## Multi-Field Frequency Modulation Spectroscopy

Optics Express, Vol. 16, Issue 9, pp. 6081-6097 (2008)

http://dx.doi.org/10.1364/OE.16.006081

Acrobat PDF (753 KB)

### 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.

© 2008 Optical Society of America

## 1. Introduction

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

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]

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]

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]

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

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]

13. 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]

^{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

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

*f*) of

_{hfs}^{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*) with the carrier and one side band [11

_{hfs}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]

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]

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]

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

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]

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]

^{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

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

### 2.1. Conventional 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]

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]

*E*

_{(t)}be an electromagnetic field with an amplitude

*A*and a carrier frequency

*ω*:

_{c}*I*

_{(t)}∝|

*Ẽ*(

*t*)|

^{2}with

*I*(

*t*) being the field intensity. For a pure sinusoidal frequency modulation, the field (eq. 1) becomes:

*ω*is the frequency modulation rate, Δ

_{m}*ω*is the maximum frequency deviation from the carrier and

*J*

_{n(M)}denotes the

*n*’th order Bessel function of the first kind. The instantaneous frequency is:

*N*.

*T*

_{(ω)}:

*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)}and its corresponding intensity,

*I*

_{(t)}, contain information about the complex transfer function

*T*.

*J*

_{n(M)}terms for |

*n*|>1 can be neglected [2

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

*ω*. Demodulation at

_{m}*ω*yields two signals which are proportional to the amplitude of the two orthogonal components: cos (

_{m}*ω*) and sin (

_{m}t*ω*). These phase sensitive detection components are defined as the in-phase (i-p) and quadrature (quad) components, respectively and are given by:

_{m}t*δ*-

_{n}*δ*|≪1 and |

_{l}*ϕ*-

_{n}*ϕ*|≪1,

_{l}*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

**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]

**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

^{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*of the interacting atoms: 3.417GHz. The two first sidebands of the modulated diode serve as the two required coherent fields.

_{hfs}*E*

_{(t)}be the total electromagnetic field comprising two spectral components,

*E*

_{1(t)}and

*E*

_{2(t)}:

*M*, where

_{i}*i*={1,2}. When the two components are derived from the same source,

*ω*

_{1}=-Δ

*ω*

_{2}; consequently:

*M*

_{1}=-

*M*

_{2}≡

*M*for

*E*

_{1(t)}and

*E*

_{2(t)}, are:

*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, Δ*ω*, 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.

*i*=1 and 2, respectively.

*ω*is the optical carrier and

_{o}*ω*is the RF modulation frequency,

_{µ}*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*is around 780nm (384THz),

_{o}*i*={1,2}, respectively and

*ω*is of the order of 1kHz.

_{m}#### 2.2.1. The medium transfer function

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

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]

23. I. Ben-Aroya, M. Kahanov, and G. Eisenstein, “A CPT based ^{87}Rb 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. 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]

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]

*a*is the peak value of the absorption function

*δ*

^{(i)}

_{(ω)}.

*ω*is the resonance frequency and Γ is the Lorentzian resonance width (FWHM).

_{res}*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”

#### 2.2.3. First order analysis and comparison to conventional FM spectroscopy

*M*≪1, all

*J*(

_{n}*M*) terms for |

*n*|>1 can be neglected. The resulting expression enables a clearer description of DFFMS.

*Ẽ*

_{(t)}|

^{2}:

*ω*, since

_{µ}*i*=1 and 2, respectively. These two terms can be neglected. Each of the two first, self-beating, terms includes spectral components around DC, 2

*ω*and

_{m}*ω*with the first two being rejected by the demodulating Lock-in amplifier.

_{m}*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:

*A*

_{1}=

*A*

_{2}, both contributions of each component are equal since

*η*=-

_{2}*η*,

_{1}**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

**C**

_{nl}and

**S**

_{nl}convention as:

**C**

^{(i)}

_{n0}and

**S**

^{(i)}

_{n0}can be interpreted as contributing a spectral feature at ±

*nω*therefore the DFFMS output includes peaks near ±2

_{m}*ω*. The last term in the in-phase component of the DFFMS output (eq. 17) enhances the peaks at ±

_{m}*ω*with respect to the peaks near ±2

_{m}*ω*. Also, eliminating the FM modulation in one of the two carriers, for example at

_{m}*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.

*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 ±2

*ω*. Notice the different energy distributions which the two FM spectroscopy methods yield due to the enhancement term in the DFFMS in-phase component.

_{m}## 3. Experimental results versus the analytic model

*m*=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

_{F}23. I. Ben-Aroya, M. Kahanov, and G. Eisenstein, “A CPT based ^{87}Rb 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).

**15**, 15060–15065 (2007). [CrossRef] [PubMed]

^{87}

*Rb*

*D*

_{2}transition). The RF signal frequency scans around half the

*f*of the ground states, namely, 3.417GHz. The RF current is FM modulated at a low-frequency.

_{hfs}^{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 50

*µ*T 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).

*f*/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

_{hfs}*X*and

*Y*components, respectively.

*X*and

*Y*components, respectively.

## 4. Residual amplitude modulation

*α*-parameter [34

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

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

### 4.1. Theoretical analysis

*i*={1,2}

*R*and

_{i}*ψ*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

_{i}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]

**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]

*R*are much smaller than one and therefore the square root can be replaced by its first order approximation as presented in eq. 20.

_{i}*R*’s are very small compared with the relevant Bessel functions,

_{i}*J*(

_{n}*M*).

*A*

_{1}=

*A*

_{2}) and RAM of the two components are equal but opposite in sign, namely,

*ψ*=

_{1}*ψ*=0 and

_{2}*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

**2**, 1510–1526 (1985). [CrossRef]

### 4.2. Experimental observations of the RAM

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]

*R*,

*θ*) of the two dimensional signal space rather than the orthogonal (

*X*,

*Y*) representation of the amplitudes of the cos(

*ω*) and sin (

_{m}t*ω*) 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 (

_{m}t*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.

*R*≡

*R*

_{1}-

*R*

_{2}=1·10

^{-5}and Δ

*ψ*=

*ψ*-(

_{1}*ψ*-

_{2}*π*)=0.003

*π*

*rad*≈0.5°;

*ψ*=0).

_{1}*X*and

*Y*components are almost zero.

## 5. Summary

*ω*. In particular, demodulation at odd harmonics provides different output signals which are important is some reported cases [37

_{m}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]

## Acknowledgments

## References and links

1. | G. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. |

2. | G. Bjorklund, M. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy,” Appl. Phys. B |

3. | J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. |

4. | E. A. Whittaker, M. Gehrtz, and G. C. Bjorklund, “Residual amplitude modulation in laser electro-optic phase modulation,” J. Opt. Soc. Am. B |

5. | M. Gehrtz, G. Bjorklund, and E. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B |

6. | W. Lenth, “Optical heterodyne spectroscopy with frequency- and amplitude-modulated semiconductor lasers,” Opt. Lett. |

7. | W. Lenth, “High frequency heterodyne spectroscopy with current-modulated diode lasers,” IEEE J. Quantum Electron. |

8. | D. Cassidy and J. Reid, “Harmonic detection with tunable diode lasers — Two-tone modulation,” Appl. Phys. B |

9. | G. R. Janik, C. B. Carlisle, and T. F. Gallagher, “Two-tone frequency-modulation spectroscopy,” J. Opt. Soc. Am. B |

10. | D. E. Cooper and R. E. Warren, “Frequency modulation spectroscopy with lead-salt diode lasers: a comparison of single-tone and two-tone techniques,” Appl. Opt. |

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 |

12. | A. Nagel, C. Affolderbach, S. Knappe, and R. Wynands, “Influence of excited-state hyperfine structure on ground-state coherence,” Phys. Rev. A |

13. | E. Arimondo and G. Orriols, “Nonabsorbing Atomic Coherences by Coherent Two-Photon Transitions in a Three-Level Optical Pumping,” Lett. Nouvo Cim. |

14. | E. Arimondo, “Coherent population trapping in laser spectroscopy,” in |

15. | A. Taichenachev, V. Yudin, R. Wynands, M. Stahler, J. Kitching, and L. Hollberg, “Theory of dark resonances for alkali-metal vapors in a buffer-gas cell,” Phys. Rev. A |

16. | N. Cyr, M. Têtu, and M. Breton, “All-optical microwave frequency standard: a proposal,” IEEE Trans. Instrum. Meas. |

17. | J. Vanier, A. Godone, and F. Levi, “Coherent population trapping in cesium: Dark lines and coherent microwave emission,” Phys. Rev. A |

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 |

19. | J. Vanier, “Atomic clocks based on coherent population trapping: a review,” Appl. Phys. B |

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. |

21. | S. Knappe, P. Schwindt, V. Shah, L. Hollberg, J. Kitching, L. Liew, and J. Moreland
, “
A chip-scale atomic clock based on |

22. | R. Lutwak, P. Vlitas, M. Varghes, M. Mescher, D. K. Serkland, and G. M. Peake, “The MAC-A miniature atomic clock,” in |

23. | I. Ben-Aroya, M. Kahanov, and G. Eisenstein, “A CPT based |

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 |

25. | M. O. Scully and M. Fleischhauer, “High-Sensitivity Magnetometer Based on Index-Enhanced Media,” Phys. Rev. Lett. |

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. |

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. |

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 |

29. | J. A. Silver, “Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods,” Appl. Opt. |

30. | J. M. Supplee, E. A. Whittaker, and W. Lenth, “Theoretical description of frequency modulation and wavelength modulation spectroscopy,” Appl. Opt. |

31. | R. Wynands and A. Nagel, “Inversion of frequency-modulation spectroscopy line shapes,” J. Opt. Soc. Am. B |

32. | A. Yariv, |

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. |

35. | X. Zhu and D. T. Cassidy, “Modulation spectroscopy with a semiconductor diode laser by injection-current modulation,” J. Opt. Soc. Am. B |

36. | A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous Interaction of Spectral Modes in a Semiconductor Laser,” IEEE J. Quantum Electron. |

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 |

**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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