A series of practical tests of slow light (light with reduced group velocity) in saturable absorbers is proposed. These include experimental tests for saturable absorption, which can mimic slow light effects in saturable media, the dependence of slow light on the mutual coherence of pump and probe, since both slow and fast light effects can be simulated with incoherent sources, and the influence of polarization. The principal requirements for practical observation of spectral hole burning are reviewed and shown to be achievable for a wide range of saturable media with the narrow line sources now available.
©2010 Optical Society of America
Theoretical analysis of the dynamic response of saturable media has shown that the phase shift and time delay associated with signal transmission by a saturable absorber can be equally interpreted as a consequence of a temporal variation of the absorption or a reduced group velocity (‘slow light’) associated with coherent population oscillations (CPO) [1,2]. This equivalence can be traced to the perturbation treatment of wave interactions in saturable absorbers, in which the hole-burning and saturation terms are of equal magnitude, the former having a Lorentzian frequency profile, the latter being independent of frequency . As a consequence, the phase shift and modulation gain of the transmitted signal – together with hole burning and power broadening in the frequency modulation spectrum  – observed in single-beam experiments utilizing an intensity modulated laser source [5–8], can be analyzed in terms of the delayed response of the saturable absorber [9,10]. Only when independent pump and probe beams are employed is there any practical means of distinguishing the two. When this is done, the probe can be tuned to scan the homogeneously broadened absorption profile and reveal the coherent hole created within it via the coherent population oscillations driven by the beating of pump and probe . The same method can also be used to test whether hole burning is necessary to produce the observed signal delay. A recent experiment on erbium doped optical fiber (EDF) provides a clear counter-example in showing that the modulation phase may be either delayed or advanced by simply changing the relative phase of the modulated pump and probe for two counter-propagating beams at widely separated wavelengths within the broad absorption band . These observations are fully consistent with saturable absorption theory, but difficult to reconcile with ‘slow light’, given the enormous beat frequencies involved. The simple explanation offered, that the pump saturates the homogeneously broadened absorption band as a whole, the modulated probe undergoing a phase shift (delay or advance) determined by saturable absorption theory , seems the obvious choice in this case. Given the differing interpretations of single beam experiments in saturable absorbers, some practical pump and probe tests are presented here in the hope of stimulating a more critical approach to the experimental investigation of slow light. The theoretical results for coherent hole burning are outlined in the first part of the paper, and compared with the corresponding expressions for saturable absorption. This is followed by a discussion of the principal requirements placed on the sources used for slow light experiments i.e. line-width, coherence and polarization, and some simple methods proposed for testing the dependence of slow light effects on these. Benchmark tests for saturable absorption based on the method described in  are discussed next with reference to a number of saturable media and sources for testing their response. The more demanding requirements for practical observation of spectral hole burning are considered in the following section; these can be met for a wide range of saturable absorbers with the narrow line sources now available, illustrated by selecting appropriate sources for performing benchmark hole burning experiments. In the dynamic regime, the different outcomes predicted for simultaneous and sequential application of pump and probe pulses are discussed as an alternative method of distinguishing slow light and saturable absorption. The paper concludes with a discussion of the similarities and the differences between slow light, a linear effect, and the inherently non-linear nature of saturable absorption [1,9].
Perturbation analysis of the density matrix equations for the non-linear interaction of a strong wave of angular frequency ω1 and a weak wave of angular frequency ω2 in a two-level saturable absorber leads to a reduction in the power absorbed from the weak wave3]. The frequency dependent term represents the coherent hole of width Δωh ~2/τs formed by the beating of the two waves coupled via the non-linear interaction [Fig. 1 ]. The hole in the absorption line leads to phase dispersion and a reduction of the group velocity . The constant term represents partial saturation of the absorption and is independent of frequency. The perturbation terms for hole burning and saturable absorption are of equal magnitude , which gives rise to the close similarity the two effects have on signal transmission by a saturable absorber. To distinguish them requires the use of narrow line-width sources and phase sensitive detection to scan the coherent hole and map the phase dispersion . The magnitude of the perturbation depends on the relative polarization of pump and probe, but not on propagation direction. The hole depth is reduced by a factor of three when the pump and probe beams are orthogonally polarized .
Saturable absorption, modeled as an intensity dependent incoherent interaction with the non-linear medium, can be analyzed exactly at all levels of saturation. The transmission of a weakly modulated beam of intensity I(t) = I0(1 + mcosωt) and modulation amplitude m<<1 produces a phase shift φ and modulation gain |K|>1 in the transmitted signal: Itr(t) = TsI0[1 + m|K|cos(ωt–φ)], where Ts is the steady state transmission. The complex gain factor K(ω) (amplitude and phase) is given by 5,10]. Pulse transmission by a saturable absorber can be analyzed via the dynamic transmission equation uτ + u = βf(τ)[1-T(τ)] , where u(τ) = ln(T(τ)/T0), T(τ) is the time dependent transmission with initial value T0, f(τ) is the pulse envelope and τ = t/τs. This has the analytic solution u(τ) = e-τ ∫-∞ βf(τ')eτ'dτ' in the low transmission limit Ts<<1 and has been used to analyze ‘slow light’ pulse propagation in saturable media , for which there is no equivalent theoretical treatment based on CPO.
3. Source characteristics
The original CPO slow light experiments utilized multimode laser sources modulated at audio frequencies to create coherent holes of frequency width comparable to the inverse relaxation time i.e. a few tens of Hz in ruby  and Er-fiber , which could only be inferred indirectly from time delays in the transmitted signals, and were thus indistinguishable from saturable absorption [1,9]. Benchmark tests of slow light require both pump and probe sources to meet this restriction on frequency width, the probe frequency being precision tuned to sweep across the locked pump frequency to scan the coherent hole created by their mutual interaction in the nonlinear medium . When the laser pump frequency is subject to rapid fluctuations, its power spectrum has a narrow central peak of frequency width Δν ≈<Ω2>/q, where <Ω2> and q are respectively the mean square amplitude and the rate of frequency jumps. Thus the peak can be much narrower than the frequency excursion when q is large [15,16].
It has been pointed out that the nonlinear medium can only respond to beat frequencies of the order of the inverse relaxation time of the absorption, the delayed response of the medium itself meeting the narrow frequency requirement, such that narrow line sources are not necessary for CPO slow light experiments . However, this raises practical and theoretical difficulties for demonstrating the existence of the coherent hole and is not considered here.
A further issue to be addressed is the coherence of the pump source, usually an Ar-ion laser . The high spectral brightness multimode output constitutes an intense, spectrally broad, partially coherent light source subject to both phase and frequency fluctuations. The coherence of the source could be significantly improved by ensuring single axial mode operation of the pump laser, thereby reducing the bandwidth to that determined by the modal frequency and phase fluctuations. Phase coherence of pump and probe is a necessary condition for coherent hole burning [3,11]. This is automatically ensured in modulated single beam experiments, where the frequency fluctuations of the pump are also present in the probe, but the pump source would need to satisfy the restriction on frequency width Δωp ≈Δωh ≈2/τs for a benchmark hole-burning experiment. Diverting part of the incident pump beam to form the probe would enable independent control of modulation amplitude and phase, while retaining mutual coherence. It also allows independent control of polarization and choice of interaction geometry. The coherence range of pump and probe sources used for benchmark tests on saturable absorbers with ns lifetimes could be scanned in a tabletop interferometer, enabling the dependence of group velocity reduction on the mutual coherence of pump and probe to be quantified.
Phase coherence is not necessary for saturating the absorption, as exemplified by the flash-lamp pumped ruby laser , while ‘fast light’ has been simulated in photochromic glass using an incoherent source (halogen lamp) to induce photo-darkening . Shin et al have demonstrated reduced pulse distortion for a laser pulse superimposed on a mutually incoherent background , thereby confirming the intensity dependent nature of the effect . These results underscore the need for experimental tests of the dependence of slow light phenomena on source coherence to distinguish between ‘slow light’ and saturable absorption.
Perturbation analysis of a weak probe wave beating with a strong pump wave in a saturable absorber shows that the magnitude of the hole-burning effect depends on the relative polarization of pump and probe; the hole depth is greater by a factor of three for parallel polarizations than perpendicular, and can differ by an order of magnitude at high pump saturation . This does not appear to have been tested in any CPO experiment thus far. In the definitive multiple quantum well (MQW) slow light experiment, where the coherent hole was created by combining the output of a tunable diode laser with a single mode Ti:sapphire pump and displayed against the broad saturable absorption background, parallel polarizations of pump and probe were used . The effect of selecting orthogonal polarizations should be clearly visible via a much reduced hole depth in this case, resulting in a diminished ‘slow light’ effect. Where the coherent hole is not probed directly, the phase shift of the modulated probe signal for orthogonal polarizations of pump and probe will be smaller than for parallel polarizations in the presence of a coherent hole, again amenable to experiment.
4. Benchmark experiments
Following the detailed analyses presented in  and  demonstrating the close similarities of dynamic saturable absorption and CPO ‘slow light’ phenomena, some practical tests aimed at distinguishing the two are described below.
4.1 Saturable absorption
There seems no simple way of validating the original CPO ‘slow light’ experiments in ruby, alexandrite and bacteriorhodopsin (bR) [5–7], because the frequency width of the coherent hole for slow light in these media is a few tens of Hz (sub-Hz for bR film ) – too narrow to observe directly – while the multimode pump sources used have spectral widths in the GHz range. In the absence of independent evidence, these experiments could be said to simulate slow light rather than demonstrate it, the results being indistinguishable from saturable absorption . They could be repeated using the same pump source (Ar ion or Ar-Kr laser) and an independent probe e.g. a modulated diode laser, to test this . An etalon placed in the pump laser cavity would limit its frequency width by ensuring single longitudinal mode (SLM) operation. Ruby, alexandrite and bR can be tested by the method previously applied to erbium fiber, using an independent probe e.g. modulated diode laser, operating at a different wavelength to the modulated pump source within the homogeneously broadened absorption line . Achieving a signal advance τadv ~1 sec in bR-doped polymer film in such an experiment would be particularly impressive, given the claim of ‘snail pace’ light for this medium . ‘Fast light’ phenomena arising from reverse saturable absorption (RSA) in alexandrite  and fullerene  could be similarly investigated.
4.2 Coherent hole burning
4.2.1 Source line-width
The recent development of a <1 kHz line-width, injection locked Ti:sapphire laser  enables some crucial slow light experiments to be conducted in saturable absorbers with sub-ms lifetimes, the pump source spectrum being of comparable width to the coherent hole. This is a minimum condition for performing single beam experiments with an amplitude-modulated pump [5–8], but is not sufficient to validate slow light, a tunable probe of similar width being required for observing the coherent hole . Tunable diode lasers have been used in CPO slow light experiments in semi-conductor quantum well structures to scan the coherent hole in the optical spectrum . The relaxation times of semi-conductors lie in the ns range for inter-band transitions, enabling slow light experiments to be conducted with GHz width pump and probe beams. The phase shifts of the frequency-shifted sidebands can be detected independently via optical filtering . The <1 kHz line-width Ti:sapphire laser in conjunction with a narrow line dye laser (~200 Hz)  would be suitable for probing the anti-hole in alexandrite, with 260 μs lifetime for reverse saturable absorption [6,25], the frequency doubled output of the Ti:sapphire laser being matched to the absorption band of alexandrite. The same combination could be used for probing the anti-hole in fullerene, responsible for ‘fast light’ . However, a lower power frequency stabilized source e.g. narrow line dye laser , could be used in place of the Ti:sapphire pump because of the large absorption cross-section  and relatively low saturation intensity of fullerene. Hole-burning experiments in saturable media with relaxation times of several ms e.g. ruby, Er-fiber, require light sources with line-widths of just a few tens of Hz. Here we may look to the optical frequency comb, with component line-widths Δν<40 Hz , and sub-Hz line-width lasers locked to ultra-stable reference cavities [28,29]. The 100 kHz line-width required for probing the coherent hole in Cr4+:YAG, commonly used as a Q-switch in solid state laser systems , could be met by an extended cavity diode laser (ECDL ), but a high intensity pump source will be required to saturate the absorption e.g. single-mode Ti:sapphire laser. Cr4+:YAG also undergoes excited state absorption (ESA), which could be probed with this experimental setup. Spectral hole-burning observed in early experiments on intra-cavity dye cells used as passive laser Q-switches  could be studied using a repetitively Q-switched Ti:sapphire laser as the pump source, combined with a tunable diode laser probe.
4.2.2 Mutual coherence
Because of the coherent nature of the wave interaction in a saturable absorber, which gives rise to coherent population oscillations in the nonlinear medium, quantitative measurements of the dependence of phase shift and time delay on the mutual coherence of pump and probe sources are called for, as noted in Sec 3.2. Phase dispersion in semiconductor quantum well slow light experiments has been measured utilizing a Mach-Zehnder interferometer  and practical tests involving an incoherent source  and mutually incoherent sources  have been reported, but no systematic determination of the dependence on the degree of mutual coherence Γ12(τ) = 〈E1(t)E2*(t + τ)〉, where τ is the time delay between the pump and probe fields E1, E2, has been presented thus far. For sources with coherence lengths in the metre range, appropriate for experiments on nonlinear media with ns lifetimes, such as semiconductor quantum wells , a Michelson type interferometer is proposed for scanning the range of mutual coherence of pump and probe. Measurement of phase shift or time delay vs. the difference in arm length ΔL would reveal the dependence on mutual coherence Γ12(τ) of the sources directly. Where a clear dependence on ΔL is recorded, this will support the coherent population oscillation model of ‘slow light’. If no dependence is found, the intensity dependent saturable absorption model is favored over the field dependent coherent interaction . For pulsed operation, the pulse width τp should equal or exceed the coherence time τcoh ≈τs to allow the spatial overlap of pump and probe pulses to be scanned over the full coherence range Lcoh = cτcoh ≈cτs.
4.3 Sequential pump and probe
Coherent hole burning in the homogeneously broadened absorption line requires the simultaneous presence of pump and probe fields, the coherent hole vanishing when either of these is removed . Sequential application of a saturating pump pulse and delayed probe pulse to the non-linear medium could be used to distinguish between saturable absorption and slow light. When this is done, saturable absorption theory predicts an advance in the transmitted probe pulse, rather than the delay observed when pump and probe are applied simultaneously. For the benchmark experiment, a broadband pump can be used to saturate the absorption, but the probe should satisfy the line-width requirement for coherent hole burning. The results for simultaneous and sequential application of pump and probe can then be compared with confidence that an observed delay for simultaneous application corresponds to slow light, while an advance in the case of sequential application corresponds to saturable absorption. When pump and probe fields are applied separately, they cannot interact, and the result should be independent of polarization (which can also be tested), whereas it will be polarization dependent for simultaneous application. It is convenient to analyze the case of a rectangular pump pulse of sufficient duration to achieve steady state saturation. The pump action ceases abruptly at the end of the pulse, when τ = τ' say. Thereafter the logarithmic transmission function will decay exponentially: u(τ) = use-(τ-τ') and the transmission T(τ) = T0eu(τ) super-exponentially as the population returns to the ground state . For a weak probe signal s(τ) of intensity Is(τ) = βss(τ), with βs<<1, we may approximate the transmitted signal as Is'(τ) ≈βss(τ)T(τ), for negligible modulation of the absorption by the probe. The transmission of a modulated signal satisfying these conditions is shown in Fig. 2 , the initial cycle showing a large fractional advance, which decreases rapidly for succeeding cycles of the transmitted signal. The frequency range over which the advance occurs is determined by the absorption relaxation time: Δω ~1/τs. For reverse saturable absorption, the probe pulse will experience a delay in transmission, rather than the advance normally observed [6,21,35].
The results of the great majority of ‘slow light’ experiments involving saturable absorbers are indistinguishable from saturable absorption phenomena, the observed phase shifts in transmitted signals and delays in pulse transmission being entirely consistent with the slow response of the non-linear medium [1,9,10]. Where independent pump and probe beams are used, the coherent hole burned in the homogeneously broadened absorption is revealed when the narrow linewidth requirement is met ; by contrast, a probe operating at a different wavelength to the pump shows broad saturation of the absorption and a phase advance or phase delay in the transmitted signal, depending on the relative phase of the modulated pump and probe beams . Clearly, some care is required in selecting appropriate sources and experimental conditions for the observation of slow light. In addition to the line-width requirement, the mutual coherence and polarization state of the sources are pertinent to the hole burning process; for example, both slow and fast light effects can be simulated with an incoherent source, such as a halogen lamp , while the depth of the coherent hole created by the interaction of coherent pump and probe sources – and the associated reduction in group velocity – is determined by the relative polarization of the pump and probe beams, being greatly reduced when these are orthogonally polarized . A dynamic test of saturable absorption is provided by abruptly terminating the pump, the transmitted signal showing a phase advance during the subsequent recovery of the absorption instead of the expected delay (Fig. 2). Saturable absorption is amenable to rate equation analysis, which treats it as an intensity dependent, incoherent process, independent of spectral width. However, a saturable absorption term occurs in the perturbation solution of the density matrix equations, which treat the wave interactions with the non-linear medium . Here too it is spectrally flat, but it is dependent on polarization, an aspect that could easily be tested.
Because of the inverse relation between relaxation time and coherent hole width , the potential reduction of group velocity is greatest in saturable media with the longest excited state lifetime, but for the smallest signal bandwidth. Conversely, a large signal bandwidth implies a smaller reduction of group velocity, but the coherent hole is more easily observed . Hence, the lowest ‘slow light’ speeds have been inferred indirectly from phase shifts observed in the transmitted signal, which can equally well be interpreted as a saturable absorption effect i.e. not slow light, but the slow response of the non-linear medium . This may seem of little consequence for many practical applications, but the two phenomena are nevertheless physically distinct . In the frequency domain, the finite relaxation time associated with saturable absorption gives rise to a non-linear response in the modulation frequency spectrum (up to ~300 Hz in ruby ), whereas the slow light effect arises from the coherent hole of width 2/τs burned in the optical absorption spectrum (centered at ν0 ~1015 Hz in ruby). It is this distinction which the benchmark experiments are intended to address.
Given the ambiguity in the interpretation of ‘slow light’ experiments on saturable absorbers - and recent counter-demonstrations in EDF (erbium-doped fiber)  and photochromic glass  - more definitive experiments on signal transmission are needed to establish the reality of low group velocity ‘slow light’ via coherent population oscillations in these media. Benchmark tests along the lines proposed may help resolve this ambiguity for a wide range of saturable absorbers.
I am indebted to Valeriĭ Zapasskiĭ for first bringing the ambiguity of slow light in saturable absorbers to my attention and for his valued comments on this subject. I also wish to thank Bruno Macke for sharing his insights on slow light and for kindly providing reference material related to motional narrowing.
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