FIELD OF THE INVENTION

[0001]
This invention generally relates to twodimensional distributed feedback (DFB) lasers. At least some embodiments of the invention relate to surfaceemitting photonic crystal distributed feedback lasers and selfpumping selffocusing mechanisms.
BACKGROUND OF THE INVENTION

[0002]
An ideal semiconductor laser that would emit high power into a single spectral mode with diffractionlimited output profile is of great interest for a number of commercial and military applications, including spectroscopy, infrared countermeasures, wavelength multiplexing in telecommunications, and diode pump arrays. However, known laser configurations have been unable to provide the desired singlemode characteristics without sacrificing power output, and vice versa. For example, the distributedfeedback (DFB) configuration with its onedimensional (1D) diffraction grating parallel to the laser facets routinely yields high spectral purity as long as the waveguide is narrow enough (with typical widths of 25 microns for lasers emitting at 0.81.55 microns wavelengths) to considerably lower the gain of higherorder lateral modes with respect to the fundamental mode and thereby prevent their excitation. However, scaling up the stripe width for increased power leads to a loss of phase coherence across the DFB laser stripe, primarily owing to the selfmodulation of the refractive index in the active region by nonuniformly distributed carriers. The output then has a broadened spectral profile, ultimately limited by the width of the gain spectrum, and a rapidly diverging, often doublelobed, farfield pattern.

[0003]
An ideal semiconductor laser would produce a diffractionlimited circular output beam with high optical power and a single spectral mode. In broadarea distributedfeedback lasers lateral coherence is established by defining a grating in the epitaxial device structure. In spite of the great promise of edgeemitting Photoniccrystal distributed feedback (PCDFB) lasers, drawbacks include severe ellipticity of the output beam having a fast angular divergence along the growth axis and a slow divergence along the laser stripe. In prior SE PCDFB lasers, the use of electrical pumping obscured identification of the lasing mode(s). The impact of PCDFB grating structure choice on device performance, and the relation to onedimensional SE DFB lasers was not understood. The absence of a general theoretical description of SE PCDFB lasers precluded any systematic analyses of the limits to their singlemode operation, output power, and brightness.

[0004]
The basic concept of employing 2D DFB gratings was proposed by Wang and Sheem (U.S. Pat. No. 3,970,959). However, there was no guidance provided concerning the proper choice of the device parameters or the 2D lattice structure.

[0005]
Other prior approaches considered only two superimposed 1D gratings (rather than actual 2D gratings), in which only two diffraction processes are possible. Furthermore, realistic device geometries were not considered, and the critical role played by the linewidth enhancement factor (LEF) was not considered.
SUMMARY OF THE INVENTION

[0006]
At least some embodiments of the invention relate to surfaceemitting photonic crystal distributed feedback lasers and selfpumping selffocusing mechanisms.

[0007]
According to one embodiment, a surfaceemitting photonic crystal distributed feedback laser apparatus configured to output an optical beam of light is described. The apparatus includes a laser waveguide bounded by a top and bottom optical claddings, an active region configured to produce optical gain upon receiving optical or electrical pumping to inject electrons and holes into the active gain region, a periodic twodimensional grating having an order higher than the fundamental, the grating configured to induce modulation of a modal refractive index and a lateral pumped gain area contained within an area covered by the grating and configured to produce gain in one or more lasing modes and the modal index of the one or more lasing modes is modulated by the periodic twodimensional grating, the lateral pumped gain area having a substantially circular shape of diameter D, and wherein the pumped gain area is enclosed by an unpumped region contained within the area covered by the grating but not receiving the optical or electrical pumping.

[0008]
According to another embodiment of the invention, a method of producing a diffractionlimited beam using a surface emitting photonic crystal distributed feedback laser. The method comprises providing a waveguide in a laser cavity bounded by top and bottom optical claddings; configuring an active region to produce optical gain upon receiving optical or electrical pumping; providing a periodic twodimensional grating having an order higher than a fundamental, the grating configured to modulate a modal refractive index; confining a lateral pumped gain area to within an area covered by the grating, the gain area configured to produce gain in at least one lasing mode having a modal index modulated by the periodic twodimensional grating, the lateral pumped gain area having a substantially circular shape of diameter D; and enclosing the gain area by an unpumped region contained within the area covered by the grating but not receiving the optical or electrical pumping.

[0009]
According to an additional embodiment of the invention, a method of producing a diffractionlimited beam using a surface emitting photonic crystal DFB laser, comprising a) selecting one or more parameters for a twodimensional grating to operate at a predetermined wavelength; b) calculating coupling coefficients κ_{1}, κ_{2}, and κ_{3 }using at least the grating parameters; c) calculating output power and beam quality of the laser; d) determining if the output power corresponds to a desired quantum efficiency; and e) determining if the beam quality is in a desired range relative to the diffraction limit.

[0010]
Other aspects of the invention are disclosed herein as is apparent from the following description and figures.
DESCRIPTION OF THE DRAWINGS

[0011]
Preferred embodiments of the invention are described below with reference to the following accompanying drawings.

[0012]
[0012]FIG. 1a is a schematic of the 2D photonic crystal distributed feedback (PCDFB) laser in accordance with one embodiment of the present invention.

[0013]
[0013]FIG. 1b is a reciprocalspace diagram showing the Brillouinzone boundaries for the Γpoint of a hexagonal lattice of the 2D grating. Diffraction by this crystal structure produces surface emission by the photonic crystal distributed feedback (PCDFB) laser in accordance with one embodiment of the present invention.

[0014]
[0014]FIG. 1(c) is a reciprocalspace diagram showing the Brillouinzone boundaries for the Γpoint of a square lattice of the 2D grating.

[0015]
[0015]FIGS. 2a2 b show nearfield profiles for the 6lobed and 2lobed out of phase modes of a hexagonallattice SE PCDFB with TE polarization;

[0016]
[0016]FIG. 2c shows a singlelobed inphase mode of a hexagonallattice surface emitting (SE) PCDFB with TE polarization. The diagrams in the lower righthand corner illustrate schematically the signs of the six field components displaced from the origin in their propagation directions.

[0017]
[0017]FIG. 3 shows farfield profiles along one of the axes for the modes depicted in FIG. 2b (dashed curve) and FIG. 2c (solid curve).

[0018]
[0018]FIG. 4 is a graph showing coupling coefficients κ_{1 }(dotted), κ_{2 }(dashed), κ_{3 }(dashdot), and κ_{0 }(solid) as a function of circular feature diameter for a hexagonallattice (m=2) SE PCDFB with TE polarization. The modulation of the modal index is Δn=0.06.

[0019]
[0019]FIG. 5 is a graph showing external differential quantum efficiency divided by the etendue (i.e., normalized brightness) as a function of internal loss for hexagonallattice TEpolarized SE PCDFBs with circular feature diameter d=0.684 Λ and index modulation Δn=0.06. Results are given for LEF=0.05 (dotted), LEF=0.5 (dashed), LEF=1.5 (solid), and LEF=4 (dashdot).

[0020]
[0020]FIG. 6 is a graph showing normalized brightness as a function of pumpspot diameter D for hexagonallattice TEpolarized SE PCDFBs with circular feature diameters d=0.684 Λ, LEF=1.5, and an internal loss of 5 cm^{−1}. The solid curve represents reoptimization of the index modulation for each pumpspot diameter, whereas the dashed curve holds the modulation fixed at Δn=0.06 (optimized for D=800 μm).

[0021]
[0021]FIG. 7 is a graph showing spectral characteristics of hexagonallattice TEpolarized SE PCDFBs with fixed LEF (1.5), internal loss (5 cm^{−1}), and pumpspot diameter (800 μm). Spectra are shown for Δn=0.052 (solid), 0.069 (dotted), and 0.086 (dashed), where in each case the peak intensity is normalized to unity rather than to the total power.

[0022]
[0022]FIG. 8 is a graph showing spectra for hexagonallattice TEpolarized SE PCDFBs with fixed internal loss (5 cm^{−1}), pumpspot diameter (800 μm), and index modulation (0.06), at a series of LEF: 1.5 (solid), 2.5 (dashed), and 3.5 (dotted). The peak intensities are normalized to unity.

[0023]
[0023]FIG. 9a shows the nearfield profile for a typical inphase mode, and FIG. 9b shows the nearfield profile for a typical outofphase mode of TMpolarized squarelattice SE PCDFBs.

[0024]
[0024]FIG. 10 is a graph showing normalized brightness as a function of internal loss for squarelattice TMpolarized SE PCDFBs with circular features of diameter d=0.70 Λ and index modulation Δn=0.043. Results are given for LEF=0.05 (dotted curve), LEF=0.5 (dashed), and LEF=1.5 (solid).

[0025]
[0025]FIG. 11 is a graph showing normalized brightness as a function of pumpspot diameter D for squarelattice TMpolarized SE PCDFBs with circular features of diameter d=0.70 Λ, LEF=0.1, and α=15 cm^{−1}. The solid curve represents reoptimization of the index modulation for each value of the pumpspot diameter, whereas the dashed curve holds Δn fixed at 0.043 (optimized for D=800 μm).

[0026]
[0026]FIG. 12 is a graph showing spectral characteristics of squarelattice TMpolarized SE PCDFBs with D=800 μm, Δn=0.043, and LEF=0.05, for internal losses of 10 cm^{−1 }(dashed curve with points), 20 cm^{−1 }(solid), and 40 cm^{−1 }(dotted). The peak intensities are normalized to unity.

[0027]
[0027]FIG. 13 is a high level block diagram of a computer system for implementing the methodology of the present invention in one embodiment.

[0028]
[0028]FIG. 14 is a methodology for producing a neardiffractionlimited beam using a surface emission photonic crystal DFB laser in one embodiment.
DETAILED DESCRIPTION OF THE INVENTION

[0029]
This disclosure of the invention is submitted in furtherance of the constitutional purposes of the U.S. Patent Laws “to promote the progress of science and useful arts” (Article 1, Section 8).

[0030]
In one embodiment, the SE PCDFB uses a timedomain FourierGalerkin (TDFG) numerical solution of a reduced wave equation. A parameter space having high differential quantum efficiency is used in combination with singlemode operation and a neardiffraction limited output beam. A twodimensional square lattice or hexagonallattice grating without any phase shifts and only refractive index modulation is found to be sufficient to produce a symmetric mode that is coherent over a wide device area.

[0031]
Referring to FIG. 1a, there is shown a crosssectional schematic 100 of the surfaceemitting photonic crystal distributed feedback (SE PCDFB) laser in accordance with one embodiment of the present invention. Laser 100 includes a laser waveguide comprised of an active region 106 configured to produce optical gain upon receiving optical or electrical pumping, the active region 106 having a lateral pumped gain area 109. The laser 100 also includes a top and bottom optical claddings 102, 104, respectively, and a periodic twodimensional grating 110 having an order higher than the fundamental, the grating 110 configured to induce modulation of a modal refractive index. In the example embodiment shown in FIG. 1a, the grating 110 which extends over an area 111 is formed in a layer 112 in the waveguide. Other arrangements of forming gratings in multiple layers of materials are also possible. The lateral pumped gain area 109, contained within the grating area 111, is pumped either optically or electrically to produce gain in one or more lasing modes whose modal index is modulated by the grating 110. It will be appreciated that the lateral pumped gain area 109 is the lateral extent of a gain region whose vertical extent is the active region 106.

[0032]
Electrodes 116, 118 are provided for connecting the laser 100 to a voltage source. In an exemplary embodiment, the electrode 116 is an optically opaque electrode configured to apply an input voltage to the laser apparatus, the opaque electrode covering a portion of the lateral pumped gain area, and the electrode 118 is transparent. In one embodiment, the lateral pumped gain area 109 has a nearly circular shape of diameter D, and wherein the gain area 109 is enclosed by an unpumped region 108 contained within the grating area 111, the unpumped region 108 configured so as to not receive the optical or electrical pumping. In one embodiment, etch features of the grating 110 are substantially circular with a diameter of about 6070% of the period of the hexagonal lattice.

[0033]
[0033]FIG. 1b is a reciprocalspace diagram of the hexagonal lattice of the 2D grating which produces surface emission in accordance with one embodiment of the present invention. The primary symmetry points Γ, X, and J are illustrated in the figure along with the boundary of the first Brillouin zone shown as a thin line. Since diffraction processes taking place at this zone boundary do not produce any surface emission, one of the higher zone boundaries must be employed for this purpose. Both m=1 and m=2 zone boundaries (in the notation defined below) are shown in the figure. It will be demonstrated that the greatest flexibility and the optimized designs are obtained using diffraction at m=2.

[0034]
TimeDomain FourierGalerkin Formalism

[0035]
In the TDFG, Bloch expansion of the field and dielectricconstant components are substituted into a wave equation for a TM polarized electric field. The modifications for TEpolarized light are addressed as below:
$\begin{array}{cc}\begin{array}{c}\sum _{{G}^{\prime}}^{\text{\hspace{1em}}}\ue89e\text{\hspace{1em}}\ue89e\stackrel{\u22d2}{\kappa}\ue8a0\left(G{G}^{\prime}\right)\ue89e\lfloor {\uf603k+{G}^{\prime}\uf604}^{2}\ue89e{a}_{k+{G}^{\prime}}\uf74e\ue8a0\left(2\ue89ek+G+{G}^{\prime}\right)\xb7{\stackrel{\ue232}{\nabla}}_{{a}_{k+{G}^{\prime}}}\ue89e{\nabla}_{{a}_{k+{G}^{\prime}}}^{2}\rfloor =\\ \text{\hspace{1em}}\ue89e\left[\frac{{\omega}_{0}^{2}}{{c}^{2}}\ue89e{n}_{0}^{2}\uf74e\ue89e\frac{{\omega}_{0}}{c}\ue89e{n}_{0}\ue89e\hat{G}+\uf74e\ue89e\frac{2\ue89e{\omega}_{0}}{{c}^{2}}\ue89e{n}_{0}^{2}\ue89e\frac{\partial \text{\hspace{1em}}}{\partial t}\right]\ue89e{a}_{k+G}\end{array}& \left(1\right)\end{array}$

[0036]
where ∂_{k}(r,t) are optical field components with wave vector k, κ(G) represents the diffractive coupling corresponding to a reciprocal lattice vector G, n_{0 }is the average value of the modal refractive index n_{m}(r), and ω_{0}=k_{0}n_{0}/c is the center frequency. The carrier contribution may be written out explicitly as: Ĝ=Γ_{c}g(1−iα)−α_{i}, where Γ_{c }is the activeregion optical confinement factor, g(r,N) is the material gain, N is the carrier density calculated from the appropriate rate equation, α=−4 π/λ(dn_{m}/dN)/(dg/dN) is the linewidth enhancement factor (LEF), which measures the focusing/defocusing induced by carrierdensity perturbations, and α is the internal loss. The product of the threshold gain and the LEF (Γ_{c}gthα) is a figure of merit that governs the optical coherence properties of highpower singlemode lasers. In edgeemitting semiconductor lasers, a low value of that product is nearly always desired.

[0037]
Retention of the explicit time dependence in Eq. (1) allows modeling of the spectral properties. In one embodiment, the time derivatives can be treated without approximation (apart from the discretization of the spatial and time intervals). For a particular PC lattice symmetry, Eq. (1) can be simplified considerably by limiting the expansion to a small number of reciprocal lattice vectors G.

[0038]
For example, consider a square lattice with period A as shown in FIG. 1c, the four equivalent propagation directions are labeled P_{1}, P_{2}, −P_{1}, and −P_{2}. Two sets of highsymmetry propagation angles are possible: (1) with P_{1 }oriented along the diagonal of the primitive unit cell; and (2) with P_{1 }bisecting the side of the primitive unit cell. Since at a given wavelength the period is larger by a factor of {square root}2 for the first choice (P_{1}={1,1}, P_{2}={−1,1}, −P_{1}={−1,−1}, and −P_{2}={1,−1}, where the {l,m} represents components along the two axes in units of 2 π/Λ), it is adopted in one embodiment of the invention. However, the inventive concepts are equally applicable to lasers employing a second choice of orientation. The square lattice provides nonzero coupling into an orthogonal inplane propagation direction only for TMpolarized light. For notational simplicity, the coordinate system is rotated such that the zaxis points along the diagonal of the unit cell {1,1}. Also consider k_{0}n_{0}=mπ{square root}2/Λ, i.e., Λ=mλ_{c}/({square root}2n_{0}), where λ_{c }is the grating's resonance wavelength.

[0039]
In an embodiment with a square lattice, the grating is defined on the square lattice with four equivalent propagation directions spaced at 90° angles, and the grating is configured to produce three distinct inplane diffraction processes from one of said equivalent propagation directions into another one of the equivalent propagation directions, the first of said diffraction processes being reflection by an angle of 180° and having a coupling coefficient κ_{1}, and the second and third of said diffraction processes being diffraction by an angle of 90° and having a coupling coefficient of κ_{2}, and an outofplane diffraction process from one of said equivalent propagation directions into a direction perpendicular to all four of said directions and having a coupling coefficient κ_{0}.

[0040]
It would be difficult to start from a scalar wave equation and then treat the surfaceemitting terms on the same footing as the inplane coupling terms. Scalar equations are generally written for the field component polarized in a direction that is not considered further, whereas all three spatial dimensions are to be considered in the present invention. Rather than attempt a full vectorial rederivation from Maxwell's equations, the inventors introduce the surfaceemitting terms as a perturbation, along with the terms that reduce to the previously treated 1 D form. This approximation should be valid as long as the grating is sufficiently “weak”, which applies to the simulations that follow in the sense that both Δn/n_{0 }and the dielectric contrast at the grating interface are small. On the other hand, the weakgrating assumption may break down in such cases as a metallized grating with a large coefficient coupling the guided modes to the field radiating normal to the waveguide.

[0041]
The inventors have observed that strong coupling to the surface emission is not a necessary condition for obtaining highefficiency coherent emission from a large device area. In fact, it was found to be undesirable for the surfaceemitting coefficient to be either substantially larger or substantially smaller than the internal loss.

[0042]
The inplane coupling coefficients, κ_{1 }and κ_{2}, are introduced by following the discussion disclosed in the publication “PhotonicCrystal DistributedFeedback quantum Cascade Lasers”, IEEE J. Quantum Electronics, June 2002, the entire disclosure of which is incorporated by references. It is well known that in addition to the inplane diffraction, Bragg reflections with m>1 also produce losses associated with emission out of the plane. That emission, into a layer with n_{1}<n_{0}, occurs at an angle φ with respect to normal whenever sin φ=n_{0}(P_{1}−G)/n_{1}P_{1}<1 for any G. Focusing on the m=2 case, the surfaceemitting terms can be added as a perturbation of the equations to treat edgeemitting PCDFB lasers. For “weak” gratings, the Green's function solution of the ridgewaveguide problem leads to the introduction of a surface outcoupling coefficient, κ_{0}, defined more precisely as below.

[0043]
In one exemplary embodiment, it may not be necessary to include more than four components, since the net power in the higherorder components is expected to be lower by two orders of magnitude. For example, it is assumed that all of the field components should couple to the radiating wave at the same rate, provided that the etched features do not have a lower symmetry than the lattice (e.g., when the features are circular). Thus all four components experience an outcoupling loss proportional to the product of κ_{0 }and the radiatingfield magnitude, which is given by the coherent sum of the field components.

[0044]
The resulting final form of the propagation equations for a square lattice with TM polarization is:
$\begin{array}{cc}\begin{array}{c}\frac{\partial {a}_{1}}{\partial t}=\frac{c}{{n}_{0}}[\frac{\partial {a}_{1}}{\partial z}+\frac{\Gamma \ue89e\text{\hspace{1em}}\ue89eg\ue8a0\left(1\uf74e\ue89e\text{\hspace{1em}}\ue89e\alpha \right){\alpha}_{i}2\ue89e{\kappa}_{0}}{2}\ue89e{a}_{1}+\frac{\uf74e}{2\ue89e{k}_{0}\ue89e{n}_{0}}\ue89e{\nabla}^{2}\ue89e{a}_{1}+\\ \text{\hspace{1em}}\ue89e\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{1}{\kappa}_{0}\right)\ue89e{\stackrel{\_}{a}}_{1}+\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{2}{\kappa}_{0}\right)\ue89e\left({a}_{2}+{\stackrel{\_}{a}}_{2}\right)]\end{array}& \left(2\right)\\ \begin{array}{c}\frac{\partial {a}_{2}}{\partial t}=\frac{c}{{n}_{0}}[\frac{\partial {a}_{2}}{\partial x}+\frac{\Gamma \ue89e\text{\hspace{1em}}\ue89eg\ue8a0\left(1\uf74e\ue89e\text{\hspace{1em}}\ue89e\alpha \right){\alpha}_{i}2\ue89e{\kappa}_{0}}{2}\ue89e{a}_{2}+\frac{\uf74e}{2\ue89e{k}_{0}\ue89e{n}_{0}}\ue89e{\nabla}^{2}\ue89e{a}_{2}+\\ \text{\hspace{1em}}\ue89e\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{1}{\kappa}_{0}\right)\ue89e{\stackrel{\_}{a}}_{2}+\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{2}{\kappa}_{0}\right)\ue89e\left({a}_{1}+{\stackrel{\_}{a}}_{1}\right)]\end{array}& \left(3\right)\end{array}$

[0045]
Similar equations hold for {overscore (a)}_{1 }and {overscore (a)}_{2}, except for a reversal of the sign in front of the first derivative and removing the bars from the field components in the first coupling term. Note the close analogy of Eqs. (2) and (3) to the field equations derived previously for 1D DFB structures in a copending application having application Ser. No. 10/385,165 and filed on Mar. 7, 2003, the entire contents of which are incorporated herein by reference, with the important difference (aside from the doubling of dimensions and components) being retention of the secondorder derivatives throughout our formalism. Since the κ_{2 }coefficient vanishes for TE polarization, the square lattice was found to be not a viable choice for largearea coherent emission in that polarization.

[0046]
Examining the coupling coefficients, κ
_{0}, κ
_{1}, and κ
_{2}, more closely, it is observed that for the m=2 coupling order, the wavevector and reciprocal lattice vectors are defined: k={0,0}, G
_{1}={1,1}, G
_{−1}={−1,−1}, G
_{2}={−1,1}, and G
_{−2}={1,−1}. For a relatively small index modulation (Δn), the following expressions for the inplane coupling coefficients κ
_{1 }and κ
_{2 }are obtained:
$\begin{array}{cc}{\kappa}_{1}=\frac{2\ue89e\pi \ue89e\text{\hspace{1em}}\ue89e\Delta \ue89e\text{\hspace{1em}}\ue89en}{\lambda}\ue89e\frac{1}{{a}_{L}}\ue89e\int {\int}_{R}^{\text{\hspace{1em}}}\ue89e\text{\hspace{1em}}\ue89e\uf74cx\ue89e\uf74cz\ue89e\text{\hspace{1em}}\ue89e\mathrm{exp}\ue8a0\left[\uf74e\ue8a0\left({G}_{1}{G}_{1}\right)\xb7r\right]& \left(4\right)\\ {\kappa}_{2}=\frac{2\ue89e\pi \ue89e\text{\hspace{1em}}\ue89e\Delta \ue89e\text{\hspace{1em}}\ue89en}{\lambda}\ue89e\frac{1}{{a}_{L}}\ue89e\int {\int}_{R}^{\text{\hspace{1em}}}\ue89e\text{\hspace{1em}}\ue89e\uf74cx\ue89e\uf74cz\ue89e\text{\hspace{1em}}\ue89e\mathrm{exp}\ue8a0\left[\uf74e\ue8a0\left({G}_{1}{G}_{2}\right)\xb7r\right]& \left(5\right)\end{array}$

[0047]
where i=1, 2, ∂_{L }is the area of the reciprocallattice primitive cell, G_{1}−G_{−1}=G_{2}−G_{−2}, and G_{1}−G_{2}=G_{−1}−G_{−2}=G_{1}−G_{−2}=G_{−1}−G_{2}. Considering indexmodulation regions R that are symmetric with respect to reflection along the propagation axes, the imaginary parts of Eqs. (4) and (5) vanish. For example, for simplicity, the present discussion is confined to features R (e.g., circles, squares oriented along one of the axes etc.) that have identical arguments in the exponent of Eqs. (4) and (5) for all the equivalent couplings, although a certain amount of asymmetry can be tolerated in practice and is included in the scope of the present invention.

[0048]
With these assumptions, the coefficient κ
_{1 }accounts for the reflectionlike diffraction into the counterpropagating wave, e.g., from P
_{1 }into −P
_{1 }or from P
_{2 }into −P
_{2}. The coefficient κ
_{2 }represents diffraction into the perpendicular inplane component, e.g., from P
_{1 }into P
_{2 }or −P
_{2}, or from P
_{2 }into P
_{1 }or −P
_{1}. The corresponding expression for the surfacecoupling coefficient, κ
_{0}, which quantifies diffraction from all four waves P
_{1}, −P
_{1}, P
_{2}, and −P
_{2 }into the radiating wave, is given by
$\begin{array}{cc}{\kappa}_{0}=\frac{{d}_{g}}{{\Gamma}_{g}}\ue89e{\uf603\frac{2\ue89e\pi \ue89e\text{\hspace{1em}}\ue89e\Delta \ue89e\text{\hspace{1em}}\ue89en}{\lambda}\ue89e\frac{1}{{a}_{L}}\ue89e\int {\int}_{R}^{\text{\hspace{1em}}}\ue89e\text{\hspace{1em}}\ue89e\uf74cx\ue89e\uf74cz\ue89e\text{\hspace{1em}}\ue89e\mathrm{exp}\ue8a0\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{G}_{1}\xb7r\right)\uf604}^{2}& \left(6\right)\end{array}$

[0049]
where d_{g }is the thickness of the grating layer and Γ_{g }is its confinement factor. For m=2, G_{1}−G_{−1}=2G_{1}, and in the weakmodulation limit κ_{1 }and κ_{2 }are proportional to Δn whereas κ_{0}∝Δn^{2}. Equations (4)(6) are generalized to the case of complex coupling by setting the imaginary part lm{Δn}=−(λ/4 π)Δg or (λ/4 π)Δα.

[0050]
Since the gain is in principle a spatiotemporal function that depends on the complex field components via the carrier rate equation, Eqs. (2) and (3) are in fact secondorder nonlinear differential equations in 3 dimensions. In the following a splitstep FourierGalerkin solution of these equations is described, subject to Dirichlet boundary conditions in the two spatial dimensions based on approximate linearization for short time intervals Δt. In order to perform the linearization, in Eqs. (2) and (3) the gain term is separated from the diffractioncoupling terms, transforming the latter into the frequency domain using fast Fourier transformations (FFT). Without the gain term, Eqs. (2) and (3) then read:
$\begin{array}{cc}\frac{\partial {\stackrel{~}{a}}_{1}}{\partial t}=\frac{c}{{n}_{0}}\ue8a0\left[\uf74e\ue89e\frac{2\ue89e\pi \ue89e\text{\hspace{1em}}\ue89e{l}_{z}}{{L}_{z}\ue89e\Delta \ue89e\text{\hspace{1em}}\ue89ez}\ue89e{\stackrel{~}{a}}_{1}\frac{\uf74e}{2\ue89e{k}_{0}\ue89e{n}_{0}}\ue89e\left(\frac{4\ue89e{\pi}^{2}\ue89e{l}_{x}^{2}}{{{L}_{x}^{2}\ue8a0\left(\Delta \ue89e\text{\hspace{1em}}\ue89ex\right)}^{2}}+\frac{4\ue89e{\pi}^{2}\ue89e{l}_{z}^{2}}{{{L}_{z}^{2}\ue8a0\left(\Delta \ue89e\text{\hspace{1em}}\ue89ez\right)}^{2}}\right)\ue89e{\stackrel{~}{a}}_{1}+\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{1}{\kappa}_{0}\right)\ue89e{\stackrel{~}{\stackrel{\_}{a}}}_{1}+\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{2}{\kappa}_{0}\right)\ue89e\left({\stackrel{~}{a}}_{2}+{\stackrel{~}{\stackrel{\_}{a}}}_{2}\right)\right]& \left(7\right)\\ \frac{\partial {\stackrel{~}{a}}_{2}}{\partial t}=\frac{c}{{n}_{0}}\ue8a0\left[\uf74e\ue89e\frac{2\ue89e\pi \ue89e\text{\hspace{1em}}\ue89e{l}_{x}}{{L}_{x}\ue89e\Delta \ue89e\text{\hspace{1em}}\ue89ex}\ue89e{\stackrel{~}{a}}_{2}\frac{\uf74e}{2\ue89e{k}_{0}\ue89e{n}_{0}}\ue89e\left(\frac{4\ue89e{\pi}^{2}\ue89e{l}_{x}^{2}}{{{L}_{x}^{2}\ue8a0\left(\Delta \ue89e\text{\hspace{1em}}\ue89ex\right)}^{2}}+\frac{4\ue89e{\pi}^{2}\ue89e{l}_{z}^{2}}{{{L}_{z}^{2}\ue8a0\left(\Delta \ue89e\text{\hspace{1em}}\ue89ez\right)}^{2}}\right)\ue89e{\stackrel{~}{a}}_{2}+\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{1}{\kappa}_{0}\right)\ue89e{\stackrel{~}{\stackrel{\_}{a}}}_{2}+\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{2}{\kappa}_{0}\right)\ue89e\left({\stackrel{~}{a}}_{1}+{\stackrel{~}{\stackrel{\_}{a}}}_{1}\right)\right]& \left(8\right)\end{array}$

[0051]
where the components with tildes have been obtained using the discrete Fourier transform of the original variables on a mesh (Δx,Δz) in the laser plane. Since the lattice is square, consider Δx=Δz. In Eqs. (7) and (8), the indices corresponding to the spatial frequencies are labeled I_{x }and I_{z }(0≦I_{x}≦L_{x}−1, 0≦I_{z}≦L_{z}−1). Thus for each time step, Eqs. (7) and (8) can be solved using the fourthorder RungeKutta method applicable to ordinary differential equations of the first order. Numerical simulations by the inventors confirm that the FourierGalerkin method described above offers a much higher precision than other possible approaches. For example, the finitedifference approximation was found to be unsuitable owing to a substantial phase error.

[0052]
The second part of the splitstep TDFG algorithm involves evaluation of the opticalgain and refractiveindexfluctuation terms from the rate equation for the carrier density and the appropriate gain vs. density model. The field components are backtransformed to real space and integrated at each spatial mesh point, for example, using the RungeKutta method. The procedure may be repeated until turnon transients are eliminated. The output spectra are obtained from the power spectrum of the surfaceemitting field. The etendue determined from the 2D nearfield and farfield profiles is defined as the geometric average of the figures of merit measured along the two orthogonal inplane directions.

[0053]
In one embodiment, the figure of merit identified herein is in most cases equivalent to the M^{2 }figure of merit and is proportional to the product of the standard deviation of the nearfield profile and the average angle of the farfield profile. The standard deviation is not employed for the farfield profile in order to minimize the experimental measurement error at large angles. The result for the geometric average of the figures of merit is then normalized to the etendue of a circular Gaussian beam with diffractionlimited beam quality. In most cases, the symmetry of the device structure leads to a symmetric output beam.

[0054]
The TDFG algorithm of the present invention does not include spontaneous emission. Generally the spontaneous emission magnitude can only be roughly estimated, and the inventors have found that it has negligible influence on the operation far above threshold. While the primary role of spontaneous emission is to provide seeding for the initial laser turnon dynamics, simulations by the inventors confirm that any initial distribution (obtained with a pseudorandom number generator) leads to the same steadystate solutions described below.

[0055]
The power emitted from the surface is calculated from the expression:
$\begin{array}{cc}P=\frac{c}{{n}_{0}}\ue89e{\kappa}_{0}\ue89e\int \int \uf74cx\ue89e\uf74cz\ue89e{\uf603{a}_{1}+{\stackrel{\_}{a}}_{1}+{a}_{2}+{\stackrel{\_}{a}}_{2}\uf604}^{2}& \left(9\right)\end{array}$

[0056]
where the coherent sum of the field components is proportional to the amplitude of the surfaceemitting field. An important difference exists between the intensity of the waveguideconfined laser light, which is proportional to the sum of the squares of the individual field components because carrier diffusion washes out effects on the scale of λ_{c}/n_{0}, and the intensity of the light emitted normal to the laser plane, which is proportional to the square of the sum of these components.

[0057]
A similar TDFG algorithm as described above is applicable to the hexagonal lattice with period Λ. For example, consider the sixfold coupling that takes place at the Γ point with propagation vectors pointing along the Γ−X directions. Defining β=(4{square root}3/3)mπ/Λ [Λ=(2{square root}3/3)mλ
_{c}/n
_{0}, m=1, 2, 3, . . .], the relevant reciprocal lattice vectors become: G
_{1}=β(0,1), G
_{−1}=β(0,−1), G
_{2}=β({square root}3/2, ½), G
_{2}=β(−{square root}3/2, −½), G
_{3}=β(−{square root}3/2, ½), and G
_{−3}=β({square root}3/2, −½). There are now six equivalent propagation directions [P
_{1}=(0,1); P
_{2}=({square root}3/2, ½); and P
_{3}=(−{square root}3/2, ½)]. In one embodiment, for circular features (and other shapes that preserve the distance from the origin to the intersection of the reciprocallattice vectors with the Brillouinzone boundary), three distinct coupling coefficients can be calculated. The inventors use a convention whereby κ
_{1 }is the DFBlike coefficient corresponding to distributed reflection by 180° (e.g., from P
_{1 }to P
_{−1}), κ
_{2 }is the coefficient for diffraction by a 60° angle (e.g., P
_{1 }to P
_{2}, P
_{1 }to P
_{3, }or P
_{−2 }to P
_{3}) and κ
_{3 }is the coefficient for diffraction by 120° (e.g., P
_{1 }to P
_{−2}, P
_{1 }to P
_{−3}, or P
_{2 }to P
_{−1}). The coupling coefficients κ
_{1 }and κ
_{2 }are computed using Eqs. (4) and (5), and the coupling coefficient κ
_{3}from:
$\begin{array}{cc}{\kappa}_{3}=\frac{2\ue89e\pi \ue89e\text{\hspace{1em}}\ue89e\Delta \ue89e\text{\hspace{1em}}\ue89en}{\lambda}\ue89e\frac{1}{{a}_{L}}\ue89e\int {\int}_{R}^{\text{\hspace{1em}}}\ue89e\text{\hspace{1em}}\ue89e\uf74cx\ue89e\uf74cz\ue89e\text{\hspace{1em}}\ue89e\mathrm{exp}\ue8a0\left[\uf74e\ue8a0\left({G}_{1}{G}_{2}\right)\xb7r\right]& \left(10\right)\end{array}$

[0058]
The surfaceemitting coefficient κ
_{0 }is again calculated from Eq. (6). Evaluations of the coupling coefficients corresponding to circular features can be found in the above incorporated reference. In place of Eqs. (7) and (8), the following formulas are employed:
$\begin{array}{cc}\frac{\partial {\stackrel{~}{a}}_{1}}{\partial t}=\frac{c}{{n}_{0}}\ue8a0\left[\uf74e\ue89e\frac{2\ue89e\pi \ue89e\text{\hspace{1em}}\ue89e{l}_{z}}{{L}_{z}\ue89e\Delta \ue89e\text{\hspace{1em}}\ue89ez}\ue89e{\stackrel{~}{a}}_{1}\frac{\uf74e}{2\ue89e{k}_{0}\ue89e{n}_{0}}\ue89e\left(\frac{4\ue89e{\pi}^{2}\ue89e{l}_{x}^{2}}{{{L}_{x}^{2}\ue8a0\left(\Delta \ue89e\text{\hspace{1em}}\ue89ex\right)}^{2}}+\frac{4\ue89e{\pi}^{2}\ue89e{l}_{z}^{2}}{{{L}_{z}^{2}\ue8a0\left(\Delta \ue89e\text{\hspace{1em}}\ue89ez\right)}^{2}}\right)\ue89e{\stackrel{~}{a}}_{1}+\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{1}{\kappa}_{0}\right)\ue89e{\stackrel{~}{\stackrel{\_}{a}}}_{1}+\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{2}{\kappa}_{0}\right)\ue89e\left({\stackrel{~}{a}}_{2}+{\stackrel{~}{a}}_{3}\right)\ue89e\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{3}{\kappa}_{0}\right)\ue89e\left({\stackrel{~}{\stackrel{\_}{a}}}_{2}+{\stackrel{~}{\stackrel{\_}{a}}}_{3}\right)\right]& \left(11\right)\\ \frac{\partial {\stackrel{~}{a}}_{2}}{\partial t}=\frac{c}{{n}_{0}}\ue8a0\left[\uf74e\ue89e\frac{\pi \ue89e\text{\hspace{1em}}\ue89e{l}_{z}}{{L}_{z}\ue89e\Delta \ue89e\text{\hspace{1em}}\ue89ez}\ue89e{\stackrel{~}{a}}_{2}\uf74e\ue89e\frac{\sqrt{3}\ue89e\pi \ue89e\text{\hspace{1em}}\ue89e{l}_{x}}{{L}_{x}\ue89e\Delta \ue89e\text{\hspace{1em}}\ue89ex}\ue89e{\stackrel{~}{a}}_{2}\frac{\uf74e}{2\ue89e{k}_{0}\ue89e{n}_{0}}\ue89e\left(\frac{4\ue89e{\pi}^{2}\ue89e{l}_{x}^{2}}{{{L}_{x}^{2}\ue8a0\left(\Delta \ue89e\text{\hspace{1em}}\ue89ex\right)}^{2}}+\frac{4\ue89e{\pi}^{2}\ue89e{l}_{z}^{2}}{{{L}_{z}^{2}\ue8a0\left(\Delta \ue89e\text{\hspace{1em}}\ue89ez\right)}^{2}}\right)\ue89e{\stackrel{~}{a}}_{1}+\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{1}{\kappa}_{0}\right)\ue89e{\stackrel{~}{\stackrel{\_}{a}}}_{2}+\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{2}{\kappa}_{0}\right)\ue89e\left({\stackrel{~}{a}}_{2}+{\stackrel{~}{\stackrel{\_}{a}}}_{3}\right)\ue89e\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{3}{\kappa}_{0}\right)\ue89e\left({\stackrel{~}{a}}_{3}+{\stackrel{~}{\stackrel{\_}{a}}}_{1}\right)\right]& \left(12\right)\\ \frac{\partial {\stackrel{~}{a}}_{3}}{\partial t}=\frac{c}{{n}_{0}}\ue8a0\left[\uf74e\ue89e\frac{\pi \ue89e\text{\hspace{1em}}\ue89e{l}_{z}}{{L}_{z}\ue89e\Delta \ue89e\text{\hspace{1em}}\ue89ez}\ue89e{\stackrel{~}{a}}_{3}+\uf74e\ue89e\frac{\sqrt{3}\ue89e\pi \ue89e\text{\hspace{1em}}\ue89e{l}_{x}}{{L}_{x}\ue89e\Delta \ue89e\text{\hspace{1em}}\ue89ex}\ue89e{\stackrel{~}{a}}_{3}\frac{\uf74e}{2\ue89e{k}_{0}\ue89e{n}_{0}}\ue89e\left(\frac{4\ue89e{\pi}^{2}\ue89e{l}_{x}^{2}}{{{L}_{x}^{2}\ue8a0\left(\Delta \ue89e\text{\hspace{1em}}\ue89ex\right)}^{2}}+\frac{4\ue89e{\pi}^{2}\ue89e{l}_{z}^{2}}{{{L}_{z}^{2}\ue8a0\left(\Delta \ue89e\text{\hspace{1em}}\ue89ez\right)}^{2}}\right)\ue89e{\stackrel{~}{a}}_{1}+\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{1}{\kappa}_{0}\right)\ue89e{\stackrel{~}{\stackrel{\_}{a}}}_{3}+\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{2}{\kappa}_{0}\right)\ue89e\left({\stackrel{~}{a}}_{1}+{\stackrel{~}{\stackrel{\_}{a}}}_{2}\right)\ue89e\left(\uf74e\ue89e\text{\hspace{1em}}\ue89e{\kappa}_{3}{\kappa}_{0}\right)\ue89e\left({\stackrel{~}{a}}_{2}+{\stackrel{~}{\stackrel{\_}{a}}}_{1}\right)\right]& \left(13\right)\end{array}$

[0059]
For example, the hexagonal lattice can also be used to obtain surface emission with TEpolarized light. In that case, the following substitutions should be made in Eqs. (11)(13): κ_{1}→−κ_{1}, κ_{2}→κ_{2}/2, and κ_{3}→−κ_{3}2. In the following, the values of κ_{2}and κ_{3 }for TE polarization are divided by a factor of 2 as compared to the definitions in Eqs. (5) and (10). The sign change in two of the three coupling coefficients leads to different optimized designs for TE and TMpolarized surfaceemitting PCDFB lasers.

[0060]
In one embodiment of the SE PCDFB laser using the disclosed selfpumping/selffocusing (SPSF) mechanism, the pumped area 109 (which may be pumped either optically or electrically) is circular and has a tophat injection profile, although similar results are obtained for a Gaussian profile, and small deviations from circularity are not of significance. In the exemplary embodiment shown in FIG. 1a, the pumped spot 109 (FIG. 1a) is taken to be far away from any facets and/or discontinuities in the grating 110 (FIG. 1a). The properties of the activeregion 106 are assumed to be uniform over the entire simulation area, although this is not required by the TDFG formalism. In an exemplary case, for definiteness, the optical or electrical pump injection level is assumed to be 10 times the threshold value.

[0061]
In a preferred embodiment, the gain spectrum is considered to be much broader than any spectral features associated with the grating 110 (FIG. 1a) and the peak of the gain spectrum is aligned with the cavity resonance wavelength. The inventors have observed that the lattice temperature increase does not substantially detune the gain peak from the cavity resonance for pulsedmode operation.

[0062]
Since the SE PCDFB relies on diffraction from higher grating orders, the index modulation Δn is preferred to be rather large, e.g., several percent of the modal refractive index. In the preferred embodiment, the grating 110 (FIG. 1a) is realized by etching holes into the grating layer 112 and epitaxially regrowing a lowerindex cladding material 102 (FIG. 1a) In other embodiments, a metallic grating may be deposited, a dielectric material with index lower than that of the waveguide core is deposited, or the holes are left empty. In the case where a metallic grating is deposited, the inventors have observed that modulations of the imaginary part of the refractive index generally dominate and there is a large dielectric contrast across the interface.

[0063]
For the regrowth approach, the dielectric deposition approach, or the approach in which the holes are left empty, the modulation of Δn is primarily real, with only a small imaginary contribution arising from variations of the optical confinement factor in the active region. For example, in the following sample calculations the modulation is taken to be purely real, although the results were found to be unchanged when the simulation includes an imaginary component of the expected magnitude.

[0064]
Since for most interband semiconductor lasers the optical gain is greater for TE polarization than for TM, in the sample calculations a hexagonal lattice was used. In one embodiment, for definiteness, the inventors have used a parameter set characteristic of the midIR “W” active region with 40 periods emitting at λ_{c}=4.6 μm. However, the results would be the same for any active region with the same differential gain of 10^{−15 }cm^{2 }and optical confinement factor per period of ≈1%. Assuming that the midIR test structure includes a GaSb grating layer on top of the active region, we obtain d_{g}/Γ_{g}=1.5 μm. Since κ_{0 }scales quadratically with Δn whereas the other coupling coefficients scale linearly (at relatively small Δn), the results observed by the inventors are expected to hold at least qualitatively when d_{g}/Γ_{g }varies somewhat.

[0065]
The TDFG simulations show that a singlemode SE PCDFB laser can display at least three distinct nearfield patterns. Although two of those correspond to undesirable farfield profiles, the third (preferred) mode may be selected by following a set of design rules that will be presented below.

[0066]
[0066]FIG. 2(a) illustrates that the nearfield characteristic for one of the three modes form an annular pattern, with six peaks spaced almost evenly around the perimeter of the annulus. The diagrams in the lower righthand corner illustrate schematically the signs of the six field components displaced from the origin in their propagation directions. For this mode, the phases of the six field components alternate as a function of azimuthal angle, so that no two adjacent components separated by 60° experience constructive interference.

[0067]
For the second mode structure illustrated in FIG. 2(b), the phase is flipped as one crosses a given axis. This produces a nearfield structure that has only two lobes. Both of the modes shown in FIGS. 2(a) and 2(b) may be classified as outofphase, since pronounced destructive interference in the surfaceemitted component substantially reduces the output power and also degrades the beam quality. The simulations show that a small amount of gain coupling (on the order of 10% of the index modulation) leads to stabilization of the sixlobe pattern FIG. 2(a) in those cases where the twolobe pattern FIG. 2(b) would otherwise dominate.

[0068]
In one embodiment, the PC DFB laser is configured to maximize the output power of the optical beam from the laser so that a differential quantum efficiency is no smaller than about 3%, and the laser is further configured to optimize the beam quality of the optical beam so that it is no worse than about three times the diffraction limit.

[0069]
A third mode, which can have the lowest threshold gain under certain conditions, is symmetric and singlelobed because all six components have the same phase. FIG. 2(c) shows that the symmetric nearfield pattern for this inphase mode is close to a circular Gaussian profile.

[0070]
[0070]FIG. 3 shows that the corresponding farfield pattern (solid curve) is also singlelobed, and quite narrow. On the other hand, the second outofphase mode is doublelobed in the farfield (dashed curve of FIG. 3) as it was in the nearfield (FIG. 2b), along the same phaseflip axis. The net angular divergence is clearly much smaller for the inphase mode, resulting in a beam quality that is typically no more than 2030% greater than the diffraction limit (DL) when the nearfield and farfield characteristics are combined.

[0071]
In one embodiment, the present invention relates to means of stabilizing the inphase mode in FIG. 2(c) over the outofphase modes in FIG. 2(a) and FIG. 2(b) by means of reducing its threshold gain relative to all the other modes. From simulations that explored a broad range of the available design space, the inventors have determined that operation in the symmetric (inphase) single mode can be reliably selected as long as the following exemplary criteria are met (except as noted, the conditions for TM polarization are similar):

[0072]
(1) The mode must preferably be allowed to selfpump regions somewhat beyond the nominal pump spot (the selfpumping requirement).

[0073]
(2) The linewidth enhancement factor and the internal loss are preferred to be sufficiently large, with the two minimum values being interdependent. This selffocusing condition will be expressed more precisely below.

[0074]
(3) For TE polarization, the signs of the inplane coupling coefficients are preferred to be: κ_{1}>0, κ_{2}<0, and κ_{3}>0, although singlemode operation with somewhat degraded characteristics may also be obtained with: κ_{1}<0, κ_{2}>0, and κ_{3}<0. For TM polarization this condition becomes: κ_{1}<0, κ_{2}<0, and κ_{3}<0.

[0075]
(4) The magnitudes of at least two of the coupling coefficients (taken to be κ_{2 }and κ_{3}) should range approximately from 1/D to 3/D, while the third coefficient (taken to be κ_{1}) falls between 1/(5D) and 3/D, where D is the diameter of the pump spot.

[0076]
(5) κ_{0 }is preferred not to be much larger than 1 cm^{−1}, with the exact upper limit depending on the LEF, internal loss, and pump spot diameter as described above.

[0077]
Having identified the preferred design space that assures emission into a singlelobed symmetric mode, its implementation in practice for the case of a hexagonal PCDFB lattice is shown. Assuming that Condition (1) as described above is met, first consider Conditions (3)(5). By calculating the coupling coefficients for circular etched features, optimized lattice structures for both TE and TM polarizations are deduced. One finding is that neither polarization can reach optimized performance in the lowest diffraction order m=1 [the order definitions precede Eqs. (1113)]. However, with m=2 the larger reciprocal lattice vectors afford sufficient freedom for simultaneously optimizing all four coupling coefficients.

[0078]
For TE polarization and an index modulation of Δn=0.06, FIG. 4 plots the four κ_{i }variations as a function of featurediameter/period ratio (d/Λ). The inventors have found in an exemplary case, that Conditions (3)(5) are satisfied for feature diameters falling in the range 0.6 Λ<d<0.7 Λ, with d=0.68 Λ being optimal. This assumes a pumpspot diameter (D) on the order of 1 mm. However, the same d/λ ratio tends to satisfy Conditions (3)(5) for other D as well, if Δn is varied to scale the coupling coefficients. This confirms that there is a limited, but straightforwardly accessible parameter space that may be expected to yield highefficiency SE PCDFB emission into a single inphase mode. On the other hand, for TM polarization the optimum diameter is d≈0.90 Λ, which is considerably more demanding from the fabrication standpoint. Therefore in that case a square lattice is employed in the preferred embodiment.

[0079]
Quantifying Condition (2), which follows from the observation made below that LEF=0 inevitably leads to SE PCDFB operation in an outofphase mode. The need for a minimum LEF starkly contrasts the requirements of conventional PCDFB and other edgeemitting semiconductor lasers, since those devices nearly always perform best when the product of the LEF and the threshold modal gain is as small as possible, further details of which are described in the article by 1. Vurgaftman and J. R. Meyer, “PhotonicCrystal DistributedFeedback Quantum Cascade Lasers”, IEEE J. Quantum Electronics 38, 592 (2002)], the entire contents of which are incorporated herein by reference. To some extent, one can compensate for a small LEF (e.g., <1) in the SE PCDFB by increasing the internal loss (α). However, that strategy naturally reduces the quantum efficiency and output power.

[0080]
In an exemplary illustration, FIG. 5 plots the “normalized brightness”, defined as the external differential quantum efficiency (assuming an internal efficiency of unity) divided by the etendue (in units of the DL), as a function of internal loss for several representative values of the LEF. The simulations, for example, assume a hexagonal lattice (d=0.684 Λ, κ_{1}=6.1 cm^{−1}, κ_{2}=32 cm^{−1}, κ_{3}=15 cm^{−1}, κ_{0}=0.31 cm^{−1}) with a pump spot diameter of D=800 μm. When LEF=0.05 (characteristic of optimized quantum dot lasers), inphase emission occurs only when α_{i}≧20 cm^{−1}, which imposes a limit of ≈5% on the normalized brightness. As the LEF is increased to 0.5 (characteristic of nonuniform quantum dot lasers and optimized nearIR strained quantumwell QW) diodes), inphase operation is possible down to α_{i}≈5 cm^{−1}, which boosts the maximum normalized brightness to ≈15%. For any LEF≦0.5, the SE PCDFB emits in a single mode for all considered values of the internal loss (≦40 cm^{−1}) despite the large diameter of the pump spot.

[0081]
Optimal performance is expected when LEF≈12 which is characteristic of many interband QW diode lasers operating in the visible, nearIR (12 μm) and midIR wavelength ranges. The inventors estimate that a maximum normalized brightness of at least 30%, which may be improved by up to an additional factor of 2 if the equal emission in the opposite direction can be collected. This indicates that performance on a par with optimized edgeemitting PCDFB lasers should be possible, with the added advantage of a symmetric circular output beam. However, the LEF=1.5 curve decreases rapidly above α_{i}>20 cm^{−1}, since at that point the output becomes multimode. Thus the SE PCDFB approach becomes disadvantageous for QW diodes with high losses. The inventors have found that the normalized brightness never reaches an attractive value when the LEF>2, since the operation remains multimode. This is shown in FIG. 5 for LEF=4.

[0082]
[0082]FIG. 6 shows a graph of the normalized brightness as a function of pumpspot diameter, where the solid curve corresponds to reoptimization of the etch depth at each D. The consequences of increasing D to maximize the singlemode output power are next described. Since the discrimination between inphase and outofphase modes, resulting from the imposition of additional losses by selfpumping, is greatest for a small pumpspot area, the preferred index modulation decreases as D increases. The inventors have found that the optimal value ranges from Δn=0.12 for D=300 μm to Δn=0.039 for D=2 mm. Although the maximum singlemode output power is greatest for the largest pump spot, the efficiency decreases considerably at large D. In the opposite limit of D<<300 μm, the index modulation required to make an attractive device becomes so large that it probably cannot be realized without a fundamental modification of the waveguide geometry.

[0083]
Whereas each point of the solid curve in FIG. 6 corresponds to a different etch depth, it is also of interest to examine the range of pumpspot diameters over which a single SE PCDFB device with fixed Δn may be operated. The dashed curve in FIG. 6 plots the normalized brightness for the etch depth that maximizes the singlemode power at D=800 μm (Δn=0.06). We find that such a device should operate in a single inphase mode when 500 μm <D<900 μm, although that range can be extended at the expense of slightly lower peak efficiency if the etch depth is reduced somewhat. Singlemode inphase operation is maintained for an index modulation as little as Δn=0.04.

[0084]
An advantage of the SE PCDFB lasers using the SPSF mechanism is the ability to produce a highefficiency, neardiffractionlimited, circular, spectrally pure output beam in a direction normal to the device plane from a very broad emitting aperture (a diameter on the order of 1 mm). Since verticalcavity surfaceemitting lasers (VCSELs) can produce such a beam only from a narrow aperture (a diameter on the order of a few wavelengths) and edgeemitting PCDFB lasers cannot produce a circular output beam, the above capability is unique in the semiconductorlaser field.

[0085]
Only the definition of the 2D grating is necessary for device fabrication. Unlike the edgeemitting 1D DFB and PCDFB lasers, no facets are required and no alignment of the grating (e.g., to a cleave plane) is needed. Since the SE PCDFB laser generally operates in a relatively high diffraction order, the critical dimension in the lithographical definition of the grating is not excessively small. For example, for the emission wavelength of 1.55 μm the required period of the hexagonallattice (m=2) grating is approximately 1.0 μm, and the separation between the features in the preferred embodiment ranges from 300 to 400 nm. For comparison, the commonly used edgeemitting 1D DFB devices emitting at the same wavelength require a period of approximately 220 nm and a feature size of 110 nm.

[0086]
The realization of the selfpumping mechanism is straightforward in both 2D and 1D. The only requirement is not to limit the active region to the pump spot, i.e. the region optically or electrically pumped above threshold should preferably be surrounded by unpumped material with the same gain characteristics. For example, in 1D the selfpumping requirement is realized by placing any reflective interfaces (such as mirror facets) far away (at distances on the order of several hundred microns) away from the region pumped above threshold. In 2D the selfpumping can be accomplished by pumping sufficiently far away from any discontinuities in the material such as the edges of the wafer.

[0087]
Most prior simulations of 1D SE DFB lasers have failed to account for the LEF, which according to modeling carried by the inventors can lead to the erroneous conclusion that the antisymmetric mode always has lower threshold gain. In one study that included the LEF, the potential for lasing in the symmetric mode was not observed because too high a value was employed for the surfaceemitting coupling coefficient (κ_{0}).

[0088]
For most of the known 1D SE DFB devices, a relatively high κ_{0 }(of order 10 cm^{−1}) was employed since the output power emitted from the surface nominally scales with this coupling [see FIG. 10)]. However, a large κ_{0 }makes it difficult to select the symmetric mode, since secondary effects cannot overcome such a wide gain margin. The inventors have recognized that by reducing κ_{0 }to <1 cm^{−1}, one can manipulate the relative modal losses such that the inphase mode actually has a lower threshold gain than the outofphase mode.

[0089]
In particular, this can be accomplished with selfpumping and selffocusing (governed by the LEF and the internal loss), because the outofphase mode extends farther into the unpumped regions and consequently experiences higher loss. That effect is enhanced by weaker selffocusing of the antisymmetric mode which is more extended spatially. The inventors in their TDFG simulations have observed that stronger constructive interference of the inphase mode more than compensates for the reduced outcoupling coefficient. Conditions (3) and (4) above are necessary to maintain coherence over a largearea emitting aperture.

[0090]
For example, an approximate upper bound on the normalized efficiency may be written as 6_{κ0}/(α_{i}+2_{κ0}+α_{d}), where α_{d }is the selfconsistent diffraction loss from selfpumping and the prefactor of 6 corresponds to independent contributions by the 6 constructivelyinterfering field components (the prefactor is 12 for coupling into both output directions). The upper bound can never exceed one, since by analogy with 1D SE DFB lasers the inphase operation is obtained only when α_{d}≧12_{κ0}. The prefactor is reduced to 4 for a square lattice, because then there are only four rather than six field components. This expression for the upper bound shows that a seemingly low value of κ_{0 }may nevertheless result in highefficiency operation.

[0091]
The advantages of SE PCDFB lasers employing the SPSF mechanism are reflected in the spectral characteristics of these devices as described below. FIG. 7 shows projected spectral characteristics for typical interband QW devices (LEF=1.5, α=5 cm^{−1}) with D=800 μm at several different Δn. Inphase operation is seen to occur on the longwavelength side of the resonance wavelength λ_{c}. The difference between the lasing frequency and λ_{c }is approximately proportional to Δn, although if the index modulation becomes too large there is a transition to an outofphase mode with wavelength slightly longer than λ_{c}. One may also conclude from FIG. 7 that it should not place any excessive demands on current processing technologies to stay within tolerance for the rather noncritical range of acceptable etch depths.

[0092]
[0092]FIG. 8 illustrates another limit on the spectral purity, imposed by degradation of the mode structure when the LEF is increased. Note that at LEF=2.5 the SMSR drops to 12 dB, while for LEF≧3.5 one cannot avoid having at least two competing modes with similar amplitudes.

[0093]
The general features of the SPSF mechanism as applied to the SE PCDFB are also applicable to many classes of semiconductorlaser active regions emitting at wavelengths from the visible to the farIR. Of these active regions, the preferred embodiments will be those with the LEF ranging from 0.5 to 2 and a low internal loss (<20 cm^{−1}), although the possibility of employing active regions that do not satisfy these requirements is not excluded. In one embodiment, the linewidth enhancement factor is increased to about 1 by varying electronic densities of states.

[0094]
One preferred embodiment of the SE PCDFB laser using the disclosed selfpumping/selffocusing (SPSF) mechanism is optically pumped, since current injection may under some conditions introduce compromises associated with partial blocking of the emitting aperture by electrical contacts. Nevertheless, electrically pumped devices represent one of the alternative embodiments, since they can employ the same SPSF mechanism to operate in an inphase, circular mode. The optimization of electrically pumped SE PCDFB lasers will depend on the degree to which current spreading can be used as a means for maximizing the emission area outside of the contacts (which can be either monolithic, annular, or a mesh grid). The optimization of electrically pumped SE PCDFB lasers will thus be specific to a given material system. A preferred embodiment with electrical pumping would employ transparent contacts, e.g., based on the deposition of indium tin oxide (ITO). In the limit of a completely transparent top contact that coincides with the lateral pumped gain area 109 and does not contribute to the lateral modulation of the refractive index by the grating, the electrical pumping case becomes completely analogous to optical pumping and the design optimization may be performed in the same manner.

[0095]
Although embodiments of the present invention focus on gratings fabricated by regrowth, metallic gratings defined by depositing a metal such as gold into etched holes is also possible. The SPSF mechanism is also applicable to buried gratings with primarily imaginary modulation of the refraction index (gain coupling) or to the general case of complex coupling.

[0096]
Considering the polarization of the output beam, since the output beam emerges nearly normal to the device plane, its polarization may vary in a random manner within that plane depending on small, uncontrollable errors occurring during the fabrication process. However, for some applications it may be useful to induce the output beam to have a fixed specified polarization. In an alternative embodiment, the polarization direction may be fixed by weakly perturbing the etched features to make them asymmetric (e.g., elliptical instead of circular).

[0097]
The output power has thus far been taken to be only half of the light emitted normal to the laser plane, since the other half emitted in the opposite direction is generally not collected. However, blazed gratings or extra reflectors may be used boost the output power by up to a factor of 2. For example, a distributed Bragg reflector 114 (FIG. 1a) grown below the active region 106 (FIG. 1a) can increase the efficiency of an epitaxialsideup mounted device. For epitaxialsidedown mounting, the same can be accomplished by adjusting the thickness of the regrown material to an integer number of halfwavelengths in that material and then coating the surface adjacent to the heat sink with a highreflectivity metal. Other means known to the art may also be used to reflect the beam emitted in one vertical direction such that it is added to the beam emitted in the opposite direction. In all cases, the optical path length must be adjusted so as to assure that the reflected beam constructively interferes with beam emitted in the opposite vertical direction. In one embodiment, a reflector 114 may be disposed at an opposite vertical end with respect to an output beam emitted by the laser apparatus 100 such that reflection of light from the reflector 114 constructively interferes with the output beam.

[0098]
The hexagonal lattice geometry is the preferred embodiment for the TEpolarization case that is the basis for most interband QW lasers. The square lattice cannot maintain lateral coherence for the TEpolarization, since κ_{2}=0 due to symmetry. The reduced symmetry of a rectangular lattice leads to nonzero κ_{2}, which can also be employed in conjunction with the SPSF mechanism, although the results are not expected to be as good as for the hexagonal lattice.

[0099]
The square lattice is the preferred alternative embodiment for lasers whose gain is stronger for TM polarization. This is true for intersubband lasers such as the QCL and also certain heavilystrained QW lasers with a lightholelike valence band maximum. These devices can also operate using the hexagonal and rectangular lattices. The TDFG simulations show that Conditions (1)(5) as described above needed to attain inphase rather than outofphase operation are applicable to the square lattice with TM polarization as well [for which κ_{2}=κ_{3 }in Conditions (3) and (4)], except as noted in those Conditions.

[0100]
FIGS. 9(a)9(b) show nearfield patterns for typical inphase and outofphase modes, respectively. For the outofphase mode in FIG. 9(b) the four field components lead to four lobes in the near field. Condition (3), which requires κ_{1}<0 and κ_{2}<0, leads to an additional constraint that the square lattice cannot also have square etch features. One can show using Eqs. (4) and (5) that symmetry would then lead to at least one coupling coefficient (κ_{1 }for P_{1 }oriented along the diagonal of the primitive unit cell) that is identically nonnegative because it is proportional to the product of two sines having the same argument. However, if circular features with diameter d≈0.70 Λ are used, all five Conditions can be met in the lowest diffraction order that permits surface emission (m=2 in the squarelattice notation as above). This contrasts the hexagonallattice case, in that there it was necessary to go to the secondlowest permissible diffraction order to obtain optimal results. The simulations yield that inphase operation may be expected for any circle diameter in the range 0.68 Λ<d<0.80 Λ, and for etch depths as much as 50% less than the optimal value. These ranges are rather nonrestrictive from the fabrication standpoint.

[0101]
[0101]FIG. 10 illustrates the normalized brightness that can be obtained using d=0.70 Λ and Δn=0.043, for which: κ_{1}=−17 cm^{−1}, κ_{2}=−21 cm^{−1}, and κ_{0}0.74 cm^{−1}. The simulations employed d_{g}/Γ_{g}=4 μm, an appropriate value for a QCL designed for emission at λ_{c}=4.6 μm. For example, the normalized brightness that can be achieved is seen to be quite similar to the results shown in FIG. 4 for TE polarization and a hexagonal lattice. While inphase operation with LEF=0.05 can be obtained when α_{i}≧15 cm^{−1}, for LEF=1.5 it is realized when α_{i}<25 cm^{−1}. Best results are obtained over the full range of internalloss values for LEF=0.5, which can in principle be achieved in welldesigned QW devices, although in lowloss materials LEF=1.5 is slightly preferable. Again, active regions with LEF>2 will display lower normalized brightness.

[0102]
Since most intersubband lasers have LEF<0.1, it may be difficult to design QCL SE PCDFBs that simultaneously optimize both the efficiency and the mode quality. One possibility is to use band structure engineering to alter the densities of states near the lasing transition energy, such that the LEF is increased. For a QCL with LEF=0.1, FIG. 11 plots the normalized brightness as a function of pumpspot diameter. The solid curve employs the optimized etch depth at each D (Δn ranging from 0.056 at D=300 μm to 0.021 at D=2 mm), while the dashed curve assumes a fixed etch depth optimized for D=800 μm (Δn=0.043). The smallest pump spot for which inphase operation can be expected is D≈400 μm, which places high demands on the requirements for current spreading in electrically pumped devices. However, the large transport anisotropy in a QCL (the mobility is much higher in the plane than along the currentinjection axis) is beneficial in this regard. The degradation of the normalized brightness at large pump spots is also much faster for the square lattice than for the hexagonal lattice.

[0103]
[0103]FIG. 12 shows typical spectral characteristics for TMpolarized SE PCDFB lasers employing the squarelattice geometry, where LEF=0.05 and spectra are given for several values of the internal loss. The inventors have found that while singlemode operation with a large SMSR is possible for α=20 cm^{−1}, the SMSR drops to 21 dB when α_{i}=40 cm^{−1}, and the device is essentially multimode with broadening of the main lasing line for α=10 cm^{−1}.

[0104]
The SE 1D DFB lasers may also benefit from the SPSF technique, as disclosed in embodiments of the present invention, for achieving lasing in the inphase mode, which has never been pointed out previously. The conditions (Conditions (1)(5)) that apply to SE 1D DFB lasers are quite similar to those specified above that are required for SE PCDFB lasers. Condition (3) is modified to include only κ_{1}, and Condition (4) is modified to refer to the pumped length rather than beam diameter.

[0105]
[0105]FIG. 13 is a highlevel block diagram of a computer system 130 for implementing the methodology of the present invention in one embodiment. The computer system 130 includes processing circuitry 132, storage device 134, and memory 136. The processing circuitry 132 may be arranged to execute programming instructions (e.g., software, hardware, etc.) to process input data to produce a neardiffractionlimited beam using SE PCDFB. Accordingly, in such arrangement, processing circuitry 132 may be implemented as a microprocessor of a notebook computer, personal computer; workstation or other digital signal processing arrangement. Processing circuitry 132 may also comprise a field programmable gate array or any other hardware and/or software configuration capable of producing desired output data as described above.

[0106]
Memory 136 and processing circuitry 132 are depicted separate from one another. In other possible embodiments, memory 136 and processing circuitry 132 may be embodied within a single device. Memory 136 may be arranged to store digital information and instructions and may be embodied as random access memory (RAM), read only memory (ROM), flash memory or another configuration capable of storing digital information, instructions (e.g., software or firmware instructions utilized by the processing circuitry 132), or other digital data desired to be stored within storage device 134.

[0107]
[0107]FIG. 14 is a methodology for producing a neardiffractionlimited beam using a surface emission photonic crystal DFB laser in one embodiment. At a step S2, the processing circuitry 132 (FIG. 13) is configured to select grating parameters of the 2D grating 110 (FIG. 1a), the grating parameters including for example, grating order, circular feature diameter, and index modulation achieved by etching the grating to a desired depth.

[0108]
At a step S4, coupling coefficients κ_{1}, κ_{2}, κ_{3 }are calculated using the grating parameters.

[0109]
At a step S6, output power and beam quality of the SE PCDFB laser are calculated for a sufficiently large number of time intervals Δt.

[0110]
At a step S8, an inquiry is made to determine if the output power is within predetermined limits. If yes, the method proceeds to step S10. If no, the method proceeds to step S2 where one or more grating parameters are varied and the method is iterated.

[0111]
At a step S10, an inquiry is made to determine if the beam quality is within a desired range of diffraction limit. If no, the method proceeds to step S2.

[0112]
In compliance with the statute, the invention has been described in language more or less specific as to structural and methodical features. It is to be understood, however, that the invention is not limited to the specific features shown and described, since the means herein disclosed comprise preferred forms of putting the invention into effect. The invention is, therefore, claimed in any of its forms or modifications within the proper scope of the appended claims appropriately interpreted in accordance with the doctrine of equivalents.