Cavity mode-width spectroscopy with widely tunable ultra narrow laser 1 Marcin Bober,1 ´ Agata Cygan,1,∗ Daniel Lisak,1 Piotr Morzynski, Michał Zawada,1 Eugeniusz Pazderski,2 and Roman Ciuryło1 1 Institute 2

of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5, 87-100 Torun, Poland Torun Center for Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5, 87-100 Torun, Poland *[email protected]

Abstract: We explore a cavity-enhanced spectroscopic technique based on determination of the absorbtion coefficient from direct measurement of spectral width of the mode of the optical cavity filled with absorbing medium. This technique called here the cavity mode-width spectroscopy (CMWS) is complementary to the cavity ring-down spectroscopy (CRDS). While both these techniques use information on interaction time of the light with the cavity to determine absorption coefficient, the CMWS does not require to measure very fast signals at high absorption conditions. Instead the CMWS method require a very narrow line width laser with precise frequency control. As an example a spectral line shape of P7 Q6 O2 line from the B-band was measured with use of an ultra narrow laser system based on two phase-locked external cavity diode lasers (ECDL) having tunability of ± 20 GHz at wavelength range of 687 to 693 nm. © 2013 Optical Society of America OCIS codes: (120.6200) Spectrometers and spectroscopic instrumentation; (140.3425) Laser stabilization; (140.3570) Lasers, single-mode; (140.3600) Lasers, tunable; (300.1030) Absorption; (300.3700) Linewidth; (300.6320) Spectroscopy, high-resolution.

References and links 1. R. D. van Zee and J. P. Looney, Experimental Methods in the Physical Sciences: Cavity-Enhanced Spectroscopies (Elsevier Science, 2002). 2. B. A. Paldus and A. Kochanov, “An historical overview of cavity-enhanced methods,” Can. J. Phys. 83, 975-999 (2005). 3. G. Berden and R. Engeln, Cavity Ring-Down Spectroscopy: Techniques and Applications (Wiley-Blackwell, 2009). 4. J. T. Hodges, H. P. Layer, W. M. Miller, and G. E. Scace, “Frequency-stabilized single-mode cavity ring-down apparatus for high-resolution absorption spectroscopy,” Rev. Sci. Instr. 75, 849-863 (2004). 5. J. T. Hodges and R. Ciuryło, “Automated high-resolution frequency-stabilized cavity ring-down absorption spectrometer,” Rev. Sci. Instrum. 76, 023112 (2005). 6. D. A. Long, G.-W. Truong, R. D. van Zee, D. F. Plusquellic, and J. T. Hodges, “Frequency-agile, rapid scanning spectroscopy: absorption sensitivity of 2 × 10−12 cm−1 Hz−1/2 with a tunable diode laser,” Appl. Phys. B, in press, DOI 10.1007/s00340-013-5548-5. 7. D. Long, A. Cygan, R. van Zee, M. Okumura, C. Miller, D. Lisak, and J. Hodges, “Frequency-stabilized cavity ring-down spectroscopy,” Chem. Phys. Lett. 536, 1-8 (2012). 8. A. Cygan, D. Lisak, S. W´ojtewicz, J. Domysławska, J. T. Hodges, R. Trawi´nski, and R. Ciuryło, “High signalto-noise ratio laser technique for accurate measurements of spectral line parameters,” Phys. Rev. A. 85 022508 (2012). 9. K. Nakagawa, T. Katsuda, A.S. Shelkovnikov, M. Delabachelerie, and M. Ohtsu, “Highly sensitive detection of molecular absorption using a high finesse optical cavity,” Opt. Commun. 107, 369-372 (1994).

#195746 - $15.00 USD Received 13 Aug 2013; revised 4 Oct 2013; accepted 24 Oct 2013; published 25 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029744 | OPTICS EXPRESS 29744

10. G.-W. Truong, K. O. Douglass, S. E. Maxwell, R. D van Zee, D. F. Plusquellic, J. T. Hodges, and D. A. Long, “Frequency-agile, rapid scanning spectroscopy,” Nature Photon. 7, 532-534 (2013). 11. D. Lisak, A. Cygan, K. Bielska, M. Piwi´nski, F. Ozimek, T. Ido, R. S. Trawi´nski, and R. Ciuryło, “Ultra narrow laser for optical frequency reference,” Acta Phys. Pol. A 121, 614-621 (2012). 12. M. Takamoto, F.L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435, 321324 (2005). 13. T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 18081812 (2008). 14. R. Le Targat, L. Lorini, Y. Le Coq, M. Zawada, J. Gu´ena, M. Abgrall, M. Gurov, P. Rosenbusch, D. G. Rovera1, B. Nag´orny, R. Gartman, P.G. Westergaard, M.E. Tobar, M. Lours, G. Santarelli, A. Clairon, S. Bize, P. Laurent, P. Lemonde, and J. Lodewyck, “Experimental realization of an optical second with strontium lattice clocks,” Nat. Commun. 4, 2109 (2013). 15. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97-105 (1983). 16. T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nature Photon. 6, 687692 (2012). 17. C. Daussy, M. Guinet, A. Amy-Klein, K. Djerroud, Y. Hermier, S. Briaudeau, Ch. J. Bord´e, and C. Chardonnet, “Direct determination of the Boltzmann constant by an optical method,” Phys. Rev. Lett. 98, 250801 (2007). 18. M. Kumagai, H. Kanamori, M. Matsushita, and T. Kato, “Development of phase-lock system between two singlemode lasers for optical-optical double resonance spectroscopy,” Jpn. J. Appl. Phys. 38, 6102-6106 (1999). 19. T. H. Loftus, T. Ido, M. M. Boyd, A. D. Ludlow, and J. Ye, “Narrow line cooling and momentum-space crystals,” Phys. Rev. A 70, 063413 (2004). 20. T. Zelevinsky, M. M. Boyd, A. D. Ludlow, T. Ido, J. Ye, R. Ciuryło, P. Naidon, and P. S. Julienne, “Narrow Line Photoassociation in an Optical Lattice,” Phys. Rev. Lett. 96, 203201 (2006). 21. S. W´ojtewicz, D. Lisak, A. Cygan, J. Domysławska, R. Trawi´nski, and R. Ciuryło, “Line-shape study of selfbroadened O2 transitions measured by Pound-Drever-Hall-locked frequency-stabilized cavity ring-down spectroscopy,” Phys. Rev. A 84, 032511 (2011). 22. A. Foltynowicz, Fiber-laser-Based Noise-Immune Cavity-Enhanced Optical Heterodyne Molecular Spectrometry, Doctoral Thesis, Ume˚a University, Sweden 2009. 23. W. Demtr¨oder, Laser Spectroscopy, Vol. 1: Basic Principles (Springer-Verlag, 2008). 24. K. G. Libbrecht and M. W. Libbrecht, “Interferometric measurement of the resonant absorption and refractive index in rubidium gas,” Am. J. Phys. 74, 1055-1060 (2006). 25. L. S. Rothman, I. E. Gordon, A. Barbe, D. Chris Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. ˇ Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Simeˇ ckov´a, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database” J. Quant. Spectrosc. Radiat. Transf. 110, 533-572 (2009). 26. A. Cygan, D. Lisak, P. Masłowski, K. Bielska, S. W´ojtewicz, J. Domysławska, R. S. Trawi´nski, R. Ciuryło, H. Abe, and J. T. Hodges, “Pound-Drever-Hall-locked, frequency-stabilized cavity ring-down spectrometer,” Rev. Sci. Instrum. 82, 063107 (2011). 27. S. W´ojtewicz, K. Stec, P. Masłowski, A. Cygan, D. Lisak, R. S. Trawi´nski, and R. Ciuryło, “Low pressure lineshape study of self-broadened CO transitions in the (3 ← 0) band,” J. Quant. Spectrosc. Radiat. Transf., DOI: 10.1016/j.jqsrt.2013.06.005 (2013). 28. D. A. Long, S. W´ojtewicz, and J. T. Hodges, “Effects of incomplete light extinction in frequency-agile, rapid scanning spectroscopy,” Proc. SPIE 8726, 87260O (2013).

1.

Introduction

Cavity enhanced spectroscopy techniques became standard methods for measurement of weak absorption in gas phase and plenty of applications was found in both, basic and applied research, see e.g. [1, 2, 3]. Among these techniques the cavity ring-down spectroscopy (CRDS) is particularly interesting because of its unique properties. In the continuous-wave CRDS (CWCRDS) the absorption coefficient α of a sample placed inside the optical cavity is determined from the decay of light wave leaking out of the cavity after the incident wave was switched off rapidly. In the case when only one mode of the cavity is excited the decay is described by pure exponential function and absorption coefficient α is directly related to the time constant τ

#195746 - $15.00 USD Received 13 Aug 2013; revised 4 Oct 2013; accepted 24 Oct 2013; published 25 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029744 | OPTICS EXPRESS 29745

of this decay. Therefore resulting α is independent of the incident laser power. The absorption spectrum of a sample can be measured by tuning the laser frequency to the consecutive cavity modes or also by tuning the cavity mode frequencies. Stabilization and precise control of the cavity length leads to the frequency-stabilized CRDS (FS-CRDS) [4, 5] which is widely used to the most precise spectral line shapes investigation with low uncertainty, extremely high sensitivity [6], resolution limited only by the width of the cavity modes, very good reproducibility [7], and the highest of all spectroscopic methods signal-to-noise ratio of the spectrum exceeding 2 × 105 [8]. In this paper we explore cavity enhanced spectroscopy technique in which an absorption spectrum is determined from the dependence of a high finesse cavity TEM00 mode widths on absorption coefficient at given frequency. It is well known that the spectral width of the cavity mode in the frequency domain is mathematically related to the cavity mode ring-down decay in the time domain. Therefore, the approach discussed here can be recognized as a complementary technique to CRDS. It was reported by Nakagawa et al. [9] that higher intracavity absorption coefficient is accompanied by larger cavity mode width. Very recently experimental spectrum determined from cavity mode widths measurement was demonstrated by Long et al. [6] with use of the FARS (frequency-agile, rapid scanning spectroscopy) technique [10]. 3 − As an example the line shape of P7 Q6 O2 transition from the b1 Σ+ g (v = 1) ← X Σg (v = 0) band (B-band) was measured using the cavity mode-width spectroscopy (which will be called here CMWS). In this kind of spectroscopy it is necessary to measure the spectral width of the cavity mode resonance at kHz level. Therefore precise laser frequency control is required. In contrast to experimental scheme used by Long et al. [6] we employed widely tunable laser phase-locked with frequency offset to ultra-narrow narrow laser [11] and the cavity was passively stabilized. Development of ultra-narrow line width lasers is mainly motivated by probing strongly forbidden transitions of atoms trapped in the optical lattice [12] or single trapped ions [13] for new optical time and frequency standards expected in the near future [14]. It was demonstrated that using the Pound-Drever-Hall technique [15] of locking the laser frequency to the resonant mode of the ultra stable high-finesse optical cavity one can achieve line widths below 1 Hz. In the best case so far a laser line width below 40 mHz was obtained [16]. Wide tunability of laser radiation in tens of GHz range with subkilohertz accuracy can be achieved by using an electro-optical modulator (EOM) which generates sideband frequencies [17, 10] which was used by Long et al. [6]. The other scheme is to use a second laser phase-locked with variable frequency offset [18, 19] to the ultra narrow laser. Such approach allows to obtain single-frequency optical source with wide tunability and is very useful in precise spectroscopy [20]. With recent development of ultra narrow and stable lasers this method of spectrum measurement may be competitive to other cavity enhanced methods, such as cavity ring-down spectroscopy (CRDS). Similarly to CRDS the CMWS is almost insensitive to wavelength dependence of laser power and contrary to CRDS measurement of mode width of the cavity by direct scan of the narrow laser through it does not require fast detection system with flat frequency response [21]. 2.

Cavity mode-width spectroscopy

Similarly to the cavity ring-down spectroscopy (CRDS) the idea of cavity mode-width spectroscopy (CMWS) is based on variation of interaction time of the laser light with the optical cavity. While this interaction time can be measured directly from the ring-down decay time constant it has also influence on spectral width of the cavity resonant modes and their shift due to dispersion. With decreasing interaction time, due to absorption by intracavity medium the mode width increases. Scheme of an experiment is presented in Fig. 1. A slave laser with spec-

#195746 - $15.00 USD Received 13 Aug 2013; revised 4 Oct 2013; accepted 24 Oct 2013; published 25 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029744 | OPTICS EXPRESS 29746

tral line width much lower than the cavity modes is scanned through consecutive longitudinal modes of the cavity filled with absorbing medium. From measurement of the mode half width δ νm an absorption spectrum can be determined, as described in the next subsection. Similarly to CRDS the spectral resolution of this method is limited to the cavity mode widths.

Fig. 1. Scheme of the cavity mode-width spectroscopy experiment. νMS and νOFL are frequencies of the master and the offset-frequency locked slave lasers, respectively. δ νBeat is a frequency offset between lasers, and δ νm is the mode width of the cavity, which increases with laser light absorption α (ν ) (red curve on the graph) in the cavity.

3.

Theory

Propagation of monochromatic radiation field E with frequency ω and wave vector in a vacuum k along a z axis through a medium can be described as E(t, z) = E0 ei(2πν t−knz)

(1)

where knz describes a phase shift of the electric filed in the medium without absorption and dispersion and k = 2πν /c. Near the resonance transition, the absorption and dispersion of the field E can be described by an imaginary and real part of the complex refractive index na (ν ) = n (ν ) − iκ (ν ).

(2)

The expression for the field E at given time t and displacement z in medium having an optical resonance can be given, as in [22], replacing n by n + na (ν ) − 1: 

E(t, z) = E0 e−kκ (ν )z ei{2πν t−k[n+n (ν )−1]z}

(3)

The transmission function of this field through an optical cavity along its z axis leads to a well known Airy equation (see e.g. [23]) far from the absorption resonance. When Eq. (2) is taken into account the transmitted field ET can be written as 

ET = t 2 E0 e−kκ (ν )L ei{2πν t−k[n+n (ν )−1]L}

1  1 − r2 e−2kκ (ν )L e−i2k(n+n (ν )−1)L

,

(4)

#195746 - $15.00 USD Received 13 Aug 2013; revised 4 Oct 2013; accepted 24 Oct 2013; published 25 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029744 | OPTICS EXPRESS 29747

where L is the cavity length, and t and r are the cavity mirrors transmission and reflection coefficients for field E, respectively. When introducing absorption coefficient α (ν ) = 2kκ (ν ), and writing the phase shift in terms of the free spectral range of the cavity (νFSR (ν ) = c/[2L(n + n (ν ) − 1)]) as 2k(n + n (ν ) − 1)L = 2πν /νFSR the transmission function of the radiation intensity IT ∼ ET ET∗ is given by the equation IT (ν ) = I0

(1 − R)2 e−α (ν )L (1 − R e−α (ν )L )2 (1 − R e−α (ν )L )2 (1 − R e−α (ν )L )2 + 4R e−α (ν )L sin2 ( ν

πν

FSR (ν )

)

,

(5)

where R is the intensity reflectivity of the cavity mirrors. Dependence of the cavity free spectral range on frequency νFSR (ν ) can be calculated using known absorption coefficient α (ν ) and the Kramers-Kr¨onig relation [24]. In case when absorption coefficient α (ν ) is described by Lorentzian line-shape the resonance contribution to the refractive index can be given in the following form α (ν ) (ν − ν 0 ) , (6) n (ν ) = 1 + kγ where ν0 and γ are the center frequency and half width (FWHM) of the absorption line, respectively. Using this relation νFSR can be expressed as

νFSR (ν ) = 4.

c 2Ln[1 −

α (ν )c(ν −ν0 ) ] 2π nνγ

.

(7)

Experimental setup

Fig. 2. Experimental setup. Master and Slave are ECDL lasers, EOM is a 20-MHz electrooptic modulator to generate sidebands for the PDH lock of the Master laser to the reference cavity, λ /4 is a quarter-wave plate, Pol is a linear polarizer, Gen1 and Gen2 are signal generators, PBS are polarizing beam splitters, DetPDH and DetBN are detectors for PDH lock signal and for beat note signal between both lasers, ⊗ are RF mixers. The probe cavity is high-finesse optical cavity filled with an absorbing gas. The detector DetCMWS registers OFL laser light transmitted through the cavity.

The scheme of the experimental setup is presented in Fig. 2. Both the ”master” and the ”slave” lasers are external cavity diode lasers (ECDL) (Toptica, DL-pro) with spectral line widths of emitted radiation below 200 kHz and output power of about 20 mW. The master laser is locked with the Pound-Drever-Hall (PDH) scheme [15] to a high-finesse (F ∗ = 117000) reference optical cavity made of the ultra-low expansion glass. Shape of the cavity is optimized #195746 - $15.00 USD Received 13 Aug 2013; revised 4 Oct 2013; accepted 24 Oct 2013; published 25 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029744 | OPTICS EXPRESS 29748

for insensitivity of resonant mode frequencies to mechanical vibrations. To minimize influence of temperature variations and acoustic noise in laboratory on resonant frequencies of the cavity it is placed in a temperature-stabilized vacuum chamber. The chamber and the master laser head are placed on vibration-isolation platform damping vibrations down to 0.5 Hz (Minus-K, BM-1). The platform is placed inside acoustic-noise isolation box (Novascan, 40 dB isolation), see [11] for details. As was demonstrated in [11] the line width of the master laser locked to the reference cavity was 8 Hz (FWHM) measured with resolution bandwidth of 1 Hz. The transmitted light from the reference cavity is superimposed on the beam of the slave laser and a beat note signal is measured by a 20-GHz bandwidth detector (NewFocus 1434). To generate the error signal for the phase lock of the slave to the master laser, in a ±20 GHz range of their relative detuning, a beat note signal is first down-converted to 20 MHz by mixing it with a stable signal from Gen1 generator (Agilent E8257D). We should note here that the fast mixer (Marki Microwave M1R-0726) works in a nominal range of input (LO and RF) frequencies of 7 − 26.5 GHz, but we found that stable phase-lock of the master to the slave laser can be achieved for LO and RF frequencies down to 3 GHz. Moreover the nominal output frequency (IF) is DC − 8 GHz, but we found that the phase lock is more stable when IF signal is about 20 MHz. Therefore a second stage mixer is used. The signal from the output of the fast mixer is mixed with stable 20 MHz signal from Gen2 generator in a DC coupled mixer (Minicircuits ZP-3). The output of this mixer is an error signal used to phase-lock the slave laser to the master with optical frequency difference equal to the sum of frequencies of Gen1 and Gen2 generators. The error signal is filtered by fast PID controller (Toptica FALC) and the output of PID controls the laser diode current. Additional slow integrator is used to tune slave laser frequency by adjusting its diffraction grating what allows to maintain the fast diode-current tuning signal near zero. The relative linewidth of the slave laser to the master laser is below 150 mHz [11]. The laser system described above allows wide tunability (range ±20 GHz) of ultranarrow spectrum (width 8 Hz) of laser radiation. The tunable and Hz-level-linewidth offset-frequency lock (OFL) laser was then used as a probe laser in cavity mode-width spectroscopy. The finesse and free spectral range of the probe cavity was 10500 and 203.970(3) MHz, respectively, what corresponds to resonance mode widths below 20 kHz. The cavity was filled with 20.4 Torr of O2 and the spectral region was chosen to cover absorption spectrum of the P7 Q6 line (14504.791333 cm−1 ), from the B-band, having intensity of 4.841 × 10−25 cm−1 /(molecule cm−2 ). Change of the laser frequency was achieved by switching Gen1 generator frequency, what can be done via Ethernet and takes time of 1 ms. The frequency tuning can be realized in maximum steps of 0.5 MHz, at which the phase lock loop remains closed. In the transmission spectrum of the probe cavity measured across the P7 Q6 line (see Fig. 3), the frequency axis corresponded to detuning of the slave laser from the master laser and frequency steps were 30 kHz between TEM00 modes (transmission peaks) and 2 kHz in a range ±100 kHz around the modes. Data acquisition of the whole spectrum took about 1 hour, and the frequency axis was corrected for temperature drift of the cavity modes in such a way that difference between consecutive TEM00 modes was equal to the measured free spectral range. 5.

Analysis of experimental results

Demonstration of the cavity mode-width spectroscopy with the use of phase-locked laser system was done using cavity filled with pure oxygen at pressure 20.4 Torr. Transmission spectrum with three individual modes enlarged in the insets was shown in Fig. 3. These modes correspond to three different absorption coefficients α (ν ) near P7 Q6 transition from the O2 B-band. Clearly mode width increases with higher absorption. It should be noted that noise in the experimental spectrum is mainly associated with frequency axis and not the transmission

#195746 - $15.00 USD Received 13 Aug 2013; revised 4 Oct 2013; accepted 24 Oct 2013; published 25 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029744 | OPTICS EXPRESS 29749

signal and it is caused by acoustic noise of the cavity length. It was shown [11] that if ultra stable cavity is used the noise of measured shape of the cavity mode can be significantly reduced. Other way of elimination of this kind of the noise is to lock the master laser to one of the cavity modes. It assures that relative tuning of the slave laser to the other cavity mode will not be affected by mechanical vibration of the cavity. The width δ νm of the cavity modes were determined from the least-squares fit of the Lorentzian profiles to experimental data. The Lorentzian profile is a good approximation of a single mode spectrum for high finesse cavity. From the dependence of the width δ νm of the δνm = (53.13±1.47) kHz

0.1

0.6

δνm = (28.00±1.00) kHz

1.2

δνm = (18.53±0.61) kHz

0.8 0.3

0.05

0.4

(b)

(c)

-100

0

100

δν (kHz)

-100

1.2

0

δν (kHz)

-100

100

0

δν (kHz)

100

Experiment Lorentz fit

(a) Normalized intensity

(d) 0

0

0

1 0.8 0.6 0.4 0.2 0 -0.2 11

11.5

12

12.5

13

13.5

14

14.5

15

15.5

16

16.5

17

Beat note (GHz)

Fig. 3. Transmission spectrum of the high finesse cavity filled with 20.4 Torrs of O2 near the absorption line P7 Q6 from the O2 B-band. Three individual cavity modes are enlarged in the insets with fitted Lorentzian profiles, from which the mode widths δ νm are determined.

cavity mode on the absorption coefficient at a mode central frequency νm an absorption spectrum α (ν ) can be calculated. Exact mode half width can be found from solution of equation IT (ν ) = IT (νm )/2 with IT given by Eq. (5) which depends on R, νFSR (ν ), and α (ν )L. Alternatively, some approximations can be done to obtain simple analytical formulas. Equation (5) can further reduced assuming constant νFSR and constant α within the range of the cavity mode c ; α ≈ α (ν m ) . νFSR ≈ (8) 2Ln We should note here that the width of the investigated O2 absorption line is over 900 MHz, while the cavity mode widths do not exceed 60 kHz. We have found from simulations that relative difference between the exact (5) and approximated (8) mode shape is up to 1.2 × 10−5 . This approximation allows to rewrite Eq. (5) with an effective cavity mirrors reflectivity Reff = Re−α L which take into account absorption between the mirrors and obtain the Airy equation: IT = I0

(1 − Reff )2 . (1 − Reff )2 + 4Reff sin2 ( 2ννFSR )

(9)

In case of high finesse cavity, a single cavity mode profile described by Eq. (9) is approaching Lorentzian profile. Its full width at half maximum δ νm can be written in the following form

δ νm =

1 √ 2π nτeff Reff

(10)

#195746 - $15.00 USD Received 13 Aug 2013; revised 4 Oct 2013; accepted 24 Oct 2013; published 25 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029744 | OPTICS EXPRESS 29750

(10-6 cm-1)

0.0012

Δ(δν)/δν

α

0.0008

8

ν0 ν1 ν2

N=1

6 4

Eqs. (10), (11) Eqs. (13), (14)

ν0

ν9 ν20

2 0 -6

-4

-2

0

δν (GHz)

2

4

6

ν1

0.0004

ν2 ν20 ν9

0

0

1

2

3

4

5

N Fig. 4. Relative differences σ δ νm /δ νm between mode widths δ νm calculated from Eq. (5) and from approximated formulas: (10) and (11) - solid line, and (13) and (14) - dashed line. Horizontal axis corresponds to increasing absorption α , which for N = 1 is equal to our experimental conditions shown in the inset plot.

where

τeff =

L c(1 − Reff )

(11)

is an effective cavity mode ring-down time. The effective intensity loss per unit length of the cavity defined using τeff or Reff can also be given in terms of mode width δ νm  [2 + (πδνm /νFSR )2 ]2 − 4 − (πδνm /νFSR )2 1 1 − Reff = (12) = cτeff L 2L using Eqs. (10) and (11). Moreover, in case of high finesse cavities with weakly absorbing medium where R approaches unity and α L is close to zero Reff is well described by approximated relation: Reff ≈ R− α L. Similarly, two other quantities δ νm , τeff can be written as follows: 1 , 2π nτeff

(13)

L . c(1 − R + α L)

(14)

δ νm ≈ τeff ≈

These relations lead to expression for effective loss per unit length in the form commonly used in CRDS 1 ≈ αbg + α (15) cτeff where αbg = (1 − R)/L describes empty cavity losses. In Fig. 4 relative differences σ δ νm /δ νm between mode widths δ νm calculated from exact Eq. (5) and from approximated formulas: (10) with τeff given by (11) - solid line, and with Eqs. (13) and (14) - dashed line. The horizontal axis corresponds to increasing absorption coefficient α (νm ) in units of the absorption of our experimental line shape at given mode number indicated in the inset plot of α (ν ). The relative systematic errors σ δ νm /δ νm caused by both approximations decrease with frequency detuning |νm − ν0 | from the absorption line center ν0 . Moreover #195746 - $15.00 USD Received 13 Aug 2013; revised 4 Oct 2013; accepted 24 Oct 2013; published 25 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029744 | OPTICS EXPRESS 29751

the sign of these systematic errors can be different for different |νm − ν0 |. It is worth to note that approximation (13) is related to R and therefore it introduces systematic error even for the case of no absorption α = 0. In conclusion, to obtain a cavity mode-width spectrum with relative accuracy of a few thousands, as can be typically achieved in the FS-CRDS [7] experiments, both the absorption α (ν ) and the dispersion νFSR (ν ) cannot be treated as constant within the cavity mode profile. Experimental spectrum of the P7 Q6 line from the O2 B-band is presented in Fig. 5. A sum of O2 absorption coefficient α (ν ) and cavity losses α0 (ν ) was calculated from the cavity TEM00 mode widths using Eq. (12). It should be noted that in case of presented spectrum any systematic errors introduced by used equations are negligible comparing to the random noise. The measured line profile was fitted with the Voigt profile with Gaussian width constrained to the Doppler width of measured line γD = 960.33 MHz at temperature 304 K of the cavity. For comparison the Voigt profile of this line is also simulated with use of the HITRAN [25] data. Both the line width and line area of our experimental spectrum agree with HITRAN simulation to within their standard uncertainties. Presented spectrum was measured with the frequency step equal to νFSR . Density of spectrum points can be increased by changing the cavity length in a controlled way, as was earlier demonstrated in frequency-stabilized CRDS by Hodges et al. [4]. 14

P7 Q6 O2 line

12

α + α0 (10-6 cm-1)

Experiment Voigt fit HITRAN simulation

wavenumber: 14504.79133 cm-1 pressure: 20.4 Torr temperature: 304 K

10 8 6 4 2 -5

-4

-3

-2

-1

0

δν (GHz)

1

2

3

4

5

Fig. 5. Absorption spectrum of the P7 Q6 line from the O2 B-band determined from the cavity mode widths measurement. Solid line corresponds to the Voigt profile fit to the experimental data. Dashed line is a simulated Voigt profile using the HITRAN data.

6.

Discussion

Simple and well known idea of cavity mode width spectroscopy (CMWS) seems to be another very promising technique for obtaining spectra of weak transitions. For CMWS as well as other cavity enhanced techniques it is convenient to introduce relative precision of absorption coefficient σ (α )/α assuming an ideal laser source with constant power and infinitely narrow laser linewidth, perfectly stable cavity and taking into account only statistical noise of a detection system. In real conditions only CRDS and CMWS techniques are considered as methods insensitive to laser power fluctuations and any background radiation. From Eqs. (13), (14) we developed simple models of relative precision of absorption coefficient for CRDS and CMWS given by the formulas (σ (α )/α )CRDS = (σ (τeff )/τeff )τbg /(τbg − τeff ),

#195746 - $15.00 USD Received 13 Aug 2013; revised 4 Oct 2013; accepted 24 Oct 2013; published 25 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029744 | OPTICS EXPRESS 29752

(σ (α )/α )CMWS = (σ (δ νm )/δ νm )δ νm /(δ νm − δ νbg ), respectively. Here τbg and δ νbg correspond to effective decay time constant and cavity mode width for an empty cavity. Statistical uncertainties σ (τeff ), σ (δ νm ) were expressed as:

σ (τeff ) =

σ (τbg ) I0 1 − τmin /τbg τeff τbg IT (νm ) 1 − τmin /τeff

(16)

and 

σ (δ ν m ) =

σ (δ ν ) I0 δ νm δ νbg IT (νm )

2

+ (σfreq )2 ,

(17)

respectively. In Eq. (16) τmin describes the shortest decay time constant that can be measured in CRDS without systematic error. Uncertainty of mode width determination, not including frequency jitter, and statistical noise of frequency axis in CMWS are denoted in Eq. (17) as σ (δ ν ) and σfreq , respectively. Relative uncertainty σ (τbg )/τbg = 0.002 corresponds to typical value of decay time constant uncertainty obtained by us in CRDS experiments [26, 27]. The similar value σ (δ ν )/δ νbg = 0.002 is also reasonable in case of CMWS where it describes contribution of amplitude noise of the laser source and detector on the precision of mode width determination. In [11] we have shown that we are able to measure the mode width of the well acoustic-isolated cavity with precision better than 0.1 kHz. IT (νm)

100

CRDS τmin = 0 us

1

CRDS τmin = 0.5 us CRDS τmin = 2 us CMWS σfreq = 5 kHz

0.1

CMWS σfreq = 0.05 kHz

IT (νm)

Δα/α

10

1 0.01 0.1

0.01

0.001 0

2

4

6

8

10

12

14

α/αbg Fig. 6. Dependence of relative precision of absorption determination σ (α )/α on absorption α for CRDS and CMWS techniques - solid lines. Dependence of transmission signal IT (νm ) for the cavity mode center νm on absorption - dashed line. See text for details.

Simulated relative uncertainties (σ (α )/α )CRDS and (σ (α )/α )CMWS dependent on α were shown and compared in Fig. 6. As a reference for our comparison we can use the case in which τeff measurement in CRDS method is not limited by τmin which is set to zero. It is marked as the pink curve (mostly covered by green curve) in Fig. 6. It should be emphasized that exactly the same results are obtained for CMWS method with no noise on frequency axis, σfreq = 0. The dashed line in Fig. 6 demonstrates decrease of transmission signal IT (νm ) for the cavity mode center with increase of absorption. Now we will discuss the influence of τmin and σfreq on relative uncertainty of the absorption coefficient σ (α )/α . To avoid any systematic errors the beginnings of ring-down decays are usually trimmed by the value of τmin . Blue and violet curves in Fig. 6 shows (σ (α )/α )CRDS for τmin equal to 0.5 us and 2 us, respectively. As seen the precision of the absorption coefficient α determination decreases with increasing #195746 - $15.00 USD Received 13 Aug 2013; revised 4 Oct 2013; accepted 24 Oct 2013; published 25 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029744 | OPTICS EXPRESS 29753

α the faster the greater τmin is. It should be noted that in our typical experimental conditions τmin = 2 us limits the ring-down duration [21]. In this case maximal absorption coefficients that we can measure are only two times bigger than αbg . However, bigger range of absorption can be covered by CMWS technique. While in CRDS both the amplitude and the decay time of the ring-down signal decrease with increasing α in the CMWS decrease of signal amplitude is partially compensated by larger mode width δ νm . The crucial thing in CMWS is to get a very stable frequency axis. Red and green curves in Fig. 6 correspond to (σ (α )/α )CMWS with frequency axis noise σfreq equal to 5 kHz and 0.05 kHz, respectively. As seen reduction of frequency axis noise by two orders of magnitude in CMWS, what is still possible to achieve, improves precision of α determination to value similar to CRDS for α < 2αbg , moreover for α > 2αbg precision of CMWS becomes even higher than in case of real CRDS system. It is clearly shown that CMWS in comparison to CRDS significantly increases dynamic range of absorption measurement. The common way to minimize σfreq is to use stable, ultranarrow laser systems and acoustic-isolated cavities [11]. Moreover, if the master laser will be PDH-locked to the probe cavity and the phase-locked slave laser will be used to measure cavity mode widths any acoustic noise of the cavity length would be transferred to the slave laser and would not contribute to frequency noise σfreq in the measurement of the narrow spectrum of the cavity mode. Such a system could be used complementary to CRDS to measure absorption spectrum at conditions in which absorption is too high for precise ring-down time determination. It is worth to mention that tuning of the laser sideband generated by the fast EOM [10], as was demonstrated by Long et al. [6] also provides a convenient scheme of switching between CRDS and CMWS without any serious modification of the experimental setup. In the range of absorption coefficients where CMWS has lower relative uncertainty σ (α )/α than CRDS the transmitted signal is still measurable by high-gain detectors. The CMWS does not require fast light detectors and therefore detectors with very high gain can be used and if necessary their gain can be varied during spectrum measurement to further increase dynamic range of absorption measurement. On the other hand increase of the detector gain also decreases its bandwidth what is crucial for accurate ring-down time measurements in CRDS [21]. Moreover, a natural limitation for precise cavity ring-down time measurement is that it must be significantly longer than the laser beam switching off time [28]. Finally, accuracy of absorption measurement in both CRDS and CMWS is also limited by frequency dependence of absorption coefficient in the range of the cavity mode. Acknowledgments The authors would like to thank Dr. Joseph T. Hodges for sending us paper [6] before its publication. We also thank Dr. Piotr Masłowski and Dr. J´erˆome Lodewyck for valuable discussions and help in laser linewidth narrowing. The research was supported by the Polish MNiSW project No. N N202 1489 33, the Polish National Science Centre, project No. DEC2011/01/B/ST2/00491, project No. NCN 2012/07/B/ST2/00235 and the Foundation for Polish Science TEAM Project co-financed by the EU European Regional Development Fund and is part of the program of the National Laboratory FAMO in Toru´n, Poland. A. Cygan is partially supported by the Foundation for Polish Science START Project.

#195746 - $15.00 USD Received 13 Aug 2013; revised 4 Oct 2013; accepted 24 Oct 2013; published 25 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029744 | OPTICS EXPRESS 29754

Cavity mode-width spectroscopy with widely tunable ultra narrow laser.

We explore a cavity-enhanced spectroscopic technique based on determination of the absorbtion coefficient from direct measurement of spectral width of...
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