Performance-enhanced superluminescent diode with surface plasmon waveguide

Mehdi Ranjbaran; Xun Li

doi:10.1364/OE.17.023643

1. Introduction

Superluminescent Diode (SLD) is a semiconductor light source with the output power and spectral width of emitted light typically in between those of laser diode (LD) and light emitting diode (LED). This feature of having both high power and broad spectral width is desired in a number of applications such as fiber-optic gyroscopes [1] and incoherent medical imaging [2].

As in LD, the key to achieve high power in SLD is to amplify the spontaneously generated light in a region with gain through the stimulated emission process. However, unlike LD, optical feedback in SLD must be eliminated in order to avoid spectral width shrinking or even lasing. Therefore, while the injected current is pushed up, lasing oscillation caused by optical feedback is suppressed. To increase the spectral width, several techniques have been developed to broaden the gain bandwidth itself. They include implementing asymmetric multiple quantum wells (MQW) with different well thicknesses and/or material compositions [3,4].

A one dimensional analysis of SLD in [5] gives simple analytical expressions relating the output power and the spectral width to several operational and structural parameters under the assumptions of parabolic gain profile and zero facet reflectivities. Although these assumptions make the resulting expressions not applicable to all SLDs with different structures, the general implications understood are qualitatively valid. It was shown that the output power is characterized by a peak value

P_{p} = \frac{2 β n_{s p} h c^{2}}{λ_{p}^{3}} \exp {- α L + G (N)}

and a Gaussian-like spectrum with a full-width at half-maximum (FWHM) of

Δ λ_{F W H M} \propto \frac{λ_{w} (N)}{\sqrt{G (N)}}

Parameters in Eq. (1) are, h the Planck constant, c the speed of light in vacuum,βportion of the spontaneously emitted light coupled to the guided wave,

n_{s p}

the population inversion factor,

λ_{p}

the peak-gain wavelength,

λ_{w}

half width of the parabolic material gain profile (full width defined by zero crossing points), αthe modal loss, L the device length, N the carrier density and G the total device gain given by

G (N) = L Γ g_{0} (N)

in which Γis the confinement factor and $g_{0}$ is the peak material gain.
According to Eqs. (2)-(3) an attempt to increase the output power by increasing the device length will end up with the cost of a reduced spectral width. When the injection current (and consequently carrier density) is increased however, two counter acting effects take place, the total device gain G(N) increases because material gain increases; Also, the gain profile becomes broader due to the so called band-filling effect. The first effect, as seen from Eq. (2), tends to reduce the spectral width $Δ λ_{F W H M}$ while the second one increases it. Whichever effect wins the competition determines whether $Δ λ_{F W H M}$ will increase or decrease. As being such, for an optimized material gain profile if the spectral width sets an upper limit on total device gain (according to Eq. (2)) increasingβseems to be the only way to achieve higher output power. Also, from Eqs. (1)-(2) the output power is proportional toβ and increasing it has no harmful effect on the spectral width.
Photons generated by the spontaneous recombination of carriers assume all directions and a wide range of wavelength. They act as seeds whose number subsequently grows as they propagate along the active region. In the conventional dielectric waveguides only a small fraction of these photons couple to the guided modes resulting in a smallβ. For a surface plasmon waveguide (SPWG) however,β as will be shown, is much greater. Exploiting this advantage, a new device is proposed in which dielectric waveguide is replaced with an SPWG [6]. The performance of the new device will be compared with that of a conventional one in terms of the output power, its spectral width and their product. The latter which is sometimes called the power-linewidth product is commonly used as a figure of merit for SLDs.
This paper is organized as follows; Section 2 gives a brief introduction to two types of SPWGs and different aspects of the selected one when embedded in the proposed new SLD structure including loss, confinement factor, facet reflectivity and coupling of the spontaneous emission to different modes of SPWG. Section 3 discusses the numerical method used to simulate the new and conventional devices and compares performance of the two types of device and finally a conclusion is made in section 4.
2. SPWG parameters
2.1 Propagation constant and loss
The simplest type of an SPWG is the planar interface between a half space filled with a metal (with dielectric constant $ε_{2}$ ) and a dielectric material filling the rest of space (with dielectric constant $ε_{1}$ ) provided that $Re (ε_{1} + ε_{2}) < 0$ . Such interface supports only one bounded mode (of TM nature) [7]. The propagation constant of the mode is given by
$γ = k_{0} \sqrt{\frac{ε_{1} ε_{2}}{ε_{1} + ε_{2}}}$ where $k_{0} = ω / c$ is the free space wavenumber. Metals at optical frequencies have considerable loss. To get an estimate of the magnitude of loss if $ε_{1} = 11.2$ (typical of InGaAsP materials) and $ε_{2} = - 116.38 + 11.1 i$ (silver at $λ = 1.55 μ m$ [8]) the loss coefficient is $α = 2 Im (γ) = 1434 c m^{- 1}$ which is huge and makes it impossible to achieve a net gain in an active device with the values of gain today's technology has to offer. Another type of SPWGs, a thin metallic slab, however, as will be shown has a reasonable loss provided it is thin enough.
A thin slab of metal surrounded by two dielectric materials as shown in the inset of Fig. 1(a) supports up to one symmetric and one antisymmetric nonradiative bounded TM mode. The two modes are referred to as s_b and a_b, respectively. The complex propagation constants ( $γ = β_{r} + i β_{i}$ ) of these modes are obtained by numerically solving the dispersion equation [9]:
$\tanh (S_{2} h) (ε_{1} ε_{3} S_{2}^{2} + ε_{2}^{2} S_{1} S_{3}) + ε_{2} S_{2} (ε_{1} S_{3} + ε_{3} S_{1}) = 0$ where $S_{i}^{2} = γ^{2} - ε_{i} k_{0}^{2}$ for i = 1,2,3. Figure 1 shows the real and imaginary parts of the propagation constants of the two bounded modes as a function of film thickness for a symmetric SPWG (i.e. $ε_{1} = ε_{3}$ ) with parameters given in the figure caption. At small film thicknesses s_b mode, as seen from Fig. 1(b), has a significantly lower loss than a_b has, hence the names long-range and short-range for the two modes, respectively. At a thickness of 10 nm the loss coefficient for the s_b mode is about 15 cm⁻¹. In the limit of zero film thickness the loss for a symmetric structure becomes that of the surrounding dielectric media.
For an asymmetric structure where $ε_{1} \neq ε_{3}$ there is a film thickness below which the s_b mode will be cut off. The exact value of the cutoff thickness depends on the level of asymmetry in the structure. This means that for semiconductor devices using thin film SPWG the material above the waveguide cannot be air. With variable $ε_{1}$ , Fig. 2 shows the maximum allowable asymmetry defined by $| ε_{3} - ε_{1} | / ε_{3} \times 100$ so that the structure still supports the s_b mode. Since we are only interested in the long-range (s_b) mode we must keep the structure asymmetry low especially for thin films. Instead of a very thick $ε_{1}$ layer one can use a finite thickness superstrate layer above the metal film [10] with a dielectric constant similar to that of the material below the metal film ( $ε_{3}$ ) as shown schematically by the inset of Fig. 3 . Figure 3 shows the minimum thickness of the superstrate layer required to keep the s_b mode from cutoff. As the film thickness increases the need for a superstrate layer decreases. In the limit of very thick film the two metallic interfaces are decoupled each acting as a single interface SPWG. Also shown in the inset of Fig. 3 is the H_y component of the s_b mode for the structure with superstrate. Graphs in Figs. (1) -(3) have been obtained by first finding the characteristic equation for each structure using the transfer matrix method (TMM) [11,12] and then solving the equation using the argument principle method [12,13].

Fig. 2 Maximum allowable asymmetry vs. metal thickness for the structure of Fig. 1 to support the s_b mode. $ε_{3} = 11.2$ while $ε_{1}$ is reduced to make the SPWG asymmetric.
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Fig. 3 Minimum superstrate layer thickness required to support the s_b mode as a function of film thickness. $ε_{1} = ε_{3} = 11.2$ and $λ = 1.55 μ m$ ; inset: structure with superstrate layer (not to be scaled) and the profile of H_y component of the symmetric mode.
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Fig. 1 (a) Real and (b) imaginary parts of the propagation constants of bounded symmetric (s_b) and antisymmetric (a_b) modes of a silver slab SPWG. $λ = 1.55 μ m$ , . $ε_{2} = - 116.38 + i 11.1$ ., $ε_{1} = ε_{3} = 11.2$ .
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2.2 Confinement factor
In an SLD with SPWG the active region is embedded in the dielectric region at one side of the metal slab and close to it where modal fields decay exponentially away from the slab. The overlap of the active region with the tail of modal field which determines the confinement factor depends on the active region thickness, its separation from the metal slab and the depth of field penetration into the dielectric regions. Using the modal field expressions in [9] the confinement factor is calculated from
$Γ = \frac{Re (\int_{A . R .} E \times H^{*} \cdot \hat{z} d x)}{Re (\int_{A . S .} E \times H^{*} \cdot \hat{z} d x)}$ where the integration in the numerator is taken over the active region and that in the denominator is taken over the whole lateral extent of the modal fields. z is the propagation direction while x is the normal direction to the planar structure. As an example, for a 10 nm thick active region separated by 10 nm from a 10 nm thick silver slab waveguide with parameters similar to those given in the caption of Fig. 1 the confinement factor from Eq. (6) is about 1%.
2.3 Facet reflectivities
The reflection of surface plasmon modes from an interface with a uniform dielectric region has been investigated both theoretically and numerically [14,15]. It is concluded that the SPWG mode can radiate very efficiently (more than 98%) out of the SPWG provided that the refractive index of the dielectric material above the SPWG is close to that of the uniform dielectric region. Further suppression of multiple reflections should be possible through techniques commonly used in the conventional SLDs and semiconductor optical amplifiers (SOA) including incorporating an unpumped loss section at one end of the device, tilting the waveguide relative to the facets and coating facets with anti-reflective layers.
2.4 Coupling of dipoles in the active region to the SPWG
In broadband devices such as SLD and SOA a typical value for the spontaneous emission coupling factor,β, is 0.01 [16] (This should be compared to a typical value of 10⁻⁴ in a LD where βrepresents the portion of the spontaneous emission which not only couples to the waveguide but also has the wavelength of the lasing mode. In an SLD/SOA, on the other hand, the latter restriction is naturally eliminated). This means most of the spontaneously generated photons are wasted because they just cannot be captured by the waveguide. The reason for such a small value in dielectric waveguides is two fold; among the spontaneously generated photons incident on the core-cladding interface only those with angles greater than the critical angle will have a chance to remain inside the waveguide (total internal reflection condition). Moreover, guided waves supported by the dielectric waveguide must satisfy the transverse resonance condition [17]. This further limits the acceptable angles of incidence to a set of discrete values. SPWG, on the other hand, does not require either of the two conditions be satisfied. It is therefore, expected that SPWG have a much larger spontaneous emission coupling factor. With a largeβ, output power increases without the spectral width being reduced, because the increased power does not come from a greater device gain (G).
The problem of coupling of electron-hole pairs to a nearby SPWG is similar to that of a radiating molecular dipole near planar metallic interfaces which has been extensively studied theoretically and in experiments [18,19]. When a radiating dipole is placed near a metallic interface new radiation decay channels open up besides the familiar photon emission associated with an isolated dipole. The new decay mechanisms are attributed to surface plasmon coupling and intraband electron-hole excitations inside metal, the latter sometimes being called “lossy surface waves” [20,21]. The coupling efficiency to these channels at a certain wavelength depends on the geometrical structure, dipole distance from the metallic interface and the dipole orientation (polarization). The length scale of dipole-metal separation for coupling to these different decay channels is not the same. Generally, at long distances ( $> > λ$ ) coupling to surface plasmon modes and lossy surface waves is negligible. At short distances ( $~ λ$ ) dipole energy mostly goes to the surface plasmon modes. At very short distances ( $~ λ / 100$ ) however, coupling to the lossy surface waves becomes dominant [21].
An electron-hole pair (exciton) in a semiconductor material close to a metallic surface, as indicated by Gontijo et. al. [22], has an inherent difference with a molecular point dipole; there is no dissipation due to lossy surface waves because, such excitons do not have high enough wave vector components necessary to excite intraband transitions in the metal Fermi sea. As a result, a semiconductor quantum well can be placed very close to the SPWG to increase surface plasmon mode coupling factor without any concern of energy coupling to lossy surface waves as opposed to molecular radiators.
An electric dipole in a uniform dielectric region has a decay constant (inverse lifetime) b ₀ and when placed near a metallic structure has a decay constant b. The normalized decay constant $\hat{b} (= b / b_{0})$ is given by [18]
$\hat{b} = 1 + \frac{3 n_{1}^{2}}{2 μ_{0} k_{1}^{3}} Im (E_{0})$
where n ₁ and $k_{1} (= ω n_{1} / c)$ are the refractive index and the wave number in the medium in which dipole is embedded, $μ_{0}$ is the vacuum permeability and E ₀ is related to the amplitude of the electric field reflected from the metallic structure at the dipole position. $\hat{b}$ is written as
$\hat{b} = {\hat{b}}_{r} + {\hat{b}}_{n r}$
where the (normalized) radiative decay constant, ${\hat{b}}_{r}$ , is due to the plane waves for which the component of wavevector parallel to the metal interface is less than k ₁. These plane waves cannot couple to the surface plasmon modes but propagate in the dielectric media surrounding the metallic film. Nonradiative decay constant, ${\hat{b}}_{n r}$ , accounts for the portion of dipole energy which is transferred to the metal. From the CPS (Chance, Prock, Silbey) theory [18], the normalized decay rate of dipoles near any planar structure made up of metallic/dielectric materials can be obtained using the dyadic Green's function method [23]. Here, we consider a metallic film ( $ε_{2}$ ) of thickness d ₂ separating two dielectric half spaces ( $ε_{1}$ and $ε_{3}$ ). The dipole is embedded in the medium $ε_{1}$ at a distance of d ₁ away from the film. For the special cases of dipoles oriented perpendicular (⊥) and parallel ( $| |$ ) to the planar interfaces one finally obtains the following results
${\hat{b}}_{n r}^{⊥} = - \frac{3}{2} Im \int_{1}^{\infty} I^{⊥} (u) d u$ ${\hat{b}}_{r}^{⊥} = 1 - \frac{3}{2} Im \int_{0}^{1} I^{⊥} (u) d u$ ${\hat{b}}_{n r}^{∥} = \frac{3}{4} Im \int_{1}^{\infty} I^{∥} (u) d u$ ${\hat{b}}_{r}^{∥} = 1 + \frac{3}{4} Im \int_{0}^{1} I^{∥} (u) d u$
where
$I^{⊥} (u) = (\frac{R_{12}^{∥} + R_{23}^{∥} e^{- 2 l_{2} {\hat{d}}_{2}}}{1 + R_{12}^{∥} R_{23}^{∥} e^{- 2 l_{2} {\hat{d}}_{2}}}) e^{- 2 l_{1} {\hat{d}}_{1}} \frac{u^{3}}{l_{1}}$ $I^{∥} (u) = [\frac{R_{12}^{∥} + R_{23}^{∥} e^{- 2 l_{2} {\hat{d}}_{2}}}{1 + R_{12}^{∥} R_{23}^{∥} e^{- 2 l_{2} {\hat{d}}_{2}}} (1 - u^{2}) + \frac{R_{12}^{⊥} + R_{23}^{⊥} e^{- 2 l_{2} {\hat{d}}_{2}}}{1 + R_{12}^{⊥} R_{23}^{⊥} e^{- 2 l_{2} {\hat{d}}_{2}}}] e^{- 2 l_{1} {\hat{d}}_{1}} \frac{u}{l_{1}}$
and
${\hat{d}}_{j} = k_{1} d_{j}$
for j = 1,2. Also,
$R_{12}^{⊥} = \frac{l_{1} - l_{2}}{l_{1} + l_{2}}$ $R_{12}^{∥} = \frac{ε_{1} l_{2} - ε_{2} l_{1}}{ε_{1} l_{2} + ε_{2} l_{1}}$
are the Fresnel reflection coefficients for the perpendicular and parallel polarizations, respectively. $R_{23}^{⊥}$ and $R_{23}^{∥}$ are obtained similarly with the appropriate index changes. For an isotropic distribution of dipoles the decay constants are given by
$\hat{b} = \frac{2}{3} {\hat{b}}^{∥} + \frac{1}{3} {\hat{b}}^{⊥}$
Figure 4 shows the calculated nonradiative decay rate as a function of d₂ / d₁ for a silver film bounded by dielectric material with $ε_{1} = ε_{3} = 11.2$ . The wavelength is 1.55 $μ m$ . As indicated in [18], the peak occurs when $d_{2} / d_{1} = \frac{4}{3} (ε_{1} / | ε_{2} |)$ . At large d₂ / d₁ the nonradiative decay rate is proportional to d₁ ^- ³ whereas, for small d₂ / d₁ it has a d₁ ^- ⁴ behavior.

Fig. 4 Normalized nonradiative decay rate of an isotropic dipole near a silver film as a function of normalized distance (film thickness divided by the dipole distance from the film).
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Surface plasmon coupled energy divides between the long-range and short-range modes. In view of device operation energy coupled to the short-range mode is lost. Energy going into each mode is proportional to the square of modal fields amplitude at the dipole position [22]. In the surrounding dielectric regions field decays exponentially away from the film and the penetration depth depends on the real part of the modal propagation constants. Therefore, as seen from Fig. 1, the long-range mode penetrates deeper into the dielectric region. From the total energy coupled to the metal (i.e. to both long-range and short-range modes) the percentage that goes into the long-range mode is shown in Fig. 5 by the dotted line as a function of dipole distance from the film. Because of the exponential behavior of field decay, as the distance increases energy is distributed more in favor of the long-range mode. For the nonradiative decay rate calculated from Eqs. (9a), (9c) and (12), however, the trend is the other way around as seen from the dashed curve in Fig. 5. Product of these two curves gives the coupling to the long-range mode,β, and is shown in Fig. 5 with solid line. As a result, there is an optimal dipole-film distance at whichβ is maximum. For the structure parameters used in Fig. 5, the optimal distance is 30 nm, and maximumβis 75%.

Fig. 5 Coupling percentage versus dipole distance, dashed line: percentage of dipole power coupled to surface plasmon modes; Dotted line: percentage of surface plasmon coupled energy that goes to the long-range mode and solid line: the product of the previous two curves for a 5 nm thick silver film at 1.55 $μ m$ embedded in dielectric with $ε = 11.2$ .
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2.5 Device structure
In the proposed SLD the SPWG is placed close to the quantum well active region. This implies no electrode can be placed above the SPWG. Therefore, charge carriers should be injected into the active region laterally. Lateral injection of carriers has been used previously for example by Ahn and Chuang for a gain switched laser [24]. In their device a metallic layer was deposited as an electrode to apply electric field pulses across the active region but, in our case the metallic film plays the role of an SPWG (see Fig. 6 ).

Fig. 6 Schematic cross section of the proposed SLD structure with SPWG and lateral current injection
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3. Numerical analysis and discussion
We have used a one-dimensional mixed-frequency-time domain method [25] to obtain the output power spectrum and the carrier density distribution as a function of the longitudinal position inside an SLD with a single transverse mode. In this method the device length (along the z direction) as well as the spectral range of spontaneous emission is divided into many spatial and spectral subsections, respectively. Two photon rate equations for each spectral subsection (one for each forward and backward propagating waves) and one carrier rate equation for the whole spectrum are solved simultaneously:
$\frac{\partial P_{f, r} (z, t, λ_{k})}{v_{g} \partial t} \pm \frac{\partial P_{f, r} (z, t, λ_{k})}{\partial z} = [Γ g (z, t, λ_{k}) - α_{s}] P_{f, r} (z, t, λ_{k}) + P_{_{f, r}}^{s} (z, t, λ_{k})$ $\frac{\partial N (z, t)}{\partial t} = \frac{I}{q d w L} - [A + B N (z, t) + C N^{2} (z, t)] N (z, t) - R_{s t i m} (z, t)$ with $P_{f, r}$ being the forward/backward propagating power wave of the SPWG mode, $λ_{k}$ the center wavelength of the kth spectral subsection and $P_{_{f, r}}^{s}$ the spontaneous emission power coupled into an individual spectral subsection at a certain time and position along the device given by $P_{f, r}^{s} (z, t, λ_{k}) = β R_{s p} (z, t, λ_{k}) h v_{k} w d$ In Eq. (15) $R_{s p}$ is the spontaneous emission rate [26], w and d are the active region width and thickness, respectively and $h ν_{k}$ is the photon energy at the center of the spectral subsection k. Other parameters in Eqs. (13), (14) have their usual definitions [25]. This method enables us to model the dynamic behavior, longitudinal effects such as the longitudinal spatial hole burning (LSHB) and spectral characteristics of the device operation.
Figure 7(a)-(c) shows the simulation results for a single quantum well SLD with a conventional dielectric waveguide emitting at 1.55 $μm$ where the output facet power, FWHM linewidth of the output ASE (amplified spontaneous emission) and power-linewidth product versus the device length are drawn, for two values of waveguide loss. Simulation parameters are, I = 100 mA, . $β = 0.01$ . and $Γ = 0.01$ . Facet reflectivities have been set to zero. The nonradiative, bimolecular and Auger recombination rates are. $A = 2.8 \times 10^{8} (1 / s)$ ., $B = 1 \times 10^{- 10} ({cm}^{3} / s)$ and $C = 3.5 \times 10^{- 29} ({cm}^{6} / s)$ , respectively. As seen from Fig. 7(a), the output power reaches its maximum at an optimal device length. If the length is further increased device gain decreases due to the carrier dilution effect whereas if the length is decreased from the optimal value the total device gain given by Eq. (3) reduces leading, for both cases, to a decreased output power. Linewidth, on the other hand, decreases monotonically as the device length increases. The power-linewidth product also has a peak at a device length which is different from the optimal device length for the maximum output power.

Fig. 7 (a)-(c), respectively, facet power, spectral width and their product in an SLD with dielectric waveguide for two values of loss (cm⁻¹); (d)-(f), corresponding graphs for a device with SPWG with different values of loss.
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Figures 7(d)-7(f) shows similar graphs for an SLD with SPWG for which $β = 0.63$ (obtained from the theory of Sec. 2.4 corresponding to an 8 nm thick SPWG). Curves are plotted for four different modal loss values. Other parameters remain unchanged. The first advantage of the SLD with SPWG is a higher output power as can be seen by comparing Fig. 7(a) and Fig. 7(d). The maximum output power obtained from the SLD with SPWG is obtained at a lower device length which reveals another advantage for this device when observing that the emission linewidth increases monotonically with the decrease of the device length: a higher power-linewidth product which is a common measure of SLD performance. The maximum power-linewidth product achievable from the SPWG-SLD with a modal loss of 50 cm⁻¹ is, in this case, higher than that of a conventional SLD with a modal loss of 10 cm⁻¹.
The optimal device lengths for maximum output power and maximum power-linewidth product are different. Therefore, an optimum length can be selected based on whether the SLD is being used in an application which needs a high power or both high power and broad spectral width. For example, consider two SLDs, one with dielectric waveguide with a modal loss of 10 cm⁻¹ (device 1) and the other with SPWG and a loss of 20 cm⁻¹ (device 2). From Fig. 7 for a maximum output power of 18.4 mW, device 1 has a length of 3300 $μ m$ , emission linewidth of 25 nm and power-linewidth product of 0.45 mW-nm. Device 2, on the other hand, has a maximum output power of 27.2 mW at a length of 1020 $μ m$ , linewidth of 90 nm and power-linewidth product of 3.5 mW-nm showing considerable improvement in all three parameters. On the other hand if device 1 was to be designed for the maximum power-linewidth product, the length would be 2250 $μ m$ , the output power 15 mW, the linewidth 48 nm and the power-linewidth product 0.75 mW-nm. The corresponding numbers for device 2 are 480 $μ m$ , 20.2 mW, 170 nm and 3.5 mW-nm, respectively, showing again a remarkable improvement with a shorter device length. For thinner waveguides loss is lower and at the same time spontaneous coupling factor is higher making the difference in performance of the two devices even more significant.
4. Conclusion
The use of metallic slab surface plasmon waveguide is proposed in superluminescent diodes in place of the conventional dielectric waveguides. With the new waveguide electrons and holes injected into the active region tend to recombine and transfer their energy to the SPWG mode rather than to produce photons with random propagation directions from which only a small portion will contribute to the waveguide mode. This results in a much greater spontaneous emission coupling factor. Simulation results show improvement in the facet power, spectral width of emission and power-linewidth product achieved with a shorter device.
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Performance-enhanced superluminescent diode with surface plasmon waveguide

Abstract

1. Introduction

2. SPWG parameters

2.1 Propagation constant and loss

2.2 Confinement factor

2.3 Facet reflectivities

2.4 Coupling of dipoles in the active region to the SPWG

2.5 Device structure

3. Numerical analysis and discussion

4. Conclusion

References and links

Cited By

Figures (7)

Equations (23)

Optics Express