Appl. Phys. B 32, 145-152 (1983) Applied physics Physics B and Laser Chemisb'y 9 Springcr-Verlag1983 Frequency Modulation (FM) Spectroscopy Theory of Lineshapes and Signal-to-Noise Analysis G. C. Bjorklund and M. D. Levenson IBM Research Laboratory, San Jose, CA 95193, USA W. Lenth MIT Lincoln Laboratory, P.O. Box 73, Lexington, MA02173, USA C. Ortiz Instituto de Optica, Serrano 121, Madrid, Spain Received 30 June 1983/Accepted 31 July 1983 Abstract. Frequency modulation (FM) spectroscopy is a new method of optical heterodyne spectroscopy capable of sensitive and rapid measurement of the absorption or dispersion associated with narrow spectral features. The absorption or dispersion is measured by detecting the heterodyne beat signal that occurs when the FM optical spectrum of the probe wave is distorted by the spectral feature of interest. A short historical perspective and survey of the FM spectroscopy work performed to date is presented. Expressions describing the nature of the beat signal are derived. Theoretical lineshapes for a variety of experimental conditions are given. A signal-to-noise analysis is carried out to determine the ultimate sensitivity limits. PACS: 07.65 Frequency modulation (FM) spectroscopy is a new method of optical heterodyne spectroscopy capable of sensitive and rapid detection of absorption or dispersion features [1, 21. As shown in Fig. 1, the output of a single-axial-mode dye laser oscillating at optical frequency coc is passed through a phase modulator driven sinusoidally at radio frequency e)m to produce a pure FM optical spectrum consisting of a strong carrier at frequency e) with two weak sidebands at frequencies coc_+com. The beam is next passed through the sample containing the spectral feature of interest and the impinges on a fast square law photodetector. The Fourier component of the photodetector electrical signal at frequency c% is detected using standard rf techniques. A key concept is that co,, can be large compared to the width of the spectral feature of interest, so that, as illustrated in Fig. 2, the spectral feature can be probed by a single isolated sideband and the full spectral resolution of the original laser source is maintained. Both the absorption and dispersion associated with the spectral feature can be separately measured by monitoring the phase and amplitude of the rf heterodyne beat signal that occurs at frequency ~% when the FM optical spectrum is distorted by the effects of the spectral feature. In contrast to traditional wavelength modulation techniques, the signal occurs at rf frequencies large compared to the linewidth of the laser source. Since single-mode dye lasers have little noise at radio frequencies, these beat signals can be detected with a high degree of sensitivity. Furthermore, the entire lineshape of the spectral feature can be scanned by tuning either e)c or com. It should be noted that the use of sidebands for spectroscopy is not new. The Pound microwave frequency stabilizer [3] worked on this principle and the N M R techniques developed by Smaller [4] and Acrivos [5] are closely related. Standard (non- 146 G.C. Bjorklund et al. E 1 (t) E2(t) E3(t) Fast Photodetector h S(t) I DC Signal Fig. 1. A typical experimental arrangement for FM spectroscopy ! A8 0 o_ ~ C - - COrrl co c Optical Frequency c o c + oo m , Fig. 2. Frequency domain illustration of FM spectroscopy heterodyne) laser absorption spectroscopy with sidebands has been performed by Corcoran et al. [6], Mattick et al. E7], and Magerl et al. E8]. Heterodyne laser spectroscopy with amplitude modulated (AM) sidebands has been accomplished by Szabo [9] and Erickson [10]. Sideband techniques have recently been employed with great success to heterodyne detect signals from resonant degenerate four-wave mixing experiments [11-14]. Wavelength modulation spectroscopy with lasers, as has been done by Hinkley and Kelley E15] and Tang and Telle [16], can be viewed as heterodyne spectroscopy with very closely spaced FM sidebands. Harris et al. E17] recognized that the appearance of rf beats provides a sensitive indication of distortion of the output of an FM laser. However, the use of widely separated FM sidebands for optical heterodyne spectroscopy has only recently been accomplished and exploitation of the attendant advantages of zero-background signal, rapid response to transients, and laser limited resolution has only just begun. Since the initial experiments with cavity resonances [1,2-1, FM laser spectroscopy has been utilized to measure saturation holes in 12 vapor [18, 19], persistent photochemical holes burned in color-center zero phonon lines [20-22], time dependent laser gain [23], two-photon absorption [24], stimulated Raman gain [25], frequency offset between a fixed frequency laser and an absorption line [26], and atomic absorption in atmospheric pressure flames [27]. Generalizations of the theory to include nonlinear effects have been performed by Hall et al. [19], Agarwal [28], Schenzle et al. [29], and Shirley [30]. Whittaker and Bjorklund [31] have extended the theory to cover arbitrarily large modulation index. Research on the use of the technique for interferometer stabilization [32] and laser frequency stabilization has continued. Several groups have reported short term laser linewidths on the order of 100 Hz [33, 34]. Recently, FM spectroscopy with widely spaced sidebands has been performed using direct injection current modulation of GaA1As diode lasers [35]. Finally, the time resolution capability of FM spectroscopy has been demonstrated in the microsecond regime using acousto-optic chopping of a cw dye laser beam [20] and in the nanosecond regime by Gallagher et al. [36] who used a high-power pulsed dye laser of relatively standard design. The significance of the work of Gallagher et al. is that FM spectroscopy can be performed with relatively broad-band laser sources so long as the rf modulation frequency is greater than the laser bandwidth and that nonlinear optical techniques can be used to translate the high power frequency modulated laser radiation to the uv or ir spectral region. This later point has recently been demonstrated by Tran e t al. [37]. Basic Principles and Theory of Lineshapes In the limit that the modulation index M<I, the electric field EE(t) emerging from the modulator shown in Fig. 1 is described by E2(t) = 1/2/~2(t) + c.c. where exp iI o 1 M + exp (icoct)+ ~- exp [i(coc + co,")t]/ (1) and E o is the electric field amplitude of the original laser beam. This is a pure FM optical spectrum with sidebands at frequencies ~o +~o,". Figure 2 shows the power spectrum and illustrates the case where the coc+ co,. sideband probes a Lorentzian spectral feature. The beam emerging from the modulator is next passed through the sample containing the spectral feature. The sample is assumed to be of length L and to have intensity absorption coefficient c~ and index of refraction n which are functions of the optical frequency. It is convenient to define the amplitude transmission, attenuation, and phase shift for each spectral corn- Frequency Modulation (FM) Spectroscopy 147 Tj = e x p ( - 3j - iCj), 3j = c~jL/2, and Cj=njL(co~+jcom)/c, where j = 0 , _+1 denotes the components at coc, and co _+co~, respectively. Thus cSj ponent describes the amplitude attenuation and Cj describes the optical phase shift experienced by each component. The transmitted field is E3(t ) =/~3(t)/2 + c.c. with /~3(t) = E o - T_ 1 ~- exp [i(co~- corn)t] + T o exp (ico~t) + T1 ~- exp [i(co + co,,)tl . (2) The slowly varying intensity envelope, Ia(t), of the beam impinging on the photodetector is given by I3(t)=dEa(t)[2/8rc. Dropping terms of order M 2 and and assuming that [6o-31[, I~o-~_~l, Ir162 1r162 are all ~1, I3(t) = e-2~~ +(g)_l-6l)Mcoscom t + (r 1+ r 1 - 2r o)M sin to.J]. (3) The photodetector electrical signal is proportional to 13(0, and thus will contain a beat signal at the rf modulation frequency con if c5_1-614:0 or if r162162 Such a signal is easily detected using standard phase sensitive rf detection techniques, as illustrated in Fig. 2. The cosco"t component of the beat signal is proportional to the difference in amplitude loss experienced by the upper and lower sidebands, whereas the sine)mr component is proportional to the difference between the phase shift experienced by the carrier and the average of the phase shifts experienced by the sidebands. If co,, is small compared to the width of the spectral feature of interest, then the coscomt component is proportional to the derivative of the absorption and the sinc%t component is proportional to the second derivative of the dispersion. On the other hand, the absorption or dispersion lineshape of the spectral feature can be directly measured if com is large enough that the spectral feature is probed by a single isolated sideband, as shown in Fig. 2. The sideband can be scanned through the spectral feature by tuning coc or com. In either case, the losses and phase shifts experienced by the carrier and lower sideband remain essentially constant. Thus 6_ t = 3 o = 3 and r 2= r = r where 3 and r are the constant background loss and phase shift, respectively. If the quantities A6 and Ar are defined to express the deviations from the background values caused by the spectral feature, then _ 23,I3(t) = ~cE~ -e ti-A3Mcoscomt+dCMsincomt). (4) The cos%,t and sincomt components of the beat signal are thus, respectively, proportional to the absorption and dispersion induced by the spectral feature. The rf beat signal arises from a heterodyning of the FM sidebands and thus the signal strength is proportional to the geometrical mean of the intensities of one sideband and of the carrier. Thus the signal strength is proportional to E~M, while the intensity of the probing sideband is I = cE~M2/8rc. Because of the different M dependence, arbirarily large signal strengths can be achieved for arbitrarily low sideband intensities by properly adjusting the values orE 0 and M. The perturbing effects of the probing sideband on the spectral feature can thus be minimized. The null signal that occurs when the FM spectrum is not distorted, can be thought of as arising from a perfect cancellation of the rf signal due to the upper sideband beating against the carrier with the rf signal arising from the lower sideband beating against the carrier. The high sensitivity to the phase or amplitude changes experienced by one of the sidebands results from the disturbance of this perfect cancellation. For many applications, the spectral feature of interest has a Lorentzian or near Lorentzian lineshape. The lineshapes of the corresponding FM spectroscopy signals depend critically on the ratio of sideband spacing to the Lorentzian linewidth. It is convenient to define the spectral dependence of the dimensionless attenuation 3 and phase shift r by the relations 3 1 ~(co)= peak(R2(CO)+1)' r = ~peak(R(co) \R2(co)+ 2)' (5) (6) where ~peak is the peak attenuation at line center, f2 is the line center frequency, AO is the full width at half maximum, and R is a normalized frequency scale defined by cO--Q R(CO)- (AO/2 " (7) Thus, 3(0)=Speak and r Equations (5-7) define 3j = ~5(coj)and by = r for each spectral component of the FM optical spectrum. Again, the subscript j = 0, ___1 denotes the values at frequencies % and coo-+co,., respectively. Substitution of these results in (3) gives a complete specification of~the FM signal lineshape obtained when cot or co,, is scanned. It is often experimentally convenient to scan the sidebands through the spectral feature by tuning the laser carrier frequency coc with the rf modulation frequency co,, held constant. Figures 3 and 4 show the various possible heterodyne beat signals as function of the normalized parameter R 0=R(coc) which denotes S1 - AR = 0.05 S2 S3 T T i - - I i B AR = 0.1 0 t t [ - 0,5 -/,R = 0 -0.5 1.0 _-- A R = 0 . 8 0.5 ~ , 0 - -0.5 ,= 1.0 - AR = 1.6 -AR = 2.5 0.5 0 -0.5 -o -1.0 +J 1.0 E < 0.5 C: C~ 03 ..... / 0--- I F--- V -0.5 m -1.0 1.0 -AR =3.0 0.5 0---- -0.5 V -1.0 1.0 ~~4~i - r m _ AR = 4.0 0.5 O-- -- -0.5 -1.0 I I I I I I i I i I -16 -8 0 8 16 I I -16 I I -8 ., I I 0 I I 8 R0-----~ Fig. 3. Heterodyne beat signals $1, $2, and S 3, vs. R o for 0.05-< AR <-4.0 I I 16 I I I I I I I I I I I I -16 -8 0 8 16 S3 S2 S1 0.5 i- AR = 0,1 f ,-~-. 0 -0.5 1.0 t + L ~176 f A~:~o -0.5 L -1.0 T 1.0[0.5 2~R--2.5 0 i i i ~ -0.5 -1.0 l if 0. 2~R= 5.0 ~ ~Lt ~-o~ I "o -1.0 +.,, E < AR = 10.0 0. ._m -0.5 -1,0 ~__ E 0.5 1"0I 0 ,~R = 2_~ ~ _~ + -0.5 - -1.0 Vi l~I 0.5 0 -0,5 -1.0 l~I --y 0.5 0 AR = 50.0 t -0.5 -1.0 I I I I -80-40 I it11] 0 40 80 l l l l l -80-40 I I I I I 0 40 80 ,qv-- R0.-.-.~ Fig. 4. Heterodyne beat signals S 1, $2, and S a vs. R o for 0.1-<AR-<50.0 I I I I I I I I I I -80-40 0 40 80 I 150 the relative position of the carrier frequency with respect to the resonance. Here each curve corresponds to R o being smoothly tuned through the vicinity of the spectral feature with a constant sideband spacing described in terms of the parameter AR=~o'/(AO/2) and with 6pe,k equal tO unity. When R o = - A R , the coc+ c0,, sideband is resonant with the spectral feature ; when Ro=0, the % carrier is resonant; and when R o =AR, the co -co,, sideband is resonant. The column denoted $1 shows the amplitude of the absorption (cosco't) signal, the column denoted S z shows the amplitude of the dispersion (sinco't) signal, and column S 3 = ~ shows the modulus of the total beat signal. S~ and S 2 corresponds to dc signals which could be directly observed using the exact set-up shown in Fig. 1, provided that the phase adjuster is set properly. Intermediate settings of the phase adjuster produce mixed lineshapes. S 3 corresponds to the signa! which could be directly observed using a phase insensitive detector such as a spectrum analyzer to process the photodiode signal. The evolution from the wavelength modulation to the F M spectroscopy limits can clearly be seen. For AR<0.1, S 1 is very close to the derivative of the absorption and S 2 is very weak. For 0.1 < A R < 1.6, S~ reaches its full strength and becomes a somewhat distorted derivative lineshape, while S 2 grows rapidly in strength. It is interesting to note that, aside from polarity, in this region the shape of S 2 resembles that of S 1. For 3 < A R < 5 , S 1 begins to show separate resonances as the upper and lower frequency sidebands probe the spectral feature. However, S 2 and S 3 do not yet show resolved resonances, although for AR-~ 4, the curves show a pronounced "flattening" at the extrema. For AR >5, the Sa resonances become very close to the Lorentzian absorption lineshape. S 2 begins to show resolved dispersion shaped resonances as the upper sideband, carrier, and then the lower sideband probe the feature. S 3 also begins to show resonances at these locations. Since both the S 2 and S 3 resonances fall off linearly with detuning, substantial overlap occurs even for AR as large as 50. The S 3 lineshape always dips sharply to exact zero when R 0 = 0 corresponding to the carrier tuned to exact line center. In fact, for all values of AR, S~=S 2=S 3=0 when R o=0. This property of F M spectroscopy is useful in frequency locking applications. Signal-to-Noise Analysis For simplicity, we consider the case where a purely absorptive spectral feature is probed with a single isolated sideband and no background absorption is present. Then, from (4), with A~b=3=0, the slowly G . C . B j o r k l u n d et al. varying envelope Pa(t) of the optical power incident on a photoconductor of area A is given by P3(t) = Po(1 - A 5M cos c%t), (8) where P3(t)= AI3(t ) and P0 = ACE2~8re is the total laser power. The current i(t) generated by a photodetector of quantum efficiency tl and gain g is i(t)={+ is(t) where the dc photocurrent is given by P0 7= getl he)., (9) and the beat signal photocurrent is P0 is(t) = - get1~ - A (~M cos corot. (10) tlco c Thus the rms power of the beat signal is given by i2(t )=1~g 2e 2~12/Po~: I T ) A62M2" (11) Since 1If amplitude noise is insignificant at rf frequencies for single-mode lasers, the dominant sources of noise are thermal noise and shot noise generated at the photodetector. The rms noise power is given by (12) where the shot noise power is ,-~N=2eg-{Af=2gZe2~(~)Af, (13) and the thermal noise power is Here A f is the bandwidth of the detection electronics, k is Boltzman's constant, T is the temperature in K, and R is the input impedance of the detection electronics. The ratio of signal-to-noise (S/N) is S i2(t) N ~ 1 2 2 2{Po12A32M 2 ~g e ~ \hco~] 2gZe2tl Af+ -- (15) Af From (15), it can be seen that it is always advantageous to increase the total laser power Po, to increase the modulation index M, to have t/near unity, and to work with narrow band detection electronics. When Po>Pomi,=2kThcoc/tlg2e2R, shot noise predominantes over thermal noise. For photomultipliers, typical values are hcoc=3x10-19J, r/=0.1, g = 1 0 s, R = 50 f2, and T = 300 K, yielding Po rain= 2 X 10- ~2 W. For photodiodes, typical values for q and 9 are both 1, yielding Pomln=2 X 10 .3 W. Under these shot noise Frequency Modulation (FM) Spectroscopy 151 limited conditions, S N tl dt~2M2 4Af (16) and the minimum absorption, AcSmin, detectable with unity S/N in an integration time z = 1/Af is A6mi= 2 qM z Po z-1/2 The quantity M2(Po/ho~c)z is four times the total number of photons in a sideband arriving during interval z. Assuming t / = l , P o = 5 X 1 0 - a W , and M = 0 . 1 , a value of A6min= 1.5 x 10 .7 should be detectable for z = 1 s and AC]min=0.005 should be detectable f o r "c = 1 0 - 9 s. Actually achieving quantum noise limited detection sensitivity requires careful attention to experiment detail. Several additional effects can degrade the sensitivity, but correct experimental practice can reduce or eliminate their impact. In the following, we discuss a few of these effects on a phenomenological basis. It is important that the rf oscillator producing the modulation frequency c% be as stable as possible. Fluctuations in the phase or frequency of the oscillator produce noise at the double balanced mixer when the propagation times between the oscillator and the mixer for the optical signal differs from that for the local oscillator. The mixer then multiplies an electrical signal produced by the oscillator at one time with that produced at a different time, and possibly with a different phase. The result is a noisy dc signal. One characteristic of this kind of noise is that its amplitude depends on the phase shift introduced by the phase adjuster shown in Fig. 1. There is always one phase where the noise is minimized. Another strategy for minimizing this oscillator instability noise is to add delay lines between the oscillator and the double balanced mixer until the propagation paths for signal and local oscillators are equal. A second noise source results from the fact that available optical phase modulators do not produce the pure F M spectrum illustrated by (1). Rather, there is always a small component at com that results from a small imbalance in the amplitudes of the sidebands or a relative shift in phase which prevents the beat frequency from vanishing exactly. This residual AM can be detected by the photodiode and introduces a nonzero baseline. More seriously, the residual AM amplitude is modulated by the power fluctuations of the laser, and thus the level of the baseline fluctuates and introduces noise in the dc output signal. This noise can be minimized by carefully aligning the input and output polarizations of the modulator to balance the sidebands and by adjusting the relative phase of the local oscillator and signal to minimize the offset due to residual AM. Acceptable F M signals can then be obtained even with a relatively noisy laser and an unbalanced phase modulator. Finally, at the low signal levels described by (1'7) with a 1-s integration time ~, variations in the rf pickup due to movements in the laboratory can introduce drifts and noise. It is important to shield the detection apparatus from electromagnetic interference radiated by the relatively strong amplifier necessary to drive the modulator. With due care, quantum noise limited sensitivity, to modulated absorptions can readily be approached with F M spectroscopy [25, 38]. Conclusions Sensitive and rapid detection of narrow spectral features can be accomplished using F M spectroscopy. Since the heterodyne beat signals occur at rf frequencies where single-mode dye lasers are relatively noise free, shot noise limited detection is possible. The entire spectral feature can be scanned by tuning either the laser carrier frequency or the rf modulation frequency. The detailed lineshape of the F M spectroscopy signal depends on the phase of the rf detection electronics and on the ratio of the rf modulation frequency to the with of the spectral feature. Acknowledgement.The authors wish to thank L. Pawlowiczand J.R. DeLany for help in preparing the figures. References 1. G.C. Bjorklund: IBM Invention DisclosureSA 8790135(March 1979) G.C. Bjorklund: Opt. Lett. 5, 15 (1980) G.C. Bjorklund: U.S. Patent 4,297,035 (November 1981) 2. This technique was independentlysuggested by R.W.P. Drever as a means for servo-locking a tunable laser to a high finesse optical cavity. It was first experimentallyimplemented for this purpose by R.W.P. Drever, J.L. Hall, F.V. Kowalski, J. Hough, G.M. 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