The process of two-photon absorption (TPA) in semiconductors plays an important role at frequencies below the bandgap where it can dominate other absorption processes. TPA has been observed and studied in semiconductor materials of different composition and dimensionality^1–3. It may be useful in such devices as optical memories⁴ and switches⁵. Developing an understanding of the macroscopic effects of such a nonlinearity is essential for the operation of a variety of optical devices. However, more often than not TPA can severely limit the performance of electro-optic devices, especially at high intensities. For example, due to TPA, the nonlinear loss of saturable absorbers and other optical switches can become prohibitively high for intra-cavity applications requiring minimal losses.

In this paper we will discuss the role of TPA in one of the more common intracavity applications of semiconductors – that of mode-locking of femtosecond Cr:LiSrAlF lasers⁶. It has become commonplace to use this medium to generate pulses as short as 70 fs. However, one of the shortcomings of Cr:LiSrAlF is its low gain (a few percent per pass) which makes it very sensitive to intracavity loss. Hence, extreme care must be exercised when adding components inside the cavity such as mode-lockers. This has long been the argument used against the introduction of semiconductor devices into the cavities of high peak power, ultrafast lasers. The expected effects of TPA were presumed to overwhelm the low gain of the medium. However, Semiconductor saturable absorbers have been successfully used in such lasers⁶ without any detrimental effects to the operation of the laser.

The saturable Bragg reflector (SBR) used in ref.(6) is simply a DBR with a single quantum well (QW) in the first quarter-wavelength layer that acts as a saturable absorbers. Using this scheme as a mode-locker and reflector combination, strong mode-locking was achieved with the average mode-locked power approximately equal to the CW power (~ 100 mW). This indicates a very low level of intracavity loss introduced by the SBR and its reflectivity is deduced to be close to 99%. Yet, if one considers the high intracavity intensity (~5 GW/cm²) focused on the SBR, then one would expect to see the effects of two-photon absorption in GaAs. Indeed, with a TPA coefficient of β = 0.05cm/MW¹ and a DBR length of 3.4 μm, one would expect to observe a loss as large as 15% for a round trip through the reflector, in clear contrast to experimental results. This has led us to perform a detailed study of the effects of TPA in DBRs, taking the effect of nonreciprocity into account. To the best of our knowledge, this is the first time that TPA has been rigorously addressed in saturable DBRs.

We have modeled TPA in DBRs using the transfer matrix method⁷ to calculate the electric and magnetic fields in the optical waves as they propagate into the structure. The method was modified to accommodate the presence of TPA in the structure. The intensity was assumed to be constant within a given layer. This approximation is justified by the small thickness of the layers as compared to the anticipated nonlinear absorption length. The transfer matrix for one of the layers relates the total electric and magnetic fields at interface j to the fields at interface j + 1 according to

where c is the speed of light in vacuum, E and B are the electric and magnetic fields normalized by the incident electric field, ${\mathbf{E}}_0=\sqrt{2\eta I_0}$, and d is the layer thickness which is a quarter wavelength. Our analysis took into account the nonreciprocity between the intensities of the counter-propagating fields. This effect occurs because the presence of one field changes the value of TPA felt by the other⁸. This change in TPA is nonreciprocal because of the standing wave pattern which couples the two fields. The absorption coefficients are

where {i, r} refer to the incident and reflected intensities which are normalized by the initial incident intensity. l_a is the nonlinear absorption length defined as l_a = 1/β I ₀.

 

Fig. 1 The total optical energy density inside the DBR normalized by the incident energy density. The device structure is overlaid on the curve. The total length of the device is 3.432 μm.

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Figure 1 shows the distribution of optical field energy density inside the DBR. The device structure is overlaid on the figure. The energy density shown in the figure is normalized by the incident energy density. It is clear from the figure that the field penetration depth in the DBR is rather small. This small penetration depth limits the presence of significant TPA to the first few layers of the device. Figure 2 shows the reflectivity as a function of intensity of the AlGaAs/AlAs DBR at a wavelength of 850 nm. It is clear that the reflectivity of the DBR is unaffected by TPA for a wide range of intensities, obviously due to the small penetration depth. However, as absorption increases in the first few layers of the DBR, its reflectivity eventually drops to the Fresnel value for the front face, ((n′ - 1)/(n′ + 1))², where n′ is the index of refraction of GaAs which is modified by the high incident intensity. This final value is shown in figure 3 which will be explained in more detail later on. Naturally, the Fresnel value is related to the reflection from the front surface only with an index of refraction that is modified by the nonlinear effects. In other words, all the reflections from the interfaces of the DBR are absorbed and the medium behaves as if it were semi-infinite with only one boundary present.

 

Fig. 2 Reflectivity of the DBR vs. the input intensity in comparison with a similar piece of bulk GaAs in front of an ideal reflector. The operating point of 5 GW/cm² is shown.

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For comparison, we also plot the reflectivity of a more conventional saturable absorber mode-locker, consisting of a bulk cell of thickness equal to that of the DBR placed in front of an ideal reflector. It is easy to see that its reflectivity decreases drastically at intensities two orders of magnitude below that of the saturable DBR. The value of the critical intensity in the DBR where the sharp decrease in reflectivity takes place can be found to first order by considering the TPA length in the material which is defined as l_a = 1/βI ₀. When the TPA length is equal to twice the field penetration depth, then one would expect only a few reflections to make it back to the front face bringing the reflectivity to its Fresnel value. Simple calculations show that I_cr = 166 GW/cm², i.e., well in excess of 5 GW/cm² as confirmed by Fig. 2. Figure 3 illustrates the connection between the reflectivity and the absorption length. We estimate the penetration depth in the DBR to be 0.6μm which clearly corresponds to the region on figure 3 where the reflectivity drops.

 

Fig. 3 Reflectivity of the DBR as a function of the absorption length which is defined as 1/(βI ₀) where I ₀ is the incident intensity.

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In conclusion, we have shown theoretically that in a saturable DBR the detrimental effects of TPA are dramatically reduced in comparison to a conventional saturable absorber. While it is intuitively understood that the reduction in TPA is the result of the small penetration depth in the SBR, the main result of this work is that we obtain a good numerical estimate of the maximum intensities obtainable using the SBR for passive mode-locking. This theoretical result explains the experimental data of ref. (6) and shows that even higher intracavity powers can be used in that scheme.

We acknowledge the Air Force Office of Scientific Research for supporting this work.