Design and analysis of single mode Fabry-Perot lasers with high speed modulation capability

Yu Li; Yanping Xi; Xun Li; Wei-Ping Huang

doi:10.1364/OE.19.012131

1. Introduction

Since the recent development of single mode Fabry-Perot (FP) lasers [1–4], much attention has been paid to this type of structure [5–7] for its extraordinary single mode lasing performance under various operation conditions [8]. By etching a few shallow slots on the ridge of an FP laser, constructive interference of the cavity can be established and utilized to manipulate the gain threshold and lasing condition to suppress unwanted FP side modes. As demonstrated in a series of experiments [1,4,8], this laser showed excellent performance with characteristics, such as high side-mode suppression ratio (SMSR), high speed dynamic modulation response, and low-degree feedback sensitivity, etc. Therefore, the properly designed and re-growth-free single mode FP lasers can now serve as a promising low-cost alternative to the more expensive distributed feedback (DFB) lasers in applications such as the optical access networks. Following in Fig. 1 are the simplified 2D sketches of the slotted FP and DFB lasers, respectively, where the red-strap layer represents the active region.

Fig. 1 Simplified 2D sketches of (a) a single-mode edge-emitting FP laser with non-periodic slots etched on top of the FP cavity; and (b) a conventional single-mode DFB laser. z is the longitudinal propagation direction and x is the epitaxial growth direction.

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The non-periodic nature of the longitudinal structure inherent in single-mode FP laser, however, presents some challenges to the modeling and design optimization for such devices. The conventional time-domain coupled mode theory [9] commonly used for the DFB lasers can no longer be applicable for this structures. More complicated quantum mechanical model [10,11] was applied only to the single-slot FP laser, without explicit consideration of the device temperature, which is critical for the operation of such devices under practical situations. For those reasons, comprehensive simulation of the dynamic performance of single-mode FP lasers under large signal direct modulation and/or strong optical feedback has not yet been reported to our best knowledge. Further, the necessity of using a time domain model to characterize the laser is for design optimization purposes. Although the powerful threshold gain approaches [1,4,7] can provide certain rules and guidance for single-mode FP laser designs, they cannot be readily extended to account for the above-threshold or dynamic characteristics, under high level external feedback conditions. To perform those optimizations with respect to the key design parameters, such as the strength and distribution of slots along the laser cavity, a large signal model is essential.

In this work, we develop a rigorous and straightforward time-domain model based on the transfer matrix method [12,13], to explicitly account for the thermal and feedback effects [14,15] on single-mode FP laser under both static and dynamic situations. Comprehensive simulation studies of the laser are carried out in a systematic manner. Design optimization with respect to the slot index contrast and number is performed. With the help of the comprehensive simulation method developed in this work, a more powerful design optimization procedure can be envisaged.

2. Model governing equations and simulation method

By combining the conventional transfer matrix method and the time domain evolution of electrical and optical fields, the time-domain transfer matrix method (TD-TMM) [12,13] have been developed as a cascade of elementary transfer matrices, i.e., the scattering and propagation matrices, to describe the time-space evolution of the counter-propagating waves along the laser cavity.

2.1 Governing equations for TD-TMM

As in Ref [12], we can write the forward and backward optical fields at the two boundaries of each structural section k at time t as follows

[\begin{matrix} E_{f} (t + d t, k + 1) \\ E_{b} (t, k + 1) \end{matrix}] = [A_{k} (t)] [\begin{matrix} E_{f} (t, k) \\ E_{b} (t + d t, k) \end{matrix}] = [\begin{matrix} a_{11} (t, k) & a_{12} (t, k) \\ a_{21} (t, k) & a_{22} (t, k) \end{matrix}] [\begin{matrix} E_{f} (t, k) \\ E_{b} (t + d t, k) \end{matrix}],

where

[A_{k} (t)]

is the propagation matrix P as

P = [\begin{matrix} e^{- j β l} & 0 \\ 0 & e^{j β l} \end{matrix}] .

If the section k covers a uniform medium of length l, the parameter β is the effective propagation constant. If the section contains an index jump, e.g., from n _i to n _j, then,

[A_{k} (t)]

contains the scattering matrices T as

T_{i j} = \frac{1}{2 n_{j}} [\begin{matrix} n_{j} + n_{i} & n_{j} - n_{i} \\ n_{j} - n_{i} & n_{j} + n_{i} \end{matrix}] .

From Eq. (1), the field at a future-step can be further written in terms of the previous-step fields and the matrix elements as

\begin{array}{l} E_{b} (t + d t, k) = [E_{b} (t, k + 1) - a_{21} (t, k) E_{f} (t, k)] / a_{22} (t, k), \\ E_{f} (t + d t, k + 1) = a_{11} (t, k) E_{f} (t, k) + a_{12} (t, k) E_{b} (t + d t, k) . \end{array}

At two ends of the waveguide (

z = 0

and

z = L

), we have to require that the fields are connected by the left and the right end-facet reflectivities

r_{l}

and

r_{r}

as

\begin{array}{l} E_{f} (t + d t, 0) = r_{l} E_{b} (t, 0), \\ E_{b} (t + d t, L) = r_{r} E_{f} (t, L) . \end{array}

The carrier equation that governs the electron-photon relations can be expressed as

N (t + d t, k) = N (t, k) + d t [η \frac{J (t, k)}{e d} - R_{s p} (N (t, k)) - v_{g} g (t, k) S (t, k)],

in which the parameters in the equation are the injection current density J, the injection efficiency η, the active region thickness d, the spontaneous recombination rate

R_{s p}

, the material gain g and the photon density S, respectively. The detailed expressions for some of these parameters can be found in Ref [12,13].

2.2 Additional effects considered in TD-TMM

a) Index change due to injection current: Due to the photon-electron interaction mechanism, the laser’s refractive index n of section k (as noted in subscript) will change, when current is injected into the active region, as

n_{k} (N_{k}) = n_{k, t r} - \frac{1}{4 π} \frac{d g}{d N} \ln (N_{k} / N_{t r}) \cdot α λ,

where α is the linewidth enhancement factor. The subscript tr represents quantities at the transparency condition.

b) Wavelength shift due to injection current: The second effect aside from the index change under current injection is the lasing wavelength shift, due to alteration of the cavity resonance condition. This can be captured by two terms: one is the shift of reference wavelength $Δ λ_{r e f}$ as

Δ λ_{r e f} = \frac{\bar{Δ n}}{n_{t r}} λ_{r e f},

where

\bar{Δ n}

is an average taken over the whole structure as

\bar{Δ n} = \frac{1}{M} \sum_{k} Δ n_{k} (N_{k})

with

Δ n_{k} = - \frac{1}{4 π} \frac{d g}{d N} \ln (N_{k} / N_{t r}) \cdot α λ .

The other term is the relative shift of lasing peak

Δ λ_{p}

away from the reference wavelength, obtained by minimization of the laser’s overall matrix determinant in each time step [12] as

\det [A_{t o t a l} (t)] = a_{11} (λ) - r_{l} r_{r} a_{22} (λ) - r_{l} a_{21} (λ) + r_{r} a_{12} (λ),

where

A_{t o t a l} (t)

is the cascade of all section matrices

A_{k}

at time t.

c) Finite gain profile: To consider the impact of finite bandwidth for the material gain profile on wavelength selections in simulation, as occurred in the actual situations, the flat gain profile used in Ref [12,13]. can be replaced by a transfer function $H (λ)$ to account for the material gain selection [16] as

g (λ) = g (λ_{g}) H (λ),

where

g (λ_{g})

is the gain at chosen central wavelength

λ_{g}

and

H (λ)

can be selected as a Lorentzian response function. The forward and backward optical fields also have to be modified by the first-order infinite impulse response (IIR) filter as

\begin{array}{l} E_{f} (t + d t, k) = A E_{f} (t + d t, k) + (1 - A) E_{f} (t, k) \\ E_{b} (t + d t, k) = A E_{b} (t + d t, k) + (1 - A) E_{b} (t, k) \end{array}

with A the weighting parameter defined in Ref [16].

d) Temperature effect: When the device temperature changes under different working conditions and external environments, refractive index of the material can be affected [14] as

Δ n_{e f f, T} = \frac{d n_{e f f}}{d T} Δ T .

Higher injection can also increase the temperature as

Δ T = T - [T_{0} + f (I^{2}) + g (I - a P_{o u t})],

where f and g are functions of the device structure as expressed in Ref [14];

T_{0}

is the temperature of the substrate viewed as a constant heat sink, a is the conversion coefficient and

P_{o u t}

is the output power. At the same time, differential gain and transparent carrier density in the gain formula have to be modified to include the temperature factor as

g_{k} (N_{k}, S_{k}) = \frac{\frac{d g}{d N} e^{- \frac{Δ T}{T_{g}}} \ln (e^{- \frac{Δ T}{T_{n}}} N_{k} / N_{t r})}{1 + ε S_{k}},

where

T_{g}

and

T_{n}

are the characteristic temperature for

d g / d N

and

N_{t r}

, respectively. This will affect the resonance and lasing threshold conditions of the structure, especially the L-I curve. Parameters needed for those approximations can be extracted from the experiment already done on the same materials.

e) Optical feedback interferences: To model the optical feedback’s interference effect on laser’s performance by the time-domain transfer matrix method, we can add one term to the field as a reflected portion of the previous-time output from the facet, with an extra phase shift and time delay [15] as

E_{b} (t, L) = r_{r} E_{f} (t, L) - [r_{f e e d b a c k} e^{- j ω τ}] \sqrt{1 - r_{r}^{2}} E_{f} (t - τ, L) .

Here, we have assumed the optical network to be connected to the right end-facet of the laser, with a reflectivity

r_{f e e d b a c k}

and time delay τ for the feedback from the external system.

3. Simulation results

3.1 Single-mode FP laser design and TD-TMM verification

Single-mode FP lasers are designed by using the inverse scattering method [1,4] to guide synthesis of the shallowly etched slots onto the FP structure, which can enforce desired threshold gain target function and accurately matched phase condition at chosen wavelengths.

Briefly, considering slot perturbations to the lasing threshold equation, Eq. (18), as

a_{11} (λ) - r_{l} r_{r} a_{22} (λ) - r_{l} a_{21} (λ) + r_{r} a_{12} (λ) = 0,

we can expand its solution in terms of different orders of the small slot-index contrast

Δ n

, whose zero-th order is the flat threshold gain for FP cavity. Then, a single-mode FP laser design would require the first order solution to the equation to follow a specific shape of spectrum in frequency domain. For example, it can be a sinc function that has a sharp drop at the desired lasing wavelength and relatively higher threshold levels at all other non-lasing ones. This predetermined first-order threshold gain spectrum can be converted by Fourier analysis to a certain distribution of slots. For the explanation of this reasoning, please refer to Ref [1,4]. for more details.

Based on this theory, we have reproduced both types of the single-mode FP structures, i.e., the symmetric and the asymmetric configurations, to be examined in the following section. However, to exam our TD-TMM approach, we first applied the method to a DFB structure [17] and obtained the static and dynamic performances as in Fig. 2 .

Fig. 2 (a) L-I curve and (b) large signal modulation of a DFB laser verified from the time-domain transfer matrix method.

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From the above L-I curve and large signal modulation by 0.5GHz square wave-forms, we can see that the TD-TMM can reproduce reference results with high accuracy. For certain increase of the environment temperature, lasing behavior in terms of the threshold current and L-I curve slope are significantly changed. The saturation of output power can also be observed, as indicated by the decrease of slope at higher injections in Fig. 2(a).

3.2 Single-mode FP laser simulations

After validation of the simulation method by way of the above DFB laser example, we apply it to simulations of single-mode FP lasers. For demonstration purposes, we examined two types of designs, namely, the symmetric and asymmetric structures, whose SMSR can both reach 50 dB, but with different manufacturing complexities.

a) Symmetric single-mode FP laser structure: From threshold analysis using the inverse scattering method of Ref [1,4], we obtained the slot distribution for longitudinal structure of a FP laser and calculated its threshold gain profile as shown in Fig. 3(a) (insert is for the distribution of 90 slots for this laser design). Then, we used the TD-TMM model to obtain its lasing spectrum as in Fig. 3(b).

Fig. 3 (a) 90-slot symmetric single-mode FP laser structure (insert) and its threshold gain profile from inverse transfer matrix method; (b) corresponding spectrum calculated from TD-TMM.

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Lasing wavelength and SMSR with respect to the injection current at room temperature are shown in Fig. 4(a) to indicate the wavelength shift at different pumping levels. Red shift can be seen above the 20mA threshold and stable while slight improvement of SMSR can also be observed. By varying injection current at different temperatures, we obtain the light-current relation as in Fig. 4(b), which demonstrates the T-dependent threshold current and L-I curve slope of the laser. At higher temperature, $I_{t h}$ increases while the slope decreases rapidly.

Fig. 4 (a) SMSR and lasing wavelength shift, and (b) L-I curve at different temperature, of the symmetric single-mode FP laser from time-domain transfer matrix method.

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Further, for small signal analysis, we calculate the optical response at room temperature for three different bias currents as in Fig. 5(a) , from which we can see that the relaxation oscillation frequency increases from 1GHz to around 5GHz at higher injection. To examine the large-signal modulation performances, we also calculate the output power and lasing wavelength shift, as shown in Fig. 5(b), for the modulation current switched between 30mA and 50mA at 10 Gbit/s.

Fig. 5 (a) Small signal analysis for single-mode FP laser at different base current conditions; (b) large signal analysis with power and wavelength shift plots at 10 Gbit/s.

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To determine the impact of optical feedback on single-mode FP laser performance, we further include a high level backward reflection up to −25dB, i.e., $r_{f e e d b a c k} = 0.05$ , to the right facet of the laser. The output power, the lasing wavelength shift, and the overlapped spectrum taken at different simulation times are plotted in Fig. 6(a) . The reason to plot spectra on top of each other is to observe the possible mode-hopping during modulation, especially when injection current is switched between 0/1.

Fig. 6 Output power, lasing wavelength shift and spectrum for (a) single-mode FP laser under −5dB external optical feedbacks and (b) DFB laser without (upper) and with −25dB feedback (lower), at 10 GHz modulation. SMSR plot is obtained by overlapping the different spectra sampled at different time positions.

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Comparing Fig. 6(a) with Fig. 3(b) and 5(b), we can see that the laser can maintain high single-mode spectrum under strong feedback and fast modulation condition, as experimentally described in Ref [8]. Central lasing wavelength is also changing smoothly according to the injection modulation. For further comparison, a DFB laser with the same structural and material parameters, especially the same designed threshold gain difference and facet reflectivity, is calculated before and after feedback is applied, as in Fig. 6(b). From those graphs, we can see the different effects of the feedback on the two structures. It is observed that the DFB laser is sensitive to feedback mostly on its neighboring mode, i.e., mode hopping can occur to this vicinity wavelength. On the other hand, the feedback sensitivity of the single-mode FP laser is shared by all other suppressed FP modes covered by the material gain profile. This leads to the fact that total sensitivity to the external feedback is reduced, but at the price of possible mode hopping to wavelengths far away from the lasing mode.

b) Asymmetric single-mode FP laser structure: The asymmetric structure of the FP lasers is introduced for simplifying the manufacturing process as described in Ref [1,4]. In this work, we simulated the 15-slot asymmetric structure that has only one half of the FP cavity etched with slots, while the other half is un-etched and the facet is coated with highly reflective mirror (as insert of Fig. 7(a) ). This design can reduce the number of slots needed in the structure, while distributing them more evenly into the cavity, rather than putting high density of slots around the device center as in the symmetric design. This can also improve external efficiency for the useful output power as only one side of the laser is emitting light, with the other side used for sampling or monitoring.

Fig. 7 (a) 15-slot asymmetric single-mode FP laser structure (insert) and its threshold gain profile from inverse transfer matrix method; (b) corresponding spectrum calculated from TD-TMM.

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Threshold gain of the structure is calculated as shown in Fig. 7(a), while the lasing spectrum in condition of dc-injection and zero optical feedback is shown in Fig. 7(b). From these plots we can observe the high SMSR single-mode lasing spectrum, even with very small designed threshold gain difference, due to the carefully engineered slot positions that enforce constructive phase condition to be satisfied simultaneously around the desired wavelength. It should be pointed out that, for the same threshold gain difference, a DFB laser can hardly operate in single-mode condition.

The same procedure is done to the asymmetric single-mode FP laser at high modulation speed and −25dB feedback level, as in Fig. 8(a) in which the output power, the lasing wavelength change, and the overlapped spectrum sampled at different time points during the simulation are displayed. For comparison, we also included calculations in Fig. 8(b) for the conventional FP laser that has the same structural and material parameters, but no slots etched.

Fig. 8 Output power, lasing wavelength shift and spectrum for (a) 15-slot asymmetric single-mode FP laser, and (b) FP laser without (upper) and with −25dB feedback (lower), at 10 GHz large signal modulation, and −25dB external optical feedbacks.

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From the above, we see that the asymmetric structure can achieve high SMSR single-mode condition, while maintaining structural simplicity and certain feedback immunity.

To further optimize the structure to produce more stable single-mode spectrum with higher SMSR, we can manipulate the total number of slots and their index-contrast as two adjusting parameters for the design. We calculated the asymmetric single-mode FP laser by either fixing the contrast of $Δ n = 0.005$ while changing slot number from 11 to 50, to plot threshold gain difference and lasing spectrum SMSR as in Fig. 9(a) and 9(b); or by using its 15-slot case but changing slot-index contrast from 0.001 to 0.01, to plot the same quantities as in Fig. 10(a) and 10(b).

Fig. 9 (a) Difference between the lowest and second lowest threshold gain of single-mode FP lasers v.s. different number of slots etched; (b) SMSR calculated from TD-TMM for corresponding cases in (a) before and after feedback is applied.

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Fig. 10 (a) Threshold gain difference of the 15-slot asymmetric single-mode FP laser with different index-contrast slots etched; (b) SMSR calculated for corresponding cases in (a) before and after feedback is applied.

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From the plots, we can see that the threshold gain difference increases linearly with the number and/or depth of the slots along the cavity. However, more importantly, the effect of deeper slots will be leveled off at higher contrast, and their improvements over feedback will be reduced, as the more/deeper slots etched the more sensitive the structure will be to external perturbations. This also indicates that a recent development of single-mode FP laser [7] using the “pixel method” should prefer to adopt coarse meshing segments to reduce number of slots, considering the feedback perturbation effect on designed structures for use in the high speed, strong feedback optical communication networks. Therefore, the trade-off between slot number/depth and single-mode performance has to be balanced for an optimized single-mode laser.

4. Conclusion

We have analyzed the single-mode FP laser using time-domain traveling wave model and compared the device performance over their different (symmetric and asymmetric) configurations. From the simulations, we confirmed the effectiveness of the properly designed FP lasers in achieving single-mode lasing with high spectrum purity against high level of external feedbacks. We also carried out design optimization with respect to the key design parameters and revealed some interesting features about the dependence of threshold gain difference and SMSR on number and depth of slots.

Acknowledgments

Special thanks go to Dr. Stephen O'Brien and Prof. Liam Barry for their helpful comments for the feedback issue of this work. Thanks also go to Lin Han for his helpful discussions during the simulation.

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Design and analysis of single mode Fabry-Perot lasers with high speed modulation capability

Abstract

1. Introduction

2. Model governing equations and simulation method

2.1 Governing equations for TD-TMM

2.2 Additional effects considered in TD-TMM

3. Simulation results

3.1 Single-mode FP laser design and TD-TMM verification

3.2 Single-mode FP laser simulations

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (10)

Equations (18)

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