Multi-field frequency modulation
spectroscopy

doi:10.1364/OE.16.006081

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Multi-field frequency modulation spectroscopy

Ido Ben-Aroya, Matan Kahanov, and Gadi Eisenstein »View Author Affiliations

Optics Express, Vol. 16, Issue 9, pp. 6081-6097 (2008)
http://dx.doi.org/10.1364/OE.16.006081

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▸ Article Outline
– Abstract
1. Introduction
2. Theoretical analysis
3. Experimental results versus the analytic model
4. Residual amplitude modulation
5. Summary
– References and links

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Abstract

We Study a modification of classical FM spectroscopy in the cases where several electromagnetic fields are FM modulated, each in a different manner. This complex spectrum scans a multi-photon resonant atomic medium with the output detected by a phase-sensitive scheme. The demodulated output signal reveals the spectroscopic features of the probed medium. The case in which two different carriers are FM modulated at the same frequency and index but with an opposite phase with respect to each other is analyzed theoretically. This configuration is essential for probing Coherent Population Trapping (CPT) resonances induced by a directly modulated diode laser. Employing a macroscopic model to describe the physical properties of CPT leads to a superb fit between predicted and measured CPT characteristics.

1. Introduction

Frequency Modulation (FM) spectroscopy [1

G. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. 5, 15–17 (1980). [CrossRef] [PubMed]

] is a high resolution, sensitive spectroscopic technique developed by Bjorklund in the 1980’s. The technique employs an FM signal which is superimposed on an optical carrier that traverses a resonant atomic medium; at the output of which it is detected and demodulated. The resulting complex signal contains a signature of the spectroscopic features characterizing the probed medium, namely: absorption and dispersion coefficients [2

G. Bjorklund, M. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy,” Appl. Phys. B 32, 145–152 (1983). [CrossRef]

, 3

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981). [CrossRef]

]. FM modulation can be achieved by either external or internal-modulation of the optical carrier. Both have been demonstrated using electro-optic phase modulators [4

E. A. Whittaker, M. Gehrtz, and G. C. Bjorklund, “Residual amplitude modulation in laser electro-optic phase modulation,” J. Opt. Soc. Am. B 2, 1320–1326 (1985). [CrossRef]

, 5

M. Gehrtz, G. Bjorklund, and E. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 2, 1510–1526 (1985). [CrossRef]

] and by direct modulation of the injected current to a laser diode [6

W. Lenth, “Optical heterodyne spectroscopy with frequency- and amplitude-modulated semiconductor lasers,” Opt. Lett. 8, 575–577 (1983). [CrossRef] [PubMed]

, 7

W. Lenth, “High frequency heterodyne spectroscopy with current-modulated diode lasers,” IEEE J. Quantum Electron. 20, 1045–1050 (1984). [CrossRef]

]. The required detection bandwidth equals the FM rate and therefore high FM rate signals are sometimes difficult to handle. An improvement of the technique enables FM spectroscopy with high modulation rates but with narrow bandwidth detection in a scheme termed two-tone frequency-modulation spectroscopy [8–10

D. Cassidy and J. Reid, “Harmonic detection with tunable diode lasers — Two-tone modulation,” Appl. Phys. B 29, 279–285 (1982). [CrossRef]

]. FM spectroscopy can also be used to analyze multi-photon resonances which are induced by more than one electromagnetic field but with only one of the interacting fields carrying FM modulation, as presented schematically in fig. 1.a. Examples are described in [11

C. Affolderbach, A. Nagel, S. Knappe, C. Jung, D. Wiedenmann, and R. Wynands, “Nonlinear spectroscopy with a vertical-cavity surface-emitting laser (VCSEL),” Appl. Phys. B 70, 407–413 (2000). [CrossRef]

, 12

A. Nagel, C. Affolderbach, S. Knappe, and R. Wynands, “Influence of excited-state hyperfine structure on ground-state coherence,” Phys. Rev. A 61, 012504 (1999). [CrossRef]

Fig. 1. a) Optical spectrum of conventional FM spectroscopy. ; b) Optical spectrum of DFFMS. The carrier frequencies are ω _c1,2=ω_o ±ω_µ and ω_m is the FM modulation frequency. The FM modulated carriers are shown with the side bands up to second order. The green arrows represent tuning directions for an increased RF frequency (ω_µ ) as the respective spectra scan a two-photon resonance. ; c) The atomic energy structure of the ⁸⁷ Rb D ₂ transition three-level Λ-system.

However, FM spectroscopy where only one field is modulated is insufficient for describing multi-photon processes which are induced by two or more different electromagnetic fields (carriers), each being individuallyFM modulated; a condition termed Multi-Field FM Spectroscopy (MFFMS). MFFMS occurs, for example, in systems where the drive current to a diode laser has low frequency FM superimposed on a high frequency component. The emitted laser spectrum contains in this case several spectral lines, each being FM modulated in a different manner. The condition of MFFMS is addressed in this paper. We formulate, analyze, and demonstrate the process for the case of two carriers (a case termed Double-Field FM Spectroscopy, DFFMS) inducing a Coherent Population Trapping (CPT) resonance. The corresponding optical spectrum is described schematically in 1.b.

CPT is a quantum phenomenon in which a destructive interference of two separate quantum transitions can theoretically eliminate the absorption of an atomic medium in a resonant manner [13–15

E. Arimondo and G. Orriols, “Nonabsorbing Atomic Coherences by Coherent Two-Photon Transitions in a Three-Level Optical Pumping,” Lett. Nouvo Cim. 17, 333–338 (1976). [CrossRef]

]. This quantum phenomenon occurs when two phase-locked fields at specific optical frequencies interact with a three-level atom having the so called Λ configuration namely, two ground states and one excited state, as described schematically in fig. 1.c. The destructive interference between the two transitions traps the atom in its ground states. This trapping disables the one-photon absorption processes and therefore the medium becomes transparent. The CPT experiments described here employ a ⁸⁷ Rb vapor as the three-level Λ-system with its two hyperfine split ground levels serving as the ground states. The CPT resonance is induced by two co-polarized electromagnetic fields which propagate in the same direction and therefore can be generated by a single source [16–18

N. Cyr, M. Têtu, and M. Breton, “All-optical microwave frequency standard: a proposal,” IEEE Trans. Instrum. Meas. 42, 640–649 (1993). [CrossRef]

]; a diode laser which is directly modulated at half the desired resonance frequency (the hyperfine splitting frequency (f_hfs ) of ⁸⁷ Rb) where the first two side bands are used for the coherent interaction. This constellation yields a symmetric emission spectrum which narrows the resonance. A possible alternative configuration uses modulation at the desired frequency (f_hfs ) with the carrier and one side band [11

] acting as the two fields. This arrangement is inferior due to the lower diode modulation response (since the frequency is higher) which reduces in turn the CPT efficiency. Although the absorption can be theoretically eliminated, various physical processes (such as: wall collisions, spin exchange and “end-state” accumulation [19

J. Vanier, “Atomic clocks based on coherent population trapping: a review,” Appl. Phys. B 81, 421–442 (2005). [CrossRef]

, 20

Y.-Y. Jau, E. Miron, A. B. Post, N. N. Kuzma, and W. Happer, “Push-Pull Optical Pumping of Pure Superposition States,” Phys. Rev. Lett. 93, 160802 (2004). [CrossRef] [PubMed]

]) weaken the CPT process. Consequently, the change in absorption coefficient, is small (of the order of 1%) [18

S. Knappe, R. Wynands, J. Kitching, H. Robinson, and L. Hollberg, “Characterization of coherent population-trapping resonances as atomic frequency references,” J. Opt. Soc. Am. B 18, 1545–1553 (2001). [CrossRef]

CPT resonances driven by directly modulated lasers are commonly used in small scale atomic clocks [16

N. Cyr, M. Têtu, and M. Breton, “All-optical microwave frequency standard: a proposal,” IEEE Trans. Instrum. Meas. 42, 640–649 (1993). [CrossRef]

, 21–24

S. Knappe, P. Schwindt, V. Shah, L. Hollberg, J. Kitching, L. Liew, and J. Moreland , “ A chip-scale atomic clock based on ⁸⁷Rb with improved frequency stability,” Opt. Express 13, 1249–1253 (2005). [CrossRef] [PubMed]

] and sensitive small-scale magnetometers [25

M. O. Scully and M. Fleischhauer, “High-Sensitivity Magnetometer Based on Index-Enhanced Media,” Phys. Rev. Lett. 69, 1360–1363 (1992). [CrossRef] [PubMed]

, 26

P. D. D. Schwindt, S. Knappe, V. Shah, L. Hollberg, J. Kitching, L.-A. Liew, and J. Moreland, “Chip-scale atomic magnetometer,” Appl. Phys. Lett. 85, 6409–6411 (2004). [CrossRef]

]. While CPT resonances driven by diode lasers can be very narrow, their contrast is moderate. Clock implementation requires, therefore, a complex locking scheme which should be based on a highly sensitive spectroscopic tools such as FM spectroscopy [27

J. Kitching, S. Knappe, M. Vukicevic, L. Hollberg, R. Wynands, and W. Weidmann, “A microwave frequency reference based on VCSEL-driven dark lineresonances in Cs vapor,” IEEE Trans. Instrum. Meas. 49, 1313–1317 (2000). [CrossRef]

,28

I. Ben-Aroya, M. Kahanov, and G. Eisenstein, “Optimization of FM spectroscopy parameters for a frequency locking loop in small scale CPT based atomic clocks,” Opt. Express 15, 15060–15065 (2007). [CrossRef] [PubMed]

]. MFFMS is inherent to clocks that employ a single source and hence, clock performance optimization requires a detailed understanding of MFFMS [28

]. We first describe MFFMS analytically and introduce a detailed analysis for the DFFMS which is used for CPT resonance generation. The model is then compared to CPT experiments in a ⁸⁷ Rb vapor. Finally, we discuss the effect of residual amplitude modulation which stems from the inherent coupling of gain and phase changes in a semiconductor laser.

2. Theoretical analysis

The Multi-Field FM spectroscopy technique, in contrast with standard FM spectroscopy [2

G. Bjorklund, M. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy,” Appl. Phys. B 32, 145–152 (1983). [CrossRef]

], employs an interaction of several FM modulated electromagnetic fields with a resonant medium. These modulated fields traverse the medium and their overall intensity is detected by a single detector. Each field interacts with the medium individually in a manner that depends on other spectral components in the overall spectrum and on the physical processes taking place in the perturbed medium. The detected signal is demodulated at the FM modulation frequency yielding two output signals which contain a signature of the spectroscopic properties of the probed medium. An analysis of this modulation-demodulation process is described in this section. The starting point is a brief description of the conventional FM spectroscopy technique which is followed by a formalism and a detailed discussion of the important special case of DFFMS.

2.1. Conventional FM spectroscopy

Standard FM spectroscopy uses a single FM modulated electromagnetic field which interacts with a resonant medium while the carrier frequency is swept through the resonance. The spectral components of the detected intensity reveal the resonance characteristics [1

G. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. 5, 15–17 (1980). [CrossRef] [PubMed]

, 29–31

J. A. Silver, “Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods,” Appl. Opt. 31, 707–717 (1992). [CrossRef] [PubMed]

Let E _(t) be an electromagnetic field with an amplitude A and a carrier frequency ω_c :

E_{(t)} = A \cos (ω_{c} t) = \frac{1}{2} [A e^{i ω_{c} t} + c . c .] = \frac{1}{2} [{\tilde{E}}_{(t)} + c . c .]

(1)

where the last part is a phasor representation; I _(t)∝|Ẽ(t)|² with I(t) being the field intensity. For a pure sinusoidal frequency modulation, the field (eq. 1) becomes:

E_{(t)} = A \cos (ω_{c} t + M \cdot \sin (ω_{m} t)) = \dots

= \frac{1}{2} [A \sum_{n = - \infty}^{\infty} J_{n (M)} e^{i ((ω_{c} + n ω_{m}) t)} + c . c .]

(2)

where ω_m is the frequency modulation rate, Δω is the maximum frequency deviation from the carrier and

M = \frac{Δ ω}{ω_{m}}

is defined as the modulation index. J _n(M) denotes the n’th order Bessel function of the first kind. The instantaneous frequency is:

\frac{d ω}{d t} = ω_{c} + Δ ω \cos (ω_{m} t)

. For practical cases in which the modulation bandwidth is limited, the summation (eq. 2) is over a finite number N.

The medium to be probed is represented by a complex transfer function T _(ω):

T_{ω} = \bar{T} \cdot \exp {- δ_{(ω)} - i ϕ_{(ω)}}

(3)

where δ _(ω) and ϕ _(ω) are the amplitude attenuation (absorption) and optical phase shift (dispersion), respectively. T̄ represents a broadband background. Following the interaction with the medium, the FM modulated field (eq. 2) becomes:

E_{(t)} = \frac{1}{2} [A \sum_{n = - \infty}^{\infty} J_{n (M)} e^{i ((ω_{c} + n ω_{m}) t)} T_{(ω_{c} + n ω_{m})} + c . c .]

(4)

Both E _(t) and its corresponding intensity, I _(t), contain information about the complex transfer function T.

For simplicity we assume first a small modulation index. In this first order approximation, all J _n(M) terms for |n|>1 can be neglected [2

G. Bjorklund, M. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy,” Appl. Phys. B 32, 145–152 (1983). [CrossRef]

The detected intensity contains spectral components at DC and at frequencies which are multiples of ω_m . Demodulation at ω_m yields two signals which are proportional to the amplitude of the two orthogonal components: cos (ω_mt) and sin (ω_mt). These phase sensitive detection components are defined as the in-phase (i-p) and quadrature (quad) components, respectively and are given by:

\begin{matrix} {i-p : ∣ A ∣}^{2} J_{0 (M)} J_{1 (M)} [e^{- (δ_{1} + δ_{0})} \cos (ϕ_{1} - ϕ_{0}) - e^{- (δ_{0} + δ_{- 1})} \cos (ϕ_{0} - ϕ_{- 1})] \\ {quad : ∣ A ∣}^{2} J_{0 (M)} J_{1 (M)} [e^{- (δ_{1} + δ_{0})} \sin (ϕ_{1} - ϕ_{0}) - e^{- (δ_{0} + δ_{- 1})} \sin (ϕ_{0} - ϕ_{- 1})] \end{matrix}

(5)

where

δ_{n} \equiv δ_{(ω_{c} + n ω_{m})}

and

ϕ_{n} \equiv ϕ_{(ω_{c} + n ω_{m})}

. Both functions are related to the properties of the probed resonant medium by eq. 3.

Furthermore, for |δ_n -δ_l |≪1 and |ϕ_n -ϕ_l |≪1, i.e. for modulation frequencies which are low compared to the width of the main spectral features of the medium, or for T _(ω) which provides only minor changes in the absorption and dispersion coefficients around resonance (a ‘weak’ transfer function), it can be proven that the in-phase and quadrature components are proportional to the first derivative of δ _(ω) and the second derivative of ϕ _(ω), respectively [2

G. Bjorklund, M. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy,” Appl. Phys. B 32, 145–152 (1983). [CrossRef]

, 5

M. Gehrtz, G. Bjorklund, and E. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 2, 1510–1526 (1985). [CrossRef]

For large modulation indices, the first order approximation does not hold and more Bessel terms have to be considered. Keeping all Bessel orders leads to a generalized form of eq. 5:

i-p : {∣ A ∣}^{2} \sum_{n = 1}^{N} J_{n (M)} J_{n - 1 (M)} e^{- (δ_{n} + δ_{n - 1})} \cos (ϕ_{n} - ϕ_{n - 1}) - . . .

- J_{- (n - 1) (M)} J_{- n (M)} e^{- (δ_{- (n - 1)} + δ_{- n})} \cos (ϕ_{- (n - 1)} - ϕ_{- n})

quad : {∣ A ∣}^{2} \sum_{n = 1}^{N} J_{n (M)} J_{n - 1 (M)} e^{- (δ_{n} + δ_{n - 1})} \sin (ϕ_{n} - ϕ_{n - 1}) - . . .

- J_{- (n - 1) (M)} J_{- n (M)} e^{- (δ_{- (n - 1)} + δ_{- n})} \sin (ϕ_{- (n - 1)} - ϕ_{- n})

(6)

consistent with [2

G. Bjorklund, M. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy,” Appl. Phys. B 32, 145–152 (1983). [CrossRef]

, 30

J. M. Supplee, E. A. Whittaker, and W. Lenth, “Theoretical description of frequency modulation and wavelength modulation spectroscopy,” Appl. Opt. 33, 6294–6302 (1994). [CrossRef] [PubMed]

, 31

R. Wynands and A. Nagel, “Inversion of frequency-modulation spectroscopy line shapes,” J. Opt. Soc. Am. B 16, 1617–1622 (1999). [CrossRef]

2.2. Double-field FM spectroscopy

FM spectroscopy of multi-photon processes requires the modulation of at least one of the electromagnetic fields interacting with the probed medium. When more than one field is modulated, the conventional FM spectroscopy theory (eq. 6) does not describe the spectroscopic process properly. We present here an accurate model for DFFMS by analyzing CPT in a ⁸⁷ Rb vapor. We employ the D ₂ transition by using a directly modulated 780nm Vertical Cavity Surface Emitting Laser (VCSEL) type diode laser which is modulated at half the f_hfs of the interacting atoms: 3.417GHz. The two first sidebands of the modulated diode serve as the two required coherent fields.

Superimposing low rate frequency modulation on the RF drive current to the VCSEL translates into FM modulation of the optical field. The FM side bands accompanying the first two RF sidebands have opposite phases as shown schematically in fig. 1.b. Pairs of spectral components originating from the two different sets interact with the medium (presented schematically in fig. 1.c) leading to a complex overall response which includes the sum of all possible interactions.

Let E _(t) be the total electromagnetic field comprising two spectral components, E _1(t) and E _2(t):

E_{(t)} = E_{1 (t)} + E_{2 (t)} = \frac{1}{2} [{\tilde{E}}_{(t)} + c . c .]

(7)

Each one of these components is pure FM modulated at

ω_{m_{i}}

with an index M_i , where i={1,2}. When the two components are derived from the same source,

ω_{m_{1}} = ω_{m_{2}} = ω_{m}

and Δω ₁=-Δω ₂; consequently: M ₁=-M ₂≡M for

ω_{c_{1}} > ω_{c 2}

The two FM modulated components, E _1(t) and E _2(t), are:

E_{1 (t)} = A_{1} \cos (ω_{c_{1}} t + M \cdot \sin (ω_{m} t)) = . . .

= \frac{1}{2} [A_{1} \sum_{n = - N_{1}}^{N_{1}} J_{n (M)} e^{i ((ω_{c_{1}} + n ω_{m}) t)} + c . c .] = \frac{1}{2} [{\tilde{E}}_{1 (t)} + c . c .]

E_{2 (t)} = A_{2} \cos (ω_{c_{2}} t - M \cdot \sin (ω_{m} t)) = . . .

= \frac{1}{2} [A_{2} \sum_{n = - N_{2}}^{N_{2}} J_{n (- M)} e^{i ((ω_{c_{2}} + n ω_{m}) t)} + c . c .] = . . .

= \frac{1}{2} [A_{2} \sum_{n = - N_{2}}^{N_{2}} J_{n (M)} e^{i ((ω_{c_{2}} - n ω_{m}) t)} + c . c .] = \frac{1}{2} [{\tilde{E}}_{2 (t)} + c . c .]

(8)

The analysis of the interaction of the multi-line field, E _(t), differs from the analysis of a single field FM spectroscopy in two main aspects:

The transfer function of the medium for each one of the two fields is inherently different, namely, T ⁽¹⁾ _(Δω)≠T ⁽²⁾ _(Δω) where the super scripts refer to the two different components of the field, E _(t). Moreover, the transfer function depends only on the frequency detuning, Δω, between the two components and not on their absolute frequencies.
Since the spectrum of the interacting field contains two sets of spectral components and since the CPT process employs pairs of spectral lines from the two different sets, a “weighting function” needs to be added to the overall transferred field function for each CPT contribution.

Considering these features, the output field is (in the phasor representation):

{\tilde{E}}_{(t)} = A_{1} \sum_{n_{1} = - N_{1}}^{N_{1}} J_{n_{1} (M)} e^{i ((ω_{c_{1}} + n_{1} ω_{m}) t)} \cdot \sum_{n_{2} = - N_{2}}^{N_{2}} w_{n_{2}}^{(2)} T_{(2 ω_{μ} + (n_{1} - n_{2}) ω_{m})}^{(1)} + . . .

+ A_{2} \sum_{n_{2} = - N_{2}}^{N_{2}} J_{n_{2} (- M)} e^{i ((ω_{c_{2}} + n_{2} ω_{m}) t)} \cdot \sum_{n_{1} = - N_{1}}^{N_{1}} w_{n_{1}}^{(1)} T_{(2 ω_{μ} + (n_{1} - n_{2}) ω_{m})}^{(2)}

(9)

where

ω_{c_{i}} = ω_{o} \pm ω_{μ}

for i=1 and 2, respectively. ω_o is the optical carrier and ω_µ is the RF modulation frequency, w ⁽ⁱ⁾ _n is the “weighting function” of the contribution of the n’s component from the i’s set to the overall field. T ⁽ⁱ⁾ _(ω) represents the transmission of the corresponding field component from the set i. The frequency scale is such that ω_o is around 780nm (384THz),

ω_{μ_{i}} = \pm 3.417 GHz

for i={1,2}, respectively and ω_m is of the order of 1kHz.

The detected intensity is proportional to |Ẽ _(t)|² and therefore includes beating terms between all components comprising the two spectral sets. The demodulated signal at ω_m contains inphase and quadrature components. In addition, the interaction between the two spectral sets results in beating products at the RF frequency (2ω_µ ) each of which contains side bands at ±nω_m . These high frequency terms are not related to this paper and are neither discussed nor analyzed further.

2.2.1. The medium transfer function

CPT is a resonant quantum phenomenon [19

J. Vanier, “Atomic clocks based on coherent population trapping: a review,” Appl. Phys. B 81, 421–442 (2005). [CrossRef]

]. Nevertheless, the semi-classical approach was found to be sufficient to describe it properly [14

E. Arimondo, “Coherent population trapping in laser spectroscopy,” in Progress in Optics, E. Wolf, ed., vol. XXXV, pp. 257–354 (Elsevier Science Amsterdam, 1996).

]. Accordingly, the transfer function sustains the Kramers-Kroning relation [17

J. Vanier, A. Godone, and F. Levi, “Coherent population trapping in cesium: Dark lines and coherent microwave emission,” Phys. Rev. A 58, 2345–2358 (1998). [CrossRef]

, 32

A. Yariv, Optical Electronics in Modern Communications , 5th ed. (Oxford University Press, New York, 1997). Ch. 5.

]. Moreover, under the conditions where the fields are weak (small Rabi frequencies) and their frequencies are set near the one-photon transition, the ‘absorptive’ two-photon resonance has a Lorentzian lineshape, independent of the absolute frequencies and the amplitudes of the interacting fields [14

E. Arimondo, “Coherent population trapping in laser spectroscopy,” in Progress in Optics, E. Wolf, ed., vol. XXXV, pp. 257–354 (Elsevier Science Amsterdam, 1996).

The particularly small atomic medium employed in this work (a 6mm diameter glass ball) operated at a high temperature and used no paraffin wall coating. This dictates the decoherence rate (relaxation rate of the ground-state coherence) which was measured to be 170Hz [33

M. Kahanov, Electrical Engineering department, Technion, Haifa 32000, Israel. (personal communication, 2007).

]. Exploring CPT resonances narrower than 200Hz (see for example 186Hz in [23

I. Ben-Aroya, M. Kahanov, and G. Eisenstein, “A CPT based ⁸⁷Rb atomic clock employing a small spherical glass vapor cell,” in Proceedings of the 38th Annual Precise Time & Time Interval (PTTI) Systems & Applications Meeting, L. A. Breakiron, ed., pp. 259–270 (Naval Observatory, Reston, VA, USA, 2006).

,28

]) is possible when very-low laser intensities are used. In these cases, the different Rabi frequencies have a negligible effect on the CPT Lorentzian line width. Therefore the resonance width can be taken as constant for all pairs of spectral lines.

A more detailed analysis of the relevant coherence terms of the density matrix reveals that the dispersion functions of the two fields have opposite signs [17

J. Vanier, A. Godone, and F. Levi, “Coherent population trapping in cesium: Dark lines and coherent microwave emission,” Phys. Rev. A 58, 2345–2358 (1998). [CrossRef]

]. Their respective transfer functions differ from each other and are determined only by their frequency difference.

This macroscopic model yields the following transfer functions:

T_{(ω)}^{(i)} = \bar{T} \cdot e^{(- δ_{(ω)}^{(i)} - i ϕ_{(ω)}^{(i)})}; i = {1, 2}; ω_{c_{1}} > ω_{c_{2}}

δ_{(ω)}^{(1)} = δ_{(ω)}^{(2)} = a \cdot (\frac{1}{R_{(ω)}^{2} + 1})

ϕ_{(ω)}^{(1)} = - ϕ_{(ω)}^{(2)} = a \cdot (\frac{R_{(ω)}}{R_{(ω)}^{2} + 1})

R_{(ω)} = \frac{ω - ω_{res}}{\frac{Γ}{2}}

(10)

where a is the peak value of the absorption function δ ⁽ⁱ⁾ _(ω). ω_res is the resonance frequency and Γ is the Lorentzian resonance width (FWHM). R _(ω) stands for the normalized frequency. Note that a takes on negative values since it describes an Electromagnetically Induced Transparency (EIT) process.

2.2.2. The “weighting function”

In this macroscopic model, the interaction takes place only between pairs of spectral lines (one from each set). The CPT efficiency depends on the intensity of the two fields and therefore it is necessary to weight the contribution of each pair to the overall process. A pure sinusoidal modulation and the consequent Bessel function energy distributions of the two fields dictate the weighting functions (eq. 11) where energy conservation requires that the sum of all weighted component equals the unmodulated amplitudes

({∣ \frac{A_{1,2}}{2} ∣}^{2})

w_{n}^{(2)} = {∣ \frac{A_{2}}{2} J_{n (- M)} ∣}^{2}

w_{n}^{(1)} = {∣ \frac{A_{1}}{2} J_{n (M)} ∣}^{2}

(11)

2.2.3. First order analysis and comparison to conventional FM spectroscopy

In the limit of a small modulation index: M≪1, all J_n (M) terms for |n|>1 can be neglected. The resulting expression enables a clearer description of DFFMS.

In this limit, the output field (eq. 9) becomes:

{\tilde{E}}_{(t)} = {\tilde{E}}_{1}^{T} + {\tilde{E}}_{2}^{T}

{\tilde{E}}_{1}^{T} = A_{1} {∣ \frac{A_{2}}{2} ∣}^{2} {

- J_{1 (M)} e^{i ((ω_{c 1} - ω_{m}) t)} [J_{1 (- M)}^{2} T_{(2 ω_{μ})}^{(1)} + J_{0 (- M)}^{2} T_{(2 ω_{μ} - ω_{m})}^{(1)} + J_{1 (- M)}^{2} T_{(2 ω_{μ} - 2 ω_{m})}^{(1)}] + . . .

+ J_{0 (M)} e^{i (ω_{c 1} t)} [J_{1 (- M)}^{2} T_{(2 ω_{μ} + ω_{m})}^{(1)} + J_{0 (- M)}^{2} T_{(2 ω_{μ})}^{(1)} + J_{1 (- M)}^{2} T_{(2 ω_{μ} - ω_{m})}^{(1)}] + . . .

+ J_{1 (M)} e^{i ((ω_{c 1} - ω_{m}) t)} [J_{1 (- M)}^{2} T_{(2 ω_{μ} + 2 ω_{m})}^{(1)} + J_{0 (- M)}^{2} T_{(2 ω_{μ} - ω_{m})}^{(1)} + J_{1 (- M)}^{2} T_{(2 ω_{μ})}^{(1)}]}

{\tilde{E}}_{2}^{T} = A_{2} {∣ \frac{A_{1}}{2} ∣}^{2} {

- J_{1 (- M)} e^{i ((ω_{c 2} - ω_{m}) t)} [J_{1 (M)}^{2} T_{(2 ω_{μ})}^{(2)} + J_{0 (M)}^{2} T_{(2 ω_{μ} + ω_{m})}^{(2)} + J_{1 (M)}^{2} T_{(2 ω_{μ} - 2 ω_{m})}^{(2)}] + . . .

+ J_{0 (- M)} e^{i (ω_{c 2} t)} [J_{1 (M)}^{2} T_{(2 ω_{μ} - ω_{m})}^{(2)} + J_{0 (M)}^{2} T_{(2 ω_{μ})}^{(2)} + J_{1 (M)}^{2} T_{(2 ω_{μ} - ω_{m})}^{(2)}] + . . .

+ J_{1 (- M)} e^{i ((ω_{c 2} - ω_{m}) t)} [J_{1 (M)}^{2} T_{(2 ω_{μ} - 2 ω_{m})}^{(2)} + J_{0 (M)}^{2} T_{(2 ω_{μ} - ω_{m})}^{(2)} + J_{1 (M)}^{2} T_{(2 ω_{μ})}^{(2)}]}

(12)

where Ẽ ^T ₁ and Ẽ ^T ₂ describe the first and second lines in eq. 9, respectively.

The output intensity is proportional to |Ẽ _(t)|²:

I \propto {∣ \tilde{E} ∣}^{2} = {∣ {\tilde{E}}_{1}^{T} + {\tilde{E}}_{2}^{T} ∣}^{2} = {∣ {\tilde{E}}_{1}^{T} ∣}^{2} + {∣ {\tilde{E}}_{2}^{T} ∣}^{2} + ∣ {\tilde{E}}_{1}^{T} {\tilde{E}}_{2}^{T *} ∣ + ∣ {\tilde{E}}_{2}^{T} {\tilde{E}}_{1}^{T *} ∣

(13)

The first and second terms in eq. 13 include spectral components near DC while the third and fourth terms oscillate around an RF frequency: 2ω_µ , since

ω_{c_{i}} = ω_{o} \pm ω_{μ}

for i=1 and 2, respectively. These two terms can be neglected. Each of the two first, self-beating, terms includes spectral components around DC, 2ω_m and ω_m with the first two being rejected by the demodulating Lock-in amplifier.

The contribution of each of the two first terms in eq. 13 to the overall in-phase and quadrature components can be expressed separately as:

{∣ {\tilde{E}}_{1}^{T} ∣}^{2} i-p : η_{1} {J_{0 (- M)}^{4} C_{10}^{(1)} + J_{0 (- M)}^{2} J_{1 (- M)}^{2} C_{20}^{(1)} + J_{1 (- M)}^{4} C_{21}^{(1)} + J_{1 (- M)}^{4} C_{2 - 1}^{(1)} + . . .

+ J_{0 (- M)}^{2} J_{1 (- M)}^{2} [e^{- 2 δ_{1}^{(1)}} - e^{- 2 δ_{- 1}^{(1)}}]}

{∣ {\tilde{E}}_{1}^{T} ∣}^{2} quad : η_{1} {J_{0 (- M)}^{4} S_{10}^{(1)} + J_{0 (- M)}^{2} J_{1 (- M)}^{2} S_{20}^{(1)} + J_{1 (- M)}^{4} S_{21}^{(1)} + J_{1 (- M)}^{4} S_{2 - 1}^{(1)} - . . .

- {2 J}_{1 (- M)}^{4} S_{10}^{(1)}}

{∣ {\tilde{E}}_{2}^{T} ∣}^{2} i-p : η_{2} {- J_{0 (M)}^{4} C_{10}^{(2)} - J_{0 (M)}^{2} J_{1 (M)}^{2} C_{20}^{(2)} - J_{1 (M)}^{4} C_{21}^{(2)} - J_{1 (M)}^{4} C_{2 - 1}^{(2)} - . . .

- J_{0 (M)}^{2} J_{1 (M)}^{2} [e^{- 2 δ_{1}^{(2)}} - e^{- 2 δ_{- 1}^{(2)}}]}

{∣ {\tilde{E}}_{2}^{T} ∣}^{2} quad : η_{2} {J_{0 (M)}^{4} S_{10}^{(2)} + J_{0 (M)}^{2} J_{1 (M)}^{2} S_{20}^{(2)} + J_{1 (M)}^{4} S_{21}^{(1)} + J_{1 (M)}^{4} S_{2 - 1}^{(2)} - . . .

- 2 J_{1 (M)}^{4} S_{10}^{(2)}}

(14)

while using the following definitions:

η_{1} = \frac{1}{2} J_{0 (M)} J_{1 (M)} {∣ A_{1} A_{2}^{2} ∣}^{2} {∣ \bar{T} ∣}^{2}; η_{2} = \frac{1}{2} J_{0 (- M)} J_{1 (- M)} {∣ A_{1}^{2} A_{2} ∣}^{2} {∣ \bar{T} ∣}^{2}

C_{nl}^{(i)} = e^{- (δ_{n}^{(i)} + δ_{l}^{(i)})} \cos (ϕ_{n}^{(i)} - ϕ_{l}^{(i)}) - e^{- (δ_{- l}^{(i)} + δ_{- n}^{(i)})} \cos (ϕ_{- l}^{(i)} - ϕ_{- n}^{(i)})

S_{nl}^{(i)} = e^{- (δ_{n}^{(i)} + δ_{l}^{(i)})} \sin (ϕ_{n}^{(i)} - ϕ_{l}^{(i)}) - e^{- (δ_{- l}^{(i)} + δ_{- n}^{(i)})} \sin (ϕ_{- l}^{(i)} - ϕ_{- n}^{(i)})

δ_{n}^{(i)} = δ_{(2 ω_{μ} + n ω_{m})}^{(i)}; ϕ_{n}^{(i)} = ϕ_{(2 ω_{μ} + n ω_{m})}^{(i)}; i = {1,2}

(15)

Assuming J _0(M)>J _1(M), all J ⁴ _1(M) terms in eq. 14 can be neglected. Since J ² _1(-M)=J ² _1(M) eq. 14 becomes:

{∣ {\tilde{E}}_{1}^{T} ∣}^{2} i-p : η_{1} {J_{0 (M)}^{4} C_{10}^{(1)} + J_{0 (M)}^{2} J_{1 (M)}^{2} C_{20}^{(1)} + J_{0 (M)}^{2} J_{1 (M)}^{2} [e^{- 2 δ_{1}^{(1)}} - e^{- 2 δ_{- 1}^{(1)}}]}

{∣ {\tilde{E}}_{1}^{T} ∣}^{2} quad : η_{1} {J_{0 (M)}^{4} S_{10}^{(1)} + J_{0 (M)}^{2} J_{1 (M)}^{2} S_{20}^{(1)}}

{∣ {\tilde{E}}_{2}^{T} ∣}^{2} i-p : {- η}_{2} {J_{0 (M)}^{4} C_{10}^{(2)} + J_{0 (M)}^{2} J_{1 (M)}^{2} C_{20}^{(2)} + J_{0 (M)}^{2} J_{1 (M)}^{2} [e^{- 2 δ_{1}^{(2)}} - e^{- 2 δ_{- 1}^{(2)}}]}

{∣ {\tilde{E}}_{2}^{T} ∣}^{2} quad : η_{2} {J_{0 (M)}^{4} S_{10}^{(2)} + J_{0 (M)}^{2} J_{1 (M)}^{2} S_{20}^{(2)}}

(16)

For the symmetric case, A ₁=A ₂, both contributions of each component are equal since η₂ =-η₁ , C ⁽²⁾ _nl=C ⁽¹⁾ _nl and S ⁽²⁾ _nl=-S ⁽¹⁾ _nl (see eq. 10). Therefore the overall in-phase and quadrature components are

i-p : {2 η}_{1} {J_{0 (M)}^{4} C_{10}^{(1)} + J_{0 (M)}^{2} J_{1 (M)}^{2} C_{20}^{(1)} + J_{0 (M)}^{2} J_{1 (M)}^{2} [e^{- 2 δ_{1}^{(1)}} - e^{- 2 δ_{- 1}^{(1)}}]}

quad : {2 η}_{1} {J_{0 (M)}^{4} S_{10}^{(1)} + J_{0 (M)}^{2} J_{1 (M)}^{2} S_{20}^{(1)}}

(17)

The conventional FM spectroscopy first order approximation in-phase and quadrature components (eq. 5) can be also expressed using the C _nl and S _nl convention as:

i-p : {∣ A ∣}^{2} J_{0 (M)} J_{1 (M)} C_{10}

{quad : ∣ A ∣}^{2} J_{0 (M)} J_{1 (M)} S_{10}

(18)

C ⁽ⁱ⁾ _n0 and S ⁽ⁱ⁾ _n0 can be interpreted as contributing a spectral feature at ±nω_m therefore the DFFMS output includes peaks near ±2ω_m . The last term in the in-phase component of the DFFMS output (eq. 17) enhances the peaks at ±ω_m with respect to the peaks near ±2ω_m . Also, eliminating the FM modulation in one of the two carriers, for example at ω _c2, yields the conventional FM spectroscopy terms (eq. 18). Eliminating the FM modulation of ω _c2 means that J _0(M)=1 and J ₁(M)=0 in the curly brackets of the |Ẽ ^T ₁|² terms in eq. 16. |Ẽ ^T ₂|² does not contribute to the overall output, in this case, since its self-beating terms has spectral components only near DC so it is rejected by the Lock-in amplifier.

Fig. 2 compares the two FM spectroscopy methods whose optical spectra are shown in fig. 1.a and in fig. 1.b, while scanning a two-photon process. Fig. 2.a presents the case of a low M in which the two methods yield rather similar results. Larger M values cause large discrepancies as seen in fig. 2.b and in fig. 2.c. The deviation is due to additional peaks which are revealed near ±2ω_m . Notice the different energy distributions which the two FM spectroscopy methods yield due to the enhancement term in the DFFMS in-phase component.

3. Experimental results versus the analytic model

The DFFMS model was compared to an experimental characterization of a CPT resonance induced by the two transitions which couple the m_F =0 states of the ground levels (known also as the “clock transition”). The experimental setup used to characterize the CPT process is similar to the one reported in [23

, 28

] and is described in fig. 3.

A temperature stabilized single mode VCSEL is driven by a DC bias and a microwave signal emitting at 780.24nm (⁸⁷ Rb D ₂ transition). The RF signal frequency scans around half the f_hfs of the ground states, namely, 3.417GHz. The RF current is FM modulated at a low-frequency.

Fig. 2. A comparison between the conventional FM spectroscopy and the DFFMS (simulated up to the first order). A Lorentzian medium with Γ=1 Hz was used in both cases The FM parameters are: a) f_m =1 Hz, M=0.5 ; b) f_m =1 Hz, M=1 ; c) f_m =2 Hz, M=1. The horizontal axis describes the detuning between the two carriers relative to the resonance frequency: Δf=(ω _c1-ω _c2)-ω_res . The amplitude in each method was normalized separately.

The VCSEL output is collimated and its polarization is set to be circular. The light impinges a small spherical glass cell with a diameter of 6mm which contains pure ⁸⁷ Rb atoms and a buffer gas. The cell temperature is stabilized around an optimum temperature of 66 °C. A large solenoid generates a homogeneous magnetic field of 50µT pointing in a direction parallel to the optical axis in order to lift the Zeeman degeneracy. The entire system is wrapped by three layers of µ-metal which shields the environmental magnetic field. The transmitted light is detected by a large silicon detector and then demodulated by a Lock-in amplifier. The Lock-in amplifier provides two outputs which vary as the microwave frequency is tuned around the resonance frequency of 3 417 352 560Hz. The CPT resonance can also be measured directly (with no FM modulation) and was found to have a bandwidth of less than 200Hz (around 3.417GHz).

The measured signals for four specific sets of FM parameters are presented in fig. 4 as a function of the frequency detuning from resonance. The frequency values refers to the injected RF signal deviation from f_hfs /2. The blue and cyan lines represent the measured amplitude of the in-phase and quadrature components, respectively while the phase difference between the detected signal and the reference signal to the Lock-in was totally compensated. These are marked as the X and Y components, respectively.

This figure also presents the calculated amplitudes of the same components using the Double-Field FM-Spectroscopy model. The red and magenta dashed-lines represent the simulated amplitude of the same low-frequency components. An excellent fit between the two is observed; the residual errors of the fitting is presented under each graph, blue and cyan represent the X and Y components, respectively.

The model requires the values of the FM parameters in the optical domain, which cannot be obtained experimentally. The only accurate FM parameters are those of the RF drive current namely, the FM parameters in the electrical domain. These two domains are related by the semiconductor diode laser modulation transfer function. What is actually needed is the dependence of the complex electric field on a complex modulation current, a dependence which is not known. Since the FM modulation frequency is low, we assume that ω_m transfers directly from the electrical to the optical domain. As for the modulation index, it is directly translated for M less than half (small signal modulation) but the translation is questionable when M increases. For large M values, where the diode laser exhibits a nonlinear response, we use parameter fitting to extract the transformation from the electrical to the optical domain.

Fig. 3. The experimental setup (ND-Natural Density filter, Lin Pol-Linear Polarizer, λ/4-Quarter wave plate).

4. Residual amplitude modulation

Amplitude and frequency modulation are inherently coupled in a diode laser by the α-parameter [34

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982). [CrossRef]

]. The low rate Frequency modulation is accompanied therefore by a Residual Amplitude Modulation (RAM) [5

M. Gehrtz, G. Bjorklund, and E. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 2, 1510–1526 (1985). [CrossRef]

, 31

R. Wynands and A. Nagel, “Inversion of frequency-modulation spectroscopy line shapes,” J. Opt. Soc. Am. B 16, 1617–1622 (1999). [CrossRef]

]. This effect changes both fields, each in a different manner.

This section addresses the RAM addition in the context of DFFMS first theoretically than experimentally.

4.1. Theoretical analysis

In the double-field constellation, both fields are affected by the RAM. Therefore the field components which interacting with the medium (see eq. 8), become:

E_{1 (t)} = A_{1} \cos (ω_{c_{1}} t + M . \sin (ω_{m} t)) {[1 + R_{1} \sin (ω_{m} t + ψ_{1})]}^{\frac{1}{2}}

\Rightarrow {\tilde{E}}_{1 (t)} = A_{1} \sum_{n = - N_{1}}^{N_{1}} J_{n (M)} e^{i ((ω_{c} + n ω_{m}) t)} {[1 - i \frac{R_{1}}{2} e^{i (ω_{m} t + ψ_{1})} + i \frac{R_{1}}{2} e^{- i (ω_{m} t + ψ_{1})}]}^{\frac{1}{2}}

E_{2 (t)} = A_{2} \cos (ω_{c_{2}} t - M . \sin (ω_{m} t)) {[1 + R_{2} \sin (ω_{m} t + ψ_{2})]}^{\frac{1}{2}}

\Rightarrow {\tilde{E}}_{2 (t)} = A_{2} \sum_{n = - N_{2}}^{N_{2}} J_{n (- M)} e^{i ((ω_{c} + n ω_{m}) t)} {[1 - i \frac{R_{2}}{2} e^{i (ω_{m} t + ψ_{2})} + i \frac{R_{2}}{2} e^{- i (ω_{m} t + ψ_{2})}]}^{\frac{1}{2}}

(19)

Fig. 4. A comparison between the Double-Field FM Spectroscopy experimental and simulation results. The figure presents the amplitude of the two Lock-In amplifier output components, namely X and Y, versus frequency detuning from resonance for four different FM parameter sets, as appear in the bottom-left corner of each graph. The residual errors of the fittings are presented under each graph (simulation-measurement).

where for i={1,2} R_i and ψ_i are the RAM modulation index and phase delay with respect to the FM modulation for each field component, respectively. TheAM modulation term was added using its square root value since the AM modulation is defined for the intensity rather than the electric field [35

X. Zhu and D. T. Cassidy, “Modulation spectroscopy with a semiconductor diode laser by injection-current modulation,” J. Opt. Soc. Am. B 14, 1945–1950 (1997). [CrossRef]

]. This approach is somewhat different from the traditional one which refers to AM modulation of the field itself [5

M. Gehrtz, G. Bjorklund, and E. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 2, 1510–1526 (1985). [CrossRef]

, 31

R. Wynands and A. Nagel, “Inversion of frequency-modulation spectroscopy line shapes,” J. Opt. Soc. Am. B 16, 1617–1622 (1999). [CrossRef]

]. Nevertheless, the typical values of R_i are much smaller than one and therefore the square root can be replaced by its first order approximation as presented in eq. 20.

E_{1 (t)} ≃ A_{1} \cos (ω_{c_{1}} t + M \cdot \sin (ω_{m} t)) [1 + \frac{R_{1}}{2} \sin (ω_{m} t + ψ_{1})]

\Rightarrow {\tilde{E}}_{1 (t)} ≃ A_{1} \sum_{n = - N_{1}}^{N_{1}} J_{n (M)} e^{i ((ω_{c} + n ω_{m}) t)} [1 - i \frac{R_{1}}{4} e^{i (ω_{m} t + ψ_{1})} + i \frac{R_{1}}{4} e^{- i (ω_{m} t + ψ_{1})}]

E_{2 (t)} ≃ A_{2} \cos (ω_{c_{2}} t + M \cdot \sin (ω_{m} t)) [1 + \frac{R_{2}}{2} \sin (ω_{m} t + ψ_{2})]

\Rightarrow {\tilde{E}}_{2 (t)} ≃ A_{2} \sum_{n = - N_{2}}^{N_{2}} J_{n (- M)} e^{i ((ω_{c} + n ω_{m}) t)} [1 - i \frac{R_{2}}{4} e^{i (ω_{m} t + ψ_{2})} + i \frac{R_{2}}{4} e^{- i (ω_{m} t + ψ_{2})}]

(20)

The last result consolidates with the traditional approach (up to a factor of

\frac{1}{2}

for the RAM modulation indices).

Rewriting eq. 9, the output field becomes:

{\tilde{E}}_{(t)} = A_{1} \sum_{n_{1} = - N_{1}}^{N_{1}} (J_{n_{1} (M)} - i J_{n_{1} - 1 (M)} \frac{R_{1}}{4} e^{i ψ_{1}} + i J_{n_{1} + 1 (M)} \frac{R_{1}}{4} e^{- i ψ_{1}}) e^{i ((ω_{c_{1}} + n_{1} ω_{m}) t)} . \dots

. \sum_{n_{2} = - N_{2}}^{N_{2}} w_{m}^{(2)} T_{(2 ω_{m} + (n_{1} - n_{2}) ω_{m})}^{(1)} + \dots

+ A_{2} \sum_{n_{2} = - N_{2}}^{N_{2}} (J_{n_{2} (- M)} - i J_{n_{2} - 1 (- M)} \frac{R_{2}}{4} e^{i ψ_{2}} + i J_{n_{2} + 1 (- M)} \frac{R_{2}}{4} e^{- i ψ_{2}}) e^{i ((ω_{c_{2}} + n_{2} ω_{m}) t)} . \dots

. \sum_{n_{1} = - N_{1}}^{N_{1}} w_{n_{1}}^{(1)} T_{(2 ω_{m} + (n_{1} - n_{2}) ω_{m})}^{(1)}

(21)

and the “weighting functions” are given by:

w_{n}^{(2)} = {∣ \frac{A_{2}}{2} (J_{n (- M)} - i J_{n - 1 (- M)} \frac{R_{2}}{4} e^{i ψ_{2}} + i J_{n + 1 (- M)} \frac{R_{2}}{4} e^{- i ψ_{2}}) ∣}^{2}

w_{n}^{(1)} = {∣ \frac{A_{1}}{2} (J_{n (M)} - i J_{n - 1 (M)} \frac{R_{1}}{4} e^{i ψ_{1}} + i J_{n + 1 (M)} \frac{R_{1}}{4} e^{- i ψ_{1}}) ∣}^{2}

(22)

It should be noted that although the difference between these “weighting functions” and the ones given by eq. 11 is fundamental, there is but a small practical difference between them since the values of R_i ’s are very small compared with the relevant Bessel functions, J_n (M).

For the ‘ideal’ case in which the carrier amplitudes are equal (A ₁=A ₂) and RAM of the two components are equal but opposite in sign, namely, ψ₁ =ψ₂ =0 and R ₁=-R ₂≡R. It can be shown that the RAM has no trace in the output signals, since the contributions from both spectral sets cancel each other. This important cancellation does not occur in standard FM spectroscopy (where only one field is used) but can be compensated for by employing a very sophisticated setup, as presented in [5

M. Gehrtz, G. Bjorklund, and E. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 2, 1510–1526 (1985). [CrossRef]

4.2. Experimental observations of the RAM

In reality, the RAM is not ideal since the diode response for the two spectral components is unequal due to the Bogatov effect [36

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous Interaction of Spectral Modes in a Semiconductor Laser,” IEEE J. Quantum Electron. 11, 510–515 (1975). [CrossRef]

]. Although the difference is very small, it inherently distorts the signature of the resonance in the output components. This distortion manifest itself as an asymmetry around the resonance frequency as seen in fig. 5 which shows measured and calculated outputs of the Lock-in amplifier versus frequency detuning from resonance. The figure uses the polar representation (R,θ) of the two dimensional signal space rather than the orthogonal (X,Y) representation of the amplitudes of the cos(ω_mt) and sin (ω_mt) components, respectively. The calculated data in fig. 5.a, shown in red and magenta-dashed lines, does not include RAM terms. A detailed examination of the result reveals a minor but fundamental misalignment between the model with no RAM and the experimental results, especially near the resonance frequency. This is shown in fig. 5.b and is also enlarged in fig. 5.c. The last two figures present only the phase component (θ) of the calculated (magenta dashed-line) and the measured (cyan line) signals. For simplicity, phase jumps of π were eliminated when moving diagonally between different quarters of the unit circle while sweeping around the resonance frequency.

Figures 5.d, 5.e and 5.f are similar to the previous figures (fig. 5.a, fig. 5.b and fig. 5.c) but here the RAM was taken into account as appears in eq. 21. Employing the “non-ideal” RAM yields better results with respect to the experimental observations although the RAM modulation-index and phase-delay differences between the two components of the FM modulated field are minor (ΔR≡R ₁-R ₂=1·10^-5 and Δψ=ψ₁ -(ψ₂ -π)=0.003π rad≈0.5°; ψ₁ =0).

The sharp peak in the center of figures 5.b, 5.c, 5.e and 5.f is attributed to some numerical errors around the resonance where the X and Y components are almost zero.

Fig. 5. The effect of the RAM on the DFFMS. All figures present output components versus frequency detuning shown in polar representation. The simulation results presented in figures (a), (b) and (c) do not include RAM terms (i.e. R ₁=R ₂=0). a) R and θ components of the measured and simulated output. ; b) The θ component (simulated and measured) of (a) after neglecting the π jumps between quarters. ; c) An enlargements of (b) around the center. Figures (d), (e) and (f) include RAM terms R ₁=1E-3,R ₂=0.99E-3,ψ₁ =0,ψ₂ =0.997π rad. d) R and θ components of the measured and simulated output. ; e) The θ component (simulated and measured) of (d) after neglecting the π jumps between quarters.; f) An enlargements of (e) around the center.

5. Summary

We have introduced the MFFMS spectroscopy technique which is a modification of conventional FM spectroscopy. In MFFMS, several carriers are FM modulated, each in a different manner with the entire complex optical spectrum interacting with a resonant atomic medium and being detected by a single detector. This technique was analyzed and experimentally demonstrated for the case of two fields (DFFMS) probing a CPT resonance which was induced by the two first side bands of a directly modulated VCSEL. The model fits superbly to experiments. DFFMS was also found to be highly immuned to undesired RAM effects. The use of DFFMS enables an optimization of the frequency locking scheme in a CPT based atomic clock which employs FM spectroscopy resulting in improved clock performance.

The detected signal can also be demodulated at harmonics of the FM modulation frequency, ω_m . In particular, demodulation at odd harmonics provides different output signals which are important is some reported cases [37

D. Phillips, I. Novikova, C. Wang, R. Walsworth, and M. Crescimanno, “Modulation-induced frequency shifts in a coherent-population-trapping-based atomic clock,” J. Opt. Soc. Am. B 22, 305–310 (2005). [CrossRef]

]. These can be easily calculated by the presented model.

Acknowledgments

The authors acknowledge discussions about FM spectroscopy with Avinoam Stern of Accubeat, Israel and Michael Rosenbluh of Bar-Ilan University.

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24.	R. Lutwak, A. Rashed, M. Varghese, G. Tepolt, J. Leblanc, M. Mescher, D. K. Serkland, and G. M. Peake, “The Miniature Atomic Clock Pre-Production Results,” in proceedings of 2005 Joint IEEE International Frequency Control (UFFC) Symposium and the 21th European Frequency and Time Forum (EFTF), D. Coler, ed., pp. 1327–1333 (IEEE, Geneva, Switzerland, 2007).
25.	M. O. Scully and M. Fleischhauer, “High-Sensitivity Magnetometer Based on Index-Enhanced Media,” Phys. Rev. Lett. 69, 1360–1363 (1992). [CrossRef] [PubMed]
26.	P. D. D. Schwindt, S. Knappe, V. Shah, L. Hollberg, J. Kitching, L.-A. Liew, and J. Moreland, “Chip-scale atomic magnetometer,” Appl. Phys. Lett. 85, 6409–6411 (2004). [CrossRef]
27.	J. Kitching, S. Knappe, M. Vukicevic, L. Hollberg, R. Wynands, and W. Weidmann, “A microwave frequency reference based on VCSEL-driven dark lineresonances in Cs vapor,” IEEE Trans. Instrum. Meas. 49, 1313–1317 (2000). [CrossRef]
28.	I. Ben-Aroya, M. Kahanov, and G. Eisenstein, “Optimization of FM spectroscopy parameters for a frequency locking loop in small scale CPT based atomic clocks,” Opt. Express 15, 15060–15065 (2007). [CrossRef] [PubMed]
29.	J. A. Silver, “Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods,” Appl. Opt. 31, 707–717 (1992). [CrossRef] [PubMed]
30.	J. M. Supplee, E. A. Whittaker, and W. Lenth, “Theoretical description of frequency modulation and wavelength modulation spectroscopy,” Appl. Opt. 33, 6294–6302 (1994). [CrossRef] [PubMed]
31.	R. Wynands and A. Nagel, “Inversion of frequency-modulation spectroscopy line shapes,” J. Opt. Soc. Am. B 16, 1617–1622 (1999). [CrossRef]
32.	A. Yariv, Optical Electronics in Modern Communications , 5th ed. (Oxford University Press, New York, 1997). Ch. 5.
33.	M. Kahanov, Electrical Engineering department, Technion, Haifa 32000, Israel. (personal communication, 2007).
34.	C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982). [CrossRef]
35.	X. Zhu and D. T. Cassidy, “Modulation spectroscopy with a semiconductor diode laser by injection-current modulation,” J. Opt. Soc. Am. B 14, 1945–1950 (1997). [CrossRef]
36.	A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous Interaction of Spectral Modes in a Semiconductor Laser,” IEEE J. Quantum Electron. 11, 510–515 (1975). [CrossRef]
37.	D. Phillips, I. Novikova, C. Wang, R. Walsworth, and M. Crescimanno, “Modulation-induced frequency shifts in a coherent-population-trapping-based atomic clock,” J. Opt. Soc. Am. B 22, 305–310 (2005). [CrossRef]

OCIS Codes
(000.2170) General : Equipment and techniques
(020.1670) Atomic and molecular physics : Coherent optical effects
(120.3930) Instrumentation, measurement, and metrology : Metrological instrumentation
(300.6320) Spectroscopy : Spectroscopy, high-resolution
(300.6380) Spectroscopy : Spectroscopy, modulation
(140.3518) Lasers and laser optics : Lasers, frequency modulated

ToC Category:
Spectroscopy

History
Original Manuscript: January 30, 2008
Revised Manuscript: March 27, 2008
Manuscript Accepted: April 9, 2008
Published: April 15, 2008

Citation
Ido Ben-Aroya, Matan Kahanov, and Gadi Eisenstein, "Multi-field frequency modulation spectroscopy," Opt. Express 16, 6081-6097 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-9-6081

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References

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S. Knappe, P. Schwindt, V. Shah, L. Hollberg, J. Kitching, L. Liew, and J. Moreland, "A chip-scale atomic clock based on 87Rb with improved frequency stability," Opt. Express 13, 1249-1253 (2005). [CrossRef] [PubMed]
R. Lutwak, P. Vlitas, M. Varghes, M. Mescher, D. K. Serkland, and G. M. Peake, "The MAC-A miniature atomic clock," in Proceedings of 2005 Joint IEEE International Frequency Control (UFFC) Symposium and the 37th Annual Precise Time & Time Interval (PTTI) Systems & Applications Meeting, D. Coler, ed., pp. 767-773 (IEEE, Vancouver, BC, Canada, 2005).
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R. Lutwak, A. Rashed, M. Varghese, G. Tepolt, J. Leblanc, M. Mescher, D. K. Serkland, and G. M. Peake, "The Miniature Atomic Clock Pre-Production Results," in proceedings of 2005 Joint IEEE International Frequency Control (UFFC) Symposium and the 21th European Frequency and Time Forum (EFTF), D. Coler, ed., pp. 1327-1333 (IEEE, Geneva, Switzerland, 2007).
M. O. Scully and M. Fleischhauer, "High-Sensitivity Magnetometer Based on Index-Enhanced Media," Phys. Rev. Lett. 69, 1360-1363 (1992). [CrossRef] [PubMed]
P. D. D. Schwindt, S. Knappe, V. Shah, L. Hollberg, J. Kitching, L.-A. Liew, and J. Moreland, "Chip-scale atomic magnetometer," Appl. Phys. Lett. 85, 6409-6411 (2004). [CrossRef]
J. Kitching, S. Knappe, M. Vukicevic, L. Hollberg, R. Wynands, and W. Weidmann, "A microwave frequency reference based on VCSEL-driven dark lineresonances in Cs vapor," IEEE Trans. Instrum. Meas. 49, 1313-1317 (2000). [CrossRef]
I. Ben-Aroya, M. Kahanov, and G. Eisenstein, "Optimization of FM spectroscopy parameters for a frequency locking loop in small scale CPT based atomic clocks," Opt. Express 15, 15060-15065 (2007). [CrossRef] [PubMed]
J. A. Silver, "Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods," Appl. Opt. 31, 707-717 (1992). [CrossRef] [PubMed]
J. M. Supplee, E. A. Whittaker, and W. Lenth, "Theoretical description of frequency modulation and wavelength modulation spectroscopy," Appl. Opt. 33, 6294-6302 (1994). [CrossRef] [PubMed]
R. Wynands and A. Nagel, "Inversion of frequency-modulation spectroscopy line shapes," J. Opt. Soc. Am. B 16, 1617-1622 (1999). [CrossRef]
A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University Press, New York, 1997). Ch. 5.
M. Kahanov, Electrical Engineering department, Technion, Haifa 32000, Israel. (personal communication, 2007).
C. Henry, "Theory of the linewidth of semiconductor lasers," IEEE J. Quantum Electron. 18, 259-264 (1982). [CrossRef]
X. Zhu and D. T. Cassidy, "Modulation spectroscopy with a semiconductor diode laser by injection-current modulation," J. Opt. Soc. Am. B 14, 1945-1950 (1997). [CrossRef]
A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, "Anomalous Interaction of Spectral Modes in a Semiconductor Laser," IEEE J. Quantum Electron. 11, 510-515 (1975). [CrossRef]
D. Phillips, I. Novikova, C. Wang, R. Walsworth, and M. Crescimanno, "Modulation-induced frequency shifts in a coherent-population-trapping-based atomic clock," J. Opt. Soc. Am. B 22, 305-310 (2005). [CrossRef]

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