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Engineering » Electrical and Electronic Engineering » "Frontiers in Guided Wave Optics and Optoelectronics", book edited by Bishnu Pal, ISBN 978-953-7619-82-4, Published: February 1, 2010 under CC BY-NC-SA 3.0 license. © The Author(s).

Chapter 25

Surface-Emitting Circular Bragg Lasers – A Promising Next-Generation On-Chip Light Source for Optical Communications

By Xiankai Sun and Amnon Yariv
DOI: 10.5772/39537

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Overview

Surface-emitting circular Bragg lasers: (a) circular DFB laser; (b) disk Bragg laser; (c) ring Bragg laser. Laser emission is coupled out of the resonator plane in vertical direction via the Bragg gratings
Figure 1. Surface-emitting circular Bragg lasers: (a) circular DFB laser; (b) disk Bragg laser; (c) ring Bragg laser. Laser emission is coupled out of the resonator plane in vertical direction via the Bragg gratings
Two types of boundary conditions for calculating reflectivities. (a) A(0) = B(0), r1(L) = A(L)/B(L); (b) B(L) = 0, r2(x0, L) = B(x0)/A(x0)
Figure 2. Two types of boundary conditions for calculating reflectivities. (a) A(0) = B(0), r1(L) = A(L)/B(L); (b) B(L) = 0, r2(x0, L) = B(x0)/A(x0)
Modal field patterns, along with their threshold gains (gA) and frequency detuning factors (), of the circular DFB, disk, and ring Bragg lasers. All the three types of lasers have an exterior boundary radius of xb = 200. After (Sun & Yariv, 2008)
Table 1. Modal field patterns, along with their threshold gains (gA) and frequency detuning factors (), of the circular DFB, disk, and ring Bragg lasers. All the three types of lasers have an exterior boundary radius of xb = 200. After (Sun & Yariv, 2008)
Far-field intensity patterns of the fundamental mode of (a) circular DFB, (b) disk, and (c) ring Bragg lasers. After (Sun & Yariv, 2009a)
Figure 3. Far-field intensity patterns of the fundamental mode of (a) circular DFB, (b) disk, and (c) ring Bragg lasers. After (Sun & Yariv, 2009a)
Evolution of threshold gains of the 5 lowest-order modes of (a) circular DFB, (b) disk, and (c) ring Bragg lasers. The modes are labeled in accord with those shown in Table 1. The single-mode range for each laser type is marked in pink. After (Sun & Yariv, 2008)
Figure 4. Evolution of threshold gains of the 5 lowest-order modes of (a) circular DFB, (b) disk, and (c) ring Bragg lasers. The modes are labeled in accord with those shown in Table 1. The single-mode range for each laser type is marked in pink. After (Sun & Yariv, 2008)
Unnormalized quality factor of circular DFB, disk, and ring Bragg lasers. After (Sun & Yariv, 2008)
Figure 5. Unnormalized quality factor of circular DFB, disk, and ring Bragg lasers. After (Sun & Yariv, 2008)
Modal area of circular DFB, disk, and ring Bragg lasers. The top surface area of the laser resonator (πxb2) is also plotted as a reference. After (Sun & Yariv, 2008)
Figure 6. Modal area of circular DFB, disk, and ring Bragg lasers. The top surface area of the laser resonator (πxb2) is also plotted as a reference. After (Sun & Yariv, 2008)
Internal emission efficiency of circular DFB, disk, and ring Bragg lasers. After (Sun & Yariv, 2008)
Figure 7. Internal emission efficiency of circular DFB, disk, and ring Bragg lasers. After (Sun & Yariv, 2008)
Surface emission power Pem (in units of Psat) versus pump power level Ppump for the fundamental mode of circular DFB laser (xb = 200) under uniform pumping. The laser threshold Pth is defined as the pump level at the onset of surface laser emission. The external emission efficiency ex is defined as the slope of the linear fit of the simulated data points up to Pem = 10Psat.
Figure 8. Surface emission power Pem (in units of Psat) versus pump power level Ppump for the fundamental mode of circular DFB laser (xb = 200) under uniform pumping. The laser threshold Pth is defined as the pump level at the onset of surface laser emission. The external emission efficiency ex is defined as the slope of the linear fit of the simulated data points up to Pem = 10Psat.
Illustration of different pump profiles: (a) uniform; (b) Gaussian; (c) annular
Figure 9. Illustration of different pump profiles: (a) uniform; (b) Gaussian; (c) annular
Device-size-dependent pump level Ppump and frequency detuning factor  of the 3 lowest-order modes, under uniform pump profile, of (a) circular DFB, (b) disk, and (c) ring Bragg lasers. xb is the exterior boundary radius for all types of lasers. The inner disk radius x0 of the disk Bragg laser is set to be xb/2. The inner and outer edges of the annular defect of the ring Bragg laser are set to be xL = xb/2 –  and xR = xb/2 + , respectively. The modes are labeled in accord with those shown in Table 1. Dashed lines mark the values obtained at threshold and solid lines at Pem = 10Psat. After (Sun & Yariv, 2009b)
Figure 10. Device-size-dependent pump level Ppump and frequency detuning factor of the 3 lowest-order modes, under uniform pump profile, of (a) circular DFB, (b) disk, and (c) ring Bragg lasers. xb is the exterior boundary radius for all types of lasers. The inner disk radius x0 of the disk Bragg laser is set to be xb/2. The inner and outer edges of the annular defect of the ring Bragg laser are set to be xL = xb/2 – and xR = xb/2 + , respectively. The modes are labeled in accord with those shown in Table 1. Dashed lines mark the values obtained at threshold and solid lines at Pem = 10Psat. After (Sun & Yariv, 2009b)
Real part of the Green’s function G(z, z’) with –∞ < z < ∞ and z’ between the top and bottom surfaces of the laser resonator. After (Sun et al., 2008)
Figure 11. Real part of the Green’s function G(z, z’) with –∞ < z < ∞ and z’ between the top and bottom surfaces of the laser resonator. After (Sun et al., 2008)
Radiation coupling coefficient h1 and feedback coupling coefficient h2 as a function of the duty cycle dc of the Hankel-phased rectangular grating. After (Sun et al., 2007)
Figure 12. Radiation coupling coefficient h1 and feedback coupling coefficient h2 as a function of the duty cycle dc of the Hankel-phased rectangular grating. After (Sun et al., 2007)

Surface-Emitting Circular Bragg Lasers– A Promising Next-Generation On-Chip Light Source for Optical Communications

Xiankai Sun and Amnon Yariv

1. Introduction

Surface-emitting lasers have been attracting people’s interest over the past two decades because of their salient features such as low-threshold current, single-mode operation, and wafer-scale integration (Iga, 2000). Their low-divergence surface-normal emission also facilitates output coupling and packaging. Although Vertical Cavity Surface Emitting Lasers (VCSELs) have already been commercially available, their single-modedness and good emission pattern are guaranteed only for devices with a small mode area (diameter of ~m). Attempts of further increase in the emission aperture have failed mostly because of the contradictory requirements of large-area emitting aperture and single modedness, which casts a shadow over the usefulness of VCSELs in high-power applications.

A highly desirable semiconductor laser will consist of a large aperture (say, diameter larger than 20 m) emitting vertically (i.e., perpendicularly to the plane of the laser). It should possess the high efficiency typical of current-pumped, edge-emitting semiconductor lasers and, crucially, be single-moded. Taking a clue from the traditional edge-emitting distributed feedback (DFB) semiconductor laser, we proposed employing transverse circular Bragg confinement mechanism to achieve the goals and those lasers are accordingly referred to as “circular Bragg lasers.”

There have been intensive research activities in planar circular grating lasers since early 1990s. Erdogan and Hall were the first to analyze their modal behavior with a coupled-mode theory (Erdogan & Hall, 1990, 1992). Wu et al. were the first to experimentally realize such lasers in semiconductors (Wu et al., 1991; Wu et al., 1992). With a more rigorous theoretical framework, Shams-Zadeh-Amiri et al. analyzed their above-threshold properties and radiation fields (Shams-Zadeh-Amiri et al., 2000, 2003). More recently, organic polymers are also used as the gain medium for these lasers due to their low fabrication cost (Jebali et al., 2004; Turnbull et al., 2005; Chen et al., 2007).

The circular gratings in the above-referenced work are designed radially periodic. In 2003 we proposed using Hankel-phased, i.e., radially chirped, gratings to achieve optimal interaction with the optical fields (Scheuer & Yariv, 2003), since the eigenmodes of the wave equation in cylindrical coordinates are Hankel functions. With their grating designed to follow the phases of Hankel functions, these circular Bragg lasers usually take three configurations as shown in Fig. 1: (a) circular DFB laser, in which the grating extends from the center to the exterior boundary xb; (b) disk Bragg laser, in which a center disk is surrounded by a radial Bragg grating extending from x0 to xb; (c) ring Bragg laser, in which an annular defect is surrounded by both inner and outer gratings extending respectively from the center to xL and from xR to xb. Including a second-order Fourier component, the gratings are able to provide in-plane feedback as well as couple laser emission out of the resonator plane in vertical direction.

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Figure 1.

Surface-emitting circular Bragg lasers: (a) circular DFB laser; (b) disk Bragg laser; (c) ring Bragg laser. Laser emission is coupled out of the resonator plane in vertical direction via the Bragg gratings

This chapter will present a comprehensive and systematic study on the surface-emitting Hankel-phased circular Bragg lasers. It is structured in the following manner: Sec. 2 focuses on every aspect in solving the modes of the lasers – analytical method, numerical method, and mode-solving accuracy check. Sec. 3 gives near-threshold modal properties of the lasers; comparison of different types of lasers demonstrates the advantages of disk and ring Bragg lasers in high-efficiency surface laser emission. Sec. 4 discusses above-threshold modal behavior, nonuniform pumping effect, and optimal design for different types of lasers. Sec. 5 concludes this chapter and suggests directions for future research.

2. Mode solving techniques

Taking into account the resonant vertical laser radiation, Appendix A presents a derivation of a comprehensive coupled-mode theory for the Hankel-phased circular grating structures in active media. The effect of vertical radiation is incorporated into the coupled in-plane wave equations by a numerical Green’s function method. The in-plane (vertically confined) electric field is expressed as

E(x)=A(x)Hm(1)(x)+B(x)Hm(2)(x),
(1)

where Hm(1)(x) and Hm(2)(x) are the mth-order Hankel functions which represent respectively the in-plane outward and inward propagating cylindrical waves. A set of evolution equations for the amplitudes A(x) and B(x) is obtained:

dA(x)dx=u(x)A(x)v(x)B(x)e2iδx,
(2)
dB(x)dx=u(x)B(x)+v(x)A(x)e2iδx,
(3)

where

x = : normalized radial coordinate with being the in-plane propagation constant;

= (design–)/: frequency detuning factor, representing a relative frequency shift of a resonant mode from the designed value;

u(x)={gA(x)h1,ifxiswithinagratingregiongA(x),ifxiswithinano-gratingregion;
(4)
v(x)={h1+ih2,ifxiswithinagratingregion0,ifxiswithinano-gratingregion;
(5)

h1 = h1r + ih1i: grating’s radiation coupling coefficient, representing the effect of vertical laser radiation on the in-plane modes;

h2: grating’s feedback coupling coefficient, which can always be chosen real;

gA(x) = g(x) – : space-dependent net gain coefficient, the minimum value of which required to achieve laser emission will be solved analytically or numerically;

: nonsaturable internal loss, including absorption and nonradiative scattering losses;

g(x) = g0(x)/[1 + I(x)/Isat]: intensity-dependent saturated gain profile;

g0(x): unsaturated gain profile; and

I(x)/Isat: field intensity distribution in units of saturation intensity.

It should be noted that, although Eqs. (2) and (3) appear to be a set of coupled equations for in-plane waves only, they implicitly include the effect of vertical radiation due to h1. As it will become clearer in Sec. 2.3, the vertical radiation can simply be treated as a loss term during the process of solving the in-plane laser modes.

2.1. Analytical mode solving method

When solving the modes at threshold with uniform gain (or pump) distribution across the device, the net gain coefficient gA is x independent. The generic solutions of Eqs. (2) and (3) in no-grating regions are trivial:

In grating regions, by introducing A˜(x)=A(x)eiδx andB˜(x)=B(x)eiδx, Eqs. (2) and (3) become:

dA˜(x)dx=(uiδ)A˜(x)vB˜(x),
(8)
dB˜(x)dx=(uiδ)B˜(x)+vA˜(x),
(9)

whose generic solutions lead to

A(x)=A(0)eiδxsinh[S(xL)]+cosh[S(xL)]sinh[SL]+cosh[SL],
(10)
B(x)=A(0)eiδxv[(uiδ)S]sinh[S(xL)]+[(uiδ)S]cosh[S(xL)]sinh[SL]+cosh[SL],
(11)

whereS(uiδ)2v2, is a constant to be determined by specific boundary conditions, and L is a normalized length parameter (see Fig. 2). The determination of the constant in Eqs. (8) and (9) requires the specific boundary conditions be applied to the grating under investigation.

We focus on two typical boundary conditions to obtain and the corresponding field reflectivity in each case.

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Figure 2.

Two types of boundary conditions for calculating reflectivities. (a) A(0) = B(0), r1(L) = A(L)/B(L); (b) B(L) = 0, r2(x0, L) = B(x0)/A(x0)

Case I: As shown in Fig. 2(a), the grating extends from the center x = 0 to x = L. An inward propagating wave with amplitude B(L) impinges from outside on the grating. The reflectivity is defined as r1(L) = A(L)/B(L). The finiteness of E(x) at the center x = 0 requires A(0) = B(0), leading to

=(uviδ)sinh[SL]+Scosh[SL]Ssinh[SL]+(uviδ)cosh[SL]
(12)

and to the reflectivity

r1(L)=A(L)B(L)=e2iδL(uviδ)sinh[SL]+Scosh[SL](uviδ)sinh[SL]+Scosh[SL].
(13)

Case II: As shown in Fig. 2(b), the grating extends from x = x0 to x = L. An outward propagating wave with amplitude A(x0) impinges from inside on the grating. The reflectivity is defined as r2(x0, L) = B(x0)/A(x0). No inward propagating wave comes from outside of the grating, i.e., B(L) = 0. This condition leads to =S/(uiδ) and to the reflectivity

r2(x0,L)=B(x0)A(x0)=e2iδx0vsinh[S(Lx0)](uiδ)sinh[S(Lx0)]Scosh[S(Lx0)].
(14)

It should be noted that, as seen from their definitions, the above reflectivities Eqs. (10) and (11) include the propagation phase.

With the obtained reflectivities for the two types of boundary conditions, it is easy to derive the laser threshold condition for each circular Bragg laser configuration.

Circular DFB laser:

The limiting cases r1(xb) → ∞ or r2(0, xb) = 1 lead to the same result

tanh[Sxb]=Suviδ.
(15)

Disk Bragg laser:

Considering the radially propagating waves in the disk and taking the unity reflectivity at the center, the threshold condition is1e2gAx0r2(x0,xb)=1, which reads

e2(gAiδ)x0vsinh[S(xbx0)](uiδ)sinh[S(xbx0)]Scosh[S(xbx0)]=1.
(16)

Ring Bragg laser:

Considering the radially propagating waves in the annular defect, the threshold condition isr1(xL)e2gA(xRxL)r2(xR,xb)=1, which reads

(uviδ)sinh[SxL]+Scosh[SxL](uviδ)sinh[SxL]+Scosh[SxL]e2(gAiδ)(xRxL)vsinh[S(xbxR)](uiδ)sinh[S(xbxR)]Scosh[S(xbxR)]=1.
(17)

The above threshold conditions Eqs. (12), (13), and (14) govern the modes of the lasers of each type and will be used to obtain their threshold gains (gA) and corresponding detuning factors (). With these values, substituting Eqs. (4), (5), (8), and (9) into Eq. (1) and then matching them at the interfaces yield the corresponding in-plane modal field patterns. Despite their much simpler and more direct forms, these threshold conditions automatically satisfy the requirements that E(x) and E’(x) be continuous at every interface between the grating and no-grating regions (Sun & Yariv, 2009c).

2.2. Numerical mode solving method

When solving the modes at threshold with uniform gain (or pump) distribution across the device, gA is independent of x so that Eqs. (2) and (3) can have analytical solutions Eqs. (4) and (5), or (8) and (9). In the case of using a nonuniform pump profile and/or taking into account the gain saturation effect in above-threshold operation, gA becomes dependent on x and Eqs. (2) and (3) have to be solved numerically. The modes are then obtained by identifying those satisfying the boundary conditions.

As explained in Sec. 2.1, the same boundary conditions (BCs) apply to all the three types of circular Bragg lasers: (i) A(0) = B(0); (ii) B(xb) = 0; (iii) A(x) and B(x) continuous for 0 < x < xb. In Eqs. (2) and (3), g0(x) for a certain gain distribution profile can be parameterized with a proportionality constant, say, its maximal value g0.

The mode solving procedure is as follows: Having BC(i), we start with an amplitude set [A B] = A(0)[1 1] at the center, then numerically integrate Eqs. (2) and (3) along x to the exterior boundary xb, during which both A and B values are kept continuous at every interface between grating and no-grating regions to satisfy BC(iii). After the integration, we have B(xb) whose absolute value marks a contour map in the 2-D plane of g0 and . Now each minimum point in this contour map satisfies BC(ii) and thus represents a mode with corresponding g0 and . Retrieving A(x) and B(x) for this mode and substituting them into Eq. (1) give the modal field pattern.

We can also calculate the modal pump level using the obtained g0. Assuming a linear pump–gain relationship above transparency, the unsaturated gain g0(x) follows the profile of pump intensity Ipump(x), and we may define the pump level Ppump ≡ ∫ Ipump(x) ∙ 2 ∙ d = P0 ∫ g0(x) ∙ x ∙ dx, where P0 having a power unit is a proportionality constant determined by specific experimental setup. For simple g0(x) profiles, Ppump can have analytical expressions as will be shown in Sec. 4.2, otherwise, numerical integration always remains a resort.

2.3. Mode-solving accuracy check

In this subsection we derive an energy relation on which the examination of mode-solving accuracy is based. This energy relation is a direct result of the coupled-mode equations (2) and (3) combined with the boundary conditions and thus is exact.

Similar to the procedure in (Haus, 1975), multiplying Eq. (2) by A* and Eq. (3) by B*, then adding each equation to its complex conjugate, one obtains

d|A|2dx=2(gAh1r)|A|2vABe2iδxvABe2iδx,
(18)
d|B|2dx=2(gAh1r)|B|2+vABe2iδx+vABe2iδx.
(19)

Subtracting Eq. (16) from Eq. (15) yields

ddx(|A|2|B|2)=2gA(|A|2+|B|2)2h1r|Aeiδx+Beiδx|2.
(20)

Integrating Eq. (17) from x = 0 to x = xb and applying the boundary conditions A(0) = B(0) and B(xb) = 0 lead to

|A(xb)|2peripheralleakage+2h1rgrating|Aeiδx+Beiδx|2dxverticallaseremission=20xbgA(|A|2+|B|2)dxpowergeneratedinthegainmedium,
(21)

which is interpreted as the energy conservation theorem for the surface-emitting circular Bragg lasers. This equation states that, in steady state, the net power generated in the gain medium is equal to the sum of peripheral leakage power and vertical emission power. Due to its exactness, we may use this relation to monitor the accuracy of mode solving by substituting into Eq. (18) the obtained modal g0(x), , A(x), and B(x) and comparing the left-hand and right-hand sides of the equation.

As an aside, it should be noted that all the power terms in Eq. (18) are in units of a saturation power defined by

PsatEsat24D/β,
(22)

where Esat is the saturation field which relates to the saturation intensity by Isat = cn0|Esat|2/2 (c, the speed of light; n, transverse effective index; 0, the vacuum permittivity), and D is the thickness (vertical dimension) of the laser resonator.

3. Near-threshold modal properties

3.1. Threshold, frequency detuning, and in-plane modal pattern

For numerical demonstration, we assume all the lasers possess a vertical layer structure as described in (Scheuer et al., 2005a) which was designed for 1.55 m laser emission. The grating design procedure is detailed in Appendix B. The effective index neff is calculated to be 2.83 and the in-plane propagation constant = k0neff = 11.47 m–1. The circular grating is designed to follow the phase of Hankel functions with m = 0 to favor circularly symmetric modes. A quarter duty cycle is chosen to have both large feedback for in-plane waves while keeping a considerable amount of vertical emission. The coupling coefficients were found to be h1 = 0.0072 + 0.0108i and h2 = 0.0601.

Since we would like to compare the modal properties of different types of lasers with a same footprint, a typical device size of xb = 200 (corresponding to b ≈ 17.4 m) is assumed for all. For the disk Bragg laser, the inner disk radius x0 is assumed to be xb/2 = 100. For the ring Bragg laser, the annular defect is assumed to be located at the middle xb/2 = 100 and the defect width is set to be a wavelength of the cylindrical waves therein, yielding xL + xR = xb = 200 and xR – xL = 2. The calculated modal field patterns, along with the corresponding threshold gain values (gA) and frequency detuning factors (), of the circular DFB, disk, and ring Bragg lasers are listed in Table 1.

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Table 1.

Modal field patterns, along with their threshold gains (gA) and frequency detuning factors (), of the circular DFB, disk, and ring Bragg lasers. All the three types of lasers have an exterior boundary radius of xb = 200. After (Sun & Yariv, 2008)

A comparison of these modal properties concludes the following features of the three laser structures:

  1. All the displayed modes of the circular DFB laser are in-band modes on one side of the band gap (all > 0). This is due to the radiation coupling induced mode selection mechanism (Sun & Yariv, 2007). Increased gain results in the excitation of higher-order modes.

  2. All the displayed modes of the disk Bragg laser are confined to the center disk with negligible peripheral power leakage and thus possess very low thresholds and very small modal volumes as will be shown in Sec. 3.3.

  3. All the displayed modes of the ring Bragg laser, with the exception of the fundamental defect mode, resemble their counterparts of the circular DFB laser. The defect mode has a larger threshold gain than the fundamental mode of the circular DFB laser, but the former possesses a much higher emission efficiency as will be shown in Sec. 3.3.

3.2. Radiation field and far-field pattern

As mentioned earlier, by implementing a second-order circular grating design, the gratings can not only provide feedback for the in-plane fields but also couple the laser emission vertically out of the resonator plane. As derived in Appendix A, Eq. (41) relates the in-plane fields with the vertical radiation field in the grating regions. The radiation pattern at the emission surface is known as the near-field. For the grating design with m = 0, the near-field is expressed as

ΔE=(s1Aeiδx+s1Beiδx)|H0(1)|,
(23)

where s1 and s−1 at the emission surface can be obtained numerically according to Eq. (42) for a given grating structure. Following the design procedure in Appendix B, both s1 and s−1 at the emission surface were calculated to be 0.1725 − 0.0969i. Using the Huygens–Fresnel principle, the diffracted far-field radiation pattern of light from a circular aperture can be calculated under the parallel ray approximation (|r||r’|) (Hecht, 1998):

U(r)apertureΔE(ρ,φ)exp(ik|rr|)4π|rr|dreikr4πrapertureΔE(ρ,φ)exp(ik(rr^))dr=eikr4πrφ=02πρ=0ρbΔE(ρ)exp[ikρsinθcos(φϕ)]ρdρdφ=eikr2r0ρbΔE(ρ)J0(kρsinθ)ρdρ,
(24)

where

r=ρcosφx^+ρsinφy^
(25)

is the source point and

r=rsinθcosϕx^+rsinθsinϕy^+rcosθz^
(26)

is the field point. The far-field intensity pattern is then given by

I(r)=U(r)U(r)=|U(r)|2
(27)

and plotted in Fig. 3 for the fundamental mode of circular DFB, disk, and ring Bragg lasers.

In the far-field patterns, the different lobes correspond to different diffraction orders of the light emitted from the circular aperture. In the circular DFB and ring Bragg lasers, most of the energy is located in the first-order Fourier component thus their first-order diffraction peaks dominate. In the disk Bragg laser it is obvious that the zeroth-order peak dominates. These calculation results are similar to some of the experimental data for circular DFB and DBR lasers (Fallahi et al., 1994; Jordan et al., 1997).

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Figure 3.

Far-field intensity patterns of the fundamental mode of (a) circular DFB, (b) disk, and (c) ring Bragg lasers. After (Sun & Yariv, 2009a)

3.3. Single-mode range, quality factor, modal area, and internal emission efficiency

In the previous subsections we have compared the modal properties for devices with a fixed exterior boundary radius xb = 200. In what follows we will vary the device size and investigate the size dependence of modal gains to determine the single-mode range for each laser type. Within each own single-mode range limit, the fundamental mode of these lasers will be used to calculate and compare the quality factor, modal area, and internal emission efficiency. Similar to the prior calculations with a fixed xb, we still keep x0 = xb/2 for the disk Bragg laser and xL + xR = xb, xR – xL = 2 for the ring Bragg laser even as xb varies.

Single-Mode Range

In the circular Bragg lasers, since a longer radial Bragg grating can provide stronger feedback for in-plane waves, larger devices usually require a lower threshold gain. The downside is that a larger size also results in smaller modal discrimination, which is unfavorable for single-mode operation in these lasers. As a result, there exists a range of the exterior boundary radius xb values for each laser type within which range the single-mode operation can be achieved. This range is referred to as the “single-mode range.” Figure 4 plots the evolution of threshold gains for the 5 lowest-order modes as xb varies from 50 to 350. The single-mode ranges for the circular DFB, disk, and ring Bragg lasers are 50–250, 60–140, and 50–250, respectively, which are marked as the pink regions. Since single-mode

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Figure 4.

Evolution of threshold gains of the 5 lowest-order modes of (a) circular DFB, (b) disk, and (c) ring Bragg lasers. The modes are labeled in accord with those shown in Table 1. The single-mode range for each laser type is marked in pink. After (Sun & Yariv, 2008)

operation is usually preferred in laser designs, in the rest of this subsection we will limit xb to remain within each single-mode range and focus on the fundamental mode only.

Quality Factor

As a measure of the speed with which a resonator dissipates its energy, the quality factor Q for optical resonators is usually defined as /P where denotes the radian resonance frequency, the total energy stored in the resonator, and P the power loss. In our surface-emitting circular Bragg lasers, the power loss P has two contributions: coherent vertical laser emission coupled out of the resonator due to the first-order Bragg diffraction, and peripheral power leakage due to the finite radial length of the Bragg reflector.

Jebali et al. recently developed an analytical formalism to calculate the Q factor for first-order circular grating resonators using a 2-D model in which the in-plane peripheral leakage was considered as the only source of power loss (Jebali et al., 2007). To include the vertical emission as another source of the power loss, a rigorous analytical derivation of the Q factor requires a 3-D model be established. This is much more complicated than the 2-D case. However, since we are interested in comparing different laser types, a relative Q value will be good enough. Considering that the energy stored in a volume is proportional to ∫|E|2 dV and that the outflow power through a surface is proportional to ∫|E|2 dS, we define an unnormalized quality factor

Q=0Ddz02πdφ0ρb|E(ρ,z)|2ρdρgrating|ΔE(ρ,z=0)|2ρdρdφ+0Ddz02π|E(ρ=ρb,z)|2ρbdφ=0DZ2(z)dz0xb|E(x)|2xdxgrating|ΔE(x,z=0)|2xdx+0DZ2(z)dz|E(x=xb)|2βxb,
(28)

where Z(z) denotes the vertical mode profile for a given layer structure [see Eq. (32)] and D the thickness of the laser resonator. For a circularly symmetric mode, the angular integration factors are canceled out. The expressions for the in-plane field E and radiation field ΔE are given by Eqs. (1) and (20), respectively.

The unnormalized quality factor Q’ Eq. (23) is obviously proportional to an exact Q and the former is more intuitive and convenient for calculational purposes. The Q’ of the fundamental mode for the three laser types is calculated and displayed in Fig. 5. As expected, increase in the device size (xb) results in an enhanced Q’ value for all three types of lasers. Additionally, the disk Bragg laser exhibits a much higher Q’ than the other two laser structures of identical dimensions. As an example, for xb = 100, the Q’ value of the disk Bragg laser is approximately 3 times greater than that of the circular DFB or ring Bragg lasers. This is consistent with their threshold behaviors shown in Table 1.

Modal Area

Based on the definition of modal volume (Coccioli et al., 1998), an effective modal area is similarly defined:

Amodeeff=|E|2xdxdφmax{|E|2}.
(29)
media/image44.jpg

Figure 5.

Unnormalized quality factor of circular DFB, disk, and ring Bragg lasers. After (Sun & Yariv, 2008)

The modal area is a measure of how the modal field is distributed within the resonator. A highly localized mode having a small modal area can have strong interaction with the emitter. Figure 6 plots Amodeeff of the fundamental mode, within each single-mode range, for the three laser types. The top surface area of the laser resonator (πxb2) is also plotted to serve as a reference. The modal area of the disk Bragg laser is found to be at least one order of magnitude lower than those of the circular DFB and ring Bragg lasers. This is not surprising and can be inferred from their unique modal profiles listed in Table 1.

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Figure 6.

Modal area of circular DFB, disk, and ring Bragg lasers. The top surface area of the laser resonator (πxb2) is also plotted as a reference. After (Sun & Yariv, 2008)

Internal Emission Efficiency

As mentioned earlier, the generated net power in the circular Bragg lasers is dissipated by two kinds of loss: vertical laser emission and peripheral power leakage. The internal emission efficiency in is thus naturally defined as the fraction of the total power loss which is represented by the useful vertical laser emission. Figure 7 depicts the in of the fundamental mode, within each single-mode range, for the three laser types. As expected, all the lasers possess a larger in with a larger device size. Comparing devices of identical dimensions, only the disk and ring Bragg lasers achieve high emission efficiencies. This is a result of their fundamental modes being located in a band gap while the circular DFB laser’s fundamental mode is at a band edge, i.e., in a band. Band-gap modes experience much stronger reflection from the Bragg gratings, yielding less peripheral power leakage than in-band modes.

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Figure 7.

Internal emission efficiency of circular DFB, disk, and ring Bragg lasers. After (Sun & Yariv, 2008)

Summary of Comparison

In this subsection, by varying the device size we have obtained the single-mode range and compared the quality factor, modal area, and internal emission efficiency of the three types of lasers. It is demonstrated that, under similar conditions, disk Bragg laser has the highest quality factor, the smallest modal area, and the highest internal emission efficiency, indicating its suitability in high-efficiency, low-threshold, ultracompact laser design, while ring Bragg laser has a large single-mode range, large modal area, and high internal emission efficiency, indicating its wide application as a high-efficiency, large-area laser.

4. Above-threshold modal analysis

In Sec. 3 we have solved the modes and compared the near-threshold modal properties of the three types of surface-emitting circular Bragg lasers. This section focuses on an above-threshold modal analysis which includes gain saturation effect. The coupled-mode equations (2) and (3) will be solved numerically with boundary conditions. The relation of surface emission power versus pump power will be simulated. The laser threshold and external emission efficiency will be compared for these lasers under different pump profiles. Lastly, with the device size varying in a large range, the evolution curve of pump level for several lowest-order modes will be generated and the optimal design guidelines for these lasers will be suggested.

4.1. Surface emission power versus pump power relation

The numerical mode solving recipe is described in detail in Sec. 2.2. Simply put, Eqs. (2) and (3) are integrated along x from x = 0 to x = xb with the initial boundary condition [A B] = A(0)[1 1]. By identifying those satisfying the final boundary condition B(xb) = 0 one finds the modes with corresponding g0 and . The modal pump level is then given by Ppump = ∫ g0(x) ∙ x ∙ dx in units of a proportionality constant P0. Explained in Sec. 2.3, the surface emission power Pem from the laser is just the second term on the left-hand side of Eq. (18). By varying the value of A(0) at the beginning of the integration process, we are able to get the (Ppump, Pem) pairs which basically form the typical input–output relation for a laser mode.

As an example, we consider the circular DFB laser with xb = 200 and the other structural parameters the same as those used in Sec. 3. The additional parameter used in the numerical integration, the nonsaturable internal loss , is assumed to be 0.2 × 10–3 (already normalized by ) for typical III–V quantum well lasers. With the simulated (Ppump, Pem) pairs, the typical laser input–output relation is obtained for the fundamental mode and plotted in Fig. 8. The laser threshold Pth is defined as the pump level at the onset of surface laser emission. The external emission efficiency (or, energy conversion efficiency) ex is defined as the slope dPem/dPpump of the linear fit of the simulated data points up to Pem = 10Psat. As can be seen, the output power varies linearly with the pump power above threshold, which is in agreement with the theoretical and experimental results for typical laser systems [see, e.g., Sec. 9.3 of (Yariv, 1989)].

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Figure 8.

Surface emission power Pem (in units of Psat) versus pump power level Ppump for the fundamental mode of circular DFB laser (xb = 200) under uniform pumping. The laser threshold Pth is defined as the pump level at the onset of surface laser emission. The external emission efficiency ex is defined as the slope of the linear fit of the simulated data points up to Pem = 10Psat.

4.2. Nonuniform pumping effects

So far our studies on the circular Bragg lasers have assumed a uniform pumping profile and thus a uniform gain distribution across the devices. In practical situations, the pumping profile is usually nonuniform, distributed either in a Gaussian shape in optical pumping (Olson et al., 1998; Scheuer et al., 2005a) or in an annular shape in electrical pumping (Wu et al., 1994). The effects of nonuniform pumping have been investigated theoretically (Kasunic et al., 1995; Greene & Hall, 2001) and experimentally (Turnbull et al., 2005) for circular DFB lasers. In this subsection we will study and compare the nonuniform effects on the three types of surface-emitting circular Bragg lasers.

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Figure 9.

Illustration of different pump profiles: (a) uniform; (b) Gaussian; (c) annular

Let us focus on three typical pumping profiles – uniform, Gaussian, and annular – as shown in Fig. 9. The pump level Ppump can be expressed analytically in terms of the pump profile parameters:

Uniform:

g0(x)=g0,0xxb,Ppump=0xbg0xdx=12g0xb2,
(30)

Gaussian:

g0(x)=g0exp(x2wp2),x0,Ppump=0g0exp(x2wp2)xdx=12g0wp2,
(31)

Annular:

g0(x)=g0[exp((xxp)2wp2)+exp((x+xp)2wp2)],x0,
(32)
Ppump=0g0[exp((xxp)2wp2)+exp((x+xp)2wp2)]xdx=g0[wp2exp(xp2wp2)+πwpxperf(xpwp)],
(33)

where the error function erf(x)2π0xexp(t2)dt.

To compare the nonuniform pumping effects, the typical exterior boundary radius xb = 200 is again assumed for all the circular DFB, disk, and ring Bragg lasers. In addition, for the disk Bragg laser the inner disk radius is set to be x0 = xb/2, and for the ring Bragg laser the two interfaces separating the grating and no-grating regions are located at xL = xb/2 – and xR = xb/2 + . Following the calculation procedure in Sec. 4.1, the threshold pump level Pth and the external emission efficiency ex of the fundamental mode of the three types of lasers were calculated with the uniform, Gaussian, and annular pump profiles, respectively, and the results are listed in Table 2. Without loss of generality, the Gaussian profile was assumed to follow Eq. (26) with wp = xb/2 = 100, and the annular profile was assumed to follow Eq. (27) with xp = xb/2 = 100 and wp = xb/4 = 50. The numbers shown in Table 2 indicate an inverse relation between Pth and ex. The lowest Pth and the highest ex are achieved with the Gaussian pump for the circular DFB and disk Bragg lasers and with the annular pump for the ring Bragg laser.

These observations can actually be understood with fundamental laser physics: In any laser system the overlap factor between the gain spatial distribution and that of the modal intensity is crucial and proportionate. In semiconductor lasers once the pump power is strong enough to induce the population inversion the medium starts to amplify light. Thelasing threshold is determined by equating the modal loss with the modal gain, which is the

Pump profileCircular DFB laserDisk Bragg laserRing Bragg laser
PthexPthexPthex
Uniform9.7600.73696.5650.437413.1620.9278
Gaussian5.9670.99612.3730.87418.5701.379
Annular6.3820.97425.8550.73587.0101.500

Table 2.

Threshold pump level Pth (in units of P0) and external emission efficiency ex (in units of Psat/P0) of circular DFB, disk, and ring Bragg lasers under different pump profiles. After (Sun & Yariv, 2009b)

exponential gain constant experienced by the laser mode. This modal gain is proportional to the overlap integral between the spatial distribution of the gain and that of the modal intensity. Therefore if one assumes that, to the first order, the gain is proportional to the excess pump power over the transparency, then the threshold pump level Pth is inversely proportional to the above overlap integral [see, e.g., Sec. 11.3 of (Yariv, 1989)]. On the other hand, since the rate of simulated emission per electron and thus the gain are proportional to the modal intensity as seen by the electron [see, e.g., Sec. 8.3 of (Yariv, 1989)], this leads to a direct proportion between the external emission efficiency ex and the overlap integral. The bottom line is that a larger overlap between the pump profile and the modal intensity distribution results in more efficient energy conversion in the gain medium which consequently leads to a lower Pth and a higher ex.

4.3. Considerations in optimal design

To obtain the optimal design for these circular Bragg lasers, we will again vary their device size in a large range and inspect their size-dependent behavior. Like what we have done in Sec. 3.3, we will vary the exterior boundary radius xb for all the lasers while keeping x0 = xb/2 for the disk Bragg laser and xL = xb/2 – , xR = xb/2 + for the ring Bragg laser.

Figure 10 shows the dependence of the pump level Ppump and the frequency detuning factor on the device size xb for the 3 lowest-order modes, under uniform pump profile, of the three types of lasers. In each subfigure, the modes are numbered in accord with those shown in Table 1. For both Ppump and , dashed lines mark their values obtained at threshold and solid lines at Pem = 10Psat.

Seen from the upper left and right subplots of Fig. 10, the circular DFB and ring Bragg lasers still possess large discrimination between the modes even when operated in above-threshold regime (e.g., at Pem = 10Psat), which ensures them a large single-mode range of at least 50–250. Additionally, we have identified low-pump ranges for their Mode 1 at Pem = 10Psat, which are 100–160 for the circular DFB laser and 80–130 for the ring Bragg laser. The low-pump range is another important factor in designing such lasers for high-efficiency, high-power applications. The existence of this low-pump range is a result of competition between the pumped area and the required gain level: although larger devices require a larger area to be pumped, their longer radial Bragg gratings reduce the needed gain because of stronger reflection of the optical fields from the gratings.

Seen from the upper middle subplot of Fig. 10, the Ppump for the disk Bragg laser exhibits interesting behaviors: (i) at xb = 200, the order of Modes 1 and 2 exchanges from at threshold to above threshold due to the gain saturation effects; (ii) the single-mode range (for Mode 2) shifts from 60–140 at threshold to 90–175 at high surface emission level Pem = 10Psat. Therefore the single-mode range for designing the disk Bragg laser should be the overlap of these two ranges, i.e., 90–140.

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Figure 10.

Device-size-dependent pump level Ppump and frequency detuning factor of the 3 lowest-order modes, under uniform pump profile, of (a) circular DFB, (b) disk, and (c) ring Bragg lasers. xb is the exterior boundary radius for all types of lasers. The inner disk radius x0 of the disk Bragg laser is set to be xb/2. The inner and outer edges of the annular defect of the ring Bragg laser are set to be xL = xb/2 – and xR = xb/2 + , respectively. The modes are labeled in accord with those shown in Table 1. Dashed lines mark the values obtained at threshold and solid lines at Pem = 10Psat. After (Sun & Yariv, 2009b)

Seen from the lower subplots of Fig. 10, all the laser modes have overlapped dashed and solid lines, which means their frequency detuning factors are unaffected by the surface emission level. This is because of being an intrinsic property of a laser mode.

5. Conclusion and outlook

In this chapter we have described and analyzed a type of on-chip microlasers whose surface emission is very useful for many applications. The main advantage of these lasers would be the relative high (say, more than tens of mW), single-mode optical power emitted broadside and coupled directly into a fiber or telescopic optics. Other areas of applications that can benefit from such lasers include ultrasensitive biochemical sensing (Scheuer et al., 2005b), all-optical gyroscopes (Scheuer, 2007), and coherent beam combination (Brauch et al., 2000) for high-power, high-radiance sources in communications and display technology. Furthermore, a thorough investigation of such lasers may also lead to a better understanding in designing and fabricating a nanosized analogue, if a surface-plasmon approach is employed.

Throughout this work we have been trying to make a small contribution to understanding of the circular Bragg lasers for their applications as high-efficiency, high-power, surface-emitting lasers. We have covered the basic concepts, calculation methods, near- and above-threshold modal properties, and design strategies for such lasers.

We have studied three typical configurations of such circular Bragg lasers – namely, circular DFB laser, disk Bragg laser, and ring Bragg laser. Following the grating design principle for linear DFB lasers, the gratings of circular Bragg lasers have to be in sync with the phases of optical waves in a circular (or cylindrical) geometry. Since the eigensolutions of wave equation in a circular geometry are Hankel functions, this leads to a varying period of the gratings in radial direction, i.e., radially chirped gratings. To obtain efficient output coupling in vertical direction, a second-order scheme has been employed, and a quarter duty cycle has proved to be a good choice.

After a series of comparison of the modal properties, it becomes clear that disk and ring Bragg lasers have superiority over circular DFB lasers in high-efficiency surface emission. More specifically, disk Bragg lasers are most useful in low-threshold, ultracompact laser design while ring Bragg lasers are excellent candidates for high-power, large-area lasers.

Considering above-threshold operation with a nonuniform pump profile, it has been numerically demonstrated and theoretically explained that a larger overlap between the pump profile and modal intensity distribution leads to a lower threshold and a higher energy conversion efficiency. To achieve the same level of surface emission, disk Bragg laser still requires the lowest pump power, even though its single-mode range is modified because of the gain saturation induced mode transition. Circular DFB and ring Bragg lasers find their low-pump ranges at high surface emission level. These results provide us useful information for designing these lasers for single-mode, high-efficiency, high-power applications.

Looking ahead, there is still more work to be done on this special topic. For example, it would be interesting to further investigate how the grating design effects on the modal far-field pattern and what design results in a pattern having all, or almost all, of the energy located in the zeroth-order lobe with narrow divergence. This will be useful for applications which require highly-directional, narrow-divergence laser beams. On the other hand, since this chapter is mainly theoretical analysis oriented, experimental work, of course, has to develop to verify the theoretical predictions. In the field of optoelectronics, a single-mode, high-power laser having controllable beam shape and compatible with on-chip integration is still being highly sought. Due to the many salient features that have been described, it is our belief that the surface-emitting circular Bragg lasers will take the place of the prevailing VCSELs and make the ideal on-chip light source for next-generation optical communications and many other areas.

Appendix A: Derivation of comprehensive coupled-mode theory for circular grating structure in an active medium

In the case that the polarization effects due to the waveguide structure are not concerned, we can start from the scalar Helmholtz equation for the z component of electric field in cylindrical coordinates

[1ρρ(ρρ)+1ρ22φ2+k02n2(ρ,z)+2z2]Ez(ρ,φ,z)=0,
(34)

where , , and z are respectively radial, azimuthal, and vertical coordinates, k0 = /c = 2/0 is the wave number in vacuum.

For an azimuthally propagating eigenmode, Ez in a passive uniform medium in which the dielectric constant n2(, z) = r(z) can be expressed as

Ez(ρ,φ,z)=Ez(m)(ρ,z)exp(imφ)=[AHm(1)(βρ)+BHm(2)(βρ)]Z(z)exp(imφ),
(35)

with m the azimuthal mode number, = k0neff the in-plane propagation constant, and Z(z) the fundamental mode profile of the planar slab waveguide satisfying

(k02εr(z)+2z2)Z(z)=β2Z(z).
(36)

In a radially perturbed gain medium, the dielectric constant can be expressed as n2(, z) = r(z) + ii(z) + Δ(, z) where i(z) with |i(z)| r(z) represents the medium gain or loss and Δ(, z) is the perturbation profile which in a cylindrical geometry can be expanded in Hankel-phased plane wave series:

Δε(ρ,z)=Δε0l=±1,±2al(z)exp(ilΦ[Hm(1)(βdesignρ)])=Δε0l=±1,±2al(z)exp(ilΦ[Hm(1)(x)])exp(ilδx)=Δε0[a2(z)Hm(2)Hm(1)e2iδx+a2(z)Hm(1)Hm(2)e2iδx+a1(z)Hm(2)|Hm(1)|eiδx+a1(z)Hm(1)|Hm(1)|eiδx].
(37)

In the above expression, al(z) is the lth-order expansion coefficient of Δ(, z) at a given z. x is the normalized radial coordinate defined as x = . = (design−)/ (|| 1), the normalized frequency detuning factor, represents a relative frequency shift of a resonant mode from the designed value.

To account for the vertical radiation, an additional term ΔE(x, z) is introduced into the modal field so that

Ez(m)(x,z)=[A(x)Hm(1)(x)+B(x)Hm(2)(x)]Z(z)+ΔE(x,z).
(38)

Assuming that the radiation field ΔE(x, z) has an exp(±ik0z) dependence on z in free space, i.e.,

[1ρρ(ρρ)m2ρ2]ΔE=0,
(39)

substituting Eqs. (33), (34), and (35) into Eq. (29), introducing the large-radius approximations (Scheuer & Yariv, 2003)

|Hm(1,2)(x)x||dHm(1,2)(x)dx|,dnHm(1,2)(x)dxn(±i)nHm(1,2)(x),
(40)

neglecting the second derivatives of A(x) and B(x), and applying the modal solution in the passive unperturbed case, one obtains

2iZ(dAdxHm(1)dBdxHm(2))+ik02εiβ2(AHm(1)Z+BHm(2)Z)+1β2(k02εr+ik02εi+2z2)ΔE=k02Δε0β2[a2Hm(2)Hm(1)e2iδx+a2Hm(1)Hm(2)e2iδx+a1Hm(2)|Hm(1)|eiδx+a1Hm(1)|Hm(1)|eiδx]×(AHm(1)Z+BHm(2)Z+ΔE).
(41)

The phase-matching condition requires that the source and wave have close phase dependence. Grouping the terms with the same kind of Hankel functions leads to the following set of coupled equations:

2idAdxHm(1)Z+ik02εiβ2AHm(1)Z=k02Δε0β2(a2BHm(1)e2iδxZ+a1ΔE|Hm(1)|Hm(1)eiδx),
(42)
2idBdxHm(2)Z+ik02εiβ2BHm(2)Z=k02Δε0β2(a2AHm(2)e2iδxZ+a1ΔE|Hm(1)|Hm(2)eiδx),
(43)
(k02εr+2z2)ΔE=k02Δε0(a1A|Hm(1)|eiδxZ+a1B|Hm(1)|eiδxZ).
(44)

From Eq. (40), ΔE can be expressed as

ΔE=(s1Aeiδx+s1Beiδx)|Hm(1)|,
(45)

where

sl(z)=k02Δε0+al(z)Z(z)G(z,z)dz,
(46)

and G(z, z’) is the Green’s function satisfying

(k02εr(z)+2z2)G(z,z)=δ(zz).
(47)

Substituting Eq. (41) into Eqs. (38) and (39), multiplying both sides by Z(z), and integrating over z yield

dAdx=(gAh1,1)A(h1,1+ih2)Be2iδx,
(48)
dBdx=(gAh1,1)B+(h1,1+ih2)Ae2iδx,
(49)

where the gain coefficient

gAk022Pβ2+εi(z)Z2(z)dz,
(50)

the radiation coupling coefficients

h±1,±1ik02Δε02Pβ2+a±1(z)s±1(z)Z(z)dz,
(51)

the feedback coupling coefficients

h±2k02Δε02Pβ2+a±2(z)Z2(z)dz,
(52)

and the normalization constant

P+Z2(z)dz.
(53)

In the case of index grating, we can choose the phase of the grating such that a−1 = a1, a−2 = a2, then we denote h1 = h±1, ±1, h2 = h±2. By defining u = gA − h1 and ν = h1 + ih2, Eqs. (44) and (45) become

dA(x)dx=u(x)A(x)v(x)B(x)e2iδx,
(54)
dB(x)dx=u(x)B(x)+v(x)A(x)e2iδx.
(55)

Specifically, in the unperturbed (i.e., no-grating) region where Δ = 0, h1 and h2 vanish, and Eqs. (50) and (51) reduce to

dA(x)dx=gA(x)A(x),
(56)
dB(x)dx=gA(x)B(x).
(57)

Appendix B: Grating design procedure and calculations of the numerical Green’s function G(z, z'), coefficients h1, h2, and s1

Since we have previously fabricated such Hankel-phased circular Bragg lasers in InGaAsP active semiconductor material (Scheuer et al., 2005a), we will use the layer structure therein as an example for our numerical study. The target lasing wavelength 0 is 1.55 m. For simplicity, we approximate the complicated layer structure by an effective index profile comprising five layers as illustrated in Table B1. The vertical mode profile Z(z) and the effective index neff can be obtained numerically using a 1-D mode solver. The effective index neff was calculated to be 2.83 and the in-plane propagation constant = k0neff = 11.47 m–1. Numerical calculations of the mode profile and effective index of the approximated layer structure indicate negligible deviations from those of the exact one.

We focus our analysis on the case of a partially etched grating with an etch depth of 185 nm. The numerical Green’s function G(z, z’) that satisfies Eq. (43) with the given layer structure was calculated for –∞ < z < ∞ and z’ between the top and bottom surfaces of the laser resonator. A surface plot of the real part of G(z, z’) is displayed in Fig. 11.

Layer descriptionRefractive index nThickness
Upper cladding1.54
Third layer3.28160.5 nm
Second layer (active region)3.4057129 nm
First layer3.28160.5 nm
Lower cladding1.54

Table 3.

Approximated layer structure for numerical study

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Figure 11.

Real part of the Green’s function G(z, z’) with –∞ < z < ∞ and z’ between the top and bottom surfaces of the laser resonator. After (Sun et al., 2008)

To favor a circularly symmetric laser mode, m = 0 is specially chosen in the grating phase design. In such case, a Hankel-phased grating modulation with rectangular profile

Θ(Φ[Hm(1)(x)],α)={1,cos(Φ[Hm(1)(x)])α0,cos(Φ[Hm(1)(x)])<α
(58)

can be expanded in Fourier series as

arccosαπ+2πl=1sin(larccosα)lcos(lΦ[Hm(1)(x)])=arccosαπ+2π[sin(arccosα)cos(Φ[Hm(1)(x)])+sin(2arccosα)2cos(2Φ[Hm(1)(x)])+]=dc+12πsin(2πdc)[exp(i2Φ[Hm(1)(x)])+exp(i2Φ[Hm(1)(x)])]+1πsin(πdc)[exp(iΦ[Hm(1)(x)])+exp(iΦ[Hm(1)(x)])]+.
(59)

The expansion yields the coefficients

a2=a2=sin(2πdc)2π
(60)

and

a1=a1=sin(πdc)π
(61)

where

dcarccosαπ(1<α<1,0<dc<1)
(62)

is the duty cycle of the Hankel-phased rectangular grating. It should be emphasized that the duty cycle has a significant role in determining the coupling coefficients h1 and h2 (Barlow et al., 2004). Figure 12 plots h1 and h2 as a function of the duty cycle dc. A judicious choice would be dc = 0.25 where h2 is maximal and Re(h1) is not small so that we can have a large ratio of vertical emission to power leakage, hence a high emission efficiency. Based on Eqs. (47) and (48), the grating’s coupling coefficients were found to be h1 = 0.0072 + 0.0108i and h2 = 0.0601. Calculated from Eq. (42), s1 and s–1 at the emission surface (i.e., the top surface of the laser resonator) were found to be 0.1725 – 0.0969i.

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Figure 12.

Radiation coupling coefficient h1 and feedback coupling coefficient h2 as a function of the duty cycle dc of the Hankel-phased rectangular grating. After (Sun et al., 2007)

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