## Analysis of whispering-gallery microcavity-enhanced chemical absorption sensors

Optics Express, Vol. 15, Issue 20, pp. 12959-12964 (2007)

http://dx.doi.org/10.1364/OE.15.012959

Acrobat PDF (101 KB)

### Abstract

A theoretical analysis of the operation of a chemical sensor based on cavity-enhanced optical absorption is given for a system in which the cavity is a dielectric whispering-gallery microresonator. Continuous-wave input is assumed, and the detection sensitivity is characterized in terms of an effective absorption path length. In the case of tunable single-frequency input, it is shown that monitoring analyte-induced changes in the throughput dip depth enables detection with relative sensitivity greater than that of frequency-shift and cavity-ringdown methods. In addition, for the case of broadband input and drop-port output, an analysis applicable to microcavity-enhanced absorbance spectroscopy experiments is given.

© 2007 Optical Society of America

## 1. Introduction

*Q*) and low mode volume, and on the possibility of efficient coupling of light into and out of these modes [1

1. A. B. Matsko and V. S. Ilchenko, “Optical Resonators with Whispering-Gallery Modes - Part I: Basics,” IEEE J. Sel. Top. Quantum Electron. **12**, 3–14 (2006). [CrossRef]

*Q*of a WGM means that light makes many round trips in the resonator. This feature, combined with small mode volume and efficient coupling using prisms, angle-polished fibers, or tapered fibers, makes high intracavity power enhancement easy to achieve. Various applications such as filtering, lasing, modulation, nonlinear optics, sensing, and spectroscopy are enabled by these properties of WGMs [2

2. V. S. Ilchenko and A. B. Matsko, “Optical Resonators with Whispering-Gallery Modes - Part II: Applications,” IEEE J. Sel. Top. Quantum Electron. **12**, 15–32 (2006). [CrossRef]

## 2. Tunable single-frequency operation

8. M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. **271**, 124–131 (2007). [CrossRef]

*αL*, where

*α*is the loss coefficient and

*L*is the microresonator circumference, is assumed; it models the intrinsic loss (primarily surface scattering) of a WGM microresonator. The mirror reflection and transmission coefficients

*r*and

*it*, relating to the field mode amplitudes, are taken to be real and imaginary respectively without loss of generality; this corresponds to the usual choice for fiber or prism coupling [7

7. R. W. Boyd and J. E. Heebner, “Sensitive disk resonator photonic biosensor,” Appl. Opt. **40**, 5742–5747 (2001). [CrossRef]

9. M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-*Q* whispering-gallery modes,” J. Opt. Soc. Am. B **16**, 147–154 (1999). [CrossRef]

10. M. Cai, O. Painter, and K. J. Vahala, “Observation of Critical Coupling in a Fiber Taper to a Silica-Microsphere Whispering-Gallery Mode System,” Phys. Rev. Lett. **85**, 74–77 (2000). [CrossRef] [PubMed]

*t*

^{2}= 1 -

*r*

^{2}. Because

*Q*is so high, even under conditions of loading by the coupler, the intrinsic round-trip loss and the coupling loss (mirror transmissivity or probability of photon tunneling between fiber and microresonator) will always be small, so

*αL*≪ 1 and

*T*=

*t*

^{2}≪ 1 will be assumed throughout this work.

*δ*= 2

*πnL*(

*ν*-

*ν*

_{0})/

*c*is the round-trip phase accumulation due to detuning of the input frequency

*ν*from resonance

*ν*

_{0}(

*c*is the speed of light and

*n*the resonator’s index of refraction). The deviation of the throughput fraction from unity gives the dip profile:

*δ*=

*p*2

*π*,

*p*= 0, 1, 2, …. The spacing of 2

*π*between adjacent modes is the free spectral range (FSR) in phase. (Recall that the FSR in frequency is given to good approximation by

*c*/(

*nL*). Because many higher-order WGMs can be excited, the actual spacing between adjacent modes is much less than the FSR.) The full width at half-maximum of a WGM resonance is thus seen to be given by Δ

*δ*=

*T*+

*αL*, the total round-trip loss. The final expression in Eq. (2), valid when

*δ*≪ 1, shows the Lorentzian profile of a dip.

*δ*= 0) is determined by the ratio of the coupling loss to intrinsic loss,

*x*=

*T*/

*αL*, and is given by

*x*= 1, critical coupling is obtained and the dip depth attains its maximum value of 100%; the microresonator is said to be undercoupled if

*x*< 1 and overcoupled for

*x*> 1. While the coupling loss remains constant, the effective intrinsic loss can be changed by interaction of the evanescent fraction (

*f*) of the WGM with the surrounding medium. The effective loss coefficient can then be written as

*α*=

*α*+

_{i}*fα*+

_{a}*fα*, where the three terms denote true intrinsic loss, absorption (and perhaps also scattering) by the analyte, and absorption in the solvent (or ambient).

_{s}*L*can then be obtained from the dip depth dependence on the absorption coefficient of the analyte:

^{t}_{eff}*α*≪1 (or

_{a}L^{t}_{eff}*fα*≪

_{a}*α*), which is the condition for Eq. (4) to hold. Note that in the strongly undercoupled or overcoupled limits (

_{i}*x*≪ 1 or

*x*≫ 1) the relative detection sensitivity is determined by the intrinsic loss only. This can be advantageous, since having the tapered fiber in contact with the microresonator tends to produce overcoupling, especially when the system is immersed in a liquid. Thus in the strongly overcoupled case, this method has greater relative sensitivity (here, fractional change in dip depth) than the frequency-shift, mode-width, or ringdown methods, all of whose relative sensitivities are determined by the total loss. Recall that the total loss determines the linewidth and the cavity lifetime, and in the overcoupled limit coupling loss dominates. The relative frequency-shift sensitivity is measured as a fraction of the linewidth, the change in mode width is relative to the full width, and ringdown measures the fractional change in the overall lifetime.

*L*can be found by measuring the dip depth in the absence of analyte (

^{e}_{eff}*M*

_{0}) and in the presence of analyte (

*M*

_{0}+ ∆

*M*

_{0}):

*α*≪1. Comparison of experimental and theoretical effective absorption path lengths for detection of atmospheric trace gases shows good agreement [11], as discussed briefly in Section 4.

_{a}L^{e}_{eff}*fα*≫

_{s}*α*, the effective absorption path length can still be as large as 1 /

_{i}*α*. In effect, solvent absorption can shift the sensor from one sensitive regime to another - from strongly overcoupled to strongly undercoupled, enabling absolute analyte sensitivity, i.e., actual signal amplitudes, that would be difficult to achieve in single-pass direct absorption through the same effective path length.

_{s}## 3. Broadband operation

12. M. L. Gorodetsky and V. S. Ilchenko, “High-Q optical whispering-gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. **113**, 133–143 (1994). [CrossRef]

*δ*= 2

*T*+

*αL*and the resonant drop fraction is

*D*0.

*D*, the fraction of the incident power in the integration interval that is transmitted out the drop port, is given by

_{I}*δ*= 2

*T*+

*α*in the two limiting cases of large and small linewidth. The first limiting case says that when the linewidth fills the integration interval, the drop fraction equals what would be found by using a single-frequency source tuned to WGM resonance; this holds for either low

_{L}*Q*or many overlapping modes in the integration interval, so that integration introduces no additional linewidth dependence. The second limiting case is the usual one for well-separated modes, and is the same as that which results from approximating each WGM’s narrow transmitted lineshape as a Lorentzian; this holds as long as the integration interval is wide compared to the WGM linewidth.

*f*of the WGM:

*α*=

*αi*+

*fα*. The effect of the analyte on the resonant (single-frequency) drop signal

_{a}*D*

_{0a}, when analyte absorption is a small fraction of the total loss, can be written in terms of an approximate effective absorption path length

*L*as defined below:

_{eff}*L*is the effective absorption path length as defined in the low-analyte-absorption limit. However, the last expression for

_{eff}*D*

_{0a}is valid even for large analyte absorption, that is, there are no restrictions on the size of

*α*as long as

_{a}L_{eff}*fα*≪1.

_{a}L*D*, will also be given by Equation (9) in the large linewidth limit. For precessing-mode drop-port output collected without being spatially filtered by an aperture [13], this limit applies, for the following reason. The incident focused light (Fig. 2) incorporates a bundle of wavevectors and so excites precessing modes over a range of angles with respect to the equatorial plane. The round-trip distance depends on this angle in a spheroid, so the precessing modes are frequency-shifted by amounts depending on angle. The result is many modes (not just a range of different angles, but also ranges of different radial orders and different polar orders) that overlap to fill the integration interval, so when the analyte absorption broadens the linewidth and reduces the transmission of each mode, the broadening is not noticed, because the integration interval remains filled. Thus only the decrease in amplitude is observed, just as in the single-frequency resonant case. The overlapping of modes also means that the exact value of the spectrometer resolution interval does not matter. This large-linewidth-limit functional dependence of the integrated drop signal on

_{Ia}*α*as given in Eq. (9) has been tested in recent experimental work [14

_{a}L_{eff}14. S. L. Westcott, J. Zhang, R. K. Shelton, N. M. K. Bruce, S. Gupta, S. L. Keen, J. W. Tillman, L. B. Wald, B. N. Strecker, A. T. Rosenberger, R. R. Davidson, W. Chen, K. G. Donovan, and J. V. Hryniewicz are preparing a manuscript to be called “Broadband optical absorbance spectroscopy using a whispering gallery mode microsphere resonator.”

## 4. Discussion and summary

*x*= 0.05 or

*x*= 20 produce

*M*

_{0}= 0.18. In this example, in the overcoupled case, the total loss is twenty-one times the intrinsic loss, so a ~20× enhancement in relative sensitivity compared to frequency-shift or ringdown methods is obtained. For a wavelength on the order of 1000 nm and typical microresonator values, for example circumference

*L*~ 1 mm,

*Q*~ 10

_{8}, evanescent fraction

*f*~ 0.2% (in air) or

*f*~ 2% (in water or methanol), effective analyte absorption path lengths on the order of centimeters can be expected, even in strongly absorbing solvents (

*α*~ 1 cm

_{s}^{-1}). Using a cylindrical microresonator with

*Q*~ 5×10

^{6}[11], experimental effective path lengths of about 1 cm are found for trace amounts of methane, ethylene, or methyl chloride in air at atmospheric pressure. The dip depth is monitored by locking a WGM to the laser as the laser is scanned over absorption lines for wavelengths around 1.65 μm. Agreement with theoretical effective path lengths is very good.

14. S. L. Westcott, J. Zhang, R. K. Shelton, N. M. K. Bruce, S. Gupta, S. L. Keen, J. W. Tillman, L. B. Wald, B. N. Strecker, A. T. Rosenberger, R. R. Davidson, W. Chen, K. G. Donovan, and J. V. Hryniewicz are preparing a manuscript to be called “Broadband optical absorbance spectroscopy using a whispering gallery mode microsphere resonator.”

*α*) was in most cases as predicted by Eq. (9).

_{a}## Acknowledgments

## References and links

1. | A. B. Matsko and V. S. Ilchenko, “Optical Resonators with Whispering-Gallery Modes - Part I: Basics,” IEEE J. Sel. Top. Quantum Electron. |

2. | V. S. Ilchenko and A. B. Matsko, “Optical Resonators with Whispering-Gallery Modes - Part II: Applications,” IEEE J. Sel. Top. Quantum Electron. |

3. | I. Teraoka, S. Arnold, and F. Vollmer, “Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium,” J. Opt. Soc. Am. B |

4. | N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. |

5. | A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high- |

6. | A. A. Savchenkov, A. B. Matsko, M. Mohageg, and L. Maleki, “Ringdown spectroscopy of stimulated Raman scattering in a whispering gallery mode resonator,” Opt. Lett. |

7. | R. W. Boyd and J. E. Heebner, “Sensitive disk resonator photonic biosensor,” Appl. Opt. |

8. | M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, “Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere,” Opt. Commun. |

9. | M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high- |

10. | M. Cai, O. Painter, and K. J. Vahala, “Observation of Critical Coupling in a Fiber Taper to a Silica-Microsphere Whispering-Gallery Mode System,” Phys. Rev. Lett. |

11. | G. Farca, S. I. Shopova, and A. T. Rosenberger are preparing a manuscript to be called “Cavity-enhanced laser absorption spectroscopy using microresonator whispering-gallery modes.” |

12. | M. L. Gorodetsky and V. S. Ilchenko, “High-Q optical whispering-gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. |

13. | J. Zhang, B. N. Strecker, R. K. Shelton, S.-J. Ja, J. V. Hryniewicz, and A. T. Rosenberger are preparing a manuscript to be called “A broadband whispering-gallery mode microsphere absorbance spectrometer.” |

14. | S. L. Westcott, J. Zhang, R. K. Shelton, N. M. K. Bruce, S. Gupta, S. L. Keen, J. W. Tillman, L. B. Wald, B. N. Strecker, A. T. Rosenberger, R. R. Davidson, W. Chen, K. G. Donovan, and J. V. Hryniewicz are preparing a manuscript to be called “Broadband optical absorbance spectroscopy using a whispering gallery mode microsphere resonator.” |

**OCIS Codes**

(060.2370) Fiber optics and optical communications : Fiber optics sensors

(230.5750) Optical devices : Resonators

(300.1030) Spectroscopy : Absorption

(350.3950) Other areas of optics : Micro-optics

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: August 13, 2007

Revised Manuscript: September 15, 2007

Manuscript Accepted: September 17, 2007

Published: September 24, 2007

**Virtual Issues**

Vol. 2, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

A. T. Rosenberger, "Analysis of whispering-gallery microcavity-enhanced chemical absorption sensors," Opt. Express **15**, 12959-12964 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-20-12959

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### References

- A. B. Matsko and V. S. Ilchenko, "Optical Resonators with Whispering-Gallery Modes - Part I: Basics," IEEE J. Sel. Top. Quantum Electron. 12, 3-14 (2006). [CrossRef]
- V. S. Ilchenko and A. B. Matsko, "Optical Resonators with Whispering-Gallery Modes - Part II: Applications," IEEE J. Sel. Top. Quantum Electron. 12, 15-32 (2006). [CrossRef]
- I. Teraoka, S. Arnold, and F. Vollmer, "Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium," J. Opt. Soc. Am. B 20, 1937-1946 (2003). [CrossRef]
- N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White and X. Fan, "Refractometric sensors based on microsphere resonators," Appl. Phys. Lett. 87, 201107 (2005). [CrossRef]
- A. M. Armani and K. J. Vahala, "Heavy water detection using ultra-high-Q microcavities," Opt. Lett. 31, 1896-1898 (2006). [CrossRef] [PubMed]
- A. A. Savchenkov, A. B. Matsko, M. Mohageg, and L. Maleki, "Ringdown spectroscopy of stimulated Raman scattering in a whispering gallery mode resonator," Opt. Lett. 32, 497-499 (2007). [CrossRef] [PubMed]
- R. W. Boyd and J. E. Heebner, "Sensitive disk resonator photonic biosensor," Appl. Opt. 40, 5742-5747 (2001). [CrossRef]
- M. J. Humphrey, E. Dale, A. T. Rosenberger, and D. K. Bandy, "Calculation of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled microsphere," Opt. Commun. 271, 124-131 (2007). [CrossRef]
- M. L. Gorodetsky and V. S. Ilchenko, "Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes," J. Opt. Soc. Am. B 16, 147-154 (1999). [CrossRef]
- M. Cai, O. Painter, and K. J. Vahala, "Observation of Critical Coupling in a Fiber Taper to a Silica-Microsphere Whispering-Gallery Mode System," Phys. Rev. Lett. 85, 74-77 (2000). [CrossRef] [PubMed]
- G. Farca, S. I. Shopova, and A. T. Rosenberger are preparing a manuscript to be called "Cavity-enhanced laser absorption spectroscopy using microresonator whispering-gallery modes."
- M. L. Gorodetsky and V. S. Ilchenko, "High-Q optical whispering-gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers," Opt. Commun. 113, 133-143 (1994). [CrossRef]
- J. Zhang, B. N. Strecker, R. K. Shelton, S.-J. Ja, J. V. Hryniewicz, and A. T. Rosenberger are preparing a manuscript to be called "A broadband whispering-gallery mode microsphere absorbance spectrometer."
- S. L. Westcott, J. Zhang, R. K. Shelton, N. M. K. Bruce, S. Gupta, S. L. Keen, J. W. Tillman, L. B. Wald, B. N. Strecker, A. T. Rosenberger, R. R. Davidson, W. Chen, K. G. Donovan, and J. V. Hryniewicz are preparing a manuscript to be called "Broadband optical absorbance spectroscopy using a whispering gallery mode microsphere resonator."

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