Bandgap engineering and prospects for radiation-balanced vertical-external-cavity surface-emitting semiconductor lasers

Zohreh Vafapour; Jacob B. Khurgin

doi:10.1364/OE.26.012985

1. Introduction

Since their inception nearly six decades ago, lasers, particularly semiconductor ones, have undergone tremendous progress as both their operating powers and efficiencies have increased many fold, with wall-plug efficiencies of commercial semiconductor laser diodes approaching 80% [1–4]. Still, from thermodynamic considerations, a fraction of the power injected into any laser cannot be usefully extracted and normally ends up as heat in the lasing medium [5]. For high power lasers, even a few percent of power dissipated as heat requires cooling by various means - water, thermoelectric, air blower, etc. This all makes the laser less compact. Furthermore, for high power lasers it is not just the temperature rise that is harmful by itself but also the gradient which, through the thermo-optic effect, causes beam distortion and filamentation.

It would be beneficial to have a better way of cooling the laser, without a cooling system having a contact with the laser itself. This can be enabled by the concept of radiation cooling by anti-Stokes fluorescence that was originally proposed in 1929 [6]. Anti-Stokes cooling was first reported in 1981 for CO₂ gas [7], in 1995 and 1996 for organic dye solutions [8, 9], and in 1995 for ytterbium doped ZBLANP glass [10, 11]. Since then, optical cooling has been demonstrated in optical materials doped with the active ions ytterbium, erbium and thulium, which are, of course, also used in high power optically pumped lasers [12–14]. It would then be only natural to apply the radiation cooling concept to the cooling of optically pumped lasers because the same laser pump can also serve as a source of radiative cooling. As was first shown by Bowman et al. [15] a condition can be established when the amount of heat generated in the process of (necessary Stokes shifted) stimulated emission is balanced by the amount of heat removed in the process of anti-Stokes shifted spontaneous emission. This operating regime has been named “Radiation Balanced Lasing” or RBL.

Consider a simplified energy level picture of rare earth gain material consisting of two manifolds, each consisting of closely separated energy levels (see Fig. 1). The pumping photons of frequency (hν_P) excite the states in the upper manifold and vacate the corresponding states in the lower manifold (which is like generation of electron hole pairs in the semiconductor). While the upper and lower manifolds are not in thermal equilibrium with one another, the population of the states inside each manifold is in thermal equilibrium and follows a Boltzmann distribution. Because of this, when the system is pumped with frequency ν_P the fluorescence I_F(ν) occurs over a wide wavelength range with both Stokes and anti-Stokes transitions present in the spectrum. One can define the mean fluorescence frequency as $ν_{F} = \int ν I_{F} (ν) d ν$ . Obviously, if ν_F > ν_P (anti-Stokes transitions dominate), net cooling can be attained. As the states near the bottom of the upper manifold become occupied while the states at the top of the lower manifold are vacated, absorption α(ν) gradually decreases at the lower frequencies, and a net gain γ(ν) eventually develops at frequencies that are less than the transparency frequency ν_T < ν_P. Once the peak gain γ(ν_L) exceeds the cavity loss, lasing commences at the lasing frequency ν_L.

Fig. 1 The schematic quasi-two-level energy diagram with the pump, fluorescence and lasing transitions.

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In 2011, Bowman et al. [16, 17] achieved RBL in a Yb:YAG laser generating as much as 500W of average power. This laser is widely used in many walks of life, achieving 26% efficiency. The successful demonstration of RBL in this material is not surprising as the best optical refrigeration results have also been obtained with laser hosts containing Yb ions and having low phonon energies to assure a small nonradiative recombination rate [14]. The absorption and emission spectra of Yb ion doped materials typically feature a number of discrete sharp peaks as well as low “pedestals” - a fortuitous arrangement which allows us to achieve large AS shifts of fluorescence by pumping into the low pedestal and emitting from the sharp peak. As shown in this work, replicating these shapes of absorption and emission can go a long way towards attaining RBL in other, non-rare earth-based lasers.

While the choice of rare earth high power lasers as the first medium for applying the concept of RBL has been a prudent one, there are other high power lasers widely used in many walks of life that also require thermal management. Recent years have seen the rise of high power optically pumped semiconductor lasers (SLs) [18, 19], and as their use has become more and more widespread, the issue of their thermal management has become a more pressing one. For this reason, it is important to evaluate all the alternative cooling schemes [20, 21], including the optical one. It is essential to state that net laser cooling has not yet been achieved in semiconductors (except for one report by Zhang et al. in CdSe nanobelt [22]), but that does not necessarily mean that RBL in that medium is unattainable. The reason is twofold - first, the efficiency of laser refrigeration (roughly proportional to $\frac{k_{B} T}{h ν}$ ) increases with temperature; hence maintaining the laser medium at temperatures slightly above room temperature is less problematic than actual refrigeration well below the room temperature. Second, because the laser active medium always has a higher temperature than its surroundings, there is no heat coming from these surroundings that the optical refrigeration needs to balance. The only source of heat is nonradiative relaxation to the upper laser level, occurring in the laser medium itself. Furthermore, even if one can attain the more uniform temperature profile only inside the laser medium without full balance (i.e. additional cooling will still be required), that by itself would be advantageous, as it would mitigate filamentation, improving the beam quality.

Recent years have seen the rise of the alternative: high power, optically pumped, Vertical-external-cavity surface-emitting lasers (VECSELs) [23–26] shown in Fig. 2. A typical VECSEL contains a multi-quantum well (MQW) active region grown on top of a highly reflective (>99.5%) Bragg reflector placed on top of a heat sink (which may be unnecessary if RBL is successful) and an external partial reflector that serves as an output coupler. Optical pumping allows one to avoid doping and electrical contacts, hence reducing free carrier loss and the complexity of the VECSEL. The laser emission wavelength can be varied over a very wide range by changing the composition of the active region of the MQW, well beyond the capabilities of the rare-earth based lasers that can operate only on a few discrete wavelengths. Compared with edge emitting lasers and even with electrically pumped VECSELs, optically pumped VECSELs are, in principle, capable of generating very high optical powers in diffraction-limited beams. However, since the active region has the shape of a very thin (relative to diameter) disc, the beam is prone to breaking up into multiple beams, essentially causing a single laser to split into multiple lasers incoherent with one another. The splitting and the other effects degrading the beam quality are caused by non-uniform heating of the active region. Therefore, if this nonuniform heating could be mitigated, both the output power and the beam quality of the VECSEL would greatly improve [27]. This can happen if the temperatures are allowed to rise uncontrollably. Therefore, in our study we consider the VECSEL as the most relevant semiconductor system to which the RBL concept should be applied.

Fig. 2 Schematic of a VECSEL (not to scale) with a semiconductor gain chip and an external laser resonator.

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In this work, we first develop the theoretical description of RBL in the semiconductor gain medium and define the important performance characteristics - cooling efficiency, lasing efficiency, maximum laser power attainable and others. We then apply the theory to a simple MQW gain medium and show that the RBL condition is indeed difficult to achieve without engineering the density of states to provide simultaneous lasing and cooling. A few examples of such density of states engineering are then considered, and a few promising pathways for future development of RBL are identified.

2. Theory and quantum engineering of structures for semiconductor RBL

The operating principle of semiconductor RBL is shown in Fig. 3. The active medium consists of semiconductor quantum-wells (type I of QWs), i.e. multiple QWs (MQWs). The MQW is a relatively narrow gap semiconductor E_g₀ embedded into the wider bandgap semiconductor with bandgap energy E_g (see Fig. 3(a)). A QW whose thickness is typically 5-15 nm is a thin layer which can confine electron and hole states in the direction of growth, whereas the movement in the other dimensions is not restricted, leading to the formation of energy subbands as shown in Fig. 3(b) [28]. In optically pumped semiconductor lasers (i.e. VECSELs), most pump radiation is typically absorbed in the confinement layers around the QWs, and the generated carriers are then captured by the QWs with the difference between the pump and lasing photon energies, (called the “quantum defect”) is released in the form of thermal energy. Obviously, this situation is to be avoided in RBL and for that the pumping should be into the QW itself [29, 30].

Fig. 3 (a) MQWs with the wavefunctions, E_g₀ is as bulk bandgap and E_g is as the effective bandgap, (b) the laser cooling cycle, (c) the step-function density of state (blue-line) and the Gaussian broadened 2D density of state, (d) Fermi functions in Conduction and Valence bands, and (e) the carrier concentration (n₀ for electron and p₀ for hole concentration).

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The densities of state in the conduction band (CB) and valence band (VB) are represented by step functions as shown in Fig. 3(c). However, due to inhomogeneities of growth as well as scattering inside the subbands and between them, the densities of states are broadened as shown by red-line curve in Fig. 3(c). When pumping takes place the populations of electrons and holes inside the bands are created and their distributions in the energy space are described by Fermi energies associated with quasi-Fermi levels as shown in Fig. 3(d). The distributions of electrons and holes are shown in Fig. 3(e).

Next, we introduce joint density of states as shown in Fig 4(a). For the unbroadened QWs, the joint density of states is described by the sum of step functions

ρ_{2 D, 0} (ℏ ω) = \sum_{n} \frac{μ_{r}}{π ℏ^{2}} H (E_{c n} - E_{v n} - ℏ ω)

where

μ_{r} = {(m_{c}^{- 1} + m_{v}^{- 1})}^{- 1}

is the reduced mass, H is the Heaviside step function, and E_cn and E_vn are the energies of the n-th subbands in the CB and VB respectively. Since RBL usually operates with appreciable carrier density only in the lowest subbands, Fig. 4(a) shows only the joint density of states for the transition between the states E_c₁ and E_v₁ as a dashed line. In the presence of homogenous broadening due to intraband scattering as well as inhomogeneous broadening due to QW thickness variation, the joint density of states (JDS) “spreads out” as

ρ_{2 D} = \int_{E_{c 1} - E_{v 1}}^{\infty} ρ_{2 D, 0} (E) g (E - ℏ ω) d E

and illustrated by the solid line in Fig. 4(a). Furthermore, as we modify the density of states by introducing impurities, quantum dot states, excitons etc. the density of states will change but not by much. Hence it makes sense to introduce the normalized density of states as

ρ_{2 D}^{^{'}} (ℏ ω) = \frac{ρ_{2 D} (ℏ ω)}{ρ_{2 D, 0} (ℏ ω)}

.

Fig. 4 (a) The step-function 2D joint density of states (blue-dashed curve) and the Lorentzian broadend density of states (red-line curve), (b) the absorption coefficient, and (c) the spontaneous emission of the structure vs. energy.

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Next, we calculate absorption, gain and spontaneous emission. We shall start with Fermi’s golden rule to evaluate the rate of change of the 2D carrier density due to stimulated emission (absorption) of light as

\frac{d n_{2 D}}{d t} = - \frac{2 π}{ℏ} \frac{1}{2} \frac{e^{2} P_{c v}^{2}}{m_{0}^{2} ω^{2}} \frac{μ_{r}}{π ℏ^{2}} ρ_{2 D}^{^{'}} (f_{c} - f_{v}) \frac{E^{2} (ω)}{4} F_{c v}

where P_cv is the Kane’s matrix element of momentum for the interband transition. F_cv is the wavefunction overlap factor defined as

F_{c v} = | \int ψ_{c} (z) ψ_{v}^{*} (z) d z |^{2}

E(ω) is the optical field, and f_c,_v are conduction and valence Fermi functions, respectively;

f_{c, v} (E_{c, v}) = \frac{1}{e^{\frac{E_{c} - E_{F c, v}}{k_{B} T}} + 1}

where k_B is the Boltzmann’s constant, T is the temperature in Kelvin. E_Fc,_v are the quasi-Fermi level for conduction and valence band, respectively. Note that the energies in the bands E_c and E_v relative to the respective band edge energies E_c₁ and E_v₁ can be found as

E_{c (v)} = E_{c (v), 1} \pm (ℏ ω - E_{c 1} - E_{v 1}) \frac{μ_{r}}{m_{c (v)}} .

For the valence band it makes sense to count the energy downward and use the Fermi function for holes

{\bar{f}}_{v} (E_{v}) = 1 - f_{v} (E) = \frac{1}{e^{\frac{E_{v} - E_{E v}}{k_{B} T}} + 1} .

To evaluate gain γ(ℏω) or absorption α(ℏω) = −γ(ℏω) we start by introducing the intensity of light

I_{ω} = \frac{n_{r}}{2 η_{0}} E^{2} (ω),

where n_r is the refractive index and η₀ = 377Ω is the vacuum impedance. Substitute Eq. (7) into Eq. (2) to obtain

\frac{d n_{2 D}}{d t} = - \frac{η_{0}}{2 ℏ n_{r}} \frac{e^{2} P_{c v}^{2}}{m_{0}^{2} ω^{2}} \frac{μ_{r}}{ℏ^{2}} ρ_{2 D}^{^{'}} (f_{c} + {\bar{f}}_{v} - 1) F_{c v} I_{ω}

Next, we use the well-known relation between the effective mass of the conduction band and the interband momentum matrix element

m_{c}^{- 1} = m_{0}^{- 1} + \frac{2 P_{c v}^{2}}{m_{0} E_{g}}

and because reduced mass is not much different from the conduction band effective mass, we introduce a dimensionless parameter

R_{P} = \frac{2 P_{c v}^{2}}{m_{0}^{2} E_{g}} μ_{r}

that is on the order of unity for most III–V semiconductors (for instance for GaAs it is 0.87) and then obtain from Eq. (8) for the rate of energy density increase per one QW

\frac{d u_{2 D}}{d t} = ℏ ω \frac{d n_{2 D}}{d t} = - \frac{η_{0} e^{2}}{4 ℏ n_{r}} R_{P} ρ_{2 D}^{^{'}} (f_{c} + {\bar{f}}_{v} - 1) F_{c v} I_{ω} = α (ω) I_{ω} = - γ (ω) I_{ω}

Using the definition of the fine structure constant $α_{0} = \frac{e^{2} η_{0}}{4 π ℏ} \approx \frac{1}{137}$ , we finally obtain the expression for the two-dimensional gain (or absorption) per each QW (see Fig. 4(b)).

γ (ω) = \frac{π α_{0}}{n_{r}} R_{P} ρ_{2 D}^{^{'}} (ω) F_{c v} (f_{c} + {\bar{f}}_{v} - 1)

The scale of the laser gain for a typical III–V material is therefore less than 1% per one QW - hence multiple QWs are necessary to obtain lasing. The frequency of which lasing takes place ν_L is the one where the gain is maximum, i.e. $\frac{\partial γ (ω_{L})}{\partial ω} = 0$ and the transparency frequency ν_T is found from γ(ω_T ) = 0 the frequency of maximum gain.

Next, we calculate the rate of spontaneous emission. For each electron-hole pair state with transition energy ℏω the rate of spontaneous emission can be determined by simply relating the electric field of vacuum fluctuations causing spontaneous decay to the density of vacuum electromagnetic energy in the dielectric as

\frac{ε_{0} n_{r}^{2} d E {(ω)}_{v a c}^{2}}{2} = U_{ω} d ω = ℏ ω \frac{n_{r}^{3} ω^{2}}{π^{2} c^{3}} d ω .

And substituting this field into 2 to obtain the rate of spontaneous recombination for this electron-hole pair as

r_{s p}^{^{'}} = \frac{2 π}{ℏ} \frac{1}{3} \frac{e^{2} P_{c v}^{2}}{m_{0}^{2} ω^{2}} f_{c} {\bar{f}}_{v} \frac{n_{r}}{2} \frac{ω^{3}}{ε_{0} π^{2} c^{3}} F_{c v} \approx \frac{2}{3} n_{r} α_{0} R_{P} \frac{ℏ}{μ_{0}} \frac{ω^{2}}{c^{2}} F_{c v} f_{c} {\bar{f}}_{v} \approx \frac{8 π}{3} n_{r} α_{0} R_{P} \frac{ℏ}{μ_{0} λ^{2}} F_{c v} f_{c} {\bar{f}}_{v}

where λ is the average wavelength. Integrating from Eq. (14) over the density of states, we obtain the total rate of spontaneous emission per unit area from a single QW (see Fig. 4(c))

r_{s p} = \int_{E_{g}}^{\infty} r_{s p}^{^{'}} (ℏ ω) ρ_{2 D}^{^{'}} \frac{μ_{r}}{π ℏ^{2}} d (ℏ ω) = \frac{8 π}{3 ℏ λ^{2}} n_{r} α_{0} R_{P} F_{c v} \int_{E_{g}}^{\infty} ρ_{2 D}^{^{'}} f_{c} {\bar{f}}_{v} d (ℏ ω)

the radiative decay time

τ_{r a d} = \frac{n_{2 D}}{r_{s p}}

and the mean fluorescence energy

h ν_{F} = r_{s p}^{- 1} \int_{E_{g}}^{\infty} ℏ ω r_{s p}^{^{'}} (ℏ ω) ρ_{2 D}^{^{'}} \frac{μ_{r}}{π ℏ^{2}} d (ℏ ω) = \frac{\int_{E_{g}}^{\infty} ℏ ω ρ_{2 D}^{^{'}} f_{c} {\bar{f}}_{v} d (ℏ ω)}{\int_{E_{g}}^{\infty} ρ_{2 D}^{^{'}} f_{c} {\bar{f}}_{v} d (ℏ ω)}

Next, we proceed to calculate the cooling and lasing characteristics as a function of the 2D carrier density n₂_D. While the energies of lasing (maximum gain) hν_L and fluorescence hν_F have already been defined by Eq. (12) and Eq. (17) respectively, the energy of pump photons is determined hν_P from the carrier balance equation

\frac{α (ν_{P}) I_{P}}{h ν_{P}} = \frac{γ (ν_{L}) I_{L}}{h ν_{L}} + \frac{n_{2 D}}{η_{Q} τ_{r a d}}

where the radiative quantum efficiency is

η_{Q} = 1 + \frac{τ_{r a d}}{τ_{n r a d}}

and the energy balance equation is

α (ν_{P}) I_{P} = γ (ν_{L}) I_{L} + \frac{n_{2 D} h ν_{F}}{τ_{r a d}} η_{e x t} .

Combining these two immediately yields

\frac{α (ν_{P}) I_{P}}{h ν_{P}} = \frac{n_{2 D}}{τ_{r} η_{Q}} \frac{η_{Q} η_{e x t} h ν_{F} - h ν_{L}}{h ν_{P} - h ν_{L}}

and one can solve Eq. (20) for hν_P. The laser intensity inside the cavity can be found as

\frac{γ (ν_{L}) I_{L}}{h ν_{L}} = \frac{n_{2 D}}{τ_{r} η_{Q}} \frac{η_{Q} η_{e x t} h ν_{F} - h ν_{P}}{h ν_{P} - h ν_{L}}

and the RBL condition is

\frac{I_{L}}{I_{P}} = \frac{h ν_{L}}{h ν_{P}} \frac{α (ν_{P})}{γ (ν_{L})} \frac{η_{Q} η_{e x t} h ν_{F} - h ν_{P}}{η_{Q} η_{e x t} h ν_{F} - h ν_{L}} .

We can also introduce the expression for the cooling power density as the difference between the admitted fluorescence power and absorbed power P_abs = α(ν_P)I_P,

P_{c o o l} = \frac{n_{2 D} h ν_{F}}{τ_{r a d}} η_{e x t} - α (ν_{P}) I_{P}

And the cooling efficiency,

η_{c o o l} = \frac{P_{c o o l}}{I_{P}} = α (ν_{P}) \frac{P_{c o o l}}{P_{a b s}}

which is the product of the “material cooling efficiency”

η_{m a t} = \frac{P_{c o o l}}{P_{a b s}}

and the absorption coefficient. Clearly increase in η_mat requires a large anti-Stokes shift of fluorescence which occurs when the pump photon energy hν_P is low. That makes the absorption coefficient low and reduces overall cooling efficiency. Therefore, a lot of optimization needs to be performed to get acceptable values of cooling efficiency. For having more absorption, one can use QW with the extended bandtail below the bandgap (Urbach tail) or adding some resonant impurities below the bandgap. In these cases, there is a higher absorption coefficient in the lower 2D densities of state [31].

When it comes to the RBL efficiency, Eq. (22) is somewhat deceptive, as it states that the intensity inside the cavity may become very large as long as the gain is low. Therefore, the laser intensity outside the cavity needs to be calculated. Assuming that there are N_QW QWs in the cavity, the round trip gain is g_RT = 2γ(ν_P)N_QW, and this gain should be equal to round trip loss which is the sum of the mirror loss (1-R), where R is reflectivity and additional cavity loss L_C due to scattering. Since I_L is the sum of two counterpropagating waves, the relation between the output power of the laser and I_L is

P_{L} = N_{Q W} γ (ν_{L}) I_{L} η_{o u t} = \frac{η_{Q} η_{e x t} \frac{ν_{F}}{ν_{P}} - 1}{η_{Q} η_{e x t} \frac{ν_{F}}{ν_{L}} - 1} N_{Q W} α (ν_{P}) I_{P} η_{o u t} \frac{η_{Q} η_{e x t} \frac{ν_{F}}{ν_{P}} - 1}{η_{Q} η_{e x t} \frac{ν_{F}}{ν_{L}} - 1} P_{T, a b s} η_{o u t}

where P_T,_abs is the total absorbed power density and

η_{o u t} = \frac{(1 - R)}{(1 - R + L_{C})} = 1 - \frac{L_{C}}{2 N γ (ν_{L})}

is the outcoupling efficiency. The overall laser efficiency can be described in two ways, first as the ratio of output power intensity to input intensity

η_{L, 1} = \frac{P_{L}}{I_{P}} = \frac{η_{Q} η_{e x t} \frac{ν_{F}}{ν_{P}} - 1}{η_{Q} η_{e x t} \frac{ν_{F}}{ν_{L}} - 1} N_{Q W} α (ν_{P}) η_{o u t},

and the second as the ratio of the output intensity to total absorbed power

η_{L, 2} = \frac{P_{L}}{P_{T, a b s}} = \frac{η_{Q} η_{e x t} \frac{ν_{F}}{ν_{P}} - 1}{η_{Q} η_{e x t} \frac{ν_{F}}{ν_{L}} - 1} (1 - \frac{L_{c}}{2 N_{Q W} γ}) .

This expression is more universal since total absorbed power can be increased by recycling pump photons. The last term in Eq. (27) is critical as it shows explicitly whether the lasing can be achieved at all, i.e. whether using a reasonable input pump intensity I_P can achieve round trip gain 2N_QWγ(ν_L) sufficient to bring the laser to threshold, i.e. surpassing cavity loss which realistically is expected to be at least 1% and at the same time maintain sufficient anti-Stokes shift to keep the numerator in Eq. (26) or Eq. (27) positive. For this reason, it makes sense to take a gradual approach to the optimization routine, namely to investigate first whether the system is capable of simultaneously achieving cooling and maintaining sufficient gain under the assumption that the laser power is much less than that of the pump. With this assumption, the balance Eq. (18) becomes

\frac{α (ν_{P}) I_{P}}{h ν_{P}} = \frac{n_{2 D}}{η_{Q} τ_{r a d}}

and can be used to be immediately determines the value of ν_P for each value of carrier density. Substituting it in Eq. (23) one quickly obtains

P_{c o o l} = n_{2 D} \frac{η_{e x t} η_{Q} h ν_{F} - h ν_{P}}{η_{Q} τ_{r a d}}

and the cooling efficiency is

η_{c o o l} = α (ν_{P}) (η_{e x t} η_{Q} \frac{ν_{F}}{ν_{P}} - 1)

The cooling power should be sufficient to balance the heating due to quantum defect hν_P − hν_L during lasing, i.e.

P_{c o o l} = (\frac{ν_{P}}{ν_{L}} - 1) I_{L} γ (ν_{L})

So that,

I_{L} = \frac{P_{c o o l}}{γ (ν_{L}) (\frac{ν_{P}}{ν_{L}} - 1)} = \frac{α (ν_{P}) (η_{e x t} η_{Q} \frac{ν_{F}}{ν_{P}} - 1)}{γ (ν_{L}) (\frac{ν_{P}}{ν_{L}} - 1)} I_{P} .

and

P_{L} = \frac{N_{Q W} P_{c o o l}}{(\frac{ν_{P}}{ν_{L}} - 1)} η_{o u t} = \frac{(η_{e x t} η_{Q} \frac{ν_{F}}{ν_{P}} - 1)}{(\frac{ν_{P}}{ν_{L}} - 1)} (1 - \frac{L_{C}}{2 N_{Q W} γ}) P_{a b s}

As one can see Eq. (32) differs from the exact expression of Eq. (22) only in the denominator and no more than by a factor of 2. Therefore, we can use our approximate analysis to compare different RBL designs and once the optimum design has been identified, its characteristics can be determined with the help of an exact RBL analysis as described in Eq. (33) that we shall use in the subsequent analysis as it relates the pump, absorbed power, and the output power.

3. Engineering the joint density of states

Now, when we have derived all the equations relative to RBL, we shall proceed to investigate that joint density of states which can be favorable for RBL, namely, that has a large anti-Stokes shift ν_F − ν_P and a relatively modest quantum defect ν_P − ν_L, while maintaining sufficiently large absorption α(ν_P) and gain γ(ν_L). This “discovery” process consists of evaluating each of these parameters as a function of carrier density for a given realistic pump intensity and then finding the cooling and lasing power and efficiency.

3.1. Simple QWs

The JDS of QW is a step function modified with Gaussian broadening with FWHM = 10 meV to account for the Urbach tail associated with impurities and phonon-assisted absorption as shown in Fig. 5(a). The QW material is GaAs and the width is 5 nm. The barrier material is Al_0.35Ga_0.65 As and the CB and VB offsets are substantially larger than k_BT so that, only the lowest subbands are populated by the carriers.

Fig. 5 (a) The density of state of QW, spectra of the fluorescence and gain of QW GaAs with carrier density of n₂_D = 0.62 × 10¹²cm⁻². (b) The four relevant frequencies in radiation-balanced (RB)-VECSELs, and (c) cooling power, the peak gain for round-trip QWs and the outside laser power, respectively.

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In the two lower panels of Fig. 5(a), the spectral density of spontaneous emission $r_{s p}^{^{'}} (ℏ ω)$ (Eq. (14)) and the gain/absorption (Eq. (12)) are shown. Additionally, three frequencies ν_F (mean fluorescence), ν_T (transparency) and ν_L (maximum gain) for the carrier density (n₂_D = 0.62 × 10¹²cm⁻²) are identified. As far as the pump frequency goes, it should be chosen within the shadowed narrow range between ν_T and ν_F to assure that the aforementioned carrier density can be maintained with a given pump intensity $(I_{P} = 9 \times 10^{4} \frac{W}{c m^{2}})$ according to Eq. (28). These calculations are then performed for the wide range of carrier densities from n₂_D = 0.05 × 10¹²cm⁻² to n₂_D = 1.5 × 10¹²cm⁻² and the results are shown in Fig. 5(b). The first observation cooling is fundamentally possible only up to n₂_D = 1.234 × 10¹²cm⁻² because at higher densities, the mean fluorescence frequency shifts below the transparency which makes anti-Stokes shift impossible. Also, note that the range of frequencies for ν_T shrinks with the increased density as the transparency frequency moves up, and at n₂_D = 0.868 × 10¹²cm⁻², the absorption becomes insufficient to maintain the required carrier density while assuring the anti-Stokes shift. At the same time, the separation between ν_T and ν_L equal to roughly $\frac{1}{2}$ of the gain bandwidth, expands indicating the increase in the peak gain. These two contradictory trends can be seen from the two upper panels in Fig. 5(c), the cooling power increases up to the peak value of $1.26 \frac{W}{c m^{2}}$ at n₂_D = 0.62 × 10¹²cm⁻², but the round-trip gain 2N_QWγ(ω_L) is only 0.026%. This gain is insufficient to reach the unrealistic conditions for threshold since the cavity will always have loss L_C larger than that. Even if we consider a very modest cavity loss L_C = 0.25%, the actual laser output power as shown in the lowest panel of Fig. 5(c) is miniscule with efficiency $\frac{P_{L}}{I_{P}}$ of less than 1%.

This exercise shows that, achieving RBL with a decent efficiency is significantly more complicated than just attaining net cooling, especially in the virtual geometries. It is therefore necessary to investigate different semiconductor structures with more favorable JDS.

3.2. QWs with extended bandtail

As analyzed and explained in detailed in the previous section, we need to investigate different semiconductor structures with more favorable JDS to improve the absorption. In this part of our study, we have used a step function for the JDS of the QWs, modified with Gaussian broadening with an extended bandtail to resolve the low value of absorption/gain issue identified in the previous structure (see the first panel of Fig. 6(a)). In the two lower panels of Fig. 6(a) the spectral density of spontaneous emission $r_{s p}^{^{'}} (ℏ ω)$ is plotted and the three frequencies ν_F, ν_T and ν_L for the two-dimensional carrier density of n₂_D = 0.59 × 10¹² cm⁻² are identified. These calculations are then performed for a wide range of carrier densities from n₂_D = 0.05 × 10¹²cm⁻² to n₂_D = 1.5 × 10¹² cm⁻² and the results are shown in Fig. 6(b). As we know the RB-VECSELs will not occur unless ν_L < ν_T < ν_P < ν_F. It is critical that the quantum defect, i.e. hν_P − hν_L is as small as possible (to have an appropriate gain), which brings both ν_P and ν_L close to the transparency frequency ν_T. At the same time, the absorption α(ν_P) and gain γ(ν_L) should both be high (the first one to assure that the pump is absorbed and the) second one for the laser to reach threshold). The first observation is that the cooling is fundamentally possible only up to n₂_D = 1.21 × 10¹²cm⁻² because the anti-Stokes shift is possible only up to this carrier density. It can be seen that, at n₂_D = 0.832 × 10¹²cm⁻², the absorption becomes insufficient to maintain the required carrier density while assuring the anti-Stokes shift. So, we can have net cooling until a carrier density of n₂_D = 0.832 × 10¹²cm⁻². As can be seen from the two upper panels in Fig. 6(c), the cooling power increases up to the peak value of $0.95 \frac{W}{c m^{2}}$ at n₂_D = 0.59 × 10 cm⁻², but the round-trip gain 2N_g is only 0.11%. This gain is somewhat higher than for the previously considered JDS, but it is still insufficient to reach the threshold under realistic conditions since, as was explained before, the cavity will always have loss L_C larger than that. As a result, with the cavity loss L_C=0.25%, the actual laser output power as shown in the lowest panel of Fig. 6(c) is still miniscule, only about $100 \frac{W}{c m^{2}}$ , which is even less than in the case of a simple QW. Clearly, the issue here is that the JDS in the bandtail is not sufficient for achieving robust gain. Therefore, one need to look for a way to increase the JDS for lasing without affecting the anti-Stokes shift, which can be accompanied by introducing discrete states below the bandgap. These states can possibly be associated with impurities or with quantum dots (QDs) [31–35].

Fig. 6 (a) The density of state of QW with extended bandtail, spectra of the fluorescence and gain of QW GaAs with carrier density of n₂_D = 0.59 × 10¹²cm⁻². (b) The four relevant frequencies in RB-VECSELs and (c) cooling power, the peak gain for round-trip QWs and the outside laser power, respectively.

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3.3. Using impurities (or QD states) below the bandgap

In this part of study, we tried to engineer the QWs structure adding some resonant states associated with impurities just below the bandgap. By creating a large delta-function like density of states below the bandgap one can expect to increase the gain there without affecting the anti-Stokes shift too much. A similar JDS can be attained by QD material where the delta-like density of states would be associated with the discrete states in QDs and the step with the wetting layers. A typical density of states associated with impurity concentration of 10¹²cm⁻² is shown in the upper panel of Fig. 7(a). In the two lower panels of Fig. 7(a), the spectral density of spontaneous emission $r_{s p}^{^{'}} (ℏ ω)$ and gain are plotted and four frequencies ν_F, ν_P, ν_T and ν_L for the two-dimension carrier density of n₂_D = 0.76 × 10¹²cm⁻² are shown. As one can see a very large gain can be achieved due to large JDS at the impurity energy levels.

Fig. 7 (a) The density of state of QWs adding some resonant impurities (QDs) below the bandgap, spectra of the fluorescence and gain of QW GaAs with carrier density of n₂_D = 0.824 × 10¹²cm⁻². (b) The four relevant frequencies in RB-VECSELs, and (c) cooling power, the peak gain for round-trip QWs and the outside laser power, respectively.

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The dependence of all the pertinent frequencies on the carrier density is shown in Fig. 7(b). Note the sharp “step” in the value of the pump frequency ν_P occurring at n₂_D ~ 6 × 10¹¹ cm⁻². At this density, the absorption at the impurity level saturates and one is forced to pump into the band (wetting layer in case of QDs). Also note that the laser frequency ν_L remains pinned to the impurity level over the entire range of carrier densities.

As a result, the two upper panels in Fig. 7(c) show that the cooling power increases up to the peak value of $1.1 \frac{W}{c m^{2}}$ at n_2D = 0.76 × 10¹² cm⁻² while the round-trip gain is significantly higher than in previous cases, about 0.5%, sufficient to attain lasing. As shown in the lowest panel of Fig. 7(c), an output power density of more than $400 \frac{W}{c m^{2}}$ is attainable, raising the efficiency of lasing by a factor larger than two in comparison to previous cases.

3.4. The optimization case of using QWs adding some resonant impurities (or QD states) below the bandgap

As one can see from the previous section, one can significantly increase gain by introducing a sharp delta-like density of states below the bandgap, but the overall efficiency remains low because absorption at the pump frequency occurs too close to the average fluorescence frequency, making the anti-Stokes shift small. One can attempt to improve the situation by introducing an additional impurity or QD state between the “lasing” state and the band edge and have absorption taking place there. One way to implement this is to simply consider heavy hole and light hole states in the QDs with absorption taking place at the light hole transition and lasing at heavy hole transition as shown in the upper panel of Fig. 8(a).

Fig. 8 (a) The density of state of QWs adding some resonant impurities (QDs) below the bandgap in the optimized case, spectra of the fluorescence and gain of QW GaAs with carrier density of n₂_D = 0.69 × 10¹²cm⁻². (b) The four relevant frequencies in RB-VECSELs and (c) cooling power, the peak gain for round-trip QWs and the outside laser power, respectively.

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In the two lower panels of Fig. 8(a), the spectral density of spontaneous emission $r_{s p}^{^{'}} (ℏ ω)$ and gain are shown and four frequencies ν_F, ν_P, ν_T and ν_L for the two-dimensional carrier density of n₂_D = 0.69 × 10¹²cm⁻² are identified. In Fig. 8(b), one can see that the value of pump frequency undergoes two step-like transitions - from the lowest impurity level to the second one and then to the continuum. As one can see from Fig. 8(c), the cooling region extends to a relatively large carrier density and in fact coincides with the large value of gain of about 1.75%. This gain is the best amount of gain in comparison to the previous QWs structures and clearly makes lasing possible under realistic conditions. The results show that for this optimized case of the engineered QWs structure, one can achieve an appropriately high gain/absorption and a good value of net cooling.

4. Conclusion

In conclusion, we have investigated the active medium characteristics and operating conditions that can lead to RBL in a semiconductor medium and have shown that to achieve RBL, the gain medium should be engineered to create the joint density of states that simultaneously allows gain/absorption and strong anti-Stokes luminescence. Such media may incorporate bandtail states, impurities or QDs. We have provided the recipe for optimization of such band structure-engineered materials to achieve the lowest threshold and highest output power. As expected the best suited for RBL is the density of states resembling those in rare - earth ions, like YB³⁺, with separate states being responsible for lasing, absorption and luminescence. Even with all the band engineering achieving RBL in a semiconductor medium, it is still a difficult task and the overall lasing efficiency is very low, primarily due to low absorption. For this reason, pump photon recycling may be an essential condition for RBL in semiconductor lasers [36]. However, the numbers obtained in this work indicate that RBL in optically pumped semiconductor VECSELs can be attained with existing pump sources and at the current level of epitaxial growth technology.

Funding

Air Force Office of Scientific Research (AFOSR) (100000181); Award # FA9550-16-10362 -Multidisciplinary Approaches to Radiation Balanced Lasers

Acknowledgments

The authors would like to thank to Dr. A Brinton Cooper and Mr. Nathan Henry from Johns Hopkins University for the help.

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Bandgap engineering and prospects for radiation-balanced vertical-external-cavity surface-emitting semiconductor lasers

Abstract

1. Introduction

2. Theory and quantum engineering of structures for semiconductor RBL

3. Engineering the joint density of states

3.1. Simple QWs

3.2. QWs with extended bandtail

3.3. Using impurities (or QD states) below the bandgap

3.4. The optimization case of using QWs adding some resonant impurities (or QD states) below the bandgap

4. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (8)

Equations (33)

Optics Express