1. Introduction

High-power lasers and amplifiers with excellent beam quality are important to many defense and civilian applications. Power scaling of waveguide-based lasers and amplifiers is often limited by optical nonlinearity and optical damage of host materials induced by the high peak power. To mitigate these catastrophic problems while maintaining large output power, waveguides with large mode area (LMA) are highly desired. Several schemes have been reported, beyond the conventional approach of index guiding, to achieve waveguide lasers and amplifiers with ultra LMA (ULMA), i.e., a modal diameter of 100 µm or above. Lang et. al. have proposed lateral grating-confined broad-area waveguides in which one-dimensional gratings are used laterally to confine the transverse modes of waveguides [1]. Utilizing the narrow modal selectivity of gratings at oblique incident angle, waveguides with transverse gratings enable strong modal discrimination among higher-order modes (HOMs), and single HOM operation was demonstrated in waveguides with a core diameter of 300 µm. Yariv has proposed a similar configuration except the gratings contain gain to provide both lateral confinement and optical amplification for the signal [2]. Due to the finite penetration of the Bloch modes into the gratings, the effective modal area of such waveguides can be much larger than the core dimension [3].

Recently Siegman has proposed the use of optical gain in index-antiguided waveguides to achieve single-transverse-mode operation with ULMA [4]. Such a scheme has been experimentally demonstrated in a flash-lamp-pumped neodymium-doped phosphate fiber with a core diameter of a few hundreds microns [5,6]. Index antiguiding (IAG), although effectively reducing the threshold of gain guiding (GG), could trap pump light in the cladding leading to low optical gain in the core when the waveguides are end-pumped [7]. This would be an unwanted situation because end pumping is one of the most desirable attributes of waveguide lasers and amplifiers. In this article I propose a new scheme that could potentially resolve the above issue. By enabling gain guiding in transverse grating waveguides, I show that the signal could have robust single-transverse-mode operation with ULMA, and at the same time the optical pump can be confined in the same core to achieve high optical gain and conversion efficiency. This scheme can be extended to other photonic bandgap based structures, and is potentially attractive for optically pumped high-power lasers and amplifiers.

2. Principle and analysis

The working principle of gain-guided transverse grating waveguides (GG-TGW) is as follows. Consider a one-dimensional transverse grating waveguide composed of a dielectric slab sandwiched by two semi-infinite gratings layered in the y-direction that is perpendicular to light propagation direction (Fig. 1). This waveguide resembles conventional distributed-Bragg-reflector (DBR) waveguides except the DBR is applied transversely rather than longitudinally. The core has a complex refractive index n_c=n_co+in_i where n_i is related to the power gain coefficient by g=2n_ik_o and k_o=2π/λ is the vacuum wavenumber. The gratings are assumed to have very small index variation such that their optical properties can be well described using coupled mode theory in terms of average refractive index n_cl and coupling constant κ. [7] Furthermore, I consider that n_co is slightly less than n_cl to exclude the total internal reflection guiding mechanism.

It is well known that a one-dimensional grating can resonantly reflect light with wavelength λ at an incident angle θ_R, provided the grating pitch Λ meets the Bragg condition Λ=λ/(2n_clsinθ_R). At resonance the reflectivity of the grating equals to tanh²(κw), which approaches unity if the coupling constant κ is large and/or the length of grating w is long. [8] Given θ_R, the angular bandwidth of a grating resonance is approximately 2κ/(k_on_cl cosθ_R), which could be very narrow if θ_R is small. Thus, GG-TGW is designed to have a strong and sharp resonance at the pump wavelength by employing large κw and small (but finite) θ_R. In such case the pump light can be effectively trapped inside the core via Bragg resonance, even though n_co is smaller than n_cl. On the other hand, the signal we are after has a single-transverse mode (i.e., the lowest order mode with mode order equals 0) that has a grazing incident angle in a large core. Due to its large mismatch from the grating resonance, the signal experiences the grating as a slab with refractive index n_cl, which is slightly larger than n_co as previously defined. Thus, the waveguide behaves like an IAG slab waveguide for the signal. With appropriate material gain in the core via optical pumping (i.e., n_i>0), gain guiding is expected to take place and yield a single-transverse mode with ULMA [4].

Modal characteristics of the proposed structure are analyzed using the transfer matrix method, following Lang et. al with minor modification [1]. As shown in Fig. 1, the waveguide consists of three layers, including two identical semi-infinite gratings (j=0,2) and a slab (j=1). I adopt a time factor of exp(iωt) so that a forward propagating wave (traveling in the +z direction) with a complex propagation constant β_c=β+ig will have a spatial dependence of exp(-iβ_cz), where a positive g represents gain of the electric field. Within each layer, the transverse field profile can be written as a superposition of a forward-propagating wave E ⁺(y) and a backward-propagating wave E ^-(y) that are the eigenfunctions of Helmholtz equations for that layer. That is,

where a_j represents the complex field amplitude and y_j is the beginning coordinate of the j-th layer. For a slab, the eigenfunctions are the well-known plane waves

where k _⊥=[(n_ck_o)²-β_c²]^1/2 is the transverse wavevector of a plane wave traveling obliquely in the slab. For a grating, the forward and backward traveling waves are strongly coupled and can be calculated from coupled mode theory to be

where σ≡π/Λ is the grating resonant wavenumber, r≡κ/(S+k _⊥-σ), and S≡[(k _⊥-σ)²-κ ²]^1/2. In the above definition, I have chosen positive roots for k _⊥ and S to allow a solution of leaky modes that have outgoing phase and are unbounded in the waveguides [1].

 

Fig. 1. Schematic of a one-dimensional gain-guided transverse grating waveguide. The pump (blue) is confined via Bragg resonance and the signal (red) is confined by gain guiding (GG). Dashed and solid lines indicate leaky and bound rays, respectively. Positive n_i represents gain. See text for details. (Color online)

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The field amplitudes a_j ⁺ and a_j ^- are related to those of the first layer via electromagnetic boundary conditions by [1]

where d_j is the thickness of the j-th layer and M_j is the corresponding 2×2 transfer matrix. For a slab,

and for a grating,

A mode of the waveguide is a field distribution in which the fields in the boundary layers are purely outgoing, i.e., a pure backward-propagating wave in the j=0 layer and a pure forward-propagating wave in the j=2 layer. Assuming a₀ ^-=1 and solving a₂ ^- (β_c)=0 will yield β_c of the eigenmodes from which the modal index β/k_o and the modal gain g_m can be determined. The field distribution of the eigenmodes can then be calculated according to Eqs.(1)–(6).

3. Result and discussion

I consider a TGW with the following parameters: n_co=1.5, core width d=125 µm, pump wavelength λ_p=980 nm, signal wavelength λ_s=1550 nm, and κ=100 cm^-1. The grating also has an average refractive index n_cl=1.5005 and a resonant incident angle θ_R=20°. Figure 2(a) shows the loci of the complex effective indexes n_eff=β_c/k_o of the eigenmodes at 980 nm for a TGW (red curve) and a conventional IAG-slab waveguide (blue curve) with otherwise identical parameters. All modes are essentially leaky due to index antiguiding. As expected, the conventional IAG-slab waveguide exhibits a monotonically increasing loss at higher mode order. The transverse grating waveguide, on the other hand, shows a similar trend at small mode order but exhibits a sharp increase in reflectivity near θ_R. The eigenmode that resides in the stopband of the gratings has essentially zero propagation loss and is hereafter named the “gap mode”. This gap mode, whose transverse wavevector is set by the grating period to be π/Λ, features a very large mode order (~250) with nearly uniform interference fringes across the entire core as shown in the inset of Fig. 2(b). The amplitude of the gap mode decays exponentially into the gratings, as is typical for a Bloch mode in the stopband of a grating [9]. As a comparison, Fig. 2(b) also shows the fundamental mode (blue curve) of the IAG-slab waveguide. It has a relatively small loss coefficient ~0.05 cm^-1 at 980 nm due to grazing incidence; its leaky nature, however, is evident as its field amplitude is nonzero and rises gradually in the cladding.

 

Fig. 2. (a) Loci of the eigenmodes for a passive TGW and an IAG-slab waveguide at 980 nm, and for a passive TGW at 1.55 nm. (b) Field amplitudes of the gap mode for the TGW and the fundamental mode for the IAG waveguide as described in a at 980 nm. The inset shows the high-order-mode nature of the gap mode. (Color online)

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Figure 2(a) also shows the loci (green curve) of the eigenmodes at 1.55 µm in a passive TGW. The curve is smooth within the first few hundred modes since the corresponding θ_R at 1.55 µm occurs at an even larger incident angle and thus a smaller modal index. For a passive TGW, all modes are lossy (i.e., g_m<0) due to radiation loss in an index antiguiding waveguide. As the material gain g in the core increases by optical pumping, it compensates for the radiation loss and g_m increases linearly, as shown in Fig. 3(a) for the three lowest modes. For material gain g between 0 and 5 cm^-1, modal gain g_m,i equals g-g_th,i, where g_th,i is the threshold gain for the onset of gain guiding and equals 0.42, 1.66, and 3.64 cm^-1, respectively, for the first three modes. In fact, gain does more than compensating for the radiation loss; it also substantially changes the modal confinement. Figure 3(b) shows the field amplitude of the fundamental mode at various pump gains. At g<g_th,0=0.42 cm^-1, the modes are leaky and their amplitudes rise exponentially in the cladding; at g>g_th,0, the modes stay roughly the same in the core but change significantly in the cladding from diverging exponentially to converging exponentially. Behavior of this type is consistent with that in a conventional IAG-slab waveguide [10], indicating that the lower-order modes, which are off-resonant with the gratings, indeed experience the grating like a uniform slab as previously suggested.

Figure 3(a) also indicates that, for a given material gain g, the difference in modal gain between the fundamental (i=0) and first HOM (i=1) is 1.24 cm^-1, which is nearly 3 times the threshold gain of the fundamental mode g_th,0. This suggests potentially strong modal gain discrimination between the fundamental modes and other higher-order modes, and the signal can sustain robust single-transverse mode as long as g<g_th,1. The largest modal gain for a pure single-transverse mode can be obtained at the onset of the first higher-order mode, i.e., g_m,0^max=g_th,1-g_th,0. In the present example, g_m,0^max equals 1.24 cm^-1 at a material gain of 1.6 cm^-1. Even larger g_m,0^max is possible by further reducing the index contrast between the core and the gratings [10]. Thus, GG-TGW holds promise for robust single-transverse-mode operation with high gain.

 

Fig. 3. (a) Modal gain as a function of the pump gain in the core for the fundamental mode (i=0) and the first two higher-order modes (i=1, 2). (b) Field amplitudes of the fundamental mode at various material gains g up to the gain-guiding threshold of the first HOM. The wavelength is 1.55 µm. (Color online)

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The GG-TGW amplifiers proposed here could be suitable as power amplifiers in masteroscillator- power-amplifier systems where the last-stage amplifiers need to deliver very high power output while maintaining an excellent beam quality. Such devices could be potentially realized in rare-earth-ion-doped glass waveguide amplifiers. Take the present TGW amplifier as an example and assume sinusoidally modulated gratings used as cladding. A grating with κ=100 cm^-1 and θ_R=20° implies an index modulation with an amplitude of ~0.001 and a period of ~0.96 µm. Assuming the device is 5 cm long and the target total gain is 20, this requires a modal gain g_m,0=0.6 cm^-1. Given a core width of 125 µm and an index difference n_co-n_cl=-5×10^-4, this corresponds to a pump gain coefficient ~1 cm^-1 that is well below the onset of the first higher-order mode (g_th,1=1.66 cm^-1) and the device will therefore operate with a pure single-transverse mode. Such specifications can be most probably achieved in Er:Yb co-doped phosphate glasses (Er³⁺ density ~10²¹ cm^-3) where a device length up to 4 cm [11] and a net gain of more than 2 cm-1 has been experimentally demonstrated [12]. The vertical confinement of the guided modes can be achieved by the ion-exchange technique which allows precise tuning of the index contrast between the core and the gratings by controlling the salt concentration, exchange time, or temperature. [12] The transverse grating layers can be fabricated using reactive ion etching [13] or by UV-induced densification of phosphate glass followed by chemical etching. [14] Both methods are suitable for large-area patterning of uniform structures and have been applied to fabricate longitudinal DFB/DBR gratings in such materials.

The bandwidth of a Bragg grating at its resonance is Δλ=λ_p²κ/(πn_clsinθ_R), and this in principle imposes a minimum spectral separation between the pump and signal used in the proposed TGW scheme. For the present example, this corresponds to 15 nm which is smaller than the spectral separation in practical rare-earth-ion-doped laser/amplifier systems. In practice, however, the spectral separation is desired to be a few times larger than Δλ to avoid the finite reflectivity at the side lobes of the gratins. On the other hand, Δλ in principle can be made very small if a weak grating (and correspondingly a long grating length) is employed.

4. Conclusion

I propose a new optically-pumped waveguide laser amplifier based on gain guiding in transverse grating waveguides to support an ultra-large mode area with a high modal gain. In this configuration, the pump is guided in the core via the Bragg resonance while the signal is confined by gain guiding. Modal analysis based on the transfer matrix method confirms robust single-transverse mode operation over a large span of material gains. Such a scheme is attractive for high-power waveguide-based lasers and amplifiers.