High-speed laser modulation beyond the relaxation resonance frequency limit Wesley D. Sacher*, Eric J. Zhang, Brett A. Kruger, and Joyce K. S. Poon Department of Electrical and Computer Engineering and the Institute for Optical Sciences, University of Toronto, 10 King’s College Rd., Toronto, Ontario M5S 3G4, Canada *[email protected]

Abstract: We propose and show that for coupling modulated lasers (CMLs), in which the output coupler is modulated rather than the pump rate, the conventional relaxation resonance frequency limit to the laser modulation bandwidth can be circumvented. The modulation response is limited only by the coupler. Although CMLs are best suited to microcavities, as a proof-of-principle, a coupling-modulated erbium-doped fiber laser is modulated at 1 Gb/s, over 10000 times its relaxation resonance frequency. ©2010 Optical Society of America OCIS codes: (140.3460) Lasers; (230.0230) Optical devices; (140.4780) Optical resonators.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

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21. D. Chen, H. R. Fetterman, A. Chen, W. H. Steier, L. R. Dalton, W. Wang, and Y. Shi, “Demonstration of 110 GHz electro-optic polymer modulators,” Appl. Phys. Lett. 70(25), 3335–3337 (1997). 22. F. Koyama, and K. Iga, “Frequency chirping in external modulators,” J. Lightwave Technol. 6(1), 87–93 (1988). 23. Y. Cheng, J. T. Kringlebotn, W. H. Loh, R. I. Laming, and D. N. Payne, “Stable single-frequency traveling-wave fiber loop laser with integral saturable-absorber-based tracking narrow-band filter,” Opt. Lett. 20(8), 875–877 (1995). 24. Y. W. Song, S. A. Havstad, D. Starodubov, Y. Xie, A. E. Willner, and J. Feinberg, “40-nm-wide tunable fiber ring laser with single-mode operation using a highly stretchable FBG,” IEEE Photon. Technol. Lett. 13(11), 1167–1169 (2001). 25. K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003). 26. W. D. Sacher, and J. K. S. Poon, “Microring quadrature modulators,” Opt. Lett. 34(24), 3878–3880 (2009). 27. I. Kaminow, T. Li, and A. Willner, eds., Optical Fiber Telecommunications V B: Systems and Networks Ch. 2 (Elsevier, 2008).

1. Introduction Over the past several decades, substantial efforts have been undertaken to increase the relaxation resonance frequency of lasers to increase the maximum bandwidth at which lasers can be directly modulated; for example, through engineering the quantum confinement in the active medium [1–8], utilizing injection locking techniques [8–10], and leveraging cavity quantum electrodynamics effects [11, 12]. The direct modulation of a laser involves modulating a laser parameter, most often the pump rate, to impart information onto an optical carrier. It is commonly accepted that relaxation oscillations, which arise from the coupling between the atomic population in the gain medium and the photon density in the optical cavity, limit the modulation bandwidth of lasers [13, 14]. The relaxation resonance frequency, fR, decreases with increasing resonator quality factor, Q, and increases with the cavity photon density. In addition, as the pump rate is modulated, the gain and refractive index inside the laser cavity are modified; hence, the laser output is also unavoidably chirped [8, 13, 14]. For typical semiconductor lasers, fR is about 1-10 GHz [8, 13, 14], and for erbium-doped fiber lasers (EDFLs), fR is roughly < 1 MHz [15]. In this Letter, we propose and show that for a coupling-modulated laser (CML), in which the output coupler is modulated rather than the pump, the laser modulation bandwidth can be orders of magnitude larger than fR. Moreover, CMLs can circumvent the conventional chirp limitations and trade-offs between the laser threshold, drive power, and modulation bandwidth. In addition to large modulation bandwidths, CMLs can also achieve high extinction ratios with low drive powers, making them promising for chip-scale optical interconnects and networks. Although the technological benefits of CMLs are most obvious using microcavities, to illustrate a CML inherits the coupler modulation response, we demonstrate a coupling-modulated EDFL at 1 Gb/s, over 10000 times its relaxation resonance frequency. 2. Principle of operation 2.1 Intuitive explanation A CML implemented with a ring cavity is shown in Fig. 1(a), where the cavity is integrated with a 1x2 variable coupler. κ and σ are the field cross- and through-coupling of the coupler, respectively. The basic principle of operation can be understood simply. If we modulate the coupler and keep the pumping inside the cavity constant, the output power, Pout(t), is

Pout (t ) =| κ (t ) |2 Pin (t ),

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(1)

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Fig. 1. (a) Schematic of the coupling-modulated laser. (b) Modulation response of the intracavity field from Eq. (3) for a microcavity laser with |κ0|2 = 1%, τc = 2 ns, τph = 100 ps, fR = 1 GHz, and τ = 1.46 ps (i.e., the cavity length is 125 µm and the group index is 3.5).

where Pin(t) is the circulating power inside the resonator. The output depends on two effects: 1) The instantaneous gating |κ(t)|2 of light as it exits the resonator, and 2) the modulation of the circulating power Pin(t), which is influenced by the memory of the resonator and the response of the gain medium. If Pin(t) is approximately constant, the laser modulation is only limited by the coupler. Since Pin >> Pout in optical cavities, only a small modulation in |κ(t)|2 is required. Compared to using the coupler as an external modulator, the change in |κ(t)|2 in a coupling-modulated laser is reduced by Pin/Pout. Hence, coupling modulation leverages the large intracavity power to increase the modulation efficiency. This modulation is distinct from Q-switching and mode-locking, since the circulating power is not significantly perturbed. The condition at which Pin(t) remains essentially static is at modulation rates much greater than fR, when the photon density can no longer respond to changes in the cavity. At low modulation frequencies (less than and near fR), Pin(t) can respond strongly to changes in the output coupling ratio, which can cause significant output distortion. The range of modulation frequencies over which distortion is significant can be minimized by reducing fR, for example, by increasing the cavity finesse. Therefore, at sufficiently high modulation rates, couplingmodulation decouples the optical modulation from the intrinsic response of the gain medium and the cavity. The laser instead inherits the modulation characteristics of the coupler [16, 17]. The coupler, for example, can be a Mach-Zehnder interferometer (MZI) [16, 18–20], which can possess large modulation bandwidths [21] and be chirp-free [22]. 2.2 Rate equation analysis To mathematically illustrate the observations in the previous section, we model the intracavity power using rate equations. We assume a microcavity implementation of the laser, such that the modulation frequency, for practical purposes, does not exceed the cavity free spectral range (FSR). Neglecting spontaneous emission, imperfect waveguide confinement, and the spatial dependency of the intracavity field, the laser dynamics can be modeled by the following rate equations [13]:

dN N vg a ( N − Ntr ) Pin λτ , = R pump − − τc dt Vhc

(2a)

 dPin ln(| σ |2 )  = vg a( N − Ntr ) Pin − vg α −  Pin , dt τ  

(2b)

where N is the carrier (i.e. atomic) concentration of the upper laser level, Ntr is the transparency carrier concentration, Rpump is the pump rate, vg is the group velocity of the circulating light, a is the differential gain, V is the mode volume, λ is the laser mode #122881 - $15.00 USD

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Received 19 Jan 2010; revised 8 Mar 2010; accepted 12 Mar 2010; published 22 Mar 2010

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wavelength, τc is the carrier lifetime, τ is the cavity round-trip time, and α is the loss per length in the cavity. If the pump rate is modulated, Rpump(t), the modulation bandwidth is roughly limited to the relaxation resonance frequency, ωR = 2πfR = {vgaλPin,0 [vgατ ln(|σ|2)]/(Vhc)}1/2, where Pin,0 is the bias intracavity power [13]. Next, we perform a small-modulation-signal analysis of the coupling modulation, so Rpump is constant and κ(t) = κ0 + εκ’(t), Pin(t) = Pin,0 + εPin’(t), N(t) = N0 + εN’(t), where ε