigmatic Gaussian beam

I 1w1w2

exp

(2x2w21

+2y2w22

). (65)

This expression contains the rotational information if the x and y axes are defined by the majorand minor axes of the elliptical beamspot.

Equations (54), (58), and (59) describing the radii of curvature, the waists, and the qparameters as functions of z still hold, but the two components must be considered separately[17] [16]. For example we can write

q1 = q01 + z and q2 = q02 + z. (66)

Once q01 and q02 are known, we can solve for the waists as functions of z for each beam using(54) (58) and (59).

13

7.4 Beam Transformation by Lenses

To fully characterize the laser beam within the cell, we must also track the position of itscenter as it propagates between the mirrors. The center position of each elliptical beam spotalong the z direction is given using the ABCD ray matrix approach. Then we consider howthe q parameters and complex angle change upon reflection by an astigmatic lens. Withinformation about beam spot locations, their waists, and rotation angles, the beam patternat each cell cross section is plotted. Sample beam patterns at M1, the cell center, and M2are shown in figure (10) for separation distance d = 18.52mm, 50 passes, = 16.77rad, and = 0.05. For all laser calculations in this paper, the beam is initially circular with a waist of0.4mm, wavelength = 795 106mm, and no rotation angle . The minor and major axes ofthe elliptical spots are the beam waists and there are only 25 spots at M1 and M2 because ofcomplete beam overlap.

(a) Beam pattern at M1 (b) Beam pattern at center (c) Beam pattern at M2

Figure 10: Sample beam patterns at M1, the center of the cell, and M2, for d = 18.52mm, 50passes, = 16.77 rad, and = 0.05. The major and minor axes of the beam ellipses are thebeam waists where intensity falls to 1/e2 of the maximal value. x and y axes are in mm.

7.4.1 The ABCD Matrix and the Paraxial Approximation

We describe the laser beam as a ray propagating along the z direction. At any point, theray is described by distance from the z axis, r, and the angle it makes with it, r. The paraxialray approximation assumes small angles so sin(r) tan(r) r. This approximation is validgiven that the beam enters the cell at a small angle of inclination (see figure (7)). To derivethe ABCD matrix approach for the beam center, consider the value of r and r at two planesalong the z axis, plane 1 with (r1, r

1) and plane 2 with (r2, r

2). The approximation assumes the

linear relation [15]

r2 = Ar1 +Br1

r2 = Cr1 +Dr1

(67)

or in matrix form (r2r2

)=

(A BC D

)(r1r1

). (68)

The elements of the ABCD matrix are determined by the media the ray propagates through.As a simple example, consider the case where the ray travels through a uniform optical medium(the space between the multipass cell) of length d. The relations between the rays at the twoplanes separated by d are

r2 = r1 + d tan r1 r1 + dr1

r2 = r1.

(69)

14

The ABCD matrix M is

M =

(1 d0 1

). (70)

To generalize this approach to the position of the beam spot in the cell, we identify r as (x, y)t,the coordinates of the laser spot in the (x,y) plane, and r as (x, y)t, the projections of aunit vector along the x and y axes. Given an initial set of coordinates (x, y) and an initial setof projections (x, y), (69) describes the center position of the beam spot at any distance ifd is replaced by an arbitrary distance z. The general astigmatic Gaussian beam propagationthrough the cell may be described using this approach by writing [14]

x2y2x2y2

= (A BC D)

x1y1x1y1

. (71)where A, B, C, and D, are 2x2 matrices specific to the optical configuration. Let the subscripts(1,2) now denote the planes of (M1, M2) respectively.

Within the paraxial approximation, the ABCD matrix to describe propagation throughan optical configuration is given by the product of the matrices for each component of thesystem [14]. The matrix for one round trip between the mirrors is thus

M = M1(f1)D(d) T ()M2(f2) T ()D(d). (72)

Reading right to left, the beam goes through the hole in M1 and travels distance d, then thecoordinate system is rotated by to account for the rotation of the mirrors. Then the beamis transformed by M2 and the coordinate system is rotated by back to original coordinates.The beam travels distance d towards M1, and is finally transformed by it (the last term in (72)).All matrices in (72) are 4x4, dependent only on the parameter specified in their argument, andgiven in [14]. For N round trips between the mirrors, the ABCD matrix is simply given byMN . For example, given initial positions and projections (x0, y0, x

0, y0)t at M1 given in (47),

the parameters (x2, y2, x2, y2)t after N round trips on the back mirror are given by

x2y2x2y2

= D(d)MN1 x0y0x0y0

. (73)In the above equation, the beam passes N 1 times between the cells, ends up at M1, thentravels distance d to the back mirror M2. With (72) and initial values at M1 given in (47), thepositions of the beam center at the front and back mirrors after an arbitrary number of passescan be calculated.

With the beam center position at each mirror determined, we describe the center positionof the beam everywhere within the cell by rewriting the paraxial propagation equation (69) as(

x2y2

)=

(x1y1

)+ z

(x1y1

)(74)

where the subscripts 1 and 2 refer to any two planes orthogonal to the z axis. With the beamposition after N passes determined, we find a mirror separation distance and twist angle suchthat after N passes, the beam exits the cell through the hole in M1.

15

7.4.2 Transformation of q Parameters and Complex Angle

The ABCD matrix approach can also be used to determine how the complex beam param-eters are transformed by lenses [15] [17]. However, we use an alternate method given in [17]based upon the phase shift a thin astigmatic lens introduces into the laser field. Our cylindricalmirror with focal lengths f1 and f2 twisted by with respect to the other mirror introduces thephase shift

k

2[F1x

2 + F2y2 + tan(2)(F2 F1)xy] (75)

where

F1 =cos2

f1+

sin2

f2and F2 =

sin2

f1+

cos2

f2. (76)

Denoting with a prime the Q and parameters of the transformed beam, we write the phaseof the beam after transformation

k

2[Q1x

2 +Q2y2 + tan(2)(Q2 Q1)xy]. (77)

Inserting the phase shift given in (75) into the field equation for the beam before transformation(63), we identify

Q1 = Q1 F1 and Q2 = Q2 F2 (78)and

tan 2 =tan(2)(Q2 Q1) tan(2)(F2 F1)

(Q2 Q1) (F2 F1) . (79)

To convert back to the q parameters, (64) solved for q1,2 is

q1,2 =2

Q1 +Q2 [(Q1 Q2)/ cos(2)] . (80)

Together, the above three equations describe how the complex beam parameter q and the angle are transformed by lenses. Thus, for every reflection off the mirror, we may generate newq and values for the beam. From the analysis given in 7.3 and these initial q and values,we find the the value of the beam waists and rotation angles and thus the intensity for each zposition of each beam pass.

8 Intensity Profile Integrals

Using the techniques developed in section 7, for every xy cross section at some specified z, wehave N beam spots with known waists, rotation angles, and center positions (figure (10)). Theintensity distribution at each cross section is defined as the sum of the intensity distributionsof each beam spot

I(r) =i

Ii(r) (81)

where r = (x, y, z). Although the beam spot changes shape and orientation as it propagates,no power is dissipated, so the intensity integral in the numerator should be constant for each zslice. Indeed, integration of (65) in the xy plane gives pi2 .

To calculate the integral in the denominator of (46) numerically, for each z position, wedivide the xy plane into a grid, find the total intensity contribution from all beams at eachpoint using (65) for each beam, square it, and integrate over the xy plane. Then we integratethe result over the z direction.

The 3D numerical integration can be simplified if we ignore the intensity contribution frombeam overlap. The overlap information is contained in the cross terms of the expression (

i Ii)

2.

16

Ignoring these cross terms, the integrand in the denominator can be writen as

i I2i so the

effective area is

A = l(

i Iidxdy)2

i I2i dV

= l(

i pi/2)2

i

I2i dV

= l(Npi/2)2i

I2i dV

(82)

where N is the number of passes in the cell. The integral in the denominator can be analyticallydone in the x and y directions without regard to rotation angle as the integration covers theentire beamspot

I2i dxdy =

1

(w1iw2i)2exp(

4x2w21i

4y2

w22i)dxdy =

pi

4(w1iw2i)

1. (83)

With this simplification, calculation of the effective area is reduced from a 3 dimensional integralinto a one dimensional one as integration over each xy cross section to account for beam overlapis avoided. A is now given by

A = l(Npi/2)2

ipi4

(w1i(z)w2i(z))1dz

. (84)

We do this z integral in the denominator for each beam pass separately, then sum over all beampasses.

8.1 The High Regime of Minimal Beam Overlap

To compare the no beam overlap area calculation to the general case, we seek a multipass cellgeometry to minimize overlap. This minimization is achieved by increasing the input beamangle so the beam passes are further apart. Figure (11) shows the effect of increasing onbeam overlap at the center of the cell with 50 passes, d = 9.94mm, = 22.97 rad, and increasing values. Notice that with increasing , the area which encloses all beam spots increases toaccount for increased beam spot separation. Physically, we are limited in how high can beby mirror size. We also make the small angle (paraxial) approximation so must remain smallenough such that sin .

(a) = 0.05 (b) = 0.1 (c) = 0.2

Figure 11: Beam patterns at the cell center for 50 passes, d = 9.94, = 22.97 rad, andincreasing values. As increases, so does beam separation, decreasing overlap. x and y axesare in mm.

We compare the ratio of the area determined using general numerical calculation of (46) tothe approximate one in (84) as a function of increasing . The results are shown in figure (12)for two different cell geometries. The approximate area calculation represents a maximum valueof the area in the case of no beam overlap. This can be seen from (84) as we have ignored thepositive cross terms contribution, making the denominator smaller than the general case. As increases, the values of the approximate calculation and general one converge, in both casesexhibiting only a 5% difference when is 0.4. Note that perfect agreement is impossible, as on

17

Figure 12: Ratio of area calculation in the general case considering beam overlap to thesimplified regime ignoring it for two mirror geometries. Red data: d=18.52mm, =16.77rad.Blue data: d=9.94mm, =22.97rad. The number of passes and laser specifications are identicalfor the two cases. As increases, overlap decreases, and this area ratio approaches unity.For both cases, at = 0.4, the simplified calculation agrees with the general one within 5%.Complete agreement is impossible due to complete beam overlap at M1 and M2.

the planes of M1 and M2, the beamspots always overlap. Nevertheless, the approximate areamethod provides a simple and quick method for approximating the area in increased cases,avoiding a computationally expensive 3D integration.

9 Conclusion

This paper explains the operational principle behind atomic magnetometers and then char-acterizes the intensity profile of the probe laser beam used in field measurement. Atomicmagnetometers have demonstrated the highest magnetic field sensitivity to date so studyingthem explores the limits of magnetic sensitivity [3]. These magnetometers measure the Faradayrotation angle of the plane of polarization of probe laser light as it propogates through alkalimetal vapor. We show that this angle is proportional to the atomic spin in the direction ofmeasurement, so spin uncertainty presents a fundamental sensitivity limit. Because Faradayrotation involves excitation of the ground state electron of alkali metal atoms by the laser, wereview the energy levels of alkali metal atoms. We then explain the principle behind Faradayrotation and show that the rotation signal (t) has a decaying sinusoidal structure in timeand oscillates at the frequency L of Larmor precession of the atomic spin vector about themagnetic field. To examine magnetometer sensitivity, we quantify the effect of noise from spinuncertainty and other experimental sources on the rotation signal by examining 2(t). TheFourier transform of the rotation noise signal is a Lorentzian peaked at L which is proportionalto magnetic field strength. The area under this Lorenztian gives the total rotation noise presentin the signal.

The latter part of this paper characterizes the intensity profile of the Gaussian laser beam asit propagates through the multipass cell. This profile is essential in calculation of the effectivenumber of atoms measured and in quantifying the effect of diffusion on the noise signal [11]. Tocharacterize this profile, we use the ABCD ray matrix approach to trace the center of the laserbeam through the cell. We also find the expression for spatial intensity of the general astigmaticGaussian beam. With the intensity and center position of the beam propagation determined,we plot the beam pattern after a specified number of passes within the cell. Finally, we concludeby numerically computing integrals involving this intensity profile to find an expression for theeffective area of the photodetector used to measure the rotation signal. The number of atoms

18

measured depends on this area and we demonstrate that the approximation of ignoring beamoverlap within the cell reduces a three dimensional integral to one dimension. We find laser andoptical cell configurations so this approximation holds and show that the simpler integrationgives area values within 5% of the general calculation.

Further research should explore more efficient integration methods for calculation of atomsmeasured, as the three dimensional integral is currently done as a Riemann sum and othermethods like Monte Carlo may be less computationally expensive. To expand the frontier ofmagnetometer sensitivity, it is important to quantify or eliminate the effects of noise other thanspin noise on the rotation signal. These noise sources include photon shot noise, spin relaxation,and diffusion of atoms in and out of the beam during measurement [18]. In particular, the effectof diffusion is quantified in [11] and involves spatial intensity profile integrals. As more sources ofexperimental noise are either quantified or eliminated, we approach the fundamental quantummeasurement sensitivity limit of atomic magnetometers in the attotesla regime [5]. Indeed,current research in magnetometry is approaching attotesla sensitivity [5].

10 Acknowledgements

I thank Professor Romalis for his guidance throughout this project and Dr. Dong Sheng fordeveloping the multipass cell beamspot code and general integration method.

This paper represents my own work in accordance with University regulations.

19

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