Coherent anti-Stokes Raman scattering in a high-finesse microcavity Benjamin Petrak,1 Nicholas Djeu,2 and Andreas Muller1,∗ 1 Dept.
of Physics, University of South Florida, Tampa, FL 33620, USA 2 MicroMaterials, Inc., Tampa, FL 33637, USA ∗ [email protected]
Abstract: We report the measurement of degenerate coherent anti-Stokes Raman scattering (CARS) in a high-finesse optical microcavity, with an atmospheric gas as the nonlinear medium. Unlike the well-known bulk process in which index-dispersion compensation is required for phasematching, efficient microcavity CARS involves a resonant coupling at the Stokes, pump, and anti-Stokes frequencies which can be described using coupled-mode analysis. We show how the interaction is thereby dramatically enhanced in a microscopic sample volume and illustrate the technique for the measurement of CO2 in air. © 2014 Optical Society of America OCIS codes: (290.5860) Raman Scattering; (350.3950) Micro-optics; (270.1670) Coherent optical effects.
References and links 1. W. M. Tolles, J. W. Nibler, J. R. McDonald, and A. B. Harvey, “A Review of the Theory and Application of Coherent Anti-Stokes Raman Spectroscopy (CARS),” Appl. Spect. 4, 253-271 (1977). 2. F. El-Diasty, “Coherent anti-Stokes Raman scattering: Spectroscopy and microscopy,” Vib. Spectrosc. 55, 1-37 (2009). 3. S. Roy, J. R. Gord, A. K. Patnaik, “Recent advances in coherent anti-Stokes Raman scattering spectroscopy: Fundamental developments and applications in reacting flows,” Prog. Energy Combust. Sci. 36, 280-306 (2010). 4. B. Li, P. Borri, and W. Langbein, “Dual/differential coherent anti-Stokes Raman scattering module for multiphoton microscopes with a femtosecond Ti:sapphire oscillator,” J. of Biomed. Opt. 18, 066004 (2013). 5. D. Gachet, F. Billard, and H. Rigneault, “Coherent anti-Stokes Raman scattering in a microcavity,” Opt. Lett. 34, 1789-1791 (2009). 6. M. Marrocco, “Coherent anti-stokes Raman scattering microscopy in the presence of electromagnetic confinement,” Laser Physics 17, 935-941 (2007). 7. M. Marrocco and E. Nichelatti, “Coherent anti-Stokes Raman scattering microscopy within a microcavity with parallel mirrors,” J. Raman Spect. 40, 732-740 (2009). 8. F. Billard, D. Gachet, and H. Rigneault, “Coherent anti-Stokes Raman scattering in a Fabry-Perot cavity: A theoretical study,” J. Opt. Soc. Am. B 26, 1295-1309 (2009). 9. K. J. Vahala, “Optical microcavities,” Nature 424, 839-846 (2003). 10. T. J. Kippenberg, S. M. Spillane, D. K. Armani, and K. J. Vahala, “Ultralow-threshold microcavity Raman laser on a microelectronic chip,” Opt. Lett. 29, 1224-1226 (2004). 11. T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third harmonic generation,” Nat. Phys. 3, 430-435 (2007). 12. J. U. F¨urst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally Phase-Matched Second-Harmonic Generation in a Whispering-Gallery-Mode Resonator,” Phys. Rev. Lett. 104, 153901 (2010). 13. S. Zaitsu, H. Izaki, and T. Imasaka, “Phase-Matched Raman-Resonant Four-Wave Mixing in a DispersionCompensated High-Finesse Optical Cavity,” Phys. Rev. Lett. 100, 073901 (2008). 14. S. Zaitsu and T. Imasaka, “Intracavity phase-matched coherent anti-Stokes Raman spectroscopy for trace gas detection,” Anal Sci. 30, 75-79 (2014).
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 21999
15. B. Petrak, N. Djeu, and A. Muller, “Purcell-enhanced Raman scattering from atmospheric gases in a high-finesse microcavity,” Phys. Rev. A 89, 023811 (2014). 16. D. C. Hanna, M. A. Yuratich, and D. Cotter, “Nonlinear Optics of Free Atoms and Molecules,” (Springer, New York, 1979). 17. H. Kogelnik and T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550-1567 (1966). 18. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272-276 (2007). 19. D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. H¨ansch, and J. Reichel, “Laser micro-fabrication of concave, low-roughness features in silica,” New J. Phys. 12, 065038 (2010). 20. A. Muller, E. B. Flagg, J. Lawall, and G. S. Solomon, “Ultrahigh-finesse, low-mode-volume FabryPerot microcavity,” Opt. Lett. 35, 2293-2295 (2010). 21. B. Petrak, K. Konthasinghe, S. Perez, and A. Muller, “Feedback-controlled laser fabrication of micromirror substrates,” Rev. Sci. Instrum. 82, 123112 (2011). 22. G. Rempe, R. J. Thompson, H. J. Kimble, and R. Lalezari, “Measurement of ultralow losses in an optical interferometer,” Opt. Lett. 17, 363-365 (1992). 23. B. P. Stoicheff, “High Resolution Raman Spectroscopy of Gases: XI. Spectra of CS2 and CO2 ,” Can. J. Phys. 36, 218-230 (1958). 24. W. R. Fenner, H. A. Hyatt, J. M. Kellam, and S. P. S. Porto, “Raman cross section of some simple gases,” J. Opt. Soc. Am. 63, 73-77 (1973). 25. P. A. Roos, L. S. Meng, S. K. Murphy, and J. L. Carlsten, “Approaching quantum-limited cw anti-Stokes conversion through cavity-enhanced Raman-resonant four-wave mixing,” J. Opt. Soc. Am. B 21, 357-363 (2004). 26. X. Yang and C. W. Wong, “Coupled-mode theory for stimulated Raman scattering in high-Q/Vm silicon photonic band gap defect cavity,” Opt. Express 15, 4763-4780 (2007). 27. B. Lavorel, G. Millot, R. Saintloup, H. Berger, L. Bonamy, J. Bonamy, and D. Robert, “Study of collisional effects on band shapes of the ν1 /2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy. II. Simultaneous line mixing and Dicke narrowing in the ν1 band,” J. Chem. Phys. 93, 2185-2191 (1990). 28. C. J. S. M. Simpson, T. R. D. Chandler, and A. C. Strawson, “Vibrational Relaxation in CO2 and CO2 -Ar Mixtures Studied Using a Shock Tube and a Laser-Schlieren Technique,” J. Chem. Phys. 51, 2214-2219 (1969).
1.
Introduction
Degenerate coherent anti-Stokes Raman scattering (CARS) takes place in a medium possessing (3) a third-order optical nonlinearity, χCARS , when two waves of frequency ω p (the pump waves), and a third wave of frequency ωs (the Stokes wave) interact to generate a fourth wave of frequency ωas = 2ω p − ωs (the anti-Stokes wave). CARS has long been a powerful tool for the measurement of vibrational spectra of solids, liquids and gases with a larger signal-to-noise ratio than that provided by ordinary spontaneous Raman scattering [1–3]. Common applications are in combustion diagnostics and selective microscopy, and ultrafast CARS continues to offer ever-improving imaging capabilities in the life sciences [4]. Recently, there has been interest in taking advantage of optical resonators to localize and/or enhance the CARS process for improved microscopy and sensing. For example, Gachet et al. investigated microcavity CARS enhancement in which the anti-Stokes emission was resonant with a low-finesse planar optical microcavity [5]. Such parallel mirror arrangements are desirable for microscopy but they impose severe limitations on CARS microcavity enhancements even on purely theoretical grounds [6–8]. On the other hand, rapid development in micro-fabrication technology has made available truly three-dimensionally-confining optical resonators of high-finesse and small mode-volume [9]. In fact, in whispering gallery mode optical resonators, greatly enhanced nonlinear interactions including stimulated Raman scattering [10], four-wave-mixing, and high-harmonic generation [11, 12] have all been shown in solid media. Additionally, significant CARS enhancements have been demonstrated recently for gases using a macroscopic optical resonator with dispersion-compensated mirrors [13, 14]. We demonstrate here the enhancement of CARS in an open, microscopic high-finesse cavity for the purpose of atmospheric gas measurements. Building on recent experiments of Purcellenhanced spontaneous Raman scattering [15], we show that it is possible to tune the cavity such as to provide high-Q resonances at ω p and ωs together with a low-Q resonance at ωas . #210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22000
This yields a sizable increase in CARS intensity compared to free space while at the same time involving a minute amount of sample gas. We find good agreement between the measured CARS emission rate and the rate predicted theoretically by a coupled-mode analysis. 2.
Theoretical background
The degenerate CARS process is illustrated in the energy diagram of Fig. 1(a), in which the pump and Stokes frequencies are chosen such that their difference coincides with a rotational or vibrational transition frequency, ων , of a molecular sample gas. For degenerate CARS to occur efficiently in free space, phase matching is needed, i.e. we must have kas = 2k p − ks for the wavevectors of the anti-Stokes, pump, and Stokes waves, assumed to be planar. In contrast, for a CARS process which takes place in a resonator, the confined fields E j (r,t) (where j = p, s, as) are standing waves of the form s h¯ ω j a j (t)e−iω j t u j (r), (1) E j (r,t) = ε0V j where a j (t) are slowly time-varying amplitudes, u j (r) are spatial mode functions determined by the resonator’s type and geometry, and V j is a normalization factor given by Vj =
1 2
Z
u∗j u j d 3 r
(2)
such that |a j |2 represents the energy, in units of h¯ ω j , stored in each mode, i.e., Z
ε0 ∗ E E j d 3 r = n j h¯ ω j , 2 j
(3)
with n j = |a j |2 . For simplicity we assume unity refractive index for all waves. From Maxwell’s equations we have, for each mode, (3)
∇×∇×Ej +
1 ∂ 2E j 1 ∂ 2 PNL (ω j ) = − , c2 ∂t 2 ε0 c 2 ∂t 2
(4)
or, (3)
−∇2 E j + (a)
1 ∂ 2 PNL (ω j ) 1 ∂ 2E j = − , c2 ∂t 2 ε0 c2 ∂t 2
(5)
(b) 105
ωp
ωs
ωp
Cavity nesse
104
ωas
1000
λas λp λs
100 10 1
ων
0.1 0.7
0.8
0.9
1.0
1.1
Wavelength (µm)
Fig. 1. (a) Degenerate coherent anti-Stokes Raman scattering energy diagram. (b) Cavity finesse as a function of wavelength. The blue solid, dashed, and dot-dashed lines represent the anti-Stokes, pump, and Stokes wavelengths, respectively.
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22001
where it was assumed that all fields are polarized along the same axis. For degenerate CARS (3)
3χ
(3)
2 ∗ we have PNL (ωas ) = ε0 CARS 4 E p Es [16]. Inserting Eq. (1) with j = as into Eq. (5) then yields s s h¯ ωas h¯ ωas ∂2 − aas (t)e−iωas t c2 ∇2 uas (r) + uas (r) 2 aas (t)e−iωas t = ε0Vas ε0Vas ∂t (6) s (3) 2 3χCARS h¯ ω p h¯ ωs ∂ − u p (r)2 us (r)∗ 2 a p (t)2 as (t)∗ e−i(2ω p −ωs )t . 4 ε0Vp ε0Vs ∂t
Further using the fact that the mode functions u j (r) satisfy the scalar wave equation ∇2 u j + ω 2j u c2 j
= 0 [17], and making the slowly-varying envelope approximation we get s
(3)
3χ h¯ ω p h¯ ωas daas (t) uas (r) = i CARS ε0Vas dt 8 ε0Vp
s
h¯ ωs u p (r)2 us (r)∗ ωas a p (t)2 as (t)∗ , ε0Vs
(7)
where it was assumed that ωas = 2ω p − ωs . Finally, multiplying each side by 21 uas (r)∗ and integrating over all space, √ √ (3) 3χCARS h¯ ω p ωs ωas daas (t) =i a p (t)2 as (t)∗ , dt 4 ε0Veff
(8)
where the effective mode volume is defined by u∗as u2p u∗s d 3 r 1 . = R R R Veff u∗p u p d 3 r ( u∗s us d 3 r)1/2 ( u∗as uas d 3 r)1/2 R
(9)
For gaussian beams confined by a microcavity of length L constructed with curved mirrors (radius of curvature R), the TEM00 mode functions take the approximate form [17], u j (r) =
w j0 −r2 /w j (z)2 e sin(k jz z + φ j (z)) w j (z)
(10)
where the wavenumber k jz must assume discrete values. The psquare of the beam spot size is 2 2 2 2 2 w j (z) = w j0 (1 + z /zR ) and its waist is given by w j0 = λ j (R − L/2)L/2/π. The standing waves must satisfy the condition k jz L − ∆ϕ + ϕ j = m j π
(11)
with integer m j , where ϕ j is the (frequency-dependent) phase shift experienced by the wave at the mirror and ∆ϕ is the Gouy phase shift given by ∆ϕ = tan−1 (z2 /zR ) − tan−1 (z1 /zR ).
(12) p Here zR = L(R − L/2)/2 is the Rayleigh range defined by the microcavity, and z2 > 0 and z1 < 0 are the positions of the two mirrrors (so that z2 − z1 = L) [17]. From Eq. (8) and Eq. (9) one then sees that phase-matching for microcavity CARS is replaced by a maximization requirement of an overlap integral involving the m j ’s, analogous to the overlap integrals giving rise to selection rules in electronic transitions [12]. If we ignored the mirror phase shifts in Eq. (10) the relevant “selection rules” would be mas = 2m p − ms .
(13)
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22002
L~20 µm
Trans. Detector
R~40 µm
ωp ωs
M1
Gas in PZT
ωas
Single photon detector
Lens
PZT scan
Timing Electronics
Cavity region
Optical fiber
Fig. 2. Experimental setup for the measurement of microcavity coherent anti-Stokes Raman scattering. The cavity assembly (right) consisted of the cavity region itself (shaded area), a flexure mount for large range tuning using piezoelectric transducers (PZTs), a mode matching lens, and a temperature-controlled base. The planar substrate holding one mirror (M1) as well as the end of the optical fiber at the tip of which the other mirror was fabricated can also be seen in the photograph.
In addition, it is required that the difference between the anti-Stokes mode frequency and the pump mode frequency, as well as the difference between the pump mode frequency and the Stokes mode frequency be equal to the Raman shift, i.e. we must have ωas − ω p = ων
(14)
ω p − ωs = ων ,
(15)
and which are conditions strongly dependent upon the phase shifts experienced by the waves according to Eq. (11). Thus, a cavity configuration involving simultaneously three different resonances, i.e. a triple-resonance condition, must be met to achieve maximum CARS efficiency. While in general it is not feasible to meet this condition for three sharply peaked cavity resonances, it is always possible to be at the peaks of two sharp cavity resonances and near the peak of a third broad one. 3.
Experiments
The experimental implementation of degenerate microcavity CARS thus necessitates meeting the “selection rules” which arise from the fields of Eq. (1), to within the tolerance associated with the finite linewidth of the microcavity resonances involved. In recent work, we have demonstrated Purcell enhanced spontaneous Raman scattering in a 10 µm long microcavity constructed with mirrors of radius of curvature R ≈ 40 µm [15]. This was possible by implementing a double-resonance condition wherein Eq. (11) is satisfied for j = p and j = s and simultaneously ω p − ωs = ων holds to within the Raman linewidth. Building on these results, we use here a similar microcavity, the high finesse of which is derived from a laser microfabrication process [18–21] compatible with ultralow-loss dielectric reflective coatings [22]. In Fig. 1(b) we illustrate the spectral configuration chosen for the present CARS microcavity measurement. The choice of frequencies is constrained by the finite bandwidth over which the quarter-wave mirror stacks have high reflectivity, relative to the molecular rotational or vibrational frequency ων of the molecule being probed. We opted for a compromise so that the cavity finesse at both pump and Stokes frequencies is ≈10 000, whereas it is of order unity ( ≈3) at the anti-Stokes frequency. Without loss of generality, we focus here exclusively on the
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22003
Q-branch line in CO2 terminating in the (1 0 0) level with a frequency of ων /2πc = 1388.17 cm−1 [23], and a differential Raman scattering cross section dσ /dΩ = 4.7 × 10−32 cm2 /sr [24]. Our experimental setup is depicted in Fig. 2. A microcavity of length L ≈ 20 µm was constructed with one micromirror at the end of a single mode fiber tip and a second micromirror on a planar fused silica substrate. Both mirrors had a radius of curvature of R ≈ 40 µm. Two independently tunable continuous-wave diode lasers with a linewidth of ∼1 MHz were used for the pump and Stokes sources. The lasers were continuously monitored with a wavemeter and a scanning Fabry-Perot interferometer with a resolution of 20 MHz. The pump and Stokes laser beams were matched to TEM00 modes of the cavity through the mirror on the planar substrate using a single aspheric lens. Accounting for imperfect mode matching we estimate 0.7 mW and 3 mW were coupled into the cavity for the pump and Stokes laser beams, respectively. The cavity was hermetically sealed and gas delivered directly to the cavity region via external tubing. Piezoelectric actuators were used to scan the cavity length while the anti-Stokes wave was coupled out with a dichroic filter and detected with a single photon counting module. Instead of a permanent Pound-Drever-Hall lock as in [15], we histogrammed the time-tagged detector events while both the frequency of one laser and the cavity length were swept. In order to ensure all length and frequency combinations were realized, we scanned the laser frequency and cavity length at speeds separated by several orders of magnitude. The laser was scanned over a spectral range of 2.4 GHz in a period of 11 seconds, while the cavity length was swept by 1.42 nm in a period of 3.2 ms. In Fig. 3, we illustrate the resulting data for the case when the test gas is CO2 at atmospheric pressure. In this measurement the pump laser frequency was scanned while the Stokes laser frequency was stepped through. Each trace in the plot of Fig. 3(a) has been extracted from the raw recorded data maps of the emission rate as a function of both the cavity length and the Stokes laser frequency. The emission rate has been corrected for the known propagation and filter losses (50%) and for the detector quantum efficiency (65%). Figure 3(b) shows a closeup view of the trace from Fig. 3(a) at which the peak CARS emission rate of γCARS ≈ 4 × 106 photons/s is obtained. a) x106
b) x106
CARS emission rate (photons/s)
4.0 5.0 3.0
4.0 3.0
Sto kes
fre 304 370 qu en cy 304 360 (GH z) 304 350 304 340
2.0 1.0 0 345 960
345 970
345 980
Pump frequency (GHz)
345 990
2.0
FWHM ≈203 MHz
1.0
0
345 973.0 345 973.7 Pump frequency (GHz)
Fig. 3. (a) Measured CARS spectra. (b) Magnified view of the trace in (a) for which the peak emission occurs.
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22004
4.
Theoretical microcavity CARS emission rate
In order to quantitatively understand the features of degenerate CARS as it manifests in a microcavity, we derive an expression for the CARS emission rate starting from Eq. (8) [25, 26] √ √ (3) 3χCARS h¯ ω p ωs ωas daas (t) aas (t) a p (t)2 as (t)∗ , = − (as) + i dt 4 ε0Veff 2τc
(16)
where we have added the cavity decay term due to mirror transmission. In steady-state one then obtains for the number of anti-Stokes photons in the mode 2
nas = |aas |
(3) 2 (as) 2 |3χCARS | = (τc )
4
h¯ 2 ω p2 ωs ωas 2 n p ns , 2 ε02Veff
(17) (as)
and thus the CARS emission rate out of one end of the cavity, γCARS = nas /2τc
, is
(3)
|3χCARS |2 h¯ 2 ω p2 ωs ωas 2 γCARS = n p ns , 2 (as) ε02Veff 8∆ωc
(18) (as)
(as)
with the cavity linewidth at the anti-Stokes frequency given by ∆ωc = 1/τc . In order to (as) account for the overlap between the cavity mode and the Raman lineshape we replace 1/∆ωc with a normalized Lorentzian in Eq. (18). Further approximating a Lorentzian profile with full (3) width ∆ωR for the term |3χCARS |2 , and integrating over frequency, one obtains (3)
γCARS =
|3χCARS,0 |2
h¯ 2 ω p2 ωs ωas 2 n p ns , 2 2 (as) 8(∆ωc + ∆ωR ) ε0 Veff
(19)
(3)
The magnitude of the resonant third-order nonlinear susceptibility, χCARS,0 , can be related to the familiar spontaneous Raman differential scattering cross section, dσ /dΩ, through the relation [16] 8π 2 ε0 Nc4 dσ (3) |3χCARS,0 | = (20) h¯ ωs3 ω p ∆ωR dΩ where N is the number density of molecules in the lower vibrational (in the case of rotationally collapsed transitions such as the ν1 Q-branch of CO2 being reported here) or rotational state. Eq. (19) then evaluates to 1 c4 λs5 N 2 γCARS = 2 2 (2π)3 λas Veff
dσ dΩ
2
n2p ns (as)
(∆νc
+ ∆νR )∆νR2
.
(21)
The microcavity CARS emission rate is thus expected to be proportional to the square of the molecular density, proportional to the square of the pump power, and proportional to the Stokes power. The microcavity parameters enter in three different ways. Firstly, the microcavity CARS emission rate is inversely proportional to the square of the effective mode volume. Secondly, it is inversely proportional to the sum of the cavity linewidth of the anti-Stokes mode and the Raman linewidth. Thirdly, the cavity enhances the pump and Stokes photon number by recirculation at a resonant frequency. Specifically, in terms of the incident power, Pp,s , the pump/Stokes photon number is given by (p,s) n p,s = 2Pp,s τc /¯hω p,s (22) #210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22005
(b)
x10 6.0
Raman FWHM (MHz)
CARS emission rate (photons/s)
(a)
5.0 4.0 3.0 2.0 1.0 0
20
40 60 80 CO concentration (%)
200 150 100 50 0
100
20
40 80 60 CO concentration (%)
100
Fig. 4. (a) Peak CARS emission rate versus CO2 concentration. (b) Raman linewidth versus CO2 concentration. (p,s)
(p,s)
where τc = 1/∆ωc is the cavity photon lifetime at the pump/Stokes frequency. For the experiment of Fig. 3, the coupled input powers were Pp ≈ 0.7 mW, and Ps ≈ 3 mW. For a cavity finesse of F (p,s) ≈ 10 000, and a cavity length of L ≈ 20 µm, we then obtain n p ≈106 and ns ≈ 6 × 106 assuming a pump wavelength λ p ≈ 864 nm. The mode volume that these photons occupy can be approximated using Eq. (9) as Veff ≈ λ p
p 1 1 L2 /4 2 p ) . λs λas ( + + λ p λs λas tan−1 (L/ 2(R − L/2)L)
(23)
For L=20 µm and R=40 µm, Veff ≈ 660 µm3 . Finally, then, the theoretically expected CARS emission rates corresponding to our experimental rates of Fig. 3 are obtained by evaluating Eq. (21) wherein most parameters are known independently from experiment. Crucially, the Raman linewidth is obtained directly from Fig. 3(b) for 100 % CO2 at atmospheric pressure as ∆νR =200 MHz, in agreement with the literature value [27]. The mirror reflectivity dictates the cavity finesse at the relevant frequencies [Fig. (as) 1(b)]. In particular, we can estimate that ∆νc ≈ 15 THz (F (as) = 3). As a result, for 100% CO2 at atmospheric pressure, we compute a theoretical CARS emission rate of γCARS ≈ 1 × 107 photons/s, which is of the same order of magnitude as the measured rate of Fig. 3. We further investigated the accuracy of Eq. (21) through its scaling with specific experimental parameters. In Fig. 4(a) we present the measured CARS emission rate as a function of CO2
gCARS ä ΔnR2 (MHz2/s)
2.0 x1011 ∞N2.1
3.0 x1010 5.0 x109 1.0 x109 10
20
40
CO2 concentration (%)
100
CARS emission rate (photons/s)
(b)
(a)
x10 5.0
∞Pp ∞Ps
2.0 1.0 0.5 0.3
0.5
1.0 2.0 Power (mW)
5.0
Fig. 5. (a) Logarithmic representation of the quantity γCARS ∆νR2 as a function of concentration, exhibiting the quadratic growth described by Eq. (21). (b) Dependence of the microcavity CARS emission rate on pump and Stokes laser power.
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22006
concentration. The expected quadratic dependence [N 2 factor in Eq. (21)] is not readily seen. Instead, an approximately linear growth is observed, which can however be explained by the fact that the Raman linewidth is also varying with CO2 concentration. Indeed, as shown in the data of Fig. 4(b), the Raman linewidth is decreasing with concentration, down to only 50 MHz FWHM for 10% CO2 . The linear dependence of the Raman linewidth on CO2 concentration can be understood on the grounds that at atmospheric pressure the Q branch of the ν1 Raman transition is completely collapsed rotationally and that the collapsing effect is stronger for air than CO2 itself [27]. To our knowledge, this is the first time the collisional narrowing effect of air on the 1 388 cm−1 Raman transition in CO2 has been determined. This fortuitous decrease of the linewidth with CO2 concentration partially offsets the decrease of the CARS signal with concentration. By plotting the quantity γCARS ∆νR2 as a function of concentration, as shown in Fig. 5(a), the expected quadratic dependence on concentration is retrieved. Finally, we also verify the dependence of Eq. (21) on pump and Stokes laser power. As seen in Fig. 5(b) the observed power dependence is close to quadratic for the pump laser power as anticipated. The observed power dependence of the emission rate on Stokes power exhibits a sub-linear dependence. This can be explained by ground state depletion, since the estimated pump rate into the (1 0 0) vibrational level is 3(10)4 /s at maximum powers, and the vibrational relaxation rate is 1.3(10)5 /s [28]. 5.
Comparison to CARS in free space
An interesting question is to what extent, besides miniaturization, the generation of CARS in a microcavity is beneficial compared to its generation in free space or within a conventional macroscopic resonator. In free space and under plane-wave approximation, the intensity of anti-Stokes radiation generated by the CARS process is usually calculated as [16] Ias =
2 ωas (3) 2 2 |P | L sinc2 (∆kL/2), 8ε0 c CARS 3χ
(3)
(24)
(3)
2 ∗ where the (complex) nonlinear polarization is PCARS = ε0 CARS 4 E p Es , the wavevector mismatch is ∆k = 2k p − ks − kas , and the interaction length is L. Thus Ias can be written as
Ias =
2 ωas (3) 2 2 |3χ | I Is L2 sinc2 (∆kL/2). 16ε02 c4 CARS p
(25)
Assuming an interaction length of one Rayleigh range, i.e., L = πw2p0 /λ p and an interaction cross-sectional area of πw2p0 we then obtain the power of the radiation generated at the antiStokes frequency under phase-matched conditions (∆k = 0) as Pas =
λs3 2hcλas
2
N ∆ωR
dσ dΩ
2
Pp2 Ps .
(26)
and the corresponding CARS emission rate as γCARS, f ree = Pas /hνas . Thus, in the absence of the microcavity and using the same input laser power, we would have obtained only γCARS, f ree ≈ 6× 10−4 photons/s, that is, 9 orders of magnitude less. There is, therefore, a significant advantage in using the high-finesse microcavity for continuous-wave CARS. Nonetheless, the full potential of the microcavity enhancement is not yet realized in our present experimental configuration. In fact, using Eq. (26) with Ps and Pp corresponding to the circulating powers inside the microresonator one obtains the rate of ≈ 107 photons/s which is of the same order as the microcavity result. Thus the primary benefit of the microcavity in our
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22007
current configuration is enhancement of the input fields through recirculation at a resonant frequency. However, according to Eq. (21) the microcavity CARS emission rate scales inversely with the sum of the Raman linewidth and the linewidth of the cavity at the anti-Stokes frequency. The cavity finesse at the latter is only 3 for our present experiments. But given that (as) ∆ωc can be reduced further by at least 3 orders of magnitude before near-resonant matching becomes a challenge, there is a significant increase in CARS emission rate to be gained simply by increasing the reflectivity of the mirrors at the anti-Stokes frequency. If we assume that the cavity finesse is that of state of the art microcavities [20], i.e. F ≈100 000 at the Stokes and pump frequencies, and F ≈1000 at the anti-Stokes frequency we would expect CARS rates of order 1014 photons/s for 10 mW laser input power. There are also significant advantages in performance of microcavity enhanced CARS compared to CARS enhanced by macroscopic resonators. Consider a macroscopic resonator of length L ∼1 cm with mirrors of radii of curvature R such that the ratio R/L is the same as for the microcavity. At the same finesse (i.e., mirror reflectivity) as the microcavity, the macroscopic cavity will have a linewidth of order 1 MHz only. This means firstly that active laser stabilization will be required in practice. Secondly, for this macroscopic resonator the term (as) (∆νc + ∆νR ) in the denominator of Eq. (21) will approximately equal the Raman linewidth and thus not contribute to the enhancement. Finally, improved mechanical and thermal stability, large frequency-scanning speed and thus high frequency feedback control possibilities are other unique advantages of the microcavity approach for CARS enhancement. 6.
Conclusions
To summarize, we have explored the manifestations of coherent anti-Stokes Raman scattering in a high-finesse microcavity. We obtain good agreement between our measured CARS emission rate and that calculated using coupled-mode analysis. Due to the limited bandwidth of the dielectric reflective coating on our mirrors, only a low-Q resonance was obtained at the antiStokes frequency. However, coatings providing higher bandwidth are readily available, thus making significant increases in the microcavity CARS emission rate very realistic. Microcavity enhanced CARS could thus find practical applications in gas spectroscopy and sensing, although further investigations will be needed to determine the possible limit of detection given the well-known issues associated with non-resonant CARS backgrounds [1]. Nonetheless, the present approach, valid for any four-wave mixing process, offers new avenues for investigating strong nonlinear light-matter interactions with only microscopic gas sample volumes. Acknowledgments We acknowledge partial support from the National Science Foundation (NSF Grant No. 1254324), and thank C. K. Shih for making available a tunable laser that aided us in carrying out this work.
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22008
of Physics, University of South Florida, Tampa, FL 33620, USA 2 MicroMaterials, Inc., Tampa, FL 33637, USA ∗ [email protected]
Abstract: We report the measurement of degenerate coherent anti-Stokes Raman scattering (CARS) in a high-finesse optical microcavity, with an atmospheric gas as the nonlinear medium. Unlike the well-known bulk process in which index-dispersion compensation is required for phasematching, efficient microcavity CARS involves a resonant coupling at the Stokes, pump, and anti-Stokes frequencies which can be described using coupled-mode analysis. We show how the interaction is thereby dramatically enhanced in a microscopic sample volume and illustrate the technique for the measurement of CO2 in air. © 2014 Optical Society of America OCIS codes: (290.5860) Raman Scattering; (350.3950) Micro-optics; (270.1670) Coherent optical effects.
References and links 1. W. M. Tolles, J. W. Nibler, J. R. McDonald, and A. B. Harvey, “A Review of the Theory and Application of Coherent Anti-Stokes Raman Spectroscopy (CARS),” Appl. Spect. 4, 253-271 (1977). 2. F. El-Diasty, “Coherent anti-Stokes Raman scattering: Spectroscopy and microscopy,” Vib. Spectrosc. 55, 1-37 (2009). 3. S. Roy, J. R. Gord, A. K. Patnaik, “Recent advances in coherent anti-Stokes Raman scattering spectroscopy: Fundamental developments and applications in reacting flows,” Prog. Energy Combust. Sci. 36, 280-306 (2010). 4. B. Li, P. Borri, and W. Langbein, “Dual/differential coherent anti-Stokes Raman scattering module for multiphoton microscopes with a femtosecond Ti:sapphire oscillator,” J. of Biomed. Opt. 18, 066004 (2013). 5. D. Gachet, F. Billard, and H. Rigneault, “Coherent anti-Stokes Raman scattering in a microcavity,” Opt. Lett. 34, 1789-1791 (2009). 6. M. Marrocco, “Coherent anti-stokes Raman scattering microscopy in the presence of electromagnetic confinement,” Laser Physics 17, 935-941 (2007). 7. M. Marrocco and E. Nichelatti, “Coherent anti-Stokes Raman scattering microscopy within a microcavity with parallel mirrors,” J. Raman Spect. 40, 732-740 (2009). 8. F. Billard, D. Gachet, and H. Rigneault, “Coherent anti-Stokes Raman scattering in a Fabry-Perot cavity: A theoretical study,” J. Opt. Soc. Am. B 26, 1295-1309 (2009). 9. K. J. Vahala, “Optical microcavities,” Nature 424, 839-846 (2003). 10. T. J. Kippenberg, S. M. Spillane, D. K. Armani, and K. J. Vahala, “Ultralow-threshold microcavity Raman laser on a microelectronic chip,” Opt. Lett. 29, 1224-1226 (2004). 11. T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third harmonic generation,” Nat. Phys. 3, 430-435 (2007). 12. J. U. F¨urst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally Phase-Matched Second-Harmonic Generation in a Whispering-Gallery-Mode Resonator,” Phys. Rev. Lett. 104, 153901 (2010). 13. S. Zaitsu, H. Izaki, and T. Imasaka, “Phase-Matched Raman-Resonant Four-Wave Mixing in a DispersionCompensated High-Finesse Optical Cavity,” Phys. Rev. Lett. 100, 073901 (2008). 14. S. Zaitsu and T. Imasaka, “Intracavity phase-matched coherent anti-Stokes Raman spectroscopy for trace gas detection,” Anal Sci. 30, 75-79 (2014).
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 21999
15. B. Petrak, N. Djeu, and A. Muller, “Purcell-enhanced Raman scattering from atmospheric gases in a high-finesse microcavity,” Phys. Rev. A 89, 023811 (2014). 16. D. C. Hanna, M. A. Yuratich, and D. Cotter, “Nonlinear Optics of Free Atoms and Molecules,” (Springer, New York, 1979). 17. H. Kogelnik and T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550-1567 (1966). 18. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272-276 (2007). 19. D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. H¨ansch, and J. Reichel, “Laser micro-fabrication of concave, low-roughness features in silica,” New J. Phys. 12, 065038 (2010). 20. A. Muller, E. B. Flagg, J. Lawall, and G. S. Solomon, “Ultrahigh-finesse, low-mode-volume FabryPerot microcavity,” Opt. Lett. 35, 2293-2295 (2010). 21. B. Petrak, K. Konthasinghe, S. Perez, and A. Muller, “Feedback-controlled laser fabrication of micromirror substrates,” Rev. Sci. Instrum. 82, 123112 (2011). 22. G. Rempe, R. J. Thompson, H. J. Kimble, and R. Lalezari, “Measurement of ultralow losses in an optical interferometer,” Opt. Lett. 17, 363-365 (1992). 23. B. P. Stoicheff, “High Resolution Raman Spectroscopy of Gases: XI. Spectra of CS2 and CO2 ,” Can. J. Phys. 36, 218-230 (1958). 24. W. R. Fenner, H. A. Hyatt, J. M. Kellam, and S. P. S. Porto, “Raman cross section of some simple gases,” J. Opt. Soc. Am. 63, 73-77 (1973). 25. P. A. Roos, L. S. Meng, S. K. Murphy, and J. L. Carlsten, “Approaching quantum-limited cw anti-Stokes conversion through cavity-enhanced Raman-resonant four-wave mixing,” J. Opt. Soc. Am. B 21, 357-363 (2004). 26. X. Yang and C. W. Wong, “Coupled-mode theory for stimulated Raman scattering in high-Q/Vm silicon photonic band gap defect cavity,” Opt. Express 15, 4763-4780 (2007). 27. B. Lavorel, G. Millot, R. Saintloup, H. Berger, L. Bonamy, J. Bonamy, and D. Robert, “Study of collisional effects on band shapes of the ν1 /2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy. II. Simultaneous line mixing and Dicke narrowing in the ν1 band,” J. Chem. Phys. 93, 2185-2191 (1990). 28. C. J. S. M. Simpson, T. R. D. Chandler, and A. C. Strawson, “Vibrational Relaxation in CO2 and CO2 -Ar Mixtures Studied Using a Shock Tube and a Laser-Schlieren Technique,” J. Chem. Phys. 51, 2214-2219 (1969).
1.
Introduction
Degenerate coherent anti-Stokes Raman scattering (CARS) takes place in a medium possessing (3) a third-order optical nonlinearity, χCARS , when two waves of frequency ω p (the pump waves), and a third wave of frequency ωs (the Stokes wave) interact to generate a fourth wave of frequency ωas = 2ω p − ωs (the anti-Stokes wave). CARS has long been a powerful tool for the measurement of vibrational spectra of solids, liquids and gases with a larger signal-to-noise ratio than that provided by ordinary spontaneous Raman scattering [1–3]. Common applications are in combustion diagnostics and selective microscopy, and ultrafast CARS continues to offer ever-improving imaging capabilities in the life sciences [4]. Recently, there has been interest in taking advantage of optical resonators to localize and/or enhance the CARS process for improved microscopy and sensing. For example, Gachet et al. investigated microcavity CARS enhancement in which the anti-Stokes emission was resonant with a low-finesse planar optical microcavity [5]. Such parallel mirror arrangements are desirable for microscopy but they impose severe limitations on CARS microcavity enhancements even on purely theoretical grounds [6–8]. On the other hand, rapid development in micro-fabrication technology has made available truly three-dimensionally-confining optical resonators of high-finesse and small mode-volume [9]. In fact, in whispering gallery mode optical resonators, greatly enhanced nonlinear interactions including stimulated Raman scattering [10], four-wave-mixing, and high-harmonic generation [11, 12] have all been shown in solid media. Additionally, significant CARS enhancements have been demonstrated recently for gases using a macroscopic optical resonator with dispersion-compensated mirrors [13, 14]. We demonstrate here the enhancement of CARS in an open, microscopic high-finesse cavity for the purpose of atmospheric gas measurements. Building on recent experiments of Purcellenhanced spontaneous Raman scattering [15], we show that it is possible to tune the cavity such as to provide high-Q resonances at ω p and ωs together with a low-Q resonance at ωas . #210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22000
This yields a sizable increase in CARS intensity compared to free space while at the same time involving a minute amount of sample gas. We find good agreement between the measured CARS emission rate and the rate predicted theoretically by a coupled-mode analysis. 2.
Theoretical background
The degenerate CARS process is illustrated in the energy diagram of Fig. 1(a), in which the pump and Stokes frequencies are chosen such that their difference coincides with a rotational or vibrational transition frequency, ων , of a molecular sample gas. For degenerate CARS to occur efficiently in free space, phase matching is needed, i.e. we must have kas = 2k p − ks for the wavevectors of the anti-Stokes, pump, and Stokes waves, assumed to be planar. In contrast, for a CARS process which takes place in a resonator, the confined fields E j (r,t) (where j = p, s, as) are standing waves of the form s h¯ ω j a j (t)e−iω j t u j (r), (1) E j (r,t) = ε0V j where a j (t) are slowly time-varying amplitudes, u j (r) are spatial mode functions determined by the resonator’s type and geometry, and V j is a normalization factor given by Vj =
1 2
Z
u∗j u j d 3 r
(2)
such that |a j |2 represents the energy, in units of h¯ ω j , stored in each mode, i.e., Z
ε0 ∗ E E j d 3 r = n j h¯ ω j , 2 j
(3)
with n j = |a j |2 . For simplicity we assume unity refractive index for all waves. From Maxwell’s equations we have, for each mode, (3)
∇×∇×Ej +
1 ∂ 2E j 1 ∂ 2 PNL (ω j ) = − , c2 ∂t 2 ε0 c 2 ∂t 2
(4)
or, (3)
−∇2 E j + (a)
1 ∂ 2 PNL (ω j ) 1 ∂ 2E j = − , c2 ∂t 2 ε0 c2 ∂t 2
(5)
(b) 105
ωp
ωs
ωp
Cavity nesse
104
ωas
1000
λas λp λs
100 10 1
ων
0.1 0.7
0.8
0.9
1.0
1.1
Wavelength (µm)
Fig. 1. (a) Degenerate coherent anti-Stokes Raman scattering energy diagram. (b) Cavity finesse as a function of wavelength. The blue solid, dashed, and dot-dashed lines represent the anti-Stokes, pump, and Stokes wavelengths, respectively.
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22001
where it was assumed that all fields are polarized along the same axis. For degenerate CARS (3)
3χ
(3)
2 ∗ we have PNL (ωas ) = ε0 CARS 4 E p Es [16]. Inserting Eq. (1) with j = as into Eq. (5) then yields s s h¯ ωas h¯ ωas ∂2 − aas (t)e−iωas t c2 ∇2 uas (r) + uas (r) 2 aas (t)e−iωas t = ε0Vas ε0Vas ∂t (6) s (3) 2 3χCARS h¯ ω p h¯ ωs ∂ − u p (r)2 us (r)∗ 2 a p (t)2 as (t)∗ e−i(2ω p −ωs )t . 4 ε0Vp ε0Vs ∂t
Further using the fact that the mode functions u j (r) satisfy the scalar wave equation ∇2 u j + ω 2j u c2 j
= 0 [17], and making the slowly-varying envelope approximation we get s
(3)
3χ h¯ ω p h¯ ωas daas (t) uas (r) = i CARS ε0Vas dt 8 ε0Vp
s
h¯ ωs u p (r)2 us (r)∗ ωas a p (t)2 as (t)∗ , ε0Vs
(7)
where it was assumed that ωas = 2ω p − ωs . Finally, multiplying each side by 21 uas (r)∗ and integrating over all space, √ √ (3) 3χCARS h¯ ω p ωs ωas daas (t) =i a p (t)2 as (t)∗ , dt 4 ε0Veff
(8)
where the effective mode volume is defined by u∗as u2p u∗s d 3 r 1 . = R R R Veff u∗p u p d 3 r ( u∗s us d 3 r)1/2 ( u∗as uas d 3 r)1/2 R
(9)
For gaussian beams confined by a microcavity of length L constructed with curved mirrors (radius of curvature R), the TEM00 mode functions take the approximate form [17], u j (r) =
w j0 −r2 /w j (z)2 e sin(k jz z + φ j (z)) w j (z)
(10)
where the wavenumber k jz must assume discrete values. The psquare of the beam spot size is 2 2 2 2 2 w j (z) = w j0 (1 + z /zR ) and its waist is given by w j0 = λ j (R − L/2)L/2/π. The standing waves must satisfy the condition k jz L − ∆ϕ + ϕ j = m j π
(11)
with integer m j , where ϕ j is the (frequency-dependent) phase shift experienced by the wave at the mirror and ∆ϕ is the Gouy phase shift given by ∆ϕ = tan−1 (z2 /zR ) − tan−1 (z1 /zR ).
(12) p Here zR = L(R − L/2)/2 is the Rayleigh range defined by the microcavity, and z2 > 0 and z1 < 0 are the positions of the two mirrrors (so that z2 − z1 = L) [17]. From Eq. (8) and Eq. (9) one then sees that phase-matching for microcavity CARS is replaced by a maximization requirement of an overlap integral involving the m j ’s, analogous to the overlap integrals giving rise to selection rules in electronic transitions [12]. If we ignored the mirror phase shifts in Eq. (10) the relevant “selection rules” would be mas = 2m p − ms .
(13)
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22002
L~20 µm
Trans. Detector
R~40 µm
ωp ωs
M1
Gas in PZT
ωas
Single photon detector
Lens
PZT scan
Timing Electronics
Cavity region
Optical fiber
Fig. 2. Experimental setup for the measurement of microcavity coherent anti-Stokes Raman scattering. The cavity assembly (right) consisted of the cavity region itself (shaded area), a flexure mount for large range tuning using piezoelectric transducers (PZTs), a mode matching lens, and a temperature-controlled base. The planar substrate holding one mirror (M1) as well as the end of the optical fiber at the tip of which the other mirror was fabricated can also be seen in the photograph.
In addition, it is required that the difference between the anti-Stokes mode frequency and the pump mode frequency, as well as the difference between the pump mode frequency and the Stokes mode frequency be equal to the Raman shift, i.e. we must have ωas − ω p = ων
(14)
ω p − ωs = ων ,
(15)
and which are conditions strongly dependent upon the phase shifts experienced by the waves according to Eq. (11). Thus, a cavity configuration involving simultaneously three different resonances, i.e. a triple-resonance condition, must be met to achieve maximum CARS efficiency. While in general it is not feasible to meet this condition for three sharply peaked cavity resonances, it is always possible to be at the peaks of two sharp cavity resonances and near the peak of a third broad one. 3.
Experiments
The experimental implementation of degenerate microcavity CARS thus necessitates meeting the “selection rules” which arise from the fields of Eq. (1), to within the tolerance associated with the finite linewidth of the microcavity resonances involved. In recent work, we have demonstrated Purcell enhanced spontaneous Raman scattering in a 10 µm long microcavity constructed with mirrors of radius of curvature R ≈ 40 µm [15]. This was possible by implementing a double-resonance condition wherein Eq. (11) is satisfied for j = p and j = s and simultaneously ω p − ωs = ων holds to within the Raman linewidth. Building on these results, we use here a similar microcavity, the high finesse of which is derived from a laser microfabrication process [18–21] compatible with ultralow-loss dielectric reflective coatings [22]. In Fig. 1(b) we illustrate the spectral configuration chosen for the present CARS microcavity measurement. The choice of frequencies is constrained by the finite bandwidth over which the quarter-wave mirror stacks have high reflectivity, relative to the molecular rotational or vibrational frequency ων of the molecule being probed. We opted for a compromise so that the cavity finesse at both pump and Stokes frequencies is ≈10 000, whereas it is of order unity ( ≈3) at the anti-Stokes frequency. Without loss of generality, we focus here exclusively on the
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22003
Q-branch line in CO2 terminating in the (1 0 0) level with a frequency of ων /2πc = 1388.17 cm−1 [23], and a differential Raman scattering cross section dσ /dΩ = 4.7 × 10−32 cm2 /sr [24]. Our experimental setup is depicted in Fig. 2. A microcavity of length L ≈ 20 µm was constructed with one micromirror at the end of a single mode fiber tip and a second micromirror on a planar fused silica substrate. Both mirrors had a radius of curvature of R ≈ 40 µm. Two independently tunable continuous-wave diode lasers with a linewidth of ∼1 MHz were used for the pump and Stokes sources. The lasers were continuously monitored with a wavemeter and a scanning Fabry-Perot interferometer with a resolution of 20 MHz. The pump and Stokes laser beams were matched to TEM00 modes of the cavity through the mirror on the planar substrate using a single aspheric lens. Accounting for imperfect mode matching we estimate 0.7 mW and 3 mW were coupled into the cavity for the pump and Stokes laser beams, respectively. The cavity was hermetically sealed and gas delivered directly to the cavity region via external tubing. Piezoelectric actuators were used to scan the cavity length while the anti-Stokes wave was coupled out with a dichroic filter and detected with a single photon counting module. Instead of a permanent Pound-Drever-Hall lock as in [15], we histogrammed the time-tagged detector events while both the frequency of one laser and the cavity length were swept. In order to ensure all length and frequency combinations were realized, we scanned the laser frequency and cavity length at speeds separated by several orders of magnitude. The laser was scanned over a spectral range of 2.4 GHz in a period of 11 seconds, while the cavity length was swept by 1.42 nm in a period of 3.2 ms. In Fig. 3, we illustrate the resulting data for the case when the test gas is CO2 at atmospheric pressure. In this measurement the pump laser frequency was scanned while the Stokes laser frequency was stepped through. Each trace in the plot of Fig. 3(a) has been extracted from the raw recorded data maps of the emission rate as a function of both the cavity length and the Stokes laser frequency. The emission rate has been corrected for the known propagation and filter losses (50%) and for the detector quantum efficiency (65%). Figure 3(b) shows a closeup view of the trace from Fig. 3(a) at which the peak CARS emission rate of γCARS ≈ 4 × 106 photons/s is obtained. a) x106
b) x106
CARS emission rate (photons/s)
4.0 5.0 3.0
4.0 3.0
Sto kes
fre 304 370 qu en cy 304 360 (GH z) 304 350 304 340
2.0 1.0 0 345 960
345 970
345 980
Pump frequency (GHz)
345 990
2.0
FWHM ≈203 MHz
1.0
0
345 973.0 345 973.7 Pump frequency (GHz)
Fig. 3. (a) Measured CARS spectra. (b) Magnified view of the trace in (a) for which the peak emission occurs.
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22004
4.
Theoretical microcavity CARS emission rate
In order to quantitatively understand the features of degenerate CARS as it manifests in a microcavity, we derive an expression for the CARS emission rate starting from Eq. (8) [25, 26] √ √ (3) 3χCARS h¯ ω p ωs ωas daas (t) aas (t) a p (t)2 as (t)∗ , = − (as) + i dt 4 ε0Veff 2τc
(16)
where we have added the cavity decay term due to mirror transmission. In steady-state one then obtains for the number of anti-Stokes photons in the mode 2
nas = |aas |
(3) 2 (as) 2 |3χCARS | = (τc )
4
h¯ 2 ω p2 ωs ωas 2 n p ns , 2 ε02Veff
(17) (as)
and thus the CARS emission rate out of one end of the cavity, γCARS = nas /2τc
, is
(3)
|3χCARS |2 h¯ 2 ω p2 ωs ωas 2 γCARS = n p ns , 2 (as) ε02Veff 8∆ωc
(18) (as)
(as)
with the cavity linewidth at the anti-Stokes frequency given by ∆ωc = 1/τc . In order to (as) account for the overlap between the cavity mode and the Raman lineshape we replace 1/∆ωc with a normalized Lorentzian in Eq. (18). Further approximating a Lorentzian profile with full (3) width ∆ωR for the term |3χCARS |2 , and integrating over frequency, one obtains (3)
γCARS =
|3χCARS,0 |2
h¯ 2 ω p2 ωs ωas 2 n p ns , 2 2 (as) 8(∆ωc + ∆ωR ) ε0 Veff
(19)
(3)
The magnitude of the resonant third-order nonlinear susceptibility, χCARS,0 , can be related to the familiar spontaneous Raman differential scattering cross section, dσ /dΩ, through the relation [16] 8π 2 ε0 Nc4 dσ (3) |3χCARS,0 | = (20) h¯ ωs3 ω p ∆ωR dΩ where N is the number density of molecules in the lower vibrational (in the case of rotationally collapsed transitions such as the ν1 Q-branch of CO2 being reported here) or rotational state. Eq. (19) then evaluates to 1 c4 λs5 N 2 γCARS = 2 2 (2π)3 λas Veff
dσ dΩ
2
n2p ns (as)
(∆νc
+ ∆νR )∆νR2
.
(21)
The microcavity CARS emission rate is thus expected to be proportional to the square of the molecular density, proportional to the square of the pump power, and proportional to the Stokes power. The microcavity parameters enter in three different ways. Firstly, the microcavity CARS emission rate is inversely proportional to the square of the effective mode volume. Secondly, it is inversely proportional to the sum of the cavity linewidth of the anti-Stokes mode and the Raman linewidth. Thirdly, the cavity enhances the pump and Stokes photon number by recirculation at a resonant frequency. Specifically, in terms of the incident power, Pp,s , the pump/Stokes photon number is given by (p,s) n p,s = 2Pp,s τc /¯hω p,s (22) #210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22005
(b)
x10 6.0
Raman FWHM (MHz)
CARS emission rate (photons/s)
(a)
5.0 4.0 3.0 2.0 1.0 0
20
40 60 80 CO concentration (%)
200 150 100 50 0
100
20
40 80 60 CO concentration (%)
100
Fig. 4. (a) Peak CARS emission rate versus CO2 concentration. (b) Raman linewidth versus CO2 concentration. (p,s)
(p,s)
where τc = 1/∆ωc is the cavity photon lifetime at the pump/Stokes frequency. For the experiment of Fig. 3, the coupled input powers were Pp ≈ 0.7 mW, and Ps ≈ 3 mW. For a cavity finesse of F (p,s) ≈ 10 000, and a cavity length of L ≈ 20 µm, we then obtain n p ≈106 and ns ≈ 6 × 106 assuming a pump wavelength λ p ≈ 864 nm. The mode volume that these photons occupy can be approximated using Eq. (9) as Veff ≈ λ p
p 1 1 L2 /4 2 p ) . λs λas ( + + λ p λs λas tan−1 (L/ 2(R − L/2)L)
(23)
For L=20 µm and R=40 µm, Veff ≈ 660 µm3 . Finally, then, the theoretically expected CARS emission rates corresponding to our experimental rates of Fig. 3 are obtained by evaluating Eq. (21) wherein most parameters are known independently from experiment. Crucially, the Raman linewidth is obtained directly from Fig. 3(b) for 100 % CO2 at atmospheric pressure as ∆νR =200 MHz, in agreement with the literature value [27]. The mirror reflectivity dictates the cavity finesse at the relevant frequencies [Fig. (as) 1(b)]. In particular, we can estimate that ∆νc ≈ 15 THz (F (as) = 3). As a result, for 100% CO2 at atmospheric pressure, we compute a theoretical CARS emission rate of γCARS ≈ 1 × 107 photons/s, which is of the same order of magnitude as the measured rate of Fig. 3. We further investigated the accuracy of Eq. (21) through its scaling with specific experimental parameters. In Fig. 4(a) we present the measured CARS emission rate as a function of CO2
gCARS ä ΔnR2 (MHz2/s)
2.0 x1011 ∞N2.1
3.0 x1010 5.0 x109 1.0 x109 10
20
40
CO2 concentration (%)
100
CARS emission rate (photons/s)
(b)
(a)
x10 5.0
∞Pp ∞Ps
2.0 1.0 0.5 0.3
0.5
1.0 2.0 Power (mW)
5.0
Fig. 5. (a) Logarithmic representation of the quantity γCARS ∆νR2 as a function of concentration, exhibiting the quadratic growth described by Eq. (21). (b) Dependence of the microcavity CARS emission rate on pump and Stokes laser power.
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22006
concentration. The expected quadratic dependence [N 2 factor in Eq. (21)] is not readily seen. Instead, an approximately linear growth is observed, which can however be explained by the fact that the Raman linewidth is also varying with CO2 concentration. Indeed, as shown in the data of Fig. 4(b), the Raman linewidth is decreasing with concentration, down to only 50 MHz FWHM for 10% CO2 . The linear dependence of the Raman linewidth on CO2 concentration can be understood on the grounds that at atmospheric pressure the Q branch of the ν1 Raman transition is completely collapsed rotationally and that the collapsing effect is stronger for air than CO2 itself [27]. To our knowledge, this is the first time the collisional narrowing effect of air on the 1 388 cm−1 Raman transition in CO2 has been determined. This fortuitous decrease of the linewidth with CO2 concentration partially offsets the decrease of the CARS signal with concentration. By plotting the quantity γCARS ∆νR2 as a function of concentration, as shown in Fig. 5(a), the expected quadratic dependence on concentration is retrieved. Finally, we also verify the dependence of Eq. (21) on pump and Stokes laser power. As seen in Fig. 5(b) the observed power dependence is close to quadratic for the pump laser power as anticipated. The observed power dependence of the emission rate on Stokes power exhibits a sub-linear dependence. This can be explained by ground state depletion, since the estimated pump rate into the (1 0 0) vibrational level is 3(10)4 /s at maximum powers, and the vibrational relaxation rate is 1.3(10)5 /s [28]. 5.
Comparison to CARS in free space
An interesting question is to what extent, besides miniaturization, the generation of CARS in a microcavity is beneficial compared to its generation in free space or within a conventional macroscopic resonator. In free space and under plane-wave approximation, the intensity of anti-Stokes radiation generated by the CARS process is usually calculated as [16] Ias =
2 ωas (3) 2 2 |P | L sinc2 (∆kL/2), 8ε0 c CARS 3χ
(3)
(24)
(3)
2 ∗ where the (complex) nonlinear polarization is PCARS = ε0 CARS 4 E p Es , the wavevector mismatch is ∆k = 2k p − ks − kas , and the interaction length is L. Thus Ias can be written as
Ias =
2 ωas (3) 2 2 |3χ | I Is L2 sinc2 (∆kL/2). 16ε02 c4 CARS p
(25)
Assuming an interaction length of one Rayleigh range, i.e., L = πw2p0 /λ p and an interaction cross-sectional area of πw2p0 we then obtain the power of the radiation generated at the antiStokes frequency under phase-matched conditions (∆k = 0) as Pas =
λs3 2hcλas
2
N ∆ωR
dσ dΩ
2
Pp2 Ps .
(26)
and the corresponding CARS emission rate as γCARS, f ree = Pas /hνas . Thus, in the absence of the microcavity and using the same input laser power, we would have obtained only γCARS, f ree ≈ 6× 10−4 photons/s, that is, 9 orders of magnitude less. There is, therefore, a significant advantage in using the high-finesse microcavity for continuous-wave CARS. Nonetheless, the full potential of the microcavity enhancement is not yet realized in our present experimental configuration. In fact, using Eq. (26) with Ps and Pp corresponding to the circulating powers inside the microresonator one obtains the rate of ≈ 107 photons/s which is of the same order as the microcavity result. Thus the primary benefit of the microcavity in our
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22007
current configuration is enhancement of the input fields through recirculation at a resonant frequency. However, according to Eq. (21) the microcavity CARS emission rate scales inversely with the sum of the Raman linewidth and the linewidth of the cavity at the anti-Stokes frequency. The cavity finesse at the latter is only 3 for our present experiments. But given that (as) ∆ωc can be reduced further by at least 3 orders of magnitude before near-resonant matching becomes a challenge, there is a significant increase in CARS emission rate to be gained simply by increasing the reflectivity of the mirrors at the anti-Stokes frequency. If we assume that the cavity finesse is that of state of the art microcavities [20], i.e. F ≈100 000 at the Stokes and pump frequencies, and F ≈1000 at the anti-Stokes frequency we would expect CARS rates of order 1014 photons/s for 10 mW laser input power. There are also significant advantages in performance of microcavity enhanced CARS compared to CARS enhanced by macroscopic resonators. Consider a macroscopic resonator of length L ∼1 cm with mirrors of radii of curvature R such that the ratio R/L is the same as for the microcavity. At the same finesse (i.e., mirror reflectivity) as the microcavity, the macroscopic cavity will have a linewidth of order 1 MHz only. This means firstly that active laser stabilization will be required in practice. Secondly, for this macroscopic resonator the term (as) (∆νc + ∆νR ) in the denominator of Eq. (21) will approximately equal the Raman linewidth and thus not contribute to the enhancement. Finally, improved mechanical and thermal stability, large frequency-scanning speed and thus high frequency feedback control possibilities are other unique advantages of the microcavity approach for CARS enhancement. 6.
Conclusions
To summarize, we have explored the manifestations of coherent anti-Stokes Raman scattering in a high-finesse microcavity. We obtain good agreement between our measured CARS emission rate and that calculated using coupled-mode analysis. Due to the limited bandwidth of the dielectric reflective coating on our mirrors, only a low-Q resonance was obtained at the antiStokes frequency. However, coatings providing higher bandwidth are readily available, thus making significant increases in the microcavity CARS emission rate very realistic. Microcavity enhanced CARS could thus find practical applications in gas spectroscopy and sensing, although further investigations will be needed to determine the possible limit of detection given the well-known issues associated with non-resonant CARS backgrounds [1]. Nonetheless, the present approach, valid for any four-wave mixing process, offers new avenues for investigating strong nonlinear light-matter interactions with only microscopic gas sample volumes. Acknowledgments We acknowledge partial support from the National Science Foundation (NSF Grant No. 1254324), and thank C. K. Shih for making available a tunable laser that aided us in carrying out this work.
#210936 - $15.00 USD Received 1 May 2014; revised 21 Jun 2014; accepted 22 Aug 2014; published 3 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021999 | OPTICS EXPRESS 22008
