Frequency-Domain Model of Longitudinal Mode ... - IEEE Xplore

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Frequency-Domain Model of Longitudinal Mode Interaction in Semiconductor Ring Lasers Xinlun Cai, Ying-Lung Daniel Ho, Member, IEEE, Gábor Mez˝osi, Zhuoran Wang, Member, IEEE, Marc Sorel, and Siyuan Yu

Abstract— A general and comprehensive frequency-domain model of longitudinal mode interactions in semiconductor ring lasers (SRLs) is presented, including nonlinear terms related to third order nonlinear susceptibilities χ 3 and also linear terms due to back scattering between counter-propagating modes. The model can handle a large number of modes and complex third order nonlinear processes such as self-suppression, cross-suppression and four wave mixing occurring due to both interband and intraband effects. Every aspect of the lasing characteristics of SRLs, including lasing spectra, light-current curves and lasing direction hysteresis, can be reproduced by the model. To assess the performance and validity of the model, several miniaturized SRLs are designed, fabricated and tested. Stable unidirectional lasing in SRLs is also demonstrated by introducing asymmetric feedback from external facets. Good agreement between theoretical and experimental results is demonstrated. Index Terms— Nonlinear optics, ring lasers, semiconductor device modeling, semiconductor lasers.

I. I NTRODUCTION

S

EMICONDUCTOR ring lasers (SRLs) are attractive as potential low cost integrated single mode laser sources compared with conventional FP and DFB lasers. The cost reduction would derive from several features - they do not require cleaved facet to form a resonant cavity, thus are particularly suitable for monolithic integration and can be tested on a wafer scale before dicing, and they can achieve single mode lasing without the use of gratings and therefore are simple to fabricate, because spatial hole burning, one of the main sources of multi-longitudinal behavior in FP lasers, can be eliminated when SRL operates in unidirectional mode. The first demonstration of ring-shaped semiconductor laser dated back to 1980 [1]. However, it was not until the early 1990s that the idea of integrating SRLs received greater interest. Different cavity geometries were proposed and

Manuscript received October 31, 2011; revised December 25, 2011; accepted December 30, 2011. Date of publication January 5, 2012; date of current version February 3, 2012. This work was funded in part by the EU FP6 under Project IOLOS. X. Cai, Y.-L. D. Ho, and S. Yu are with the Department of Electrical and Electronic Engineering, Bristol University, Bristol BS8 1TR, U.K. (e-mail: [email protected]; [email protected]; [email protected]). G. Mez˝osi and M. Sorel are with the Department of Electrical and Electronic Engineering, University of Glasgow, Glasgow G12 8LT, U.K. (e-mail: [email protected]; [email protected]). Z. Wang is with the School of Optoelectronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JQE.2012.2182759

demonstrated [2]-[8], utilizing various guiding mechanism and different output coupling mechanisms [2]-[4], [9]-[11]. Although at that time their lasing characteristics were not well understood, it was generally believed that SRLs can only be forced to achieve unidirectional operation. Several forcing mechanisms were introduced to make unidirectional SRLs by providing ‘nonreciprocal’ coupling between two possible lasing directions, for example by fabricating a crossover waveguide [5], [12], tapered waveguide inside the cavity [13] or by using the feedback from a cleaved end-facet mirror outside the cavity [6], [14]. Unidirectional SRLs showed improved longitudinal mode purity and enhanced output power over bidirectional ring lasers, but the devices always suffered from intra-cavity feedback which introduces complications such as coupled cavity effects, which obscured more fundamental mechanisms behind SRL characteristics. Further experimental studies in triangular SRLs and large diameter SRLs revealed that unidirectional lasing in SRLs can be achieved without any forcing mechanisms [15], [16]. However, unlike the unidirectional SRLs mentioned above, the lasing direction for these devices is unpredictable. While the triangular lasers in [15] may still have significant intracavity back-reflection at the corners, Sorel et. al. in particular experimentally observed several operating regimes in SRLs with minimized intra-cavity back-scattering, including bidirectional continuous wave (bi-CW), bidirectional with alternate oscillations (bi-AO) and bistable unidirectional (bis-UNI). Furthermore they established a two-mode model, which theoretically predicted the various operating regimes [17], [18]. The discovery of unidirectional bistability triggered significant quantity of research in the last decade. The fact that the lasing direction of SRLs can be controlled by injected optical pulses makes SRLs ideal candidates for a number of novel functionalities such as all-optical signal processing devices and all-optical flip-flops [19]-[21]. It should be noticed that apart from the above mentioned operation regimes other regimes of SRL have been discovered recently [22]-[24]. The two-mode model, containing both linear coupling and nonlinear gain suppression between the clockwise (CW) mode and the counter clockwise (CCW) mode at a single cavity frequency, succeeds in predicting the existence of various operation regimes. It also shows that the unidirectional bistability stems fundamentally from the fact that cross-gain suppression coefficient εc is larger than self-gain suppression coefficient εs in SRLs. Moreover it reveals that the unidirectional bistability is only possible if

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CAI et al.: FREQUENCY-DOMAIN MODEL OF LONGITUDINAL MODE INTERACTION IN SRLs

the linear coupling due to back-scattering between CW and CCW modes, arising from discontinuities and imperfections in the waveguide such as sidewall roughness, is sufficiently small. Such linear coupling has significant impact on the performance of SRLs, especially small SRLs with deeply etched waveguide designs due to the higher level of scattering from the rougher waveguide sidewalls. Efforts have been made to minimize the roughness of the SRL sidewall, which leads to successful low-threshold room temperature continuous wave (RTCW) lasing and demonstration of bistable operation in semiconductor microring lasers [25], [26]. Whilst the two-mode model can explain some of the lasing characteristics, it cannot explain the periodic switching between the two unidirectional states observed with increasing current, or the stability of the lasing direction with decreasing current. Nor does it attempt to explain the influence of various mode coupling mechanisms on the spectral characteristics of SRLs. The explanation of these phenomena needs to be based on the analysis of multimode competition in SRLs. Multimode competition has been intensively studied for FP lasers [27]-[35]. Although the modes in FP lasers are standing waves, while in SRLs they are travelling waves, the principles are the same. In multimode operation cases, not only self- and crossgain suppression but also the contribution of four-wave mixing (FWM), a phase-sensitive phenomenon that leads to power exchange among different modes, need to be considered. It is well established that the net effect of self- and crossgain suppression tends to enhance the long wavelength modes and suppress the short wavelength modes (relative to the lasing mode) [30]-[32]. However this effect is partly compensated by FWM which has the opposite asymmetry with respect to wavelength [29], [36], i.e. it tends to enhance the modes on shorter wavelength side. As will be discussed in details later, in the case of SRLs the self- and cross-gain suppression happen among all the modes in both directions, but FWM can only happen among modes propagating in the same direction [36]. Therefore the existence of the intensive dominant lasing mode leads to a relatively symmetric spectrum in the lasing direction (LD) and a highly asymmetric spectrum in the nonlasing direction (NLD) – this is a profound difference from FP lasers. As the injection current is increased, the profile of the gain is shifted towards long wavelength due to joule heating in the active region and so the highest enhanced side mode will be located at the long wavelength side in the NLD. Therefore this mode is usually selected when the mode hop occurs, resulting in the reversal of lasing direction. As the current is decreased, the gain profile is shifted towards short wavelength. The suppressed short wavelength mode in NLD is unlikely to be selected during the mode hop and so the lasing direction is maintain [37]–[39]. In order to be able to correctly explain the lasing characteristics in SRLs, it is necessary to extend the two mode model to the multimode version. Previously we have derived the expressions for χ 3 , the third order nonlinear susceptibility that determines the strength of the nonlinear interactions among modes, based on a density matrix theory, taking into account the effects of carrier density pulsations (CDP), carrier heating (CH), spectral hole burning (SHB) and carrier diffusion

407

CCW

CW

Coupler Fig. 1. Schematic illustration of a SRL. The tilted output waveguides are assumed to have very low reflectivity, and coupling ratio is assumed to be small.

[37], [40]. Multimode rate equations have been derived based on the χ 3 expressions, which were solved analytically for the case of single mode operation to obtain expressions for mode spectrum. Detailed measurements were made of the lasing spectrum of SRLs, which showed good agreement with the model and were fitted to the model for parameter extraction [36]. The extracted data was further used in the model to explain the mode hysteresis with changing temperature [38]. This theory is successful in many ways in predicting the observed phenomenon and provides an intuitive picture of the underlying mechanisms. Nevertheless, it is computationally inefficient if the number of the longitudinal modes is large because the number of terms in the equations will be the number of modes cubed. Recently, a time-spatial domain model based on traveling wave description has been proposed to study the emission directionality of SRLs [41]. The main advantage of this model is that it can handle the longitudinal variation of optical field and carrier density. However, it ignored carrier heating effects, and partly relied on the interactions between different linear coupling mechanisms, such as reflections at the SRL output coupler and at the output waveguide facets, to reproduce the measured bistability and mode hopping characteristics. Such reliance on having some level of backscattering does not seem to collaborate the tendency of more clear-cut unidirectional bistability in SRLs with minimal back-reflection. In similar manners to the previous two-mode model, this may obscure the underlying physical mechanisms driving the various behavior of the SRL. In this paper, a systematic and comprehensive frequency domain model for semiconductor ring lasers suitable for numerical analysis is presented. The multi-mode rate equations, including not only χ 3 terms derived previously but also terms due to backscattering, are expressed in a concise, matrix form, from which the numerical solutions can be obtained very efficiently. Every aspect of the lasing characteristics of SRLs can be explained by the model including lasing spectra, lightcurrent (L-I) curves, and lasing direction hysteresis. To assess the validity of the model, we performed the measurement of the fabricated racetrack-shaped SRLs with similar devices parameter used in the model. Very good agreement with the theory is demonstrated. The model is then applied to simulate a type of stable unidirectional SRL, where the unidirectional lasing is achieved by introducing asymmetric optical feedback from external facets. The measurements of the unidirectional SRL, which is fabricated using Focused Ion Beam Etching (FIBE), also show very good agreement with the numerical results.

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 = ε0

CW modes



χl1 El e j (ωl t −kl z)

l

Mode index 1

M

M+1

M+2

2M+1

+

⎤



3 ∗ χlmn El E m E n e− j (kl −km +kn )z e j (ωl −ωm +ωn )t ⎦ .

l,m,n

CCW modes

(3) Mode index −1 Frequency

ω1

−M ωM

−(M+1) −(M+2) ω ωM+1/ωpk

ωM+1

−(2M+1) ω2M+1

Fig. 2. Mode referencing system. Modeled lasing spectrum consisting of a large number of discrete modes with mode index of ±i at equally spaced frequencies ωi . The central frequency at threshold is also denoted as ω pk .

The structure of the paper is as follows. In section II, the mathematical description for the coupled nonlinear equations in matrix form is derived. In section III, the numerical simulations for the lasing characterises of a SRL and a stable unidirectional SRL are presented. In section IV, the fabrication and measurement of both devices are described. Comparisons are made between the numerical and experimental results. II. M ODEL A. General Formalism A schematic diagram of the SRL is shown in Fig. 1. The end-facets of output waveguide are tilted to minimize the reflectivity and so the feedbacks from the tilted facets are ignored. The output coupler is assumed to have a low coupling ratio and so the amplitude of the field can be considered to be constant along the ring cavity. The mode referencing system is illustrated in Fig. 2. We consider an odd number 2M + 1 of modes in each direction (4M + 2 in total). The CW/CCW modes are distinguished by positive/negative numbers, and are labelled in such a way that the index M + 1 and −(M + 1) belong to the central modes at threshold. Because two modes with two different values of wave number ki can exist at each frequency, the total electric field inside the ring cavity can be expressed as a sum of all resonant modes  E i (t)e j (ωi t −ki z) (1) ε(z, t) = i

with

2mπ ωi = ω−i (i = 1, 2, 3, . . .) (2) L where Ei is the mean-field slowly varying complex amplitudes of the electric field for the i th mode at optical frequency ωi , where i ranges over positive (CW modes) and negative integers (CCW modes). The frequencies of the modes are assumed to be equally spaced by ω. Note that this scheme allows arbitrary number of modes to be modelled. Following the treatment of M. Sargent III [35], we write the total induced polarization P(z,t) as ki = ±

P(z, t) = P (1) (z, t) + P (3) (z, t)

χl1

is the first order susceptibility in the presence of pump 3 current, and χlmn is the third order susceptibility. In order to clarify the effect of the induced polarization on the individual modes, we project the total polarization onto the i th modes  1 L (1) (3) P(z, t)e− j (ωi t −ki z) dz = Pi (t) + Pi (t) Pi (t) = L 0 (4) (1)

(3)

where L is the length of the ring cavity. Pi and Pi are the first and third order nonlinear polarization to be evaluated at ωi . Under slowly varying envelope approximation, the following general expression is obtained from Maxwell equations for the i th mode in SRLs:  1E j ωi  (1) d Ei i (3) Pi (t) + Pi (t) − = . (5) dt 2 n i n g ε0 2 τp In the above equation, ni denotes the refractive index of mode i at threshold and ng is the corresponding group index. τp is the photon lifetime, which is determined by the loss factor αint as well as the coupling loss:

1 −1 c 1 τp = (αint + ln ) (6) ng L Tr where Tr is the coupling transmission ratio, and αint = αmat + αwav . αmat and αwav are the material and waveguide loss, respectively. B. First Order Susceptibility and Polarization In semiconductor laser material, the first order polarization and susceptibility are given by Pi(1) (t) = ε0 χi1 E i ni c χi1 = − [(α + j )g(N, ωi )] ωi

(7) (8)

where α is the linewidth enhancement factor, and g(N, ωi ) is the modal gain which is a function of carrier density N in the active region and the frequency of the mode. Both theoretical calculation and experimental measurements show that the linear gain spectrum in semiconductor lasers is asymmetrical with respect to its peak frequency ωpk , and the gain spectrum curve roll off more quickly on the higher frequency side than on the lower frequency side [42], [43]. In order to account for this asymmetry, a “piecewise-quadratic” frequency dependent relation is used here: 

a(N − N0 ) − ζ1 (ωi − ω pk )2 ωi ≤ ω pk

g(N, ωi ) = a(N − N0 ) − ζ2 (ωi − ω pk )2 ωi > ω pk (9)

CAI et al.: FREQUENCY-DOMAIN MODEL OF LONGITUDINAL MODE INTERACTION IN SRLs

where a is the material gain coefficient, N0 is the transparency carrier density and is the confinement factor. ζ1 and ζ2 are the quadratic coefficients on low and high frequency side respectively.

409

Self-suppression (m = n = 1) Static effects (m  n)

Pi(3) (t)

Static cross-suppression (m = n = 1)

Cross-suppression

Dynamic cross-suppression

or begatov effects

Dynamic effects (m = n)

C. Third Order Susceptibility and Polarization Semiconductor laser material has built-in mechanisms, such as CDP, CH and SHB, for modulating the carrier distributions at beating frequencies of the longitudinal modes. Beating between modes m and n can induce oscillations in the carrier distribution, which are responsible for the induced third order nonlinear polarization oscillations. The lth mode could be scattered into the i th mode by these beating-induced oscillations under certain conditions. The total third order polarization acting on the i th mode can be described by summing all the individual oscillations. Substituting (3) and (7) to (4) gives  (3) 3 ∗ χlmn El E m E n Silmn e− j (ωi −ωl +ωm −ωn )t (10) Pi (t) = ε0

Silmn

 1 L j (ki −kl +km −kn )z = e dz L  0 1 k i − kl + k m − k n = 0 = 0 ki − kl + km − kn = 0.

CW Cross-suppression Four-wave mixing

CCW

ωi (b)

Fig. 3. (a) Overview of different third order nonlinear effects in SRLs. (b) Interaction paths for ith mode in CW direction.

with (11)

Silmn is the spatial overlap integration between modes i, l, m and n. It serves as the selection rule that decides whether effective coupling can occur among the modes. As illustrated in Fig. 3(a), the combinations of i, l, m and n, leading to non-zero value of Silmn , can be divided into two groups: the case m = n yielding the static effects and m=n yielding the dynamic effects. The static effects are not associated with any beating process and can be further separated in to the case of m = n = l and the case of m=n=l, which describe the self-gain suppression and static cross-gain suppression effects, respectively. The dynamic effects only exist due to the beating of longitudinal modes. The case m = n = l yields the dynamic cross-suppression or Bogatov effects, which is the origin of the asymmetric nature of gain suppression-it works to suppress the lasing gain of the modes on the shorter wavelength side while enhance the lasing gain of the modes on the longer wavelength side. The case m=n=l yields four-wave mixing, which normally possess opposite asymmetry to the dynamic cross-gain suppression. For simplicity, both static and dynamic cross-gain suppression effects are referred to as cross-gain suppression. In SRLs, gain suppression effects occur between any longitudinal modes, while the four-wave mixing effect can only occur among modes propagating in the same direction due to the restriction of the phase matching condition in (11). Fig. 3(b) shows the interaction paths between i th mode and its neighborhood modes. 3 is given by From density matrix theory [37], [39], χlmn 3 N ch shb = χlmn (mn ) + χlmn (mn ) + χlmn (mn ) χlmn

among co-propagating modes) (a)

Self-suppression

l,m,n

where

(m = n, m  1) Four-wave mixing (m = n, m = 1,

(12)

N χlmn (mn ) = − ε N η N (α + j )

1  χ1 (1 − j mn τ N η N )(1 − j mn τ1 η1 ) pk (13) ch ch χlmn (mn ) = − εch η (αch + j ) 1 1 × χ pk ch 1 (1 − j mn τch η )(1 − j mn τ1 η ) (14) j shb 1 1 χ (15) χlmn (mn ) = − εshb η 1 − j mn τ1 η1 pk 2ε0 n pk n g (16) = h¯ ω pk ×

εN , εch and εshb give the nonlinear coefficients associated with CDP, CH and SHB. αch is the linewidth enhancement factor for CH. τN , τch and τ1 are differential carrier lifetime, electron carrier heating time, and intra-band relaxation time 1’’ is respectively. mn = ωn − ωm is the beat frequency. χpk 1 . η N , ηcd and η1 are the grating the imaginary part of χ pk visibility [37], [39] due to the carrier diffusion. The values of grating visibilities are equal to 1 for co-propagating modes and less than 1 for counter propagating modes. D. Coupled Nonlinear Equations Substitution of (8) and (10) into (5) gives the differential equation describing the evolution of the i th mode inside the ring cavity in the presence of material oscillations caused by the beating of mth and nth modes which scatters the lth mode. 1 c 1 Ei 1 ωi d Ei = (1 − j α) g(N, ωi )E i − + dt 2 ng 2 τp 2 ni ng  3 ∗ × χlmn El E m E n Silmn − K E −i + k f bk E f bk . (17) lmn

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The first term in (17) is the linear gain and phase shift experienced by the i th mode. The 3rd term is the nonlinear contribution due to the presence of mth, nth and lth modes. The 4th term describes the intra-cavity backscattering between counterpropagating modes, and K is the backscattering rate. For simplicity, the frequency dependence of K has been neglected. The last term accounts for the external feedback or injection. This term is necessary when one wants to simulate the SRLs under external optical injection or the feedback from facets. The oscillating modes in SRLs can be analysed by a combination of rate equations (17) and the following carrier density rate equation:   2ε0 n g n i I N  dN |E i |2 . (18) = − − g(N, ωi ) h¯ ωi dt eV τs i

In (18), e is the electron charge, V is the volume of active region and τs is carrier lifetime given by τs−1 = A + B N + C N 2

(19)

where A, B and C are the non-radiative, radiative and Auger recombination coefficients, respectively. It should be noticed that the carrier density rate equation (18) explicitly takes into the static change in N due account N terms where m = n to the overall intensity |Ei |2 . All χlmn are set to zero in order to prevent double counting of this population change component. The SHB and CH terms with m = n contribute to the nonlinear gain compression which is frequently modeled by introducing a factor of (1-εP0 ) [43] in the modal gain, where ε = εch + εshb . E. Generalized Equations in Matrix Form In (17), the number of terms in the summation over modes l, m and n is equal to the total number of modes cubed. This means the computation will be intensive in the situations when a large number of modes need to be considered. In order to circumvent this problem, we use here a similar treatment proposed by M. Summerfield and R. Tucker for simulating multi-wave mixing in semiconductor optical amplifiers [44]. Instead of adding up all combinations of l, m and n that contribute to the i th mode, the summation in (17) is rewritten in the form of matrix operation, and (17) is extended into a more general vector differential equation from which the numerical solution can be obtained very efficiently. 3 is the function of detuning mn =(n–m)ω = q ω χlmn (where q is an integer), and so it is rewritten as χq3 . For notational convenience, we define the four vectors, ψ, Acw , Accw , G and H, as follows: T  3 3 , . . . , χ03 , . . . , χ2M (20) ψ = χ−2M Acw = {E 1 , . . . , E M+1 , . . . , E 2M+1 }T T  Accw = E −1 , . . . , E −(M+1) , . . . , E −(2M+1)  1 c (1 − j α)g(N, ω1 ) − G= ng τp  c 1 T , . . . , (1 − j α)g(N, ω2M+1 ) − ng τp

(21)

 H =

1 ω2M+1 1 ω1 ,..., ng n1 n g n 2M+1

T .

(24)

The (2M+1)-element vector, σ cw , describing all the nonlinear interactions in CW directions can be obtained in the following way:   cw (25) θ = ψ · Acw ∗ Acw −1 ∗ A cw (26) σ = {θ2M+1 , θ2M+2 , . . . , θ4M , θ4M+1 } where θ is an intermediate vector with length of 6M+1, from which σ cw is derived. * and · imply the discrete convolution and inner product of two vectors. Acw −1 is the order reversed complex conjugate of Acw . Also we construct a (2M+1) by (2M+1) matrix X in such a way that the element Xij in i th row and j th column is given as X i j = χ03 + χ 3j −i . (27) The vector describing the cross-gain suppression experienced by CW modes due to CCW modes is given by   (28) υ ccw = X Accw · Accw∗ · Acw . Then the generalized vector differential equation for Acw can be expressed as d Acw = 0.5G · Acw + 0.5H · σ cw + 0.5H · υ ccw dt +K f bk E f bk − K Accw . (29) Similarly, the equation for Accw can be expressed as d Accw = 0.5G · Accw + 0.5H · σ ccw + 0.5H · υ cw dt +K f bk E f bk − K Acw . (30) Equations (29) and (30) are the main results of this section and represent a complete and concise description of the set of coupled equations that account for the interaction between all modes inside the SRLs. Take (29) as an example, the 1st term on the right hand side of the equation, 0.5G·Acw, is related to the linear gain. The 2nd term, 0.5H·σ cw , describes all the nonlinear interactions among CW modes. The 3rd term, 0.5H·υ ccw , describes the nonlinear cross-gain suppression experienced by the CW modes due to the existence of CCW modes. The 4th term is related to optical feedback or optical injection. The last term, –KAccw , describes the linear interaction due to the backscattering inside the SRL cavity. The average spontaneous emission for each mode is taken into account in the same way as described in [33]. A matlab program has been written to integrate the coupled equations (29), (30) and (18) using a fourth-order RungeKutta algorithm with a fixed step-size. The integration of 35 equations using 300,000 steps with time step of t=6ps takes around 10s on a PC with an i7 930 CPU under windows 7.

(22) III. N UMERICAL R ESULTS

(23)

We present below the simulation results for SRLs. The device parameters used in the simulations are given in Table 1. Some parameters, including εN , εch , εshb, ηN , ηch , η1 , τch and αch , are obtained by fitting the measurement of

CAI et al.: FREQUENCY-DOMAIN MODEL OF LONGITUDINAL MODE INTERACTION IN SRLs

TABLE I SRL PARAMETERS U SED FOR N UMERICAL S IMULATION Value

SRL cavity length Confinement factor Loss factor Coupling transmission ratio Nonrad. recombination coefficient Spont. recombination coefficient Auger recombination coefficient Transparency carrier density Gain-slope coefficient Quadratic gain coefficient Quadratic gain coefficient Group index Refractive index at gain peak Nonlinear coefficient of CDP Nonlinear coefficient of CH Nonlinear coefficient of SHB Differential carrier lifetime Electron carrier heating time Intra-band relaxation time Linewidth enhancement factor Linewidth enhancement factor for CH Backscattering rate Grating visibility for CDP Grating visibility for CH Grating visibility for SHB

L (μm) αint (m −1 ) Tr A (s−1 )

170 0.056 2700 0.995 1×108

B (m3 s−1 )

1×10−16

C (m6 s−1 )

7.5×10−41

N0 (m−3 ) a (m2 ) ζ1 (m−1 m−2 ) ζ2 (m−1 m−2 ) ng npk

2.2×1024 6.35×10−20 1.4×1018 1.8×1018 3.7 3.3

εN (m3 )

2.5×10−21

(m3 )

εch εshb (m3 )

1.9×10−23

τN ( ns) τch (ps) τ1 (ps) α

0.5 0.6 0.1 1.57

αch

2.2

K ηN ηch η1

1.8×107 +2×108 j 0.0036 / 1 0.73 / 1 0.95 / 1

Photon number

Parameter

105 CW CCW 0

40

104 I  1.11th 0

50

100

150

(a) 105

Photon number

Description

411

CW CCW 0

30

104

I  1.21th 0

50

100

150

(b)

1.7×10−23

106

Photon number

CW CCW

I  1.81th

105

104 0 0

10

50

100

150

Time (ns) (c)

the bandwidth and relative frequency detuning of the individual peak in the output spectrum by heterodyne detection methods for a SRL device [36]. Gain-slope and quadratic gain coefficients are measured using Hakki-Paoli method. The recombination parameters A, B and C are obtained by measuring the frequency response to a small signal modulation of Fabry-Perot laser fabricated with the same material. The differential carrier lifetime τ N is found from the 3-dB roll-off of the frequency response. The linewidth enhancement factor α is the calculated value from [39], and it agrees with the measured data [45]. Other parameters are chosen to the values which are generally accepted. The calculated threshold current is Ith =4.7mA. This value is somewhat lower than experimental values, which may be contributed to underestimated cavity losses or overestimated injection efficiency. However this does not affect the results presented below, as all the values of the injection current in this paper are normalized to Ith . A. Temporal Dynamics and Lasing Spectra Suppose a step-current is injected to a SRL from zero to a stationary value well above the threshold, the laser undergoes

Fig. 4. Evolution of the total photon numbers in two lasing directions at injection current of (a) 1.1I th, (b) 1.2I th, and (c) 1.8I th The inset shows the oscillations of the photon numbers at very beginning.

relaxation oscillations to a steady state. We choose the point of time, when the carrier density reaches its threshold value Nt h for the first time during the relaxation oscillations, to be the origin of our time scale. Therefore the initial condition for carrier density is N(0)=Nth. In the following we will take into account 34 modes with 17 modes in each direction and suppose that at t = 0 all modes have the same amplitudes |E j | =10−3 /3 and statistically independent phase φ j following a uniform distribution in the range of [0, 2π]. The fourth-order Runge-Kutta method using a time step of t=6ps was applied to the integration process and the integration was carried out over time period of T=500ns, which is long enough to achieve steady state for semiconductor lasers. Typical examples for time variations of the total photon numbers in two directions at regions of bi-CW, bi-AO and bis-UNI are plotted in Fig. 4 (a)–(c), which correspond to currents I = 1.1Ith , 1.2Ith and 1.85Ith, respectively. The insets

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104 3

10

105

M9 M8

M  10

M  10

M9

M  11

104

M  17

102 M  1

103

10

102 100

1.52

1.54

104 3

10

M  −8

1.56

1.58 M  −9 M  −10 M  −17

102 M  −1

Photon number

Photon number

1

CW modes 101

0

5

10

M  −9

M7

M8

50

100 M  −10

150 M  −11

104

101 103 0

10

1.52

1.54 1.56 Wavelength (μm)

1.58

Fig. 5. Lasing spectrum in CW and CCW directions in bi-CW region at I = 1.1I th.

102 CCW modes 101

0

M  −8

50

100

M  −7 150

Time (ns)

show details of the photon numbers at the beginning of the oscillations. In Fig. 4 (a), the intensities (photon numbers) in the two lasing directions reach the same intensity after relaxation oscillations. The corresponding lasing spectra are depicted in Fig. 5. The main modes are mode 9 and –9 with side mode suppression ratio of 8.6dB in both directions. The lasing spectra are obtained by calculating the time average for the photon numbers of different modes after the initial relaxation oscillations die away. In Fig. 4 (b), the intensities of both directions are periodically modulated with complementary phase at a frequency of 86 MHz. The modulation frequency is related to the specific value of the backscattering rate K, and will decrease for increasing pumping current. The oscillation behaviours of different modes are plotted in Fig. 6. For higher injection current, unidirectional operation can be obtained. As shown in Fig. 4 (c), at I = 1.85 Ith , CCW is the dominant lasing direction with more than 99% of the total power emitted from this direction, while modes in CW direction are strongly suppressed. Fig. 7 plots the time evolutions of different modes and the lasing spectra in both directions at I = 1.85Ith . As can be seen in Fig. 7, mode −10 in CCW direction is the dominate mode, and other modes are side modes which oscillate constantly due to the dynamic effects. Origins of the periodic switching between two directions can be understood with the help of the lasing spectra shown in Fig. 7 (c), (d). As discussed in section II, FWM can only occur in co-propagation modes, while gain-suppression can occur between any modes in both directions. The strong power of CCW mode –10 causes the asymmetric gain suppression to pronounce so as to enhance the modes on its long wavelength side and suppress the modes on its short wavelength side in both directions. This explains the highly asymmetric spectra in CW direction shown in Fig. 7 (c). In CCW direction, however,

Fig. 6. Evolutions of photon numbers for different modes in CW and CCW directions in bi-AO region at I = 1.1I th.

the asymmetry caused by gain suppression is compensated by FWM in which CCW mode −10 acts as the pump. This is the reason why the CCW lasing spectrum is only slightly asymmetric in contrast to CW spectrum. Therefore CW mode 11 is placed in a more advantageous position over CCW mode −11 in mode competition. As the injection current is further increased, the joule heating in active region will shift the gain spectral profile toward longer wavelength and CW mode 11 has larger chance to be selected as dominant lasing mode when the mode hop occurs, and then the lasing direction reverses. It should be pointed out that all the possible combinations of mode coupling are included in the model, not just a few modes mentioned above. B. L-I Characteristics The steady-state L-I curves for the SRL can be obtained by calculating the time domain response to a fixed current I until steady-state conditions are reached. After recording the steady value of the complex amplitude of each mode Ej and the carrier density N for that value of I. The current is increased and the calculation process is repeated using the previous calculated values of Ej and N as initial conditions. The injection current was varied in discrete steps of 1/50Ith, ranging from 0 to 4.25Ith. The thermal shift of the gain profile is assumed to be linear with respect to the injection current. The peak of the linear gain profile is assumed to be 1550 nm at I = Ith . The span of thermal shift is 13nm from threshold to the end of simulation, corresponding to 3.4 times the free spectral range (FSR) of the SRL. Fig. 8(a) displays the simulated L-I curves in both CW and CCW directions as the bias current is increased for the SRL. Three distinct operating regimes could be identified. At just above the threshold, the device operates in bi-CW

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region, where both directions lase with constant output power. From 1.1Ith to 1.3Ith , the SRL enters into bi-AO region, in which the output direction periodically oscillates at a frequency of around 100MHz. Above 1.3Ith , unidirectional operation occurs. In this region, the lasing direction is stable at a particular current, but as the current is increased the lasing direction periodically switches. Moreover, the device shows single longitudinal mode operation with side mode suppression ratio (SMSR) in the order of 20-35dB. Figure 8(b) shows the dominant lasing wavelength and the SMSR at different values of the bias current from both CW and CCW directions. The dominant lasing wavelength remains constant between directional reversals. However, a large jumps to long wavelength side is observed when the lasing direction reverses,

with a sudden drop in the SMSR of the original lasing direction because the lasing in this direction is now strongly suppressed. The L-I curves displayed in Fig. 8(a) are in very good agreement with the experimental observations [16], [25], [46] on various types of SRL devices. From this, we conclude that our model has the potential to be a very useful tool in predicting the performances of SRLs. As will be shown in section IV, the simulated results in this section closely match the measurement results from the fabricated SRL. C. Lasing Direction Hysteresis Similarly, the output intensities of the SRL with a decreasing injection current can also be calculated. Fig. 9 shows the simulated results for the intensities in CW and CCW directions as functions of increasing and decreasing injection current. As the current is increased, lasing alternates between the two directions accompanied by mode hops to the long wavelength. As the current is decreased however, the lasing direction remains stable until the mode of operation reverts to bidirectional, and the lasing wavelength sequentially steps through each cavity resonance toward shorter wavelength. Fig. 10 plots two examples of the mode dynamics for increasing and decreasing current when the mode hops happen. For increasing current, the lasing wavelength hops to longer wavelength, from 1.562 μm (CW mode 12) to 1.565 μm (CCW mode –13). The mode hop is relatively rapid with less than 10 ns transition time. Once the transition starts, the intensity of CCW mode −13 increases monotonically until

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steady state is achieved. During the transition, very little power appears in other modes. It should be noted that the mode hopping range in the increasing current case is not decided by the FSR of the SRL. It is rather related to the amount of asymmetry in the nonlinear gain spectrum. Hopping of several FSRs was observed when the FSR is small (larger SRL size), and can be readily reproduced using our model [38]. For the decreasing current case, the lasing wavelength hops to shorter wavelength, from 1.565 μm (CCW mode −13) to 1.562 μm (CCW mode –12) when the current is decreased from I = 3.13Ith to I = 3.07Ith. The mode hop takes around 15 ns. It can be observed that CW mode 11 also takes part in the mode competition besides CCW mode −12 and CCW mode −13. At I = 3.07Ith, the linear gain peak is at 1.5582 μm, which is very close to the wavelength of mode 11 (1.558 μm). This can explain why the mode 11 rises at the beginning of transition. The simulated pattern shown in Fig. 9 agrees very well with the hysteresis pattern from measurements reported in various types of SRLs [15], [16], [38]. D. Application to Stable Unidirectional SRLs The preceding sections demonstrate the effectiveness of our model in describing various lasing characteristics of SRLs. We now present an application of the model to a unidirectional SRL.

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(b) Fig. 10. Photon number dynamics during mode hops to a longer wavelength and a shorter wavelength. (a) Evolution of total photon numbers in CW and CCW during the mode hop which occurs when the injection current increased from I = 3.53I th to I = 3.61I th. (b) Evolution of total photon numbers in CW and CCW during the mode hop which occurs when the injection current decreased from I = 3.13I th to I = 3.07I th. The insects show the photon number in different modes.

The structure of the unidirectional SRL is plotted in Fig. 11. One end of the output waveguide is tilted relative to the left facet to minimize the reflectivity, while the other end is not tilted to maximize reflectivity from the right facet. The SRL would be forced to oscillate in the CW direction because the CCW modes should be strongly coupled into the CW modes by the reflection, giving the CW direction an advantage in mode competition. The rate equations should be revised to take into account the unbalanced coupling introduced by the feedback from the right facet. For the i th mode in CW direction E f bk = E −i .

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IV. E XPERIMENT In this section, we report the fabrication and measurement of the devices with similar parameters used in the previous simulation. Two steps are taken to investigate the devices. Firstly, several SRLs with racetrack shaped were designed and fabricated. Each racetrack consists of two bends and two straight waveguides with a directional output coupler. In addition to the cavity biasing contact, separate contact pads are defined on the output waveguides, which can be pumped as SOAs. The bending radius varied from 10 μm to 30 μm and the coupling length Lc varied from 5μm to 25μm (Fig. 11). The circumferences of the devices are within the range of 80-240 μm. Both ends of each output waveguide are 100 tilted. As shown later, these devices show robust directional bistability therefore are referred as bistable SRLs. The bistable SRLs are fabricated on a multi-quantum-well (MQW) InAlGaAs-InP wafer, and the process is similar to the devices reported earlier [25]. An SEM image of the devices is shown in Fig. 13. Subsequently, one end of the waveguide was cut by Focus Ion Beam Etching (FIBE) to form a perpendicular mirror facet. In order to achieve high perpendicularity and smoothness so that the maximum possible 30% reflectivity can be approached, the mirror facet is defined by deep etching through the active layer into the lower cladding in a high resolution FIBE process. It should be noticed that the mirror facets were fabricated using FIBE only for the reasons of flexibility and easy comparison. The standard cleave process provides equally

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is the roundtrip time inside the ring cavity and D=100μm is the distance between the output coupler and the facet. Fig. 12(a) plots the simulated L-I curve and Fig. 12(b) shows the SMSR and the dominant lasing wavelength. The threshold current is the same as previous device, but the SRL always lases in the direction predetermined by the position of the mirror (CW direction in the case of Fig. 11) and the intensity becomes a monotonically increasing function of the injection current except for several kinks. A comparison of the L-I curve and the lasing wavelength indicates that those kinks correspond to wavelength hopping between the longitudinal cavity modes. Although the SMSR is reduced in the vicinity of mode hops, good single longitudinal mode lasing was observed with SMSR>25dB for injection current I >2Ith .

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high quality facets. Two zoom-in pictures of the etched mirror are shown in insets of Fig. 13. All fabricated devices exhibit continuous wave lasing at room temperature (293K). The light-current L-I curves of the lasers and their lasing spectra before and after FIBE was measured. Before FIBE, the SRLs show behaviors commonly observable in similar type of devices: Periodic reversal between CW and CCW modes occurs with increasing biasing current, accompanied by longitudinal mode hops when the lasing

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direction reverses. As an example, Figure 14 (a) plots the L-I curve of a 20 μm radius racetrack with an Lc of 24μm. The waveguide contacts were left un-pumped, which means the output level can be further increased by about 10 dB by biasing the waveguides to transparency. The threshold of this device was measured to be 12 mA (corresponding to 2.16 kA/cm2 threshold current density). The lasing spectrum at 3.1Ith is plotted in the inset and the FSR is measured to be 3.8 nm. Figure 14 (b) shows the dominant lasing wavelength and the SMSR at different values of the injection current for both CW and CCW directions. These results agree very well with the simulated results shown in section III B. After FIBE, instead of periodic reversals between two directions, the SRL always lases in the predetermined direction. Fig. 15 (a) plots the L-I curve taken from the CW direction. The threshold current remains the same as the bistable device but the optical power becomes a monotonically increasing function of the injection current, which is a clear sign of stable unidirectional lasing operation. Moreover single longitudinal mode lasing was observed with SMSR>25dB for injection currents Ibias > 2.5 Ith and reaching 35 dB at Ibias > 4 Ith (in Fig. 15(b)), which confirms the stability of the unidirectional lasing operation. Again, these results are in a good agreement with the simulated results in section III D.

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There are two discrepancies between theory and experiment. The first discrepancy is the large abrupt discontinuities in the measured L-I curves are displayed in Fig. 15(a) whilst only small kinks are shown in the simulated L-I curves in Fig. 12(a). In both cases, the discontinuities and kinks are caused by mode hops, and the large discontinuities in the experiment is introduced by the wavelength dependent absorption in the output waveguide, which leads to apparent higher output power at longer wavelengths (due to mode hopping) in addition to the increase due to SRL’s own L-I characteristics. The other discrepancy is in the dominant lasing wavelength and SMSR. It can be observed in the measured results that between mode hops there is a slope in the lasing wavelength as the injection current is increased. This is due to the change in refractive index with current and temperature. Because this effect is not included in the model, the slope is not present in the simulated results. Also the increase in temperature results in broadening of the electron states in conduction band, which is equivalent to broaden the linear gain spectra. This is the reason why the measured SMSR, shown in Fig. 14 (b) and Fig. 15 (b), is reduced between mode hops when the injection current is increased.

CAI et al.: FREQUENCY-DOMAIN MODEL OF LONGITUDINAL MODE INTERACTION IN SRLs

V. C ONCLUSION We have presented a comprehensive frequency-domain numerical model of SRL that clearly links the operation of the SRL to various known nonlinearities in semiconductor laser material. The model can handle a large number of modes and situations in which the nonlinear interactions are complex and essential, such as directional reversals and associated spectral mode hops. All experimentally observed characteristics of SRL can be simulated with very good agreements. We also applied the model to stable unidirectional SRLs fabricated by introducing pre-determined external reflection, which confirms the effectiveness of the model in predicting the performances of real devices. The model should therefore provide a useful tool for further analyzing other characteristics of various SRL configurations, such as its dynamic behavior under current modulation. R EFERENCES [1] A. S.-H. Liao and S. Wang, “Semiconductor injection lasers with a circular resonator,” Appl. Phys. Lett., vol. 36, no. 10, pp. 801–803, May 1980. [2] T. F. Krauss, P. J. R. Laybourn, and J. S. Roberts, “CW operation of semiconductor ring lasers,” Electron. Lett., vol. 26, no. 25, pp. 2095– 2097, Dec. 1990. [3] J. P. Hoimer, D. C. Craft, G. R. Hadley, and G. A. Vawter, “CW room temperature operation of Y-junction semiconductor ring lasers,” Electron. Lett., vol. 28, no. 4, pp. 374–375, Feb. 1992. [4] G. Griffel, J. H. Abeles, R. J. Menna, A. M. Braun, J. C. Connolly, and M. King, “Low-threshold InGaAsP ring lasers fabricated using bi-level dry etching,” IEEE Photon. Technol. Lett., vol. 12, no. 2, pp. 146–148, Feb. 2000. [5] J. P. Hoimer and G. A. Vawter, “Unidirectional semiconductor ring laser with racetrack cavities,” Appl. Phys. Lett., vol. 63, no. 18, pp. 2457– 2459, Aug. 1993. [6] S. Oku, M. Okayasu, and M. Ikeda, “Low-threshold operation of square-shaped semiconductor ring lasers (orbiter lasers),” IEEE Photon. Technol. Lett., vol. 3, no. 7, pp. 588–590, Jul. 1991. [7] H. Han, D. V. Forbes, and J. J. Coleman, “InGaAs-AlGaAsGaAs strained-layer quantum-well heterostructure square ring lasers,” IEEE J. Quantum Electron., vol. 31, no. 11, pp. 1994–1997, Nov. 1995. [8] C. Ji, M. H. Leary, and J. M. Ballantyne, “Long-wavelength triangular ring laser,” IEEE Photon. Technol. Lett., vol. 9, no. 11, pp. 1469–1471, Nov. 1997. [9] T. F. Krauss, R. M. De La Rue, P. J. R. Laybourn, B. Vogele, and C. R. Stanley, “Efficient semiconductor ring lasers made by a simple selfaligned fabrication process,” IEEE J. Sel. Topics Quantum Electron., vol. 1, no. 2, pp. 757–761, Jun. 1995. [10] T. M. Cockerill, D. V. Forbes, J. A. Dantzig, and J. J. Coleman, “Strained-layer InGaAs-GaAs-AlGaAs buried-heterostructure quantumwell lasers by three step selective-area metalorganic chemical vapor deposition,” IEEE J. Quantum Electron., vol. 30, no. 2, pp. 441–445, Feb. 1994. [11] J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, C. W. Tu, and S. T. Ho, “Directional light output from photonic-wire microcavity semiconductor lasers,” IEEE Photon. Technol. Lett., vol. 8, no. 8, pp. 968–970, Aug. 1996. [12] J. P. Hohimer, G. A. Vawter, and D. C. Craft, “Unidirectional operation in a semiconductor ring diode laser,” Appl. Phys. Lett., vol. 62, no. 11, pp. 1185–1187, Mar. 1993. [13] J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett., vol. 70, no. 10, pp. 1192–1194, Mar. 1997. [14] J. P. Hohimer, G. A. Vawter, D. C. Craft, and G. R. Hadley, “Improving the performance of semiconductor ring lasers by controlled reflection feedback,” Appl. Phys. Lett., vol. 61, no. 9, pp. 1013–1015, Aug. 1992. [15] M. F. Booth, A. Schremer, and J. M. Ballantyne, “Spatial beam switching and bistability in a diode ring laser,” Appl. Phys. Lett., vol. 76, no. 9, pp. 1095–1097, Feb. 2000.

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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 48, NO. 3, MARCH 2012

Gábor Mez˝osi received the B.Sc. degree in electronics engineering from the Budapest University of Technology and Economics, Budapest, Hungary, and the Ph.D. degree in semiconductor ring lasers from the University of Glasgow, Glasgow, U.K., in 2005 and 2010, respectively. He is currently a Post-Doctoral Researcher with the University of Glasgow. His current research interests include semiconductor ring lasers, integrated high power lasers, mode-locked lasers, and vertical cavity surface-emitting lasers.

Zhuoran Wang (M’08) was born in Shanxi, China, in 1977. He received the B.Eng., M.Eng., and Ph.D. degrees from the Department of Electronics and Information Engineering, Tianjin University, Tianjin, China, in 2000, 2004, and 2007, respectively. He was a Research Collaborator with the Department of Electrical and Electronic Engineering, University of Bristol, Bristol, U.K., in 2005, working on an active vertical-coupler-based optical crosspoint switches for optical packet switching and optical network technology. From 2006 to 2009, he was a Research Associate with the same university, where he was engaged in the research and development of an integrated semiconductor optical amplifier based Mach-Zehnder interferometer and semiconductor ring lasers in two European projects: MUFINs and IOLOS. He is currently a Professor with the School of Optoelectronic Information, University of Electronic Science and Technology of China, Chengdu, China. He is the author or co-author of more than 50 research papers and serves as a reviewer for several IEEE journals. His current research interests include optical packet switching, integrated photonics devices, semiconductor ring lasers, all-optical signal processing, logic and memories, and optical networks.

Xinlun Cai was born in Hubei, China, in 1982. He received the B.Sc. and Masters degrees from the Department of Optoelectronics Engineering, Huazhong University of Technology, Wuhan, China, in 2004 and 2007, respectively. He is currently pursuing the Ph.D. degree with the Department of Electrical and Electronic Engineering, University of Bristol, Bristol, U.K. Prior to 2007, he worked on 3-D modeling of bend waveguide and micro-ring resonators using the full vectorial film mode matching method and coupled mode theory. His current research interests include modeling, simulation, and fabrication of semiconductor ring lasers and microcavity lasers.

Marc Sorel received the B.Sc. degree (cum laude) in electronic engineering and the Ph.D. degree in electronic engineering and computer science from Università di Pavia, Pavia, Italy, in 1995 and 1999, respectively. He joined the Optoelectronics Group, University of Glasgow, Glasgow, U.K., with a personal Marie-Curie fellowship in 1999, where he was a Lecturer in 2002 and Senior Lecturer in 2008. He has authored or co-authored more than 250 conference and journal papers. His current research interests include semiconductor laser dynamics, integrated optoelectronics, silicon photonics, and semiconductor ring lasers.

Ying-Lung Daniel Ho (S’04–M’07) received the B.Sc. degree in electrical engineering from the National Taipei University of Technology, Taipei, Taiwan, in 1999, and the Ph.D. degree from the University of Bristol, Bristol, U.K., in 2007, for his work on designing efficient single-photon sources for quantum information applications. He is currently a Researcher with the Department of Electrical and Electronic Engineering, Centre for Quantum Photonics, H. H. Wills Physics Laboratory, University of Bristol. His current research interests include modeling, simulation, and fabrication of quantum optics in wavelength scale structures, focusing on the experimental study of 3-D-nanocavities, and investigate introduction of two-level quantum systems, e.g., quantum dots and diamond nanocrystals.

Siyuan Yu received the B.Eng. degree from the Department of Electronic Engineering, Tsinghua University, Beijing, China, in 1984, and the M.Eng. degree from the Wuhan Institute of Post and Telecommunications, Wuhan, China, in 1987. He carried out his Ph.D. research at the Department of Electronics and Electrical Engineering, University of Glasgow, Glasgow, U.K., from 1993 to 1996. He was a Lecturer with the Huazhong University of Science and Technology, Hubei, China, from 1987 to 1993. He joined the Department of Electronic and Electrical Engineering, University of Bristol, Bristol, U.K., in 1996, and is currently a Professor of photonic information systems. He has published more than 140 papers in high-impact journals and international conferences. He holds nine international patents and is currently co-editing a book. His current research interests include integrated photonic components for advanced optical information systems and networks.