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Intracavity phase interferometry is a phase sensing technique using mode-locked lasers in which two intracavity pulses circulate. The beat frequency between the two output frequency combs is proportional to a phase shift to be measured. A laser gyro is a particular implementation of this device. The demonstrated sensitivity of 10−8 of these devices could be manipulated by applying a giant dispersion to each tooth of the comb. It is shown that the resonant dispersion of a Fabry-Perot inserted in the cavity couples to the modes of the frequency comb, resulting in a large change in phase response.
© 2016 Optical Society of America
1. Introduction
Mode-locked lasers can be used as differential interferometers for phase measurements of extreme sensitivity. In the method of Intracavity Phase Interferometry (IPI) [1], two pulses circulate in a laser cavity to produce two frequency combs, that are interfered on a detector. IPI can be utilized to measure many properties including, but not limited to, nonlinear indices, magnetic fields, rotation, acceleration, electro-optic coefficients, fluid velocity, and linear indices. The quantity to be measured creates a phase difference Δϕ per cavity round-trip between the two pulses, resulting in a beat note on the detector. The laser gyro is a well known particular implementation of IPI where the phase difference is the Sagnac phase shift.
It is shown here that, for the same applied phase shift Δϕ, the beat note response can be modified by introducing a giant dispersion dk/dΩ at each mode of the frequency comb, where k(Ω) is the wave vector as a function of light angular frequency. Application of a giant dispersion to each mode of the comb is demonstrated experimentally by inserting a low finesse etalon in the mode-locked laser cavity. The giant positive dispersion results in a factor 2.9 reduction of the beat note slope. Changing the sign of the dispersion would result in a considerable increase of the beat note response. Methods to achieve that goal are proposed.
IPI in this paper refers to phase interferometry inside a laser cavity as opposed to an enhancement of the phase response associated with a passive Fabry-Perot as exploited in [2], Ma et al. While the distinction may appear merely quantitative because the laser can be seen as a Fabry-Perot of extreme finesse, the difference is qualitative because the response is an optical frequency shift rather than an amplitude modulation. Another distinction has to be made with the label “active-cavity interferometer” as used by Abramovici and Vager [3], where two gain media are inserted in 2 branches of a Michelson, which result in uncorrelated spontaneous emission noise. In IPI, there is only one gain medium acting on both pulses, and the noise in the two interfering frequency combs is correlated.
2. Phase response in a mode-locked laser
In the general case, “Intracavity Phase Interferometry” (IPI) involves mode-locked lasers into which is placed a physical quantity to be measured that creates a differential phase shift Δϕ between the two pulses. Because of the resonance condition of the laser, this phase shift is translated into a difference in optical frequency [1], which is measured as a beat note produced when interfering the two frequency combs generated by the laser. The measured beat note Δω can be expressed as:
where τϕ is the round-trip time of the pulse circulating in a laser cavity of perimeter P (in the case of a linear cavity of length L, P = 2L and ΔP = 2ΔL), and ω is the average optical pulse frequency. The technique of IPI has been shown to have extreme sensitivity, with the ability to resolve phase shift differences as small as ΔΦ ≈ 10−8 radians (corresponding to a beat note bandwidth of 0.16 Hz for a cavity of τϕ = 10 ns [1, 4]). This corresponds to an optical path difference of only 0.4 fm. If applied to a square ring laser of 4 m2, the beat note bandwidth of 0.16 Hz corresponds to a sensitivity in rotation rate change of ≈ 0.2 revolution/year.The principle of the dispersion affecting the response of intracavity phase interferometry (and in particular gyro response) is to make τϕ frequency dependent through an element having a transfer function 𝒯̃ (Ω) = |𝒯̃| exp[−iψ(Ω)] with giant dispersion:
where τϕ0 = (Pnp)/c is the round-trip time without dispersive element, np is the phase index of refraction at the central carrier frequency ω0 averaged over the elements of the cavity. −ψ(Ω) is the phase of the transfer function of the dispersive optical element inserted in the cavity. By substituting Eq. (2) in Eq. (1), the beat note is thus: It should be noted that all the above considerations pertain to phase resonances and velocities. In the case of normal dispersion, dψ/dΩ|ω0 is positive, resulting in a decrease of Δω. There is amplification of the phase response if dψ/dΩ|ω0 is negative. If we consider simply propagation through a transparent medium, ψ = [k(Ω) − k0]d, where k(Ω) = Ωn(Ω)/c, is the wavevector of a medium of thickness d and index n(Ω), and k0 = k(ω0), then the second term in the denominator of Eq. (3) is: where vg is the group velocity in a dielectric. Equation (4) deals fundamentally with the phase of the light in a laser cavity, and not the average envelope velocity of a circulating pulse. Previous works [5–7] associate the dispersion induced change in gyro response with a change in pulse velocity. Even though these proposals were dealing with group velocities, they were aimed exclusively at cw lasers, where there is no propagating wave packet, and it is difficult to associate the claimed enhancement/reduction of a gyroscopic (Sagnac) response with a change in group velocity. Furthermore, as has been pointed out by Arnold Sommerfeld in 1907 already [8], the mathematical quantity group velocity does not represent the velocity of an electromagnetic signal in frequency regions with large dispersion [9].It has been demonstrated in [10] that the envelope velocity of a pulse circulating in a mode-locked laser is not related to dk/dΩ|ω0 for a k vector averaged in the cavity, but to the gain and loss dynamics inside the laser. This point will be further emphasized in the present paper, where it is shown that the envelope velocity of circulating pulses or bunches of pulses can be varied, while the IPI response remains unchanged. A dispersion related change in IPI response is demonstrated here, totally unrelated to the average velocity of the circulating pulses. Therefore it is manipulation of the phase velocities, not the envelope velocities, that can enhance or reduce the gyro response. However, it will be shown that the teeth of the frequency comb of a mode-locked laser can be coupled to the modes of an intracavity etalon, and that a large dispersion results from this coupling, with a magnitude such that is of the order of unity.
3. Challenge in achieving laser dispersion
In order to achieve the very large dispersion required to modify the phase response through Eq. (3), a very narrow-band resonant structure is required. Narrow bandwidth implies long pulses or cw radiation, where most of the research in this field has focused. For instance, theoretical estimates have found that large dψ/dΩ can be produced by two-peak gain and coupled resonators [11, 12], or by an atomic medium [13]. The latter property has been verified experimentally. These regions of large dispersion have a small bandwidth, which needs only to exceed the largest beat note to be measured. For the mode-locked laser however, a large slope of the resonant phase ψ(Ω) versus frequency has to be seen by every tooth of the comb. In a mode-locked laser gyro, as with any implementation of intracavity phase interferometry, the two circulating pulses have to meet at the same point at every round-trip [1]. As the pulses circulating in opposite direction see an optical length differential, decreased or augmented by the giant dispersion, one would expect that the crossing point cannot be maintained, if the envelope velocity were simply equal to 1/(dk/dΩ). However, it has been established that the average envelope velocity in a mode-locked laser is dominated by gain and loss dynamics of the entire cavity, and that the crossing point of the two pulses can be maintained [1,10].
3.1. Giant dispersion of an intracavity etalon
In order for the dispersion enhancement of Section 2 to apply to a mode-locked laser, there should be a resonant structure for each mode of the laser. We have recently found a method to couple the teeth of a frequency comb to the modes of a low finesse Fabry-Perot (we used a 15mm thick fused silica etalon) [10]. This etalon is a resonant structure with much larger mode spacing locked to the modes of the laser. Relevant properties of the system etalon-laser [10] are:
- In the time domain, the output of the laser consists of bursts of ≈ 20 pulses, spaced by a cavity round-trip time
- In the frequency domain, the output of the laser consists in a double frequency comb, consisting in a low frequency (νℓf) comb interwoven in a high frequency comb (νhf)
- Both frequencies νℓf and νhf vary proportionally to each other when the cavity length is changed, or when the etalon is tilted, proving the locking of the laser modes with those of the etalon.
Theory and simulation [10] explain these phenomena, as indicating that the modes of the laser and those of the etalon are linked together. Indeed, to a change in the mode spacing of the laser corresponds a change in mode spacing of the high frequency train. The fine tunability of the optical frequency as well as the number of pulse generated indicate that the etalon has acquired the finesse of the laser. This implies that the transmission and dispersion are narrow compared with the mode spacing of the laser, as sketched in Fig. 1. To the transmission peaks (blue) of the etalon correspond a dispersion curve (green). Because the modes of the laser and those of the Fabry-Perot are coupled, and because the teeth of the mode-locked comb are rigorously equally spaced, the latter comb should experience the same dispersion as the Fabry-Perot (dotted green).
Fig. 1 The tooth spacing of the mode-locked comb (top) is smaller than that of the Fabry-Perot (bottom). If, for instance, the tooth spacing on top is increased by decreasing the laser cavity length, there is a corresponding increase of the etalon tooth spacing in the bottom figure [10]. To the Fabry-Perot transmission peaks (in blue) correspond dispersion peaks (in green), which transfer to the laser comb modes (dashed green). The dashed black lines show the modes of the comb split due to the relative phase shift given to the two intracavity pulses (IPI response, Sagnac phase shift in the case of rotation). Each split comb sees a different index.
The mode-locked laser is known to create a frequency comb with equally spaced modes [14–16]. By the same mechanism explained in reference [17], the teeth of the frequency comb will be locked by the modes of the etalon (Fig. 1). The low finesse uncoated etalon, when inserted in the mode-locked cavity, acquires a high finesse determined by the laser cavity. All these points studied with a linear laser are confirmed in the ring configuration presented in the next section.
4. Experiments with a mode-locked ring Ti:sapphire laser
A Ti:sapphire ring laser mode-locked by a saturable absorber was constructed. As sketched in Fig. 2, the main components of the laser are a Ti:sapphire gain crystal pumped by a frequency doubled vanadate laser, prisms for dispersion compensation, and a saturable absorber dye jet for mode-locking and for defining the crossing point of two pulses circulating in the cavity. Such a ring laser is one of the numerous mode-locked laser gyro and IPI configurations that has been demonstrated in past research [1,18]. It has also been established that, when a saturable absorber is used, it should be in a flowing configuration (dye jet) to prevent phase coupling between the two counter-circulating pulses, by randomizing the phase of the backscattering of one pulse into the other [1].
Fig. 2 Ti:sapphire ring laser, mode-locked by the saturable absorber Hexa-Indo-Tri-Carbocyanine-Iodide (HITCI) dissolved in a jet of ethylene glycol (S). The gain medium (G) and the phase modulator (M) are located at approximately 1/4 cavity perimeter from the saturable absorber S. An output coupling is made near the other pulse crossing point, and the two output pulse trains are made to interfere on a detector Db to monitor the beat note between the two frequency combs. Instead of rotating the laser, a phase shift/round-trip is provided by a phase modulator M driven by the detector Ds at the cavity repetition rate (details in reference [1]).
The response of intracavity phase interferometry is investigated by applying a differential phase shift per round trip with a phase modulator inserted in the cavity, located preferably at 1/4 perimeter away from the pulse crossing point. The phase modulator is a 100 μm thick plate of lithium niobate, oriented at Brewster angle, with electrodes on one face to apply an electric field along the z crystallographic axis. The applied field is achieved by narrow band amplification of the signal of a detector monitoring one of the output pulse trains (detector Ds monitoring the pulse train from the clockwise circulating pulse in Fig. 2). The phase of the applied field is adjusted to be maximum for one pulse, and minimum for the countercirculating pulse.
5. Modifications of the phase response and envelope velocities with an intra-cavity Fabry-Perot etalon
5.1. Envelope velocities
A 15.119 mm thick fused silica etalon, uncoated, is inserted in the ring cavity. This material has a phase index of n = 1.4533 at the laser wavelength of 800 nm, and a group index ng = 1.4671. Similar to the situation in a linear mode-locked laser [10, 19], despite the very low finesse of this Fabry-Perot, it influences the mode-locking by creating a high frequency (close to the etalon round-trip frequency) pulse train which repeats itself at a lower frequency (close to the original laser repetition rate). Figure 3(a) shows the high repetition rate pulse train nested inside the laser pulse train, which can be explained as follows. At every round-trip, each pulse of the high frequency train adds coherently to the next one. This coherence is established through the resonance condition of the pulses within the laser cavity. Figure 3(b) shows the spectrum of the nested frequency comb associated with the double pulse train.
Fig. 3 (a) Oscilloscope trace of a high repetition pulse train created by the intracavity etalon. This picture is recorded with a fast photodiode and an 8 GHz oscilloscope. (b) Spectrum of the nested frequency comb, recorded with the same photodiode and a spectrum analyzer. The center frequency is 6.8 GHz, and the span 3 GHz. The marker is at 6.835 GHz. The baseline step is an artifact of the spectrum analyzer.
The changes in repetition rate of the laser cavity, after introduction of the Fabry-Perot, cannot be explained by the traditional group delay introduced by the etalon. The transmission function of a Fabry-Perot of thickness d and intensity reflectivity R = |r|2 (where r is the field reflectivity), at an internal angle θ with the normal, is:
The group delay is the first derivative of the phase ψ of this expression with respect to frequency: where δ = kd cos θ = ωnpd cos θ/c. This expression being correct near a resonance, we will make the approximation tan δ ≈ δ. To remain within the bandwidth of the Fabry-Perot transmission, (1 + R)/(1 − R)δ << 1, and:It has been demonstrated that the average envelope velocity of the pulse circulating in the mode-locked cavity differ considerably from the group delay of Eq. (6) and (7). The group delays are determined by dynamic gain and loss considerations. For instance, the continuous transfer of energy from each pulse of the high frequency train into the next one results in a delay of the center of gravity of that train. Saturable gain has the opposite effect of accelerating the pulse trains. The average velocities, as modified by the Fabry-Perot etalon, as function of the tilt of the etalon, have been measured and matched with theoretical simulations [10].
Figure 4 shows the repetition rate dependence of ring laser as a function of the angular tilt of the etalon with respect to the normal [Fig. 4(a)], and as a function of the cavity perimeter[Fig. 4(b)]. Figure 4(a) cannot be explained by the angular dependence of the “group delay” in Eq. (6). Furthermore, there is a change in repetition rate from 99.3072 MHz (10.069763 ns round-trip time) to 99.016 MHz (10.099378 ns round-trip time) by insertion of the Fabry-Perot, which is a change of round-trip time of 29.615 ps. This difference does not correspond to the insertion of the etalon, which should add (ng − 1)d/c = (1.4671 − 1) × 15.119/c = 23.54 ps. The measured 29.615 ps corresponds instead to the insertion of an etalon of thickness 19.021 mm, or 3.902 mm more than the inserted glass!
Fig. 4 (a) Plot of the tooth spacing of the frequency comb corresponding to the clockwise circulating group of pulses, as a function of the tilt angle of the etalon. (b) Tooth spacing of the high frequency comb (solid line) and the low frequency comb (ring cavity repetition rate - dashed line) versus cavity perimeter.
Another evidence of the coupling between the modes of the laser and those of the etalon is the plot of Fig. 4 (b) where the cavity length dependence of the low frequency and high frequency combs are compared. The pulse round-trip period in the etalon and the big ring are both linked to the perimeter of the large ring cavity. It should be noted that all these properties that have been analyzed for an intracavity etalon [10] are only observed when the pulse duration is much shorter than the (uncoated) Fabry-Perot etalon round-trip time.
5.2. Phase response
It is clear from the previous section that the modes of the laser comb and Fabry-Perot are coupled, and therefore it can be expected that the laser comb will be influenced by the dispersion of the Fabry-Perot. The previous measurements have also established that the average pulse envelope velocity in the laser is not related to dψ/dΩ as is generally taken for granted. Indeed, Figure 4 (a) has shown that the pulse envelope velocity varies significantly with the angle θ. Over that range, dψ/dΩ remains constant, and so is the phase response plotted as a function of tilt angle in Fig. 5(a). This demonstrates clearly that as the envelope velocity is manipulated by this change in angle, the gyro response remains unchanged. In Fig. 5(b), the beat note is plotted as a function of applied voltage on the lithium niobate modulator, before (solid curve) and after insertion of the Fabry-Perot. The gyro response slope switches from 2.3 kHz/V without Fabry-Perot, to 0.8 kHz/V, which implies, from Eq. (3), that:
Therefore, from Eq. (7) we deduce that the ratio of the slopes minus one is the product of the ratio of the laser to etalon optical lengths, times (1 + R)/(1 − R): The ratio (1 + R)/(1 − R) should thus be 1.9 × 68.68 = 130, which corresponds to a value of effective reflectivity R = 98%. We note that this corresponds to the reflectivity needed to create a bunch of 20 pulses.Fig. 5 (a) Slope of the beat note response as a function of Fabry-Perot angle. The slope remains at 0.8 kHz/V independently of the tilt of the Fabry-Perot. (b) Comparison of the beat note response before (solid curve) and after (dashed curve) insertion of the fabry-Perot.
5.3. Negative versus positive dispersion
In the case of an intracavity etalon, the modes of the laser couple to those of the etalon because this is the configuration of minimum losses. This is also the reason that normal dispersion (dψ/dΩ > 0) is observed, as the Kramers-Kronig correspondent of a gain line. Using the etalon in reflection would provide the negative dispersion needed for amplification of the phase response, according to Eq. (3). However, the reflection characteristic would favor operation of the laser with the modes between cavity resonances. The matching of the resonance could be forced by active stabilization of the laser modes, or by synchronous pumping (OPO configuration), which may also require active stabilization. One solution that addresses the sign of the dispersion without introducing periodic losses is to substitute a cavity mirror by a Gires-Tournois interferometer. The latter is essentially an etalon of which one face has 100% reflectivity, and the other face a field reflectivity r. Its transfer function is given by [20]:
where δ = 2kd cos θ is the phase delay, θ the internal angle. Near a resonance δ = 2Nπ, the phase shift of the device can be approximated by: Near the resonance, the group delay is approximately: which has indeed the correct sign for enhancement of the gyro response.Adding a Gires Tournois interferometer of exactly the same thickness to the present cavity will add a negative component to the denominator of Eq. (3). The Fabry-Perot with its resonances will lock the modes of the laser, which are then also locked to those of the Gires-Tournois of the same thickness. The value of the reflectivity that will make the denominator of Eq. (3) equal to zero is r = 99% (intensity reflectivity of r2 = 0.98).
6. Conclusion
This work addresses the phase response of a sensor based on Intracavity Phase Interferometry. The device is a mode-locked laser in which two pulses circulate in the cavity, and are given a phase shift relative to each other by the physical quantity to be measured. The response of the device is a beat frequency between the two frequency combs issued from the laser. The beat frequency is proportional to the phase shift (or physical parameter to be measured). The proportionality constant (between beat frequency and phase) can be modified by introducing a giant dispersion for each tooth of the frequency comb. It is demonstrated experimentally with a mode-locked ring laser that the desired coupling of a dispersion to all modes is obtained by inserting a low finesse etalon in the laser cavity. It has been shown [10] that the gain–loss dynamics force the frequency comb of the laser to coincide with the frequency comb of the inserted etalon. In particular, from the observed change of the repetition rate of the laser in dependence on the tilt (≡ optical length) of the etalon, it can be concluded that this dispersion acts on all teeth of the laser’s frequency comb. The beat note (or gyroscopic) response modification is a large reduction (by a factor 2.9), because the Fabry-Perot etalon introduces a positive (normal) resonant dispersion. It is pointed out that a resonant negative dispersion, that would enhance the beat note response, could be achieved with a Gires-Tournois interferometer. There are numerous other possibilities, for instance two photon absorption, that can be exploited to achieve a resonant dispersion affecting all modes of the frequency comb, to enhance the sensitivity of this class of devices.
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