Polarization dependent mode dynamics of metallic hybrid laser micro-resonator

Optics Communications 338 (2015) 128–132

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Polarization dependent mode dynamics of metallic hybrid laser microresonator Kai-Jun Che n, Cheng-Xu Chu, Chang-Lei Guo, Dan Zhang, Zhi-Ping Cai School of Information Science and Engineering, Xiamen University, Xiamen, China

art ic l e i nf o

a b s t r a c t

Article history: Received 15 July 2014 Received in revised form 12 August 2014 Accepted 20 October 2014 Available online 27 October 2014

Mode dynamics of metallic hybrid laser micro-resonator are investigated by the general waveguide theory. By tuning the thickness of sandwiched low refractive index dielectric layer, the lateral metal enables compression and release of the electric field into and from gain medium for both transverse electric and magnetic modes (TE/M), enhancing and weakening the coherent interaction of gain medium and quantized resonant mode. The mode quality (Q) factors depending on the geometric parameters are explored by power integration in waveguide and present different features for TE and TM polarizations. The influences of both cladding and dielectric layers on the gain threshold of passive micro-resonator are studied as well, showing that TE mode needs lower gain threshold than TM mode for compensating the overall dissipation of metal. & Elsevier B.V. All rights reserved.

Keywords: Metallic hybrid laser micro-resonator Polarization dependent Q factor Gain threshold

1. Introduction Electrically pumped semiconductor microlasers are expected as the key components in future optoelectronic circuits and have been a subject of intense interests in the past two decades [1–7], of which the circularly shaped disk laser are mostly developed due to its high Q factor whispering gallery modes and significantly low power consuming features [2–5]. Several schemes, such as double disks (one as ohmic contact and the other as active medium) [2,3] and arm air bridged contact [5], were proposed to overcome the drawbacks of air suspended active semiconductor medium (optically pumping) for electrically pumping, but the fabrication processes are complicated and not suitable for mass production. In addition to the microdisk, the electrically pumped microlasers were intuitively fabricated from the multi quantum wells wafer with weak vertical waveguide (also as carriers channel) and tight lateral air confinements [6,7], just like those FP lasers. The geometry of laser resonator was tailored by the standard photolithography, combined with dry or wet etching techniques. Nevertheless, the surrounding air functions as both confining medium and emission scattering window, the practical optical loss is actually much greater than that theoretically predicted, meaning that the laser dimension is not able to further scale down [8]. For the optoelectronic circuits, the stable operation of all the optical components is critical from the practical view, the bare laser in air n

Corresponding author. Fax: þ86 592 258 0141. E-mail address: [email protected] (K.-J. Che).

http://dx.doi.org/10.1016/j.optcom.2014.10.043 0030-4018/& Elsevier B.V. All rights reserved.

is frail to surrounding optical interferences. Thus, there is an increasing interest for developing optical insulated and electrically pumped micro- even nano- lasers (including edge and surface emitting lasers for instances) [9–11]. By introducing a metal layer to replace the low refractive index air, the surface plasmonic resonance can attain sub-wavelength mode volumes, and lasing emissions were achieved from metallically coated cavities [9,11– 14]. For the electrically pumped laser, the metal functions as both optical confining medium and electric conductor, while the high dissipative optical loss of metallic medium can be relieved by an introduced electric insulator between gain medium and metal (such as silicon nitride) [9,11–14]. In this paper, by using the general waveguide theory(GWT) (transfer matrix method for layered vertical and plane waveguide), the mode dynamics of three dimensional metallic hybrid laser micro-resonator, which is laterally confined by dielectric-metal bilayer and vertically confined by weak waveguide, are quantitatively investigated. The work is focused on the exploring the eigen values, mode Q factors and gain threshold of resonant plasmonic mode, and the polarization dependent mode dynamics are presented.

2. Modes in metallic hybrid laser micro-resonator The schematic illustration of model under study is shown in Fig. 1, including cross section and top views. The metallic hybrid laser micro-resonator is laterally confined by an electric insulator, such as silica or silicon nitride (low refractive index dielectric), and

K.-J. Che et al. / Optics Communications 338 (2015) 128–132

a

b

Fig. 1. Schematic illustration of metallic hybrid semiconductor laser resonator, (a) and (b) present the cross section view and top view of model respectively.

a metal (silver or gold), and vertically confined by cladding mediums. The top metal functions as electrode for current injection and the opened substrate is applied for lasing emission [12]. The resonator diameter of D, the thickness of dielectric d1, (the thickness of metal is set as 0.2 μm which is enough thick to completely insulate outside optical wave), the etching depth of de consisting of active layer thickness, up and bottom cladding thickness du, db, are denoted respectively. The semiconductor resonator are fabricated via combined techniques of standard photolithography and inductively-coupled-plasma dry etching [13], then the dielectric and metal are deposited sequentially on the lateral of already prepared resonator. Similar to the dielectric micro-cylinder, the resonant modes in metallic hybrid laser micro-resonator are treated as quasi-TE and -TM modes, and their electromagnetic components are defined as the groups of Hz (magnetic field) and ET (electric field), Ez (electric field) and HT (magnetic field) based on the shown coordinate. The subscripts z and T denote the vertical and transverse directions of fields, respectively. The solutions of harmonic magnetic and electric fields Hz and Ez for quasi-TE and -TM modes are separated into the vertical and lateral parts and presented as follows:

ψi(z, r , θ , t) = ϕi(z)φ(r , θ)e−iωt

(1)

ϕi(z) is the vertical electromagnetic distribution, consisting of both up and down optical waves, and is expressed as ϕi(z) = η(Ai eik ziz i + Bie−ik ziz i)

(2)

η is β/ωμ0 and β/ωεi for quasi-TE and -TM mode whose HT and ET are zero. β is transverse propagating constant or in-plane effective wave vector, ω is the eigen angular frequency, μ0 and εi are the permeability and permittivity of corresponding medium. Ai and Bi are the coefficients of up and down waves, the subscript i denotes which layer in vertical direction, kzi is the vertical wave vector. The lateral component φ(r,θ) is expressed as Bessel functions multiplied by a cosine function φ(r , θ) = cos(mθ)× ⎧ J (βr) r ≤ R1(gain) ⎪ m ⎪ CJ (n k r) + DY (n k r) R < r ≤ R (dielectric) m d 0 1 2 ⎨ m d 0 ⎪ nm r > R2(metal) ⎪ EKm( k 0r) ⎩ i

(3)

where the integer m is the azimuthal quantum number of resonant modes, the interfaces radius of gain medium and dielectric, dielectric and metal are defined as R1 and R2, nd and nm are the optical indices of dielectric and metal. Jm, Ym are Bessel functions of the first and second kinds and Km is the modified Bessel function of the second kind. Of course, the equation can also be expressed by composite of Jm and Ym uniformly [15]. The eigen

129

values of resonant mode (resonant wavelength or effective refractive index) are found by combined methods of transfer matrix and iteration calculation based on effective index method [16]. Firstly, we calculate the β of guided waves of vertical multi-layer waveguide by assuming an initial mode wavelength λ0 through transfer matrix method, and then calculate a new resonant wavelength λ1 based on the obtained β by using the Eq. (3), where the tangential electromagnetic fields are continuous along the radial direction. The calculation is cycled and terminated until the λ obtained from two adjacent calculations is less then a given value of 0.01 nm, meaning that the obtained λ is the wavelength of resonant mode. Here, we take inP/InGaAsP semiconductor laser resonator (with refractive index ng ¼3.4 for gain medium and nc ¼3.17 for cladding layer) for consideration. The metal silver(Ag) and silica(SiO2) are applied as optical and electric insulators, their optical refractive indices are set as nm ¼0.14 þ11.2i and nd ¼ 1.45 at optical communication wavelength 1550 nm band [17]. The wavelength λ of resonant mode versus the variable d1 with fixed up cladding layer du ¼1.5 μm and the variable du with fixed d1 ¼0.2 μm are shown in Fig. 2(a) and (b) respectively (Taking TE10,1 and TM10,1 as examples). Here, the substrate is assumed to be infinite from practical view, the diameter of resonator D is 2.1 μm and the thickness of gain material or active layer is set as 0.2 μm for fundamental mode confinement in vertical multi-layer waveguide. Two important features are found. Differing from that dielectric mode whose resonant wavelength increases as the volume of microcavity expanding (just like that of TM polarization), the resonant wavelength λ of TE mode here first decreases and then becomes constant as d1 ranges from 0.01 μm to 0.4 μm. This abnormal characteristic is due to the fact that there exists two kinds of modes for TE polarization, one is internal surface plasmonic mode trapped at the interface of dielectric and metal, and the other is dielectric photonic mode confined in cavity. As d1 increases, the resonant mode transits from the plasmonic state to the hybrid state of plasmonic and dielectric, and finally to the pure dielectric photonic state since λ is nearly not effected as d1 40.3 μm(the depth of evanescent wave), inducing that the internal surface plasmonic and dielectric photonic modes play the dominant role for the thin and thick dielectric respectively. To intuitively exhibit this transition, Fig. 2(c) and (d) illustrate the false color distributions of the normalized squared electric fields E2T which is comprised of radial component E2r and angular component E2θ , the maximum electric density transfers from the interface of silver and semiconductor to the internal cavity. As expected, a large discontinuity of E2T appears at radial interfaces of R1 and R2, this is because the main component E2r (E2r is much greater than E2θ as the optical wave transmits mainly along the edge of cavity) is normal to the interfaces of different mediums while the E2θ is tangential to the angular direction, it also is why the angular momentum of E2T is not strictly quantized. But for TM mode, due to the electric field Ez is parallel to the interfaces of medium(vertical direction), the smooth change of E2z along the radial direction is presented in Fig. 2(e) and (f) as d1 ¼0 and 0.2 μm respectively. On the other hand, as shown in Fig. 2(g) and (h), as d1 increases from 0.01 μm to 0.40 μm, the main component of E2r of TE10,1 moves into the internal gain cavity along the radial direction, while E2z of TM10,1 moves away from the cavity. It means that the enhancement and weakening of coherent interaction between gain medium (such as multiple quantum wells) and the quantized electromagnetic fields can be efficiently tuned by changing d1, inducing that the efficiency of photon will be improved and decreased for TE10,1 and TM10,1 as d1 increases. Namely, for the metallic hybrid laser resonator, the electromagnetic field can be artificially compressed and released by tuning the structure parameters of laser micro-resonator.

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a

c

b

d g

h e

f

Fig. 2. Resonant wavelength of TE10,1 and TM10,1 versus (a) the thickness of electric insulator silica d1 with fixed up cladding layer du ¼1.5 μm and (b) versus du with fixed d1 ¼ 0.2 μm, the dashed lines denote the cut-off du. (c) and (d) show false color distributions of the normalized squared electric fields E2T in xy plane for TE10,1 as d1 ¼ 0.0 μm and 0.2 μm. (e) and (f) show false color distributions of the normalized squared electric fields E2 z in xy plane for TM10,1 as d1 ¼0.0 μm and 0.2 μm. (g) and (h) show the shift of dominate squared electric field (normalized) of TE10,1 and TM10,1 as d1 is transferred from 0.01 μm to 0.40 μm, the dashed line shows the edge of gain disk.

3. Quality factors and gain threshold of resonant modes In addition to the eigen values, one of important parameters of resonant modes is quality (Q) factor which denotes the photon life or optical loss of resonant system. In metallic hybrid laser resonators, the optical losses are induced by several factors, such as the vertical radiation loss, the absorption loss of material and the dissipative loss of coated metal (replace the transverse radiation losses) and so on. The overall Q factor of a passive micro-resonator can be calculated by considering the main contributions

1 1 1 = + Q total Q rad Q me

(4)

Qrad is mainly related to the optical radiation emitted into the substrate, Qme results from the total optical absorption of metal. Here, the Q factors depending on structure parameters, such as the dielectric thickness d1 and the up cladding layer thickness du are calculated. For a metallic hybrid laser resonator with infinite up and below cladding layers (du ¼1.5 μm and db ¼2.0 μm for instances), the total Q factor is mainly determined by the lateral confined metal (namely Qtotal∼Qme), since the radiation loss into the substrate is much smaller than the dissipative loss produced

by lateral confining metal. By using the obtained eigen values, the Q factor is calculated by power integration of waveguide method (IWM)

Q me =

∭total ε(z, r , θ) E(z, r , θ) 2 dV εm" ∭

metal

E(z, r , θ) 2 dV

(5)

ε″ m ¼3.316 is the imaginary part of the relative permittivity of silver. For TE modes, the vertical distribution of E2T (z) is directly obtained from planar waveguide theory, the in-plane distribution E2T (r,θ) is indirectly calculated through E2T (r,θ) ¼ E2r (r,θ) þ E2θ (r,θ),

E2r (r,θ) and E2θ (r,θ) are obtained from the relations ∂Hz /∂θ = irωεEr

and −∂Hz /∂r = iωεEθ . Thus, the 3D field distribution of E2T (r,θ, z) is

obtained by E2T (z)  E2T (r,θ) and the integration of ε(r,θ, z)  E2T (r, θ, z) of TE modes can be calculated. For TM modes, the vertical

distribution of electric field Ez(z) is indirectly obtained through Ez (z) = β /ωεi × HT (z) from the transverse magnetic field. HT(z), the in-plane distribution of Ez(r,θ) is directly obtained by Eq. (3). The 3D distribution of electric field Ez(z, r,θ) is also calculated by Ez(z)  Ez(r,θ). The results are shown in Fig. 3, where both up and lateral dissipation are considered with fixed other

K.-J. Che et al. / Optics Communications 338 (2015) 128–132

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a

b

Fig. 3. The Q factors of TE10,1 and TM10,1 versus the thickness of (a) dielectric and (b) up cladding layer with fixed other parameters shown by insets respectively.

structure parameters. As expected, due to the fact that the penetration of optical power into silver is well alleviated by the introduced silica, the dissipation of silver of both TE and TM modes is significantly reduced and the Q factors of resonant modes are increased. The Q factor of TE10,1 increases firstly with great rate and then increases exponentially as that of TM10,1 which keeps an exponential rate(straight line). The reason is that the coupling between conventional photonic mode and surface plasmonic mode confined at the interface of dielectric and silver weakens as the d1 increases, and finally TE10,1 presents the characteristics of pure photonic mode(just like that of TM10,1). But for the same d1, the Q factor of TE10,1 is one order greater than that of TM10,1, meaning that TE polarization is advantageous over TM polarization for the same metallic hybrid laser micro- resonator. In addition, the Q factors of TE10,1 and TM10,1 are also dependent on the thickness of up cladding layer du. Here, we give d1 ¼0.2 μm, and other parameters are the same as above. The results are illustrated in Fig. 3(b) for TE10,1 and TM10,1. For both TE and TM modes, when d1 increases from their cut-off thickness of 0.13 μm and 0.48 μm, the total Qme firstly shows a slight decrease and gets a minimum value at du ¼0.21 μm and 0.63 μm respectively, then increases monotonously. It is reasonable since the field shape is influenced as the up cladding layer becomes thicker, there inevitably is a maximum loss point for the certain field shape. Take TE10,1 as an example, if du increases from 0.13 μm, the optical dissipation led by up metal becomes less, but the lateral field expands into the insulator layer, the optical loss led by lateral metal becomes greater, and the total optical dissipation gradually increases to a maximum point as du ¼0.21 μm. As du further increases, the field distribution in-plane is slightly influenced by a greater du, the change of optical dissipation led by lateral metal is negligible, but the optical dissipation led by up metal further decreases, so the total optical dissipation by surrounding metal decreases and the mode Q factor presents monotonous increase. Of course, for a other d1, a minimum value of Q factor can also be

Fig. 4. The gain threshold of laser resonator of TE10,1 and TM10,1 versus (a) the thickness of dielectric d1 with fixed up cladding layer du ¼ 1.5 μm, (b) the up cladding layer with fixed d1 ¼ 0.2 μm.

obtained at a corresponding du. The results obtained by IWM show that the influences on the Q induced by up cladding layer is much weaker than that induced by the introduced dielectric layer. Namely, the lateral metal dissipation plays a dominate role on the total loss of cavity. Another important parameter of laser resonator is the gain threshold, which is for compensating the total dissipative loss of metal. The expression is following [15]:

εg" =

εm" ∭

metal

E(z, r , θ) 2 dV

∭gain E(z, r , θ) 2 dV

(6)

ε″ m is the same as above mentioned, ε″ g is defined as the threshold of gain medium (the imaginary part of relative permittivity of gain medium). The relations between gain threshold ε″g of TE10,1 and TM10,1 and the thickness of silica d1 are first considered. The results are shown in Fig. 4(a) where du is fixed to be 1.5 μm. Here, due to the small mode volume and the fact that optical amplitude in up metal is much smaller than that in lateral metal, the dissipation led by up metal is productively much smaller than that led by total lateral metal. On the whole, the gain threshold of TE10,1 is less than that of TM10,1, even though the two have a slight difference for the thin dielectric. It is reasonable for that the internal plasmonic mode has great optical dissipation with small d1. The ε″g depending on du is also considered and the results are illustrated in Fig. 4(b). The effects related to both the up and the lateral silver with fixed d1 ¼0.2 μm are independently considered as ε″g  u and ε″g  l. For both TE10,1 and TM10,1, the ε″g  u is exponentially decreases with nearly the same rate from the cut-off du, since their field morphology presents exponentially decay from the interface of gain medium and up cladding layer. But with small du, the dissipation of up silver presents different characteristics for TE10,1 and TM10,1. The former is comparable with, even larger than that induced by lateral silver while the latter is much less.

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K.-J. Che et al. / Optics Communications 338 (2015) 128–132

theory. The eigen values of resonant modes are calculated by transfer matrix method based on the effective index method. The introduced electric insulator can be used for tuning the electric field distribution along radial direction of laser resonator. The compression and release of electric field of TE and TM polarizations into and from gain medium are presented and this will effectively enhance and weaken the coherent interaction of gain medium and quantized resonant mode. The Q factors and gain threshold depending on structure parameters are explored by IWM, TE modes have higher Q factor and lower threshold gain than that of TM mode, which may promote lasing of single polarization.

Acknowledgments Fig. 5. The cross section view of squared

E2r

for TE10,1.

Nevertheless, the total silver dissipation related ε″g of TE10,1 is much less than that of TM10,1, which can also be found from the Q factor as shown in Fig. 3(a) and (b), meaning that TE polarization modes have better performance than TM polarization modes. Except the above structure parameters, the smaller optical refractive index of insulator and a thicker active layer will further reduce the gain threshold of resonant modes. Finally, quasi-3D numerical simulation is performed by finite element method (FEM) for high Q TE mode with commercial available software(radio frequency module in COMSOL MULTIPHYSICS 4.4a) [18]. The 2D axisymmetric model (the inset of Fig. 5) is built and the perfect matched layers (PML) are introduced outside the calculation regions for improving the simulation accuracy. The eigen frequency of resonant mode accompanied field distribution is obtained from a computation process. The laser micro-resonator with parameters of D ¼2.1 μm, d1 ¼0.2 μm, du ¼1.5 μm and de ¼2 μm is designed. The cross section view of squared electric field component E2r (Er is the main component and quantized, but the total electric field is not quantized as shown in Fig. 2(d), thus here we give only E2r ) is shown in Fig. 5, the resonant wavelength agrees well with that obtained by general waveguide theory(the deviation is less than 0.8%). However, the mode Q factor is in less agreement with that obtained by IWM, the value for quasi-3D laser resonator is 2.45  104 which is less than 1.6  105. This deviation is due to the fact that the optical loss radiated into substrate is not taken into account in the IWM. In addition, due to the introduction of opened substrate which can be considered as a defect on infinite circular substrate, the mode morphology will have a slight deformation from that of ideal mode, the actual Q factor calculated thus deviates from that obtained by IWM. Because the direction of Er is normal to the interface of dielectric and metal, the large discontinuity of E2r is also found, which is the same as that of Fig. 2(d).

4. Conclusion In conclusion, mode dynamics of metallic hybrid laser microresonator have been investigated by general optical waveguide

The authors acknowledge the reviewers for their beneficial comments on the manuscript. This work is supported by the Fundamental Research Funds for the Central Universities under Grant no. 2011121048, the National Natural Science Foundation of China under Grant nos. 61107045 and 61107023, Specialized Research Fund for the Doctoral Program of Higher Education of china 20110121120020.

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