Whispering-gallery microcavity semiconductor lasers suitable for ...
Science in China Series E: Technological Sciences © 2009
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Whispering-gallery microcavity semiconductor lasers suitable for photonic integrated circuits and optical interconnects HUANG YongZhen†, YANG YueDe, WANG ShiJiang, XIAO JinLong, CHE KaiJun & DU Yun State Key Laboratory on Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of Sciences, PO Box 912, Beijing 100083, China
The characteristics of whispering-gallery-like modes in the equilateral triangle and square microresonators are introduced, including directional emission triangle and square microlasers connected to an output waveguide. We propose a photonic interconnect scheme by connecting two directional emission microlasers with an optical waveguide on silicon integrated circuit chip. The measurement indicates that the triangle microlasers can work as a resonance enhanced photodetector for optical interconnect. optical microcavity, microlaser, resonance mode, optical interconnect
The applications of optical interconnect are greatly demanded by the development of contemporary high-performance computing system with multiple processors and distributed shared-memory[1]. High speed optical interconnects are expected to satisfy the design requirements of delay, power, bandwidth, and delay uncertainty, which is increasingly difficult for the copper electric interconnect. To realize optoelectronic integration in Si, high efficiency luminescence in Si based materials has attracted a great attention. On the other hand, integrations of III-V devices on Si wafer are a practical solution for optoelectronic integration and optical interconnect ― on Si chip[1 3]. Optical microcavities have small cavity volume and high Q factor, the spontaneous emission of dipoles can be controlled due to the modulation of mode field distribution and mode state density in the microcavity. Ultra-low threshold current and high speed modulation can be realized in microcavity lasers due to its small cavity volume, which makes microcavity lasers an ideal light source for optoelectronic integration. In the past two decades, great progress has been made for vertical-cavity surface emitting lasers (VCSELs), whispering-gallery microcavity lasers, and photonic crystal
microcavity lasers. Low price and high performance 850 and 980 nm VCSELs have been widely applied in the short range optical interconnect. Two-dimensional VCSEL arrays are suitable for free space optical interconnect between two electric boards. Whispering-gallery microlasers with light output in the wafer plane are ideal light sources for the optical interconnect in integrated circuit chip. Optical interconnect on a silicon-on-insulator (SOI) chip was demonstrated in principle by integrating an InP based microdisk laser and a microdetector on the Si chip and coupling to a common SOI wire waveguide[4]. As the most common whispering-gallery microcavity lasers, microdisk lasers can be fabricated from edgeemitting laser wafer by plane semiconductor processing techniques, and have attracted great attention in the past two decades[5,6]. In the microdisk with circular symmetry, the mode light ray keeps the incident angle on the Received June 3, 2009; accepted July 4, 2009 doi: 10.1007/s11431-009-0306-y † Corresponding author (email: [email protected]) Supported by the National Natural Science Foundation of China (Grant Nos. 60777028, 60723002, and 60838003) and the Major State Basic Research Program of China (Grant No. 2006CB302804)
Citation: Huang Y Z, Yang Y D, Wang S J, et al. Whispering-gallery microcavity semiconductor lasers suitable for photonic integrated circuits and optical interconnects. Sci China Ser E-Tech Sci, 2009, 52(12): 3447―3453, doi: 10.1007/s11431-009-0306-y
perimeter of the microdisk, and the mode light rays with incident angle larger than the critical angle of the total internal reflection are confined in the microdisk, which is suitable for realizing high Q confined modes. But the circular symmetry and total internal reflection limit its directional output power and practical applications. Local and global deforms, and evanescent coupling to an output waveguide were applied to realize directional ― emission from microdisk lasers[7 11]. In the globally deformed microdisks, the incident angle of mode light ray varies as it transmits inside the cavity, and the mode light ray will refract out the microcavity forming a direction output as the incident angle is less than the critical angle of the total internal reflection in some positions. The mode light rays in the globally deformed microdisk usually have chaos behaviors[8]. Similar to the microdisk, equilateral polygonal optical microcavities usually have whispering-gallery type confined modes with high Q factors. We have systematically investigated the mode characteristics for equilateral triangle[12], square[13], rectangular[14], and hexagonal microresonators[15], and obtained analytical mode field patterns and mode wavelengths for equilateral triangle, square, and rectangular microresonators. In the circular symmetry microdisks, the field pattern along the perimeter of the microdisk is corresponding to longitudinal mode field distribution with a uniform envelope. But in the equilateral triangle and square microresonators, field distributions along the resonator perimeter are modulated by the longitudinal and transverse mode field patterns with the envelop distribution of transverse mode distribution. By connecting an output waveguide to the position of the microresonator with low field distribution, we can keep the confined mode with a high Q factor and realize directional emission[16,17]. Recently, we fabricated equilateral triangle and square InGaAsP/InP microlasers by standard photolithography and inductivelycoupled-plasma etching technique. Continuous-wave electrically injected operations were realized at room temperature for the 10―30 μm side ETR microlasers and 20 μm side square microlasers with a 2-μm wide output waveguide. In this paper, we will introduce the mode characteristics for equilateral triangle and square microresonators and the merits of direction emission by connecting an output waveguide, and then experimentally investigate the response behaviors of an equilateral triangle mi3448
crolaser as a resonant enhanced photodetector. An optical interconnect scheme is proposed by connecting two directional emission microlasers through optical waveguide on silicon chip.
1 Modes behaviors for an equilateral triangle resonator For a two-dimensional (2D) equilateral triangle resonator (ETR) with refractive index N > 2, a mode light ray parallel to the sides of the ETR will experience total internal reflection on the sides of the ETR, and return to the starting point after six times of the total internal reflections with the light path of the perimeter 3a, where a is the side length of the ETR. Unfolding the mode light ray inside the ETR, we find that the ETR is similar to a deformed Fabry-Pérot cavity and the mode field pattern inside the ETR can be modeled by the longitudinal and transverse mode field distributions as in the Fabry-Pérot cavity. Based on the boundary conditions of Maxwell equations, we have derived the following eigenvalue equations and the mode wavelength for the ETR resonator[12] 3a βl + 3ϕ = 2l π, (1) 2
κm
λm,l =
3a = (m + 1)π, 2 3Na 2
(2) ,
(3)
3ϕ ⎞ ⎛ 2 ⎜ 2l − ⎟ + 3(m + 1) π ⎝ ⎠ where the transverse eigenvalue equation (2) is the totally confined equation, βl and km are the longitudinal and the transverse propagation constants with the longitudinal and transverse mode indices of l and m, k0 = 2π/λ is the free space wave number, and the reflection phase shift ϕ satisfies ⎛ 3βl ⎝ 2γξ
⎞ 1 + (−1) m π, (4) ⎟⎟ + 2 ⎠ ξ is N2 and 1 for TE and TM modes, respectively. The decay constant in the external region of the ETR is
ϕ = 2 tan −1 ⎜⎜
γ=
βl 2
− k0 2 . (5) 4 There are two degenerate modes corresponding to the mode indices l and m: as l is the times of 3, the degenerate modes are accidentally degenerate modes with the
Huang Y Z et al. Sci China Ser E-Tech Sci | Dec. 2009 | vol. 52 | no. 12 | 3447-3453
same mode frequency and different Q factors; and as l is not the times of 3, the degenerate modes are perfect degenerate modes with the same mode frequency and Q factor[12]. For a 2D ETR surrounded by air with the side length of a = 4 µm and the refractive index of N =3.2, the electric field distributions of the symmetry states of the fundamental mode TM0,15 and the first order transverse mode TM1,15 are plotted in Figures 1(a) and 1(b) with the mode wavelength of 1545 and 1373 nm, respectively. The envelopes of the field distribution on the sides of the ETR have single peak and double peaks corresponding to the field behaviors of the fundamental and the first order transverse modes. Furthermore, the field distributions are very weak in the vertices of the ETR, so an output waveguide connected to one of the vertices can introduce directional emission for the ETR microlasers with high Q confined modes[16].
2 Mode behaviors for square resonator For the confined modes in a 2D square resonator with side length a and refractive index N in the air, we can approximate the field distributions in x and y directions as confined field distributions in three-layer dielectric slab waveguides independently, as the x and y axes are the symmetry axes and along the sides of the square. Thus, the magnetic field Hz for TE mode or the electric field Ez for TM mode in the square resonator can be expressed as[13] Fzp,q(x,y) = Fp(x)Fq(y), (6) F p ( x) = ⎧cos(κ x x − ϕ x ), ⎪ ⎨cos[(κ x a / 2) − ϕ x ]exp[−γ x ( x − a / 2)], ⎪cos[(−κ a / 2) − ϕ ]exp[γ ( x + a / 2)], x x x ⎩
x ≤ a / 2, x > a / 2, (7) x < − a / 2,
with Fq(y) taking the same form as (7) by replacing p and x with q and y. The mode numbers p and q denote the number of wave nodes in the x and y directions, respectively; ϕx and ϕy are zero or π/2 when the mode numbers p and q are even or odd; and κν and γν (ν = x, y) are the propagation constants in the square resonator and decay constants in the external region. They satisfy the following relations:
κ x2 + κ y2 = N 2 k02 ,
(8)
κ v2 + γ v2 = ( N 2 − 1)k02 , ν = x, y.
(9)
For the confined modes, their incident angles on the sides of the square resonator should be larger than the critical angle of total internal reflection, which is equivalent to γν > 0 in (9). Based on the continuous conditions of the tangential electric field for the TE modes or the tangential magnetic field for the TM modes on the sides of squares, we can obtain the mode eigenvalue equations as
κ x tan[(κ x a / 2) − ϕ x ] = ηγ x , κ y tan(κ y a / 2) − ϕ y ] = ηγ y ,
(10)
where η =N2 and 1 for the TE and TM modes, respectively. In the total confinement approximation, the mode propagation constants are ⎧⎪k x a = ( p + 1)π, ⎨ ⎪⎩k y a = (q + 1)π.
(11)
With the above approximation solutions (11) as the initial values, we can solve the mode wavelength as function of the mode indices p and q under an iterative process. For the square resonator, two modes with the mode indices of (p, q) and (q, p) are degenerate modes, and they are not orthogonal states because the field distribution in the corner regions is not solvable. If the mode
Figure 1 The analytical electric field distributions for the symmetry states of (a) TM0,15 and (b) TM1,15 in the equilateral triangle resonator with the side length of 4 µm.
Huang Y Z et al. Sci China Ser E-Tech Sci | Dec. 2009 | vol. 52 | no. 12 | 3447-3453
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numbers p and q are unequal and have the same evenodd characteristics, the two degenerate modes can couple together and form the following field distributions[13,14]: ⎧ e,( p ,q ) Fzp ,q + Fzq , p , = ⎪ fz ⎪ 2 (12) ⎨ p,q q, p ⎪ f o,( p , q ) = Fz − Fz , ⎪⎩ z 2 where the superscripts “e” and “o” indicates even and odd parities relative to the square diagonal mirror planes. The field patterns of fzo,(p,q) have null values at the corner of the square resonator while those of fze,(p,q) have peak values at the corner. The diffraction losses at the corners of the square are the main radiation loss for the confined modes in the square resonators, so fzo,(p,q) modes have smaller radiation losses and much higher Q factors than fze,(p,q) modes. For the rectangular resonator, two modes with the same symmetry can also result in anticrossing mode coupling as their mode wavelengths approach the same value[14]. For a square resonator surrounded by air with side length of 2.5 μm and refractive index N = 3.2, the z-directional electric field calculated by the above analytical solutions are plotted in Figure 2 for (a) TMo6,8 and (b) TMo4,8 with the mode wavelengths of 1505 and 1687 nm, respectively. Represented by the longitudinal and transverse mode indices, TMo6,8 and TMo4,8 are the fundamental and the first order transverse modes[13]. Because high Q modes are anti-symmetry to the diagonal mirror planes of the square resonator, the high Q confined modes can only couple with the high order transverse mode of an output waveguide if the output waveguide is connected to one corner of the square along the diagonal direction. However, we have found that the envelope of the field patterns along the sides of
Figure 2
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the square has one peak and double peaks for the fundamental and the first order transverse modes, respectively. The envelope of electric field pattern for the first order transverse mode TMo4,8 is a minimum point at the midpoint of the side, and its field is an even distribution relative to the midline of the square because the mode index is an even number. So by connecting an output waveguide to the midpoint of one side of the square, we can realize directional emission square microlasers based on high Q confined modes.
3 Mode field pattern for a directional emission square microlaser For a square resonator connected to an output waveguide in the midpoint of one side, finite-difference timedomain (FDTD) simulations show that the Q factor of the first order transverse mode decreases much slower with the increase of the output waveguide width than that of the fundamental transverse mode[17]. The results indicate that the output waveguide in the midpoint can result in high efficiency output and an additional mode selection. Under a narrow frequency exciting source covering only one confined mode, we can simulate the mode field patterns for a square resonator connected to an output waveguide by FDTD technique. The obtained electric field distributions are plotted in Figures 3(a) TMo6,8 and 3(b) TMo4,8, respectively, for the square resonator with the side length of 2.5 µm and an output waveguide of 0.2 µm wide. The corresponding mode wavelengths are 1507 and 1691 nm, which are 2 and 4 nm larger than the corresponding mode wavelength in Figure 2, respectively. The field distribution on the right section of Figure 3(b), with the distance to the side of the square larger than 1 µm, is magnified 4 times to
The analytical electric field distributions of (a) TMo6,8 and (b) TMo4,8 modes in the square resonator with the side length of 2.5 µm.
Huang Y Z et al. Sci China Ser E-Tech Sci | Dec. 2009 | vol. 52 | no. 12 | 3447-3453
Figure 3 The electric field distributions obtained by FDTD simulations for (a) TMo6,8, (b) TMo4,8 in the square resonator with the side length of 2.5 µm and a 0.2-µm wide output waveguide.
clearly express mode field pattern in the output waveguide. The midpoint is the maximum point for the field distribution of the fundamental transverse mode TMo6,8, so the coupling mode field pattern is very strong in the output waveguide as shown in Figure 3(a) and corresponding mode Q factor is very small. The first order transverse mode TMo4,8 can couple with the output waveguide and still have a high Q factor for realizing low threshold current microcavity lasers. For a square resonator with the side length of 20 μm and the output waveguide width of 2 μm, the simulated mode Q factor of the confined modes around 1550 nm can be larger than 104[19].
spectrum of an ETR laser with the side length of 25 μm is plotted in Figure 5, which was measured with CW injection current of 55 mA at 305 K. The laser spectrum shows
4 Optical interconnect scheme based on microcavity lasers The InP/InGaAsP ETR and square microlasers were fabricated from the common edge-emitting semiconductor laser wafer by standard photolithography and inductively― coupled-plasma etching technique[18 20]. Continuouswave (CW) electrically injected operations were realized at room temperature for the 10―30 μm side ETR microlasers with a 2-μm wide output wave- guide connected to one of the vertices of the ETR, with the highest temperature 310 K for CW operation. And square resonator microlasers with the side length of 20 μm and a 2-μm wide output waveguide were fabricated with the highest CW operation at 305 K. Recently, optical bistability was observed in the ETR microlasers, and the mode competitions of the transverse modes with different cross nonlinear gain and different output efficiencies were used to explain the bistability based on two-mode rate equations[21]. The schematic graph of the fabricated InP/GaInAsP ETR microlaser is plotted in Figure 4, and the laser output
Figure 4 Schematic graph of a directional emission ETR microlaser. The semiconductor ETR is indicated by dashed lines, which is covered by insulating SiO2 layer except in the top of the ETR for current injection, and the P-electrode.
Figure 5 Laser output spectrum of an ETR laser at 305 K with the side length being 25 µm.
Huang Y Z et al. Sci China Ser E-Tech Sci | Dec. 2009 | vol. 52 | no. 12 | 3447-3453
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single mode operation at the wavelength of 1579 nm. The directional emission microlasers have an output waveguide with the width of 2 μm. Low threshold current microlasers can be expected by reducing device size and the width of the output waveguide. If bonding two directional emission microlasers on Si chip and coupling with an optical waveguide on the Si chip, we expect to realize optical interconnect on the Si chip. The microcavity lasers can work as an emitter and a detector for bi-directional optical interconnect. By directly modulating the output of the microcavity laser and using another laser on the other side of the optical waveguide as a detector, we can set up a simple scheme of optical interconnect. In the following part, we introduce the characteristics of an ETR microlaser as a photodetector. To measure the spectral responsivity of an ETR laser, we couple the output of a tunable laser into the output waveguide of the ETR laser and measure the variation of the photocurrent with the tunable laser wavelength. The photocurrent of an ETR microlaser with the side length of 25 μm is plotted in Figure 6(a) as the solid line at the output power of the tunable laser is 1 mW. The spectral responsivity of the photo current indicates that the ETR can be a resonant enhanced photodetector, the wavelength interval between the resonant peaks is corresponding to the longitudinal mode interval of the ETR. However, the peak patterns vary with the position of coupling optical fiber to the output waveguide of the ETR laser. The peak photocurrent at 1537 nm is 47 μA and the dark current is less than 1 μA. Assuming that the coupling efficiency of the output of the tunable laser into the output waveguide of the ETR laser is 20%, we have a responsivity of 0.23 A/W at the peak value of 47 μA. The dashed lines in Figures 6(a) and 6(b) are the output laser spectra of the ETR laser at room temperature and the injection current of 10 mA, which cannot lase at room temperature. The injected current induces variations of material refractive index and temperature. So the peak positions of the photo current spectrum are not the same as those of the emission spectrum. Furthermore, the ETR with the side walls surrounded by Au layer will enhance the mode confinement. Fabry-Pérot type modes can also have high Q factor in an ETR surrounded by Au layer, and the emission spectrum can be affected by the whispering-gallery type modes and Fabry-Pérot type modes in some cases[22].
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Figure 6 The solid and the dashed lines are photocurrent spectrum and emission spectrum of an ETR micro laser with the side length being 25 µm.
5 Conclusion The mode characteristics of equilateral triangle and square resonators are introduced and are simulated for the resonators connected with an output waveguide. For the triangle and square resonators, the mode field patterns along the perimeter are modulated by the field distributions of the longitudinal and the transverse mode patterns. Connecting an output waveguide to the resonator at the position with weak mode field distribution, we can realize directional emission based on high Q confined whispering-gallery type modes in the resonator. The directional emission triangle and square microcavity lasers can be fabricated by planar semiconductor processing technique, and are a suitable light source for photonic integrated circuits. We propose to couple two directional emission microcavity lasers to an optical waveguide on Si chip to realize bi-directional optical interconnect in Si chip. The measurement indicates that the ETR microcavity lasers can work as a resonant enhanced photodetector, which proves the possibility of the optical interconnect scheme in principle.
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SCIENCE IN CHINA PRESS
www.scichina.com tech.scichina.com www.springerlink.com
Whispering-gallery microcavity semiconductor lasers suitable for photonic integrated circuits and optical interconnects HUANG YongZhen†, YANG YueDe, WANG ShiJiang, XIAO JinLong, CHE KaiJun & DU Yun State Key Laboratory on Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of Sciences, PO Box 912, Beijing 100083, China
The characteristics of whispering-gallery-like modes in the equilateral triangle and square microresonators are introduced, including directional emission triangle and square microlasers connected to an output waveguide. We propose a photonic interconnect scheme by connecting two directional emission microlasers with an optical waveguide on silicon integrated circuit chip. The measurement indicates that the triangle microlasers can work as a resonance enhanced photodetector for optical interconnect. optical microcavity, microlaser, resonance mode, optical interconnect
The applications of optical interconnect are greatly demanded by the development of contemporary high-performance computing system with multiple processors and distributed shared-memory[1]. High speed optical interconnects are expected to satisfy the design requirements of delay, power, bandwidth, and delay uncertainty, which is increasingly difficult for the copper electric interconnect. To realize optoelectronic integration in Si, high efficiency luminescence in Si based materials has attracted a great attention. On the other hand, integrations of III-V devices on Si wafer are a practical solution for optoelectronic integration and optical interconnect ― on Si chip[1 3]. Optical microcavities have small cavity volume and high Q factor, the spontaneous emission of dipoles can be controlled due to the modulation of mode field distribution and mode state density in the microcavity. Ultra-low threshold current and high speed modulation can be realized in microcavity lasers due to its small cavity volume, which makes microcavity lasers an ideal light source for optoelectronic integration. In the past two decades, great progress has been made for vertical-cavity surface emitting lasers (VCSELs), whispering-gallery microcavity lasers, and photonic crystal
microcavity lasers. Low price and high performance 850 and 980 nm VCSELs have been widely applied in the short range optical interconnect. Two-dimensional VCSEL arrays are suitable for free space optical interconnect between two electric boards. Whispering-gallery microlasers with light output in the wafer plane are ideal light sources for the optical interconnect in integrated circuit chip. Optical interconnect on a silicon-on-insulator (SOI) chip was demonstrated in principle by integrating an InP based microdisk laser and a microdetector on the Si chip and coupling to a common SOI wire waveguide[4]. As the most common whispering-gallery microcavity lasers, microdisk lasers can be fabricated from edgeemitting laser wafer by plane semiconductor processing techniques, and have attracted great attention in the past two decades[5,6]. In the microdisk with circular symmetry, the mode light ray keeps the incident angle on the Received June 3, 2009; accepted July 4, 2009 doi: 10.1007/s11431-009-0306-y † Corresponding author (email: [email protected]) Supported by the National Natural Science Foundation of China (Grant Nos. 60777028, 60723002, and 60838003) and the Major State Basic Research Program of China (Grant No. 2006CB302804)
Citation: Huang Y Z, Yang Y D, Wang S J, et al. Whispering-gallery microcavity semiconductor lasers suitable for photonic integrated circuits and optical interconnects. Sci China Ser E-Tech Sci, 2009, 52(12): 3447―3453, doi: 10.1007/s11431-009-0306-y
perimeter of the microdisk, and the mode light rays with incident angle larger than the critical angle of the total internal reflection are confined in the microdisk, which is suitable for realizing high Q confined modes. But the circular symmetry and total internal reflection limit its directional output power and practical applications. Local and global deforms, and evanescent coupling to an output waveguide were applied to realize directional ― emission from microdisk lasers[7 11]. In the globally deformed microdisks, the incident angle of mode light ray varies as it transmits inside the cavity, and the mode light ray will refract out the microcavity forming a direction output as the incident angle is less than the critical angle of the total internal reflection in some positions. The mode light rays in the globally deformed microdisk usually have chaos behaviors[8]. Similar to the microdisk, equilateral polygonal optical microcavities usually have whispering-gallery type confined modes with high Q factors. We have systematically investigated the mode characteristics for equilateral triangle[12], square[13], rectangular[14], and hexagonal microresonators[15], and obtained analytical mode field patterns and mode wavelengths for equilateral triangle, square, and rectangular microresonators. In the circular symmetry microdisks, the field pattern along the perimeter of the microdisk is corresponding to longitudinal mode field distribution with a uniform envelope. But in the equilateral triangle and square microresonators, field distributions along the resonator perimeter are modulated by the longitudinal and transverse mode field patterns with the envelop distribution of transverse mode distribution. By connecting an output waveguide to the position of the microresonator with low field distribution, we can keep the confined mode with a high Q factor and realize directional emission[16,17]. Recently, we fabricated equilateral triangle and square InGaAsP/InP microlasers by standard photolithography and inductivelycoupled-plasma etching technique. Continuous-wave electrically injected operations were realized at room temperature for the 10―30 μm side ETR microlasers and 20 μm side square microlasers with a 2-μm wide output waveguide. In this paper, we will introduce the mode characteristics for equilateral triangle and square microresonators and the merits of direction emission by connecting an output waveguide, and then experimentally investigate the response behaviors of an equilateral triangle mi3448
crolaser as a resonant enhanced photodetector. An optical interconnect scheme is proposed by connecting two directional emission microlasers through optical waveguide on silicon chip.
1 Modes behaviors for an equilateral triangle resonator For a two-dimensional (2D) equilateral triangle resonator (ETR) with refractive index N > 2, a mode light ray parallel to the sides of the ETR will experience total internal reflection on the sides of the ETR, and return to the starting point after six times of the total internal reflections with the light path of the perimeter 3a, where a is the side length of the ETR. Unfolding the mode light ray inside the ETR, we find that the ETR is similar to a deformed Fabry-Pérot cavity and the mode field pattern inside the ETR can be modeled by the longitudinal and transverse mode field distributions as in the Fabry-Pérot cavity. Based on the boundary conditions of Maxwell equations, we have derived the following eigenvalue equations and the mode wavelength for the ETR resonator[12] 3a βl + 3ϕ = 2l π, (1) 2
κm
λm,l =
3a = (m + 1)π, 2 3Na 2
(2) ,
(3)
3ϕ ⎞ ⎛ 2 ⎜ 2l − ⎟ + 3(m + 1) π ⎝ ⎠ where the transverse eigenvalue equation (2) is the totally confined equation, βl and km are the longitudinal and the transverse propagation constants with the longitudinal and transverse mode indices of l and m, k0 = 2π/λ is the free space wave number, and the reflection phase shift ϕ satisfies ⎛ 3βl ⎝ 2γξ
⎞ 1 + (−1) m π, (4) ⎟⎟ + 2 ⎠ ξ is N2 and 1 for TE and TM modes, respectively. The decay constant in the external region of the ETR is
ϕ = 2 tan −1 ⎜⎜
γ=
βl 2
− k0 2 . (5) 4 There are two degenerate modes corresponding to the mode indices l and m: as l is the times of 3, the degenerate modes are accidentally degenerate modes with the
Huang Y Z et al. Sci China Ser E-Tech Sci | Dec. 2009 | vol. 52 | no. 12 | 3447-3453
same mode frequency and different Q factors; and as l is not the times of 3, the degenerate modes are perfect degenerate modes with the same mode frequency and Q factor[12]. For a 2D ETR surrounded by air with the side length of a = 4 µm and the refractive index of N =3.2, the electric field distributions of the symmetry states of the fundamental mode TM0,15 and the first order transverse mode TM1,15 are plotted in Figures 1(a) and 1(b) with the mode wavelength of 1545 and 1373 nm, respectively. The envelopes of the field distribution on the sides of the ETR have single peak and double peaks corresponding to the field behaviors of the fundamental and the first order transverse modes. Furthermore, the field distributions are very weak in the vertices of the ETR, so an output waveguide connected to one of the vertices can introduce directional emission for the ETR microlasers with high Q confined modes[16].
2 Mode behaviors for square resonator For the confined modes in a 2D square resonator with side length a and refractive index N in the air, we can approximate the field distributions in x and y directions as confined field distributions in three-layer dielectric slab waveguides independently, as the x and y axes are the symmetry axes and along the sides of the square. Thus, the magnetic field Hz for TE mode or the electric field Ez for TM mode in the square resonator can be expressed as[13] Fzp,q(x,y) = Fp(x)Fq(y), (6) F p ( x) = ⎧cos(κ x x − ϕ x ), ⎪ ⎨cos[(κ x a / 2) − ϕ x ]exp[−γ x ( x − a / 2)], ⎪cos[(−κ a / 2) − ϕ ]exp[γ ( x + a / 2)], x x x ⎩
x ≤ a / 2, x > a / 2, (7) x < − a / 2,
with Fq(y) taking the same form as (7) by replacing p and x with q and y. The mode numbers p and q denote the number of wave nodes in the x and y directions, respectively; ϕx and ϕy are zero or π/2 when the mode numbers p and q are even or odd; and κν and γν (ν = x, y) are the propagation constants in the square resonator and decay constants in the external region. They satisfy the following relations:
κ x2 + κ y2 = N 2 k02 ,
(8)
κ v2 + γ v2 = ( N 2 − 1)k02 , ν = x, y.
(9)
For the confined modes, their incident angles on the sides of the square resonator should be larger than the critical angle of total internal reflection, which is equivalent to γν > 0 in (9). Based on the continuous conditions of the tangential electric field for the TE modes or the tangential magnetic field for the TM modes on the sides of squares, we can obtain the mode eigenvalue equations as
κ x tan[(κ x a / 2) − ϕ x ] = ηγ x , κ y tan(κ y a / 2) − ϕ y ] = ηγ y ,
(10)
where η =N2 and 1 for the TE and TM modes, respectively. In the total confinement approximation, the mode propagation constants are ⎧⎪k x a = ( p + 1)π, ⎨ ⎪⎩k y a = (q + 1)π.
(11)
With the above approximation solutions (11) as the initial values, we can solve the mode wavelength as function of the mode indices p and q under an iterative process. For the square resonator, two modes with the mode indices of (p, q) and (q, p) are degenerate modes, and they are not orthogonal states because the field distribution in the corner regions is not solvable. If the mode
Figure 1 The analytical electric field distributions for the symmetry states of (a) TM0,15 and (b) TM1,15 in the equilateral triangle resonator with the side length of 4 µm.
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numbers p and q are unequal and have the same evenodd characteristics, the two degenerate modes can couple together and form the following field distributions[13,14]: ⎧ e,( p ,q ) Fzp ,q + Fzq , p , = ⎪ fz ⎪ 2 (12) ⎨ p,q q, p ⎪ f o,( p , q ) = Fz − Fz , ⎪⎩ z 2 where the superscripts “e” and “o” indicates even and odd parities relative to the square diagonal mirror planes. The field patterns of fzo,(p,q) have null values at the corner of the square resonator while those of fze,(p,q) have peak values at the corner. The diffraction losses at the corners of the square are the main radiation loss for the confined modes in the square resonators, so fzo,(p,q) modes have smaller radiation losses and much higher Q factors than fze,(p,q) modes. For the rectangular resonator, two modes with the same symmetry can also result in anticrossing mode coupling as their mode wavelengths approach the same value[14]. For a square resonator surrounded by air with side length of 2.5 μm and refractive index N = 3.2, the z-directional electric field calculated by the above analytical solutions are plotted in Figure 2 for (a) TMo6,8 and (b) TMo4,8 with the mode wavelengths of 1505 and 1687 nm, respectively. Represented by the longitudinal and transverse mode indices, TMo6,8 and TMo4,8 are the fundamental and the first order transverse modes[13]. Because high Q modes are anti-symmetry to the diagonal mirror planes of the square resonator, the high Q confined modes can only couple with the high order transverse mode of an output waveguide if the output waveguide is connected to one corner of the square along the diagonal direction. However, we have found that the envelope of the field patterns along the sides of
Figure 2
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the square has one peak and double peaks for the fundamental and the first order transverse modes, respectively. The envelope of electric field pattern for the first order transverse mode TMo4,8 is a minimum point at the midpoint of the side, and its field is an even distribution relative to the midline of the square because the mode index is an even number. So by connecting an output waveguide to the midpoint of one side of the square, we can realize directional emission square microlasers based on high Q confined modes.
3 Mode field pattern for a directional emission square microlaser For a square resonator connected to an output waveguide in the midpoint of one side, finite-difference timedomain (FDTD) simulations show that the Q factor of the first order transverse mode decreases much slower with the increase of the output waveguide width than that of the fundamental transverse mode[17]. The results indicate that the output waveguide in the midpoint can result in high efficiency output and an additional mode selection. Under a narrow frequency exciting source covering only one confined mode, we can simulate the mode field patterns for a square resonator connected to an output waveguide by FDTD technique. The obtained electric field distributions are plotted in Figures 3(a) TMo6,8 and 3(b) TMo4,8, respectively, for the square resonator with the side length of 2.5 µm and an output waveguide of 0.2 µm wide. The corresponding mode wavelengths are 1507 and 1691 nm, which are 2 and 4 nm larger than the corresponding mode wavelength in Figure 2, respectively. The field distribution on the right section of Figure 3(b), with the distance to the side of the square larger than 1 µm, is magnified 4 times to
The analytical electric field distributions of (a) TMo6,8 and (b) TMo4,8 modes in the square resonator with the side length of 2.5 µm.
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Figure 3 The electric field distributions obtained by FDTD simulations for (a) TMo6,8, (b) TMo4,8 in the square resonator with the side length of 2.5 µm and a 0.2-µm wide output waveguide.
clearly express mode field pattern in the output waveguide. The midpoint is the maximum point for the field distribution of the fundamental transverse mode TMo6,8, so the coupling mode field pattern is very strong in the output waveguide as shown in Figure 3(a) and corresponding mode Q factor is very small. The first order transverse mode TMo4,8 can couple with the output waveguide and still have a high Q factor for realizing low threshold current microcavity lasers. For a square resonator with the side length of 20 μm and the output waveguide width of 2 μm, the simulated mode Q factor of the confined modes around 1550 nm can be larger than 104[19].
spectrum of an ETR laser with the side length of 25 μm is plotted in Figure 5, which was measured with CW injection current of 55 mA at 305 K. The laser spectrum shows
4 Optical interconnect scheme based on microcavity lasers The InP/InGaAsP ETR and square microlasers were fabricated from the common edge-emitting semiconductor laser wafer by standard photolithography and inductively― coupled-plasma etching technique[18 20]. Continuouswave (CW) electrically injected operations were realized at room temperature for the 10―30 μm side ETR microlasers with a 2-μm wide output wave- guide connected to one of the vertices of the ETR, with the highest temperature 310 K for CW operation. And square resonator microlasers with the side length of 20 μm and a 2-μm wide output waveguide were fabricated with the highest CW operation at 305 K. Recently, optical bistability was observed in the ETR microlasers, and the mode competitions of the transverse modes with different cross nonlinear gain and different output efficiencies were used to explain the bistability based on two-mode rate equations[21]. The schematic graph of the fabricated InP/GaInAsP ETR microlaser is plotted in Figure 4, and the laser output
Figure 4 Schematic graph of a directional emission ETR microlaser. The semiconductor ETR is indicated by dashed lines, which is covered by insulating SiO2 layer except in the top of the ETR for current injection, and the P-electrode.
Figure 5 Laser output spectrum of an ETR laser at 305 K with the side length being 25 µm.
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single mode operation at the wavelength of 1579 nm. The directional emission microlasers have an output waveguide with the width of 2 μm. Low threshold current microlasers can be expected by reducing device size and the width of the output waveguide. If bonding two directional emission microlasers on Si chip and coupling with an optical waveguide on the Si chip, we expect to realize optical interconnect on the Si chip. The microcavity lasers can work as an emitter and a detector for bi-directional optical interconnect. By directly modulating the output of the microcavity laser and using another laser on the other side of the optical waveguide as a detector, we can set up a simple scheme of optical interconnect. In the following part, we introduce the characteristics of an ETR microlaser as a photodetector. To measure the spectral responsivity of an ETR laser, we couple the output of a tunable laser into the output waveguide of the ETR laser and measure the variation of the photocurrent with the tunable laser wavelength. The photocurrent of an ETR microlaser with the side length of 25 μm is plotted in Figure 6(a) as the solid line at the output power of the tunable laser is 1 mW. The spectral responsivity of the photo current indicates that the ETR can be a resonant enhanced photodetector, the wavelength interval between the resonant peaks is corresponding to the longitudinal mode interval of the ETR. However, the peak patterns vary with the position of coupling optical fiber to the output waveguide of the ETR laser. The peak photocurrent at 1537 nm is 47 μA and the dark current is less than 1 μA. Assuming that the coupling efficiency of the output of the tunable laser into the output waveguide of the ETR laser is 20%, we have a responsivity of 0.23 A/W at the peak value of 47 μA. The dashed lines in Figures 6(a) and 6(b) are the output laser spectra of the ETR laser at room temperature and the injection current of 10 mA, which cannot lase at room temperature. The injected current induces variations of material refractive index and temperature. So the peak positions of the photo current spectrum are not the same as those of the emission spectrum. Furthermore, the ETR with the side walls surrounded by Au layer will enhance the mode confinement. Fabry-Pérot type modes can also have high Q factor in an ETR surrounded by Au layer, and the emission spectrum can be affected by the whispering-gallery type modes and Fabry-Pérot type modes in some cases[22].
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Figure 6 The solid and the dashed lines are photocurrent spectrum and emission spectrum of an ETR micro laser with the side length being 25 µm.
5 Conclusion The mode characteristics of equilateral triangle and square resonators are introduced and are simulated for the resonators connected with an output waveguide. For the triangle and square resonators, the mode field patterns along the perimeter are modulated by the field distributions of the longitudinal and the transverse mode patterns. Connecting an output waveguide to the resonator at the position with weak mode field distribution, we can realize directional emission based on high Q confined whispering-gallery type modes in the resonator. The directional emission triangle and square microcavity lasers can be fabricated by planar semiconductor processing technique, and are a suitable light source for photonic integrated circuits. We propose to couple two directional emission microcavity lasers to an optical waveguide on Si chip to realize bi-directional optical interconnect in Si chip. The measurement indicates that the ETR microcavity lasers can work as a resonant enhanced photodetector, which proves the possibility of the optical interconnect scheme in principle.
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