Lateral Modes in Quantum Cascade Lasers - MDPI

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Lateral Modes in Quantum Cascade Lasers Gregory C. Dente 1, * and Michael Tilton 2 1 2

*

GCD Associates, Albuquerque, NM 87110, USA The Boeing Company, Albuquerque, NM 87117, USA; [email protected] Correspondence: [email protected]; Tel.: +1-505-268-8337

Received: 18 February 2016; Accepted: 18 March 2016; Published: 22 March 2016

Abstract: We will examine the waveguide mode losses in ridge-guided quantum cascade lasers. Our analysis illustrates how the low-loss mode for broad-ridge quantum cascade lasers (QCLs) can be a higher-order lateral waveguide mode that maximizes the feedback from the sloped ridge-wall regions. The results are in excellent agreement with the near- and far-field data taken on broad-ridge-guided quantum cascade lasers processed with sloped ridge walls. Keywords: quantum cascade lasers; ridge-guided; lateral modes

1. Introduction Quantum cascade lasers (QCLs) are being used in applications requiring mid-infrared sources. For many of these applications, narrow-ridge devices can provide adequate power and beam quality. For applications requiring higher powers, broadening the ridge has led to higher-order lateral modes. These effects are displayed in recent experiments on broad-ridge quantum cascade lasers [1–3]. These near- and far-field studies of QCLs processed with sloping ridge walls have shown that as the ridge widths are increased, the lasing power is concentrated in just a few higher-order lateral waveguide modes. Consequently, the lasing lateral far-field forms a dual-lobed pattern that can remain relatively stable with nearly fixed divergence angles, independent of ridge width [2]. This paper will attempt to explain the dual-lobed far-field data taken on broad-ridge QCLs [2]. Many features could contribute to the dual-lobed far-fields. For instance, the sloping ridge walls vary with the processes used for the device fabrication. Factors such as dry- or wet-etch, wall slope and wall roughness, as well as the optical properties of the bounding regions, could all figure into a detailed analysis [4]. Here, we concentrate on one possible explanation. For our primary approximation, we analyze an ideal trapezoidal waveguide model, in which the top side has width W ´ B and the bottom side has width W ` B; the top and bottom are separated by a perpendicular distance, D. Figure 1 Photonics 2016, 3, 11 2 of 5 illustrates this ideal model for the QCL waveguide.

Figure 1. Trapezoidal waveguide model. Figure 1. Trapezoidal waveguide model.

2. Perturbation Approach to the Trapezoidal Waveguide Modes Photonics 2016, 3, 11; doi:10.3390/photonics3010011

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The ideal trapezoidal waveguide is difficult to analyze exactly. Therefore, we will use an approximate approach in which the trapezoidal waveguide is treated as a perturbation from the ideal rectangular waveguide shown in Figure 2.

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Are we justified in replacing the complicated reality of the sloped ridge side walls with the ideal trapezoidal waveguide of Figure 1? Here, we will show that the lowest-loss cavity modes for the trapezoidal waveguide can explain the two-lobed far-field data [2]. In particular, the lateral extent of the sloped ridge-wall regions, B, proves to be the critical parameter. It determines the lowest-loss mode as a function of ridge width, and most importantly, demonstrates that the angular separation of the two lobes in the far-field is essentially independent of ridge width for the broad-ridge QCLs. Figure 1. Trapezoidal waveguide model. 2. Perturbation PerturbationApproach Approachtotothe theTrapezoidal TrapezoidalWaveguide WaveguideModes Modes 2. The ideal ideal trapezoidal trapezoidal waveguide waveguide isis difficult difficult to to analyze analyze exactly. exactly. Therefore, Therefore, we we will will use use an an The approximate approach in which the trapezoidal waveguide is treated as a perturbation from the approximate approach in which the trapezoidal waveguide is treated as a perturbation from the ideal rectangular rectangular waveguide waveguide shown shown in in Figure Figure 2. 2. ideal

Figure 2. Rectangularwaveguide waveguide model. model. Cross-hatched Cross-hatched triangles triangles carry carry perturbation perturbation Figure 2.` Rectangular ˘ p2π{λq22 ¨ n22b ´ n22w . 2   nb  nw .









This rectangular thethe electric field field polarized in the epitaxial growth direction This rectangularwaveguide waveguidepropagates propagates electric polarized in the epitaxial growth (y-axis) between the back and front facets of the QCL. The ideal, unperturbed, rectangular waveguide direction (y-axis) between the back and front facets of the QCL. The ideal, unperturbed, rectangular modes are taken either waveguide modesasare taken as either sinpπyy{Dq cos UNUpx,  xyq , y “ sin D  cos  NpN  xπ x{ WpW  B`  Bqq N

(1)(1)

in in which which N N isisan an odd odd integer integer for for the the symmetric symmetric lateral lateral modes modes while while

 xyq , y “ sin D  sin  NpN  xπ x{ WpW  B`   Bqq N UNUpx, sinpπyy{Dq sin

(2)(2)

in which N is an even integer for the anti-symmetric lateral modes [5]. For either case, the in which N is an even integer for the anti-symmetric lateral modes [5]. For either case, the unperturbed unperturbed z-component of the propagation vector is z-component of the propagation vector is

 0 N b  2 n W  2   D2  N  W  B 2 .

(3) β 0 pNq “ p2 π n W {λq2 ´ pπ{Dq2 ´ pN π { pW ` Bqq2 . (3) This form of ideal rectangular waveguide mode satisfies the wave equation in the guide region This formindex of ideal waveguide the waveyequation in the nw rectangular x  guide  W region B 2 . with complex . Also, the modes equal mode zero atsatisfies the boundaries, = 0, D and with complex index n . Also, the modes equal zero at the boundaries, y = 0, D and x “ ˘pW ` Bq {2. w We could use a more exact form of mode in the y-direction, but as we will show, it has only a modest We could use calculations a more exact that formfollow. of mode in the y-direction, but as we show, it has onlymodes a modest effect on the Finally, we are assuming thatwill these waveguide all effect on the calculations that follow. Finally, we are assuming that these waveguide modes all operate operate coherently at a single wavelength, λ. coherently at a single wavelength, The propagation constant forλ.the modes in a trapezoidal waveguide can be estimated from The propagation first-order perturbationconstant theory asfor the modes in a trapezoidal waveguide can be estimated from first-order perturbation theory as » ´

β 2 pNq « β 0 2 pNq ` p2 π{λq 2 nb 2 ´ nW 2

¯ ¨–

x

triangles

UN 2 {

x

rectangle

fi UN 2 fl

(4)

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

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2

N

 0

2

 N    2



2

n

2 b

 nW

2

    U 

triangles

2 N

 U rectangle

2 N

 

(4) 3 of 5

in which U N x, y  and  0 N  are the unperturbed rectangular waveguide results. We emphasize inthat which and β 0 pNq are unperturbed rectangular waveguide We numbers, emphasizesothat theUindices refraction forthe both the waveguide and the barrier areresults. complex that N px, yqof the refraction for both the waveguide and the barrier are complex numbers, so that the theindices overallofperturbation is complex. overallThe perturbation complex. integral inisthe numerator is over the triangular cross-hatched regions shown in Figure 2, Thethe integral in the numerator is over theistriangular cross-hatched regions shown cross-section. in Figure 2, while normalization in the denominator over the entire rectangular waveguide A final afterincompleting the lateral integration, the first-order perturbation theory while theevaluation, normalization the denominator is over the entiregives rectangular waveguide cross-section. as Aestimate final evaluation, after completing the lateral integration, gives the first-order perturbation theory estimate as   N     N    2    n  n   ` ˘ β 2 pNq « β 0 2 pNq ` p2 π{λq 2 nb 2 ´ nW 2 ¨ (5) şD (5) 2   y D   y sin  f B y D   f B D   4B D  pW D` WBqq  B¨   dy d y sin pπ y{Dq ry ´ sin p f B y{Dq{ p f B {Dqs 4 pB{Dq{ pD sin 0  2

2

2

0

2

2

b

D

W

2

0

W `BBq.  . The whichf “f 22NNπ{ pW as well well as asthe thecomplex complexindices indicesofofrefraction refractionfor for ininwhich The parameters, parameters, D, W, B, as the are shown inin the figures. thewaveguide waveguideand andthe thebarrier, barrier, are shown the figures.Note Notethat thatthe theperturbation perturbationchanges changesboth boththe the real parts ofof the propagation constant forfor the N-order mode. realand andimaginary imaginary parts the propagation constant the N-order mode. 3.3.Examples Examplesof ofQCL QCLCavity CavityMode ModeCalculations Calculations The dual-lobed far-field datadata takentaken on broad-ridge QCLs isQCLs remarkable [2]. All the devices radiated The dual-lobed far-field on broad-ridge is remarkable [2]. All the devices ˝ as shown in Figure 3. Here, W defines the device 4.45 micron radiation into a pair of lobes at ˘38 radiated 4.45 micron radiation into a pair of lobes at ±38°as shown in Figure 3. Here, W defines the stripe width and n indicates the modethe number. device stripe width and n indicates mode number.

Figure 3. Far fields (Reference [2]) of broad area QCLs measured at both low (1.1 ˆ Ith ) and high Figure 3. Far fields (Reference [2]) of broad area QCLs measured at both low (1.1 × Ith) and high (2 × Ith) (2 ˆ Ith ) current in pulsed mode operation at room temperature. current in pulsed mode operation at room temperature.

Additionally, Additionally,while whilethe theindividual individuallobes lobeswere werenot notdiffraction-limited, diffraction-limited,it itisisobvious obviousthat thatonly onlya a small behavior was was observed, observed,independent independentof smallset setofofhigher-order higher-orderlateral lateralmodes modeswere wereoperating. operating. This behavior ofridge-width, ridge-width,from from50 50 micron micron wide wide devices devices out to 400 400 microns. microns. Here, Here, we we will willtry trytotoexplain explainthis this behavior perturbation result inin Equation (3). behaviorusing usingthe thefirst-order first-order perturbation result Equation (3). First, First,from fromEquation Equation(3), (3),we weobserve observethat thata astationary stationaryvalue valueofofthe theperturbation perturbationintegral integraloccurs occurs pW ` Bq, when f B{D “ 2 π {D; at this value of f “ 2 N π { the second term in the integrand integrates when f B D  2 D ; at this value of f  2 N  W  B , the second term in the integrand integrates totozero. « pW W`Bq B{B, B ,in zero.This Thiscorresponds correspondstotoN N inwhich whichN, N,the thelateral lateralmode modeorder, order,must mustbe bean aninteger. integer. Therefore, the perturbation solution suggests that the lowest-loss trapezoidal waveguide mode is Therefore, the perturbation solution suggests that the lowest-loss trapezoidal waveguide mode is of

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relatively high-order when W > B. We can insert this mode-order value, N  W  B B into the unperturbed mode expression of relatively high-order when as W > B. We can insert this mode-order value, N « pW ` Bq {B into the unperturbed mode expression as (6) U N  x, y   sin   y D  sin  N  x W  B    sin   y D  sin  x B  so that the lowest-loss waveguide dual-lobed far-field at lateral “ sin p π y{Dq sin pN πmodes x{pW `propagate Bq q « sinto p πay{Dq sin pπ x{Bq (6) UN px, yqtrapezoidal

angles given by sin     2B ; for λm and    38.5 , we calculate the lateral width of so that the lowest-loss trapezoidal waveguide modes propagate to a dual-lobed far-field at lateral the sloping ridge side-walls as B = 3.57 m . This value seems reasonable for the processing used on angles given by sinθ “ ˘λ{ p2Bq; for λ = 4.45 µm and θ “ ˘ 38.5o , we calculate the lateral width of the the Reference [2] devices. Furthermore, as long as we select the lowest-loss trapezoidal waveguide sloping ridge side-walls as B = 3.57 µm . This value seems reasonable for the processing used on the mode, these dual-lobed far-field angles and the near-field intensity periodicity do not depend on Reference [2] devices. Furthermore, as long as we select the lowest-loss trapezoidal waveguide mode, ridge width. Finally, we have defined the mode order at the base of the trapezoid. If we shift our these dual-lobed far-field angles and the near-field intensity periodicity do not depend on ridge width. reference to the mode order at the center layer, y = D/2, then N = W/B. For the Reference [2] devices, Finally, we have defined the mode order at the base of the trapezoid. If we shift our reference to the this gives perfect agreement with the data in Figure 3, with mode orders N = 14, 28, 56, 84 and 112 at mode order at the center layer, y = D/2, then N = W/B. For the Reference [2] devices, this gives perfect B = 3.57 m. agreement with the data in Figure 3, with mode orders N = 14, 28, 56, 84 and 112 at B = 3.57 µm. o

4.4.Conclusions Conclusions The result, Equation Equation(3), (3),gives givessurprisingly surprisinglygood goodagreement agreement with data The perturbation perturbation theory theory result, with thethe data on on broad-ridge QCLs. Normally, first-order perturbation theory is most reliable when we are broad-ridge QCLs. Normally, first-order perturbation theory is most reliable when we are evaluating evaluating small perturbations. The achange fromwaveguide a rectangular waveguidewaveguide to a trapezoidal small perturbations. The change from rectangular to a trapezoidal does not, waveguide does not, necessarily, seem that small. But perhaps for the broad-ridge the effects necessarily, seem that small. But perhaps for the broad-ridge QCLs, the effects at QCLs, the sloping ridge at the sloping ridge walls can be viewed as only a small part of the overall waveguide, so that the walls can be viewed as only a small part of the overall waveguide, so that the perturbation is of order perturbation is of order B/W; this appears to be the case here. An additional element of support can B/W; this appears to be the case here. An additional element of support can be developed from the be developed from the two-slab approximation to the this trapezoidal waveguide; this two-slab approximation to the trapezoidal waveguide; is illustrated in Figure 4. is illustrated in Figure 4.

Figure4.4.Two Twoslab slabapproximation. approximation. Figure

We the trapezoid trapezoid with with aa slab slab of of width width(W (W ´ − B) (W ++ B). We approximate approximate the B) centered centered over over aa slab slab of of width width (W B). We thentake takethe the lateral mode functions as sin(Px) or cos(Px) and at boundaries, the lateral We then lateral mode trialtrial functions as sin(Px) or cos(Px) and force zeroforce at thezero lateral W xB“ ˘pW  2 . The x 2 and´xBq  {2.  WThe  Blateral boundaries, lateral boundary thewidths two slab x “ ˘pW ` Bq {2and boundary conditionsconditions for the twofor slab are

 with W{B; Bthis  B ;isthis P `NBq  with W  BN widths arewhen satisfied the same lateral mode order satisfied P “when Nπ{ pW « pWN`Bq theissame lateral mode order that resulted fromfrom the perturbation theory results in Equation (3). (3). that resulted the perturbation theory results in Equation appearsthat thatfeedback feedbackfrom fromthe thesloping slopingridge ridgewalls wallsisisenhanced enhancedfor forhigher-order higher-order lateral lateral modes. modes. ItItappears This result was suggested from both first-order perturbation theory as well as from the two-slab This result was suggested from both first-order perturbation theory as well as from the two-slab approximation.AAlowest-loss lowest-losslateral lateral QCL QCL mode mode of of order order N excellent agreement agreement  B{Bisisinin excellent N«  pW W `BBq approximation. with the nearand far-field data on broad-ridge QCLs [2]. Most importantly, the resulting dual-lobed with the near- and far-field data on broad-ridge QCLs [2]. Most importantly, the resulting far-field angles and angles the near-field periodicity do not depend on ridge width. dual-lobed far-field and theintensity near-field intensity periodicity do not depend on ridge width. Author Contributions: G. Dente conceived the analysis and wrote the manuscript. M. Tilton provided the Author Contributions: G. Dente conceived the analysis and wrote the manuscript. M. Tilton provided the comparison of theory and experiment as well as manuscript preparation support. comparison of theory and experiment as well as manuscript preparation support. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest.

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