Near-infrared hybrid plasmonic multiple quantum well nanowire lasers

Jiamin Wang; Wei Wei; Xin Yan; Jinnan Zhang; Xia Zhang; Xiaomin Ren

doi:10.1364/OE.25.009358

1. Introduction

In recent years, lasers with ever smaller sizes have been realized, ranging from micron-sized vertical cavity surface emitting lasers (VCSELs) [1], single nanowires (NWs) [2] and nanoribbons [3], quantum well (QW)/dot structures [4] to subwavelength plasmonic lasers [5–14]. Among them, hybrid plasmonic lasing from a NW at subwavelength scale was demonstrated, in which a semiconductor NW was placed on a flat metal substrate separated by a nanoscale low-index dielectric layer [10–14]. NW lasers with cylindrical geometry and strong two-dimensional confinement of electrons, holes, and photons are ideal for semiconductor lasers with reduced threshold currents and narrow spectral linewidths [8, 9]. Owing to the coupling of photonic and plasmonic modes, the electromagnetic modes that can be strongly confined at subwavelength scale with low propagation loss, which contributes to the NW lasing with ultra-small diameter [15]. However, homogenous NWs with low optical gain and high propagation loss lead to degradation of laser performance. In comparison with homogeneous NWs, NW heterostructures incorporating QWs are expected to have distinct advantages for lasers such as high optical gain, low threshold, improved temperature stability, and wide-range wavelength tunability [16–20]. Moreover, radial NW/QW heterostructures decouple the gain medium and optical cavity, avoiding the reabsorption of emission during its propagation in the NW cavity. Particularly, due to the near-field coupling of photons and SPPs, the hybrid plasmonic modes have superior overlap with the MQW region in comparison with the core-shell NWs, which may lead to high optical gain and low threshold gain. In addition, the critical diameter of the hybrid plasmonic NW/QW laser is also expected to be smaller than the homogeneous NWs due to a much stronger optical confinement.

In this paper, we combine MQW heterostructures and hybrid plasmonic modes to investigate the lasing properties of MQW NW laser with diameter below diffraction limit. The plasmonic NW laser is composed of an AlGaAs/GaAs MQW NW placed on a silver substrate separated by a nanoscale MgF₂ layer. We used 3D finite-difference time-domain (FDTD) to analyze the modal distributions and lasing characteristics of the various modes of both photonic and hybrid plasmonic MQW NWs. The photonic MQW NW laser is easier to lase at large diameter, while the hybrid plasmonic MQW NW is capable of lasing at smaller diameters beyond the dimensional limit of photonic ones. The hybrid plasmonic MQW NW has the lowest threshold gain of 788 cm⁻¹ at a diameter of 130 nm, and the cutoff diameter is 80 nm, half that of the photonic lasers. In comparison with the hybrid plasmonic core-shell NW lasers, the hybrid plasmonic MQW NW lasers have distinct lower threshold gain and higher Purcell factors over a wide range of diameters due to a superior overlap between the hybrid plasmonic modes and gain medium, as well as a stronger optical confinement originating from the grating-like effect of MQW structures. Moreover, the threshold gain of hybrid plasmonic single QW (SQW) NWs has also been investigated by considering the quantum confinement effects.

2. Structure

The structure of the hybrid plasmonic MQW NW laser is depicted in Fig. 1(a). The NW and silver substrate is separated by an ultrathin layer of MgF₂ with thickness of 5 nm. The heterostructure consists of an AlGaAs core and epitaxial AlGaAs/GaAs MQWs (which serve as the effective gain medium) with the number of QW n_w = 1, 2 and 3. The GaAs well thickness is 5 nm, separated by 10 nm AlGaAs barriers with an Al composition of 32%. As shown in Fig. 1(b), due to the ultrathin thickness of the MgF₂ gap, the near-field interaction between the NW and silver can make the momentum match between photons (hω_BG) and surface plasmon polaritons (SPPs) (hω_SP), forming hybrid plasmonic modes [21]. Compared to plasmonic modes, hybrid plasmonic modes enable long propagation length under strong optical confinement [10, 16]. The contributions of hybrid plasmonic modes to lasing characteristics will be elaborated in the following sections.

Fig. 1 (a) Schematic diagram of the hybrid plasmonic MQW NW laser and magnified cross-sectional view of the MQW structure. (b) Schematic diagram of the electron-hole recombination and photons-SPPs coupling mechanism.

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3. Threshold properties of photonic and hybrid plasmonic MQW NW lasers

Due to the near-field coupling of photons and SPPs, the hybrid plasmonic modes have stronger confinement electromagnetic energy compared to photonic modes on silica substrate. Electromagnetic modal profiles and lasing characteristics of NW lasers were calculated using Lumerical FDTD Solutions with absorbing (perfectly matched layers) boundary conditions. Here, the wavelength of 785 nm was used in the calculations, which corresponds to the band gap of 5 nm thick GaAs QW [22, 23]. The material permittivity of Ag is taken from Johnson and Christy [24]. In Fig. 2, we show the profile of electric field intensity |E(x, y)| and its distribution along Y-axis of photonic modes in NW on silica substrate and hybrid plasmonic modes in NW on MgF₂-Ag substrate, respectively. The diameter of NW here is fixed at 300 nm and the NW consists 3 QWs. Three modes HEy 11, HEx 11 and TE₀₁ are observed in the photonic NWs, as shown in Fig. 2(b). The electric field is tightly confined inside the NW due to the high contrast of refractive index of NW-air and NW-substrate interfaces. The fundamental mode HE₁₁ has a twofold polarization dependency. Compared to a NW with a circular cross-section, the symmetry reduction of the hexagonal cross section of the NW together with the underneath substrate gives rise to the nonequivalent HEx 11 and HEy 11 modes. However, for the NW placed on the MgF₂-Ag substrate, the NW can support four modes – modes HEy 11, HEx 11, and TE₀₁ and one more mode HE₂₁, as shown in Fig. 2(d). Among them, modes HEy 11, TE₀₁ and HE₂₁ excite SPPs and the photons inside the NW couple with the excited SPPs, forming hybrid plasmonic modes. In a hybrid plasmonic mode, the coupling between photonic and plasmonic modes across the dielectric gap enables ‘capacitor-like’ energy storage that allows subwavelength light propagation in non-metallic regions with strong mode confinement [10]. So the electric field of modes HEy 11, TE₀₁ and HE₂₁ are strongly confined in the MgF₂ gap between the NW and Ag without high dissipation. But for mode HEx 11, the electric field keeps confined in the NW rather than forming hybrid plasmonic mode, which is similar to the NW on silica substrate. This is because that the mode HEx 11 is x-polarized and cannot excite SPPs. Besides, it is necessary to point out that there are few SPPs excited in the dielectric gap, due to the sharp corners of the NW and leaked electric field around the corners. In addition, there are some fluctuations in the electric field curves in Figs. 2(a) and 2(c), which result from the impact of the periodical refractive index difference of the MQW layers on the electric field distribution. How the existence of MQW impacts on the modal properties and subsequently lasing properties will be elaborated in the following sections.

Fig. 2 Normalized electric field distribution of mode HEy 11 along Y-axis (a) and modal profiles of modes HEy 11, HEx 11, TE₀₁ and cutoff HE₂₁ (b) for the MQW NW on silica substrate; Normalized electric field distribution of mode HEy 11 along Y-axis (c) and modal profiles of modes HEy 11, _{$H E_{11}^{x}$}, TE₀₁ and HE₂₁ (d) for the MQW NW on MgF₂-Ag substrate.

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The dependences of the real part of the effective index (Re(n_eff)), modal confinement factor (Γ), threshold gain (g_th) and Purcell factor (F_p) on the diameter of MQW NW on silica substrate and MgF₂-Ag substrate are presented in Figs. 3(a)-3(d). As shown in Fig. 3(a), with the decreasing diameter of MQW NW, the real part of the effective index Re(n_eff) decreases. For cylindrical photonic modes of the MQW NW on silica substrate, mode TE₀₁ first cuts off near the diameter of 230 nm and then mode HEx 11 and HEy 11. At last, mode HEy 11 has a smallest cutoff diameter of 160 nm. Below 160 nm, electromagnetic energy can’t be confined inside the MQW NW, and photonic mode does not exist. For mode HEx 11 of the NW on MgF₂-Ag substrate, it has a similar cutoff diameter with that of mode HEx 11 on silica substrate as it is photonic-like mode and can’t excite SPPs, as shown in Fig. 2(d). However, for hybrid plasmonic modes _{$H E_{11}^{y}$}, TE₀₁ and HE₂₁, the cutoff diameters are much smaller compared to the corresponding modes of NW on silica substrate. Among them, mode HEy 11 exists below the diameter of 160 nm and won’t vanish until 90 nm, which is attributed to the extremely strong confinement of electromagnetic energy of the hybrid plasmonic mode.

Fig. 3 (a) Effective index Re(n_eff), (b) Modal confinement factor Γ (values for hybrid plasmonic mode TE₀₁ is multiplied by 0.15 to better visualize details in the other modes), (c) Threshold gain g_th and (d) Purcell factor F_p as functions of NW diameter of the hybrid plasmonic and photonic lasers. The inset in (b) shows the Γ of photonic mode TE₀₁ and hybrid plasmonic mode HE₂₁ with diameter between 300 and 360 nm.

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The Modal confinement factor Γ represents the coupling efficiency between the gain medium and resonant modes. The Γ for semiconductor MQW NW and for metallic structure (index, m) are given as follows [13, 25, 26]:

Γ = \frac{c / n_{s}}{{\bar{v}}_{g}} \frac{\iint_{s} Re [d (ε ω) / d ω {| E |}^{2}] d x d y}{\iint_{\infty} Re [d (ε ω) / d ω {| E |}^{2} + μ_{0} {| H |}^{2}] d x d y / 2}

Γ_{m} = c ε_{0} n_{m} {\iint_{m} | E |}^{2} d x d y / \iint_{\infty} Re (E \times H) \cdot z d x d y

{\bar{v}}_{g} = \frac{1}{2} \iint_{\infty} Re (E \times H) \cdot z d x d y / \frac{1}{2} \iint_{\infty} Re [d (ε ω) / d ω {| E |}^{2} + μ_{0} {| H |}^{2}] d x d y

where c is the vacuum speed of light, _{$n_{s, m}$} are the real part of the refractive index of MQW gain medium and metal, _{${\bar{v}}_{g}$} is the average energy velocity of the mode. The second factor in Eq. (1) is the ratio of the energy inside the gain medium to the energy over the entire simulation cross-section. Note that Eqs. (1) and (2) are generally used for waveguide whose cross-section contains both semiconductor and metal. The modal confinement factors Γ as a function of D are shown in Fig. 3(b). The Γ of the photonic modes first gently ascend and then rapidly descend with decreasing diameter. The change in the curves is more like a hump rather than a peak. The reason of the drop of Γ at small diameters is that the decreased effective refractive index difference results in poor photonic modal confinement and enhanced scattering of electromagnetic energy outside the NW, which reduces the overlap between photonic modes and gain medium. At large diameters, the decrease of Γ could be attributed to a larger _{${\bar{v}}_{g}$} [8], as well as a reduced overlap between photonic mode and gain medium due to an increased mismatch between the MQW and energy-concentrated part of photonic modes. The Γ of the photonic mode TE₀₁ is much higher than those of other modes, because its doughnut-shaped intensity profile has a much better matched overlap with the MQW region. Thus, modal gain for the TE₀₁ mode is assumed to be large in MQW structure. However, the Γ of hybrid plasmonic modes are higher and exhibit quite distinct peaks compared to photonic modes, as shown in Fig. 3(b), which is attributed to the mode transition between cylinder-like and SPP-like [10]. For a large diameter, the hybrid plasmonic NW supports a low-loss cylinder-like mode with electromagnetic energy confined in the core, but has a very poor overlap with the MQW region. With the decreasing of diameter, the ascending part of Γ curve occurs for two reasons. First, the guided mode transits to SPP-like mode and most electromagnetic energy distributes towards the dielectric gap due to the strong modal confinement, which improves the modal overlap with the MQW region. Second, SPP-like modes close to the SPP resonance exhibit high metallic dissipations because there are approximately comparable electromagnetic energy density in the metallic and dielectric regions [14]. At the same time, they feature low energy velocities due to oppositely directed power flows in the metallic and dielectric regions [7]. The slow energy velocity partly mitigates the impact of high modal loss by increasing Γ in Eq. (1) together with the modal gain. As the NW diameter approaches cutoff, the small overlap with the decreased gain volume leads to a reduction of coupling efficiency. We also find that the Γ for hybrid plasmonic mode TE₀₁ is the largest ones and can surprisingly be as large as 44. This can be explained by that the material gain is sufficiently large to compensate the metal loss and thus leads to a slower _{${\bar{v}}_{g}$}. A slowed energy transport allows more energy exchange between modes and gain/loss medium, which results in huge Γ for both the NW part and metal part [26–28]. As a result, the higher Γ for hybrid plasmonic mode indicates that the electric field is highly confined in the gap region and overlaps well with the MQW gain region.

To achieve lasing, the modal loss must be overcompensated by the modal gain. For the photonic NW laser, the optical losses are dominated by mirror losses occurring at the end facets [8]. But for the hybrid plasmonic NW laser, the metallic dissipation is at the same order of magnitude as mirror loss, which cannot be ignored for the threshold calculation. Therefore, the threshold gain g_th is defined as [25, 26]:

g_{t h} = \frac{1}{Γ} (Γ_{m} α_{0}^{m} + \frac{1}{L} \ln \frac{1}{R_{1} R_{2}})

where _{$α_{0}^{m} = ω ε_{m}^{''} / (c n_{m})$} is the bulk material absorption coefficient of metal, L = 8 μm is the length of the NWs and R_1,2 are the reflectivities at the NW end facets. For the modeling, we have assumed identical cleaved end facets for the NW laser (R₁ = R₂) [22]. To investigate which mode is most likely to satisfy the lasing threshold, the g_th of all guided modes supported in the NW are calculated and shown in Fig. 3(c). Our calculations show that the photonic mode TE₀₁ has the largest modal gain due to lower loss and the highest Γ in our design. Nonetheless, owing to the combined effect of large confinement and slow energy velocity, all hybrid plasmonic modes have moderate g_th and maintain comparable or lower g_th for a large range of NW diameters. Among them, the significantly slowed-down mode TE₀₁ experiences more gain per unit length and its g_th is the lowest, even though it has a larger metallic dissipation in the meantime. Near the mode cutoff diameter, the photonic NW lasers exhibit strong dependence of the mode confinement on the NW diameter, resulting in a sharp increase in the g_th at diameter of 160 nm. Below the cutoff diameter, the photonic modes have poor overlap with the gain medium and finally scatter outside the NW. Conversely, hybrid plasmonic lasers can maintain good mode-gain overlap for diameter as small as 90 nm, at which photonic mode does not even exist.

According to the Fermi’s golden rule, the rate of spontaneous emission is proportional to the local density of optical states (LDOS) [29]. That is, by placing the light source in an environment that structure is at the scale of the wavelength, the LDOS can be spatially controlled [30]. As a result, the LDOS of an emitter can be locally increased along with the enhanced rate of spontaneous emission, which is called the Purcell effect. The Purcell factor is employed to describe the enhancement and defined as [31]

F_{p} = \frac{3}{4 π^{2}} {(\frac{λ}{n})}^{3} (\frac{Q}{V_{e f f}})

where (λ/n) is the wavelength within the material, _{$V_{e f f}$} is the effective modal volume of cavity [32] and Q is the quality factor which is inverse proportional to the total modal losses and group velocity of the guided mode [33]. Thus, high quality factor and strong field confinement result in a higher F_p. As shown in Fig. 3(d), for all photonic modes, F_p keeps lower than one due to weak confinement of modes with large modal volumes. For the hybrid plasmonic mode HEx 11, it is photonic-like and its F_p is close to that of photonic mode without enhanced spontaneous emission rate. In contrast, the F_p for hybrid plasmonic modes HEy 11, TE₀₁ and HE₂₁ are significantly enhanced more than 8 times than photonic modes due to the ultrasmall modal volume and cavity size formed by the nanolocalized electromagnetic field. In particular, the fundamental mode HEy 11 exhibits a strong increase with the decreasing diameter and obtains the maximum F_p as large as 22 when D = 130 nm, where SPP-like modes are most strongly localized. This enhancement factor corresponds to a mode that is a hundred times smaller than the diffraction limit, which agrees well with our mode size calculations (_{$A_{m} / A_{0} \approx 0.007$} for mode HEy 11). As the diameter further decreases, a sharp reduce can be seen near the cutoff diameter, because the additional leakage of energy into free space leads to large modal loss and a slight rise in the modal volume. So, we observe a broad Purcell effect arising in Fig. 3(d) for hybrid plasmonic modes resulting in the enhancement of spontaneous emission rate, which is beneficial for realizing the lasing easier.

4. Threshold properties of hybrid plasmonic MQW and core-shell NW lasers

The hybrid plasmonic MQW NW laser also possesses advantages such as low-threshold and tunable wavelength compared to the core-shell NW laser, especially for small-diameter NWs. The dependences of Re(n_eff), Γ, g_th, and F_p of mode HEy 11 on the NW diameter for MQW NW with different number of QW n_w and core-shell NW are performed in Fig. 4. Here, the core-shell NW with 10 nm thick AlGaAs capping layer emit photons at 875 nm from GaAs core, while the 5 nm thick GaAs QW emits photons at 785 nm due to the quantum-confinement-effect induced blue shift [23]. As shown in Fig. 4(a), owing to the continuously enhanced cylinder-like-feature of the hybrid plasmonic modes, the Re(n_eff) monotonically increases with the increase of NW diameter. At a diameter of 90 nm, the mode HEy 11 of core-shell NW is going to vanish and the electromagnetic energy is not well-confined in the NW but couples into the air from the NW-air and NW-substrate interfaces, resulting in a reduced Re(n_eff) and earlier cutoff, as show in the inset (i). However, for the mode HEy 11 of MQW NW, due to the grating-like effect originating from the different index of QW and barrier layers, a stronger optical confinement is created for light guiding, resulting in more electromagnetic energy confined inside the NW and few scattering from the NW-substrate interface, as shown in the inset (ii). As a result, the cutoff diameter of the MQW NW could be down to 80 nm, significantly smaller than that of the core-shell NW (about 90 nm).

Fig. 4 The dependences of (a) mode effective index Re(n_eff), (b) modal confinement factor Γ, (c) threshold gain g_th and (d) Purcell factor F_p on the NW diameter for MQW NW with different n_w and core-shell NW. The inset in (a) shows the electric field distribution of the core-shell (i) and SQW NW (ii) at D = 90nm. The inset in (c) shows the g_th for diameter between 70 and 120 nm.

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For photonic NW lasers, the Γ of MQW NWs is typically lower than that of optimized homogenous NWs as for the core-shell structure, the whole NW core acts as the gain medium and yields a large Γ value, while for the MQW NW structure, the effective gain medium is only the MQW region [19]. While for a hybrid plasmonic NW laser, the electromagnetic energy inside the NW distributes towards the NW-air interface and tightly confined in the dielectric gap. The gain region of MQW NW is MQW locating near the NW-air interface, whereas the gain region of core-shell NW is the core. The dependences of Γ for the hybrid plasmonic MQW NW and core-shell NW on the diameter are shown in Fig. 4(b). As the diameter decreases, electromagnetic energy distributes away from the core and overlaps better with the MQW region, resulting a larger Γ for the MQW NW. With the decreasing n_w of QWs, the gain volume shrinks and the Γ decreases. As noted earlier, the mode HEy 11 of MQW NW is cylinder-like for D > 130 nm and SPP-like otherwise. At D = 130 nm, the highest overlap between the hybrid plasmonic mode and gain medium is obtained, where the SPP-like modes are strongly confined and the optical capacitance of the MQW gain region reaches maximum. Below the diameter of 130 nm, the MQW NW maintains higher Γ than the core-shell NW. Moreover, the Γ for the MQW NW increases with n_w due to an increased gain region and stronger confinement.

Figure 4(c) shows the g_th of MQW NW with different n_w and core-shell NW as a function of diameter. A minimum g_th of 788 cm⁻¹ is obtained at D = 130 nm for the 3MQW NW, which is in agreement with that the 3MQW NW has the largest Γ. Below 130 nm, g_th sharply increases and rapidly cuts off at a minimum diameter of about 80 nm for the SQW and 2MQW NWs, as shown in the inset in Fig. 4(c) For the 3QW NWs, the cutoff diameter is limited by the dimension of the QW region. When the diameter exceeds 130 nm, the g_th of MQW NW slightly increases, in disagreement with the rapidly decreasing Γ shown in Fig. 4(b) which is inversely proportional to g_th. This is because that the cylinder-like mode has a very poor overlap with the MQW region and only a little energy penetrates into the metallic region, resulting in lower intrinsic loss and metallic dissipation, which mitigate the impact of low Γ on the g_th. As F_p in Eq. (5) is proportional to the emission wavelength, F_p of the core-shell NW is expected to be larger than that of the MQW due to a smaller band gap. However, the simulation results show that F_p of the MQW NW is larger than that of the core-shell NW for small diameters, as shown in Fig. 4(d). This is because that the grating-like periodical MQW layers enable more electromagnetic energy confined in the NW and gap region without scattering outside of the NW, resulting in lower modal loss and smaller effective modal volume than the core-shell NW, both of which lead to a higher F_p.

5. Quantum confinement effects

A main advantage of QWs is the quantized subbands and step-like densities of states. Energy quantization provides another degree of freedom to tune the lasing wavelength by varying the well width. In this section, we compare the threshold properties of SQW NW with different well thickness t_w to investigate the influence of quantum confinement effects. When the QW thickness t_w is comparable to the exciton Bohr radius of GaAs (~160 Å) [34, 35], quantum confinement effect is expected, resulting in a blue shift of the emission wavelength as the QW thickness decreases. Here, we investigate SQW NW with t_w = 5, 7, and 20 nm, corresponding to the room-temperature emission wavelength λ = 785, 830, and 875 nm, respectively [22]. As shown in Fig. 5, as the thickness increases, the emission wavelength redshifts, resulting in an increasing g_th due to the decreasing field confinement. However, g_th is still smaller than that of the core-shell NW, which is mainly attributed to the increased gain volume and decreased metallic dissipation. Furthermore, for a fixed dielectric gap thickness, the effective compensation area of modal gain for a QW NW is larger than that of the core-shell NW at small-diameter, where SPP-like mode dominates and overlaps better with MQW region. Therefore, the QW NW hybrid plasmonic lasers can provide higher optical gain at different wavelengths and maintain low-threshold.

Fig. 5 Threshold gain g_th for SQW NW with different emission wavelength and core-shell NW at small diameters.

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6. Conclusion

In summary, we have investigated the lasing characteristics of hybrid plasmonic AlGaAs/GaAs MQW NWs by using FDTD method. The hybrid plasmonic MQW NW has lower threshold gain over a broad diameter range in comparison with its photonic counterpart. Beyond the diffraction limit, the hybrid plasmonic MQW NW has a lowest threshold gain of 788 cm⁻¹ at a diameter of 130 nm, and a cutoff diameter of 80 nm, half that of the photonic lasers. Compared to the hybrid plasmonic core-shell NWs, the hybrid plasmonic MQW NWs exhibit significantly lower threshold gain, higher Purcell factor, and smaller cutoff diameter. Moreover, the hybrid plasmonic MQW NW has a lower threshold gain than that of the core-shell NW over a broad wavelength range. The results indicate that the hybrid plasmonic MQW NW structure is promising for ultrasmall and low-consumption near-infrared nanolasers. It should be pointed out that by introducing energy quantization and strong carrier confinement effects of QWs, which are not considered in this work, the performance of hybrid plasmonic MQW NW lasers is expected to be more excellent.

Funding

National Natural Science Foundation of China (NSFC) (61504010, 61376019); Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), P. R. China.

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Near-infrared hybrid plasmonic multiple quantum well nanowire lasers

Abstract

1. Introduction

2. Structure

3. Threshold properties of photonic and hybrid plasmonic MQW NW lasers

4. Threshold properties of hybrid plasmonic MQW and core-shell NW lasers

5. Quantum confinement effects

6. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (5)

Optics Express