Electrically pumped, solid-state distributed feedback laser with particular grating spacing
BACKGROUND OF THE INVENTION 
     Solid-state laser devices have been suggested for use in integrated optical circuits. One such laser that can be electrically pumped is described in copending U.S. Patent Application Ser. No. 499,671, filed Aug. 22, 1974, and entitled &#34;Electrically Pumped, Solid-State Distributed Feedback Laser.&#34; In that laser a grating or physical periodic structure is provided in, or adjacent to, a light wave guide layer. The spacing of the perturbations of the periodic structure are selected to be an integer number of half wavelengths of the desired light frequency within the laser, such that the perturbations produce Bragg Scattering which couples and reinforces right and left light waves traveling through the light guiding layer in a coherent manner such that reflections are in phase, thus allowing laser operation in the absence of discrete end mirrors. The degree to which the right and left going waves interact with the perturbations is described mathematically by a coupling constant, K. The magnitude of this constant affects the length of the laser gain region and/or the value of gain required for laser operation with larger values of K corresponding to shorter lengths L and/or lower gains. 
     As noted in the aforementioned application, the spacing of the perturbations of the periodic structure is calculated by utilizing the wavelength of the light frequency desired in free space, i.e., outside of the laser device, in accordance with the reflection formula Λ = m λ o  /2n where Λ is the spacing of the periodic structure, m is the Bragg diffraction order, λ o  is the free space lasing wavelength, and n is the refractive index of the light guiding layer. With the formulated spacing, laser operation is often difficult to achieve. The reason for this difficulty is believed to reside in the fact that the gratings usually do not produce large values of the coupling constant for the lowest order transverse mode. That mode, which is most tightly confined to the light guiding layer, has a propagation constant and wavelength within the laser, which is aproximately λ o  /n and therefore satisfies the reflection formula. However, in structures utilizing trapped or confined waves, such as the light guiding layer of a DFB laser, transverse modes exist which have propagation constants differing greatly from free space values. These higher order transverse modes have a higher coupling constant with the periodic structure and thus, they will lase easily, compared with the lowest order transverse mode. 
     It is also desirable in many applications to have a single mode output from a distributed feedback laser. Single mode operation is difficult to effect if the transverse mode spacing is not comparable to the spectral width of the gain of the laser. 
     OBJECTS OF THE INVENTION 
     It is therefore an object of the invention to provide an improved distributed feedback laser. 
     It is a further object of the present invention to optimize the spacing of the periodic structure of a distributed feedback laser. 
     SUMMARY OF THE INVENTION 
     In accordance with the invention, the foregoing objects are achieved by selecting the periodicity of the periodic structure to optimize the value of the coupling constant, K. This optimization will increase the interaction between the left and right going waves in the light guiding layer, thereby reducing the length of gain region and/or the value of gain needed for laser operation. Also, by regulating the coupling constant, single, higher order mode operation is more easily obtained since higher order modes have substantially larger coupling constants than do lower order modes and can be generated using grating periodicities which will not support adjacent modes. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a cross-sectional view of a single heterojunction SH DFB laser according to the invention. 
     FIG. 2 is a plot illustrating the relationship of propagation constant and guide wavelength. 
     FIG. 3 is a cross-sectional view of a double heterojunction DH DFB laser according to the invention. 
     FIG. 4 is a plot of grating height v. coupling constants for different modes of the double heterojunction structure specified. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     As noted, in a distributed feedback laser it is important to optimize the periodicity of the periodic structure since this structure serves to couple the right and left going waves in the light guiding layer. To achieve this optimization the coupling coefficients and the propagation constants of a particular laser mode must be calculated. The coupling constants, denoted by K, determine the net gain and/or length of the laser structure required to initiate lasing, and the propagation constants, denoted by B, determine the transverse mode separation and, more importantly, the required periodicity of the grating structure of the distributed feedback laser. 
     Reference is now made to FIG. 1 which shows a single heterojunction distributed feedback laser. The periodic perturbation is provided by the grating 1 of the p-type Ga Al As region 2 extending into the active gain p-type Ga As region 3 defined on the other side by n-type Ga As region 4. The refractive indicies of the regions 2, 3 and 4 can be 3.4, 3.6 and 3.58, respectively. The height g and width w of the grating teeth are shown in FIG. 1, as is the thickness τ of region 3. 
     To determine the optimal periodicity, the propagation constants B are calculated for the modes that can propagate in the device of FIG. 1. The calculations of B are in accordance with those of A. Yariv (see IEEE J. Quantum Electronics, QE-9,919 (1973) ). According to Yariv, the field component of the TE modes in the device of FIG. 1 obeys the wave equation ##EQU1## and applying equation (1) to the regions of the device of FIG. 1 using boundary conditions set forth by Yariv, yields equations 
     
         h = (n.sub.2.sup.2 k.sup.2 - B.sup.2).sup.1/2              (1a) 
    
     
         q = (B.sup.2 - n.sub.1.sup.2 k.sup.2).sup.1/2              (1b) 
    
     
         p = (B.sup.2 - n.sub.3.sup.2 k.sup.2).sup.1/2              (1c) 
    
     and 
     
         tan (ht) = h (p+q)/(h.sup.2 -pq),                          (1d) 
    
     where k=W/c. The propagation constants B of the TE modes of the device of FIG. 1 are obtained by solving equations (1a)-(1d). As can be seen from equations (1a)- (1d), the three region structure of FIG. 1, without grating considerations, determines the values of B. Thus, for a given frequency W each mode has a particular guide wavelength. For example, in Table I, the value of B for different TE modes in a single heterojunction diode with n, = 3.4, n 2  = 3.6, and n 3  = 3.58 as a function of thickness t for W = 2.2176 × 10 15  rad/sec (which corresponds to a free-space wavelength of λ o  = 8500 A) are listed, with the values computed by solving equations (1a)-(1d) by conventional methods. 
     To determine λ g  the wavelength for each mode in the guiding layer 3, we look to the propagation of the lasing waves, as shown in FIG. 2, which has a real component according to the relationship 
     
         Re{e.sup.iBZ e.sup.-.sup.iwt } = cos (BZ-wt).              (2) 
    
     Since, when Bz changes by 2π the cosine function repeats itself, which is the definition of a wavelength, i.e. z has changed by λg when BZ changes by 2π, we have 
     
         Bλ.sub.g = 2π.                                   (3) 
    
     From equation (3) we get the family of relationships between B.sup.(n) and λ g .sup.(n) as follows: B.sup.(1) =  2π /λ g .sup.(1) , B.sup.(2) = 2π /λ g .sup.(2) , B.sup.(3) = 2π /λ g .sup.(3), B.sup.(4) = 2π /λ g .sup.(4), B.sup.(5) = 2π /λ g .sup.(5) and so on where λ g .sup.(n) is the light wavelength in the light guiding layer. Since 
     
         λg = 2Λ/m                                    (4) 
    
     where m is the Bragg diffraction order, and Λ is the periodicity, equations (3) and (4) yield 
     
         Λ = πm/B,                                        (5) 
    
     which is the required periodicity of the grating of the device of FIG. 1 where m is the Bragg diffraction order. The values of λ g  and Λ for various modes, various thicknesses (t) of the guiding layer 2 and λ o  = 8500 A are given in Table I. 
     Considering now a double heterojunction diode with n 1  = 3.4, n 2  = 3.6, and n 3  = 3.4 and t = 2μm, as shown in FIG. 3. Equations (1a)- (1b) are uses to solve for values of B. Values of λ g  and Λ are then calculated for modes 1-6 using equations (3) and (4) for λ o  = 8500 A, as shown in Table II. 
     
                       TABLE I                                                     
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       Mode                                                               
t(μm)                                                                  
       Number   B (μm.sup.-.sup.1)                                     
                           λg (μm)                              
                                    Λ(A)                           
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0.5    1        26.466     0.2374   3561                                  
1.0    1        26.530     0.2368   3552                                  
1.5    1        26.564     0.2365   3548                                  
2.0    1        26.581     0.2364   3546                                  
       2        26.497     0.2371   3557                                  
2.5    1        26.590     0.2363   3544                                  
       2        26.530     0.2368   3552                                  
3.0    1        26.596     0.2362   3544                                  
       2        26.551     0.2366   3550                                  
       3        26.481     0.2373   3559                                  
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 TE MODE PROPAGATION CONSTANTS IN A SH DIODE FOR VARIOUS LAYER THICKNESS  
 AND λo = 8500A.                                                   
 
    
     
                       TABLE II                                                    
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Mode                                                                      
Number  B (μm.sup.-.sup.1)                                             
                    λg (μm)                                     
                                Λ(A)                               
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1       26.574      0.2364      3547                                      
2       26.462      0.2374      3562                                      
3       26.276      0.2391      3587                                      
4       26.017      0.2415      3623                                      
5       25.691      0.2446      3669                                      
6       25.314      0.2482      3723                                      
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 TE MODE PROPAGATION CONSTANTS IN A DH DIODE FOR t = 2μm λo =   
 8500A.                                                                   
 
    
     Clearly, for particular values of Λ each of the transverse modes resonate at a different free-space wavelength λ o . Since the actual grating period Λ is fixed in a particular laser, it is important to compute λ o  given Λ and the Bragg diffraction order m. Values of λ o  for three different grating spacing, i.e., Λ = 3547A, Λ = 3623A, and Λ = 3669A are listed in Table III. The values of λ o .sup.(i) are arrived at as follows: 
     1. Assume a trial value of the free-space wavelength λ o  denoted by λ o .sup.(t). A good trial value is given by 
     
         λ.sub.o.sup.t = 2 Λ n.sub.2 /m. 
    
     2. Using λ o   t  calculated the transverse mode propagation constants from Eqs. (1a)-(1d). Denote the constant for the i th  mode by B.sup.(i). 
     3. Calculate the guide wavelength 
     
         λ.sub.g.sup.(i) = 2π /B.sup.(i) 
    
     4. Since λ g .sup.(i) is the guide wavelength having assumed a free-space wavelength λ o   t  and since the actual guide wavelength must be almost exactly 2Λ/m, the actual free-space wavelength of the i th  transverse mode λ o .sup.(i) is given by ##EQU2## 
     5. To obtain greater accuracy one can repeat steps (2)-  (4) with λ o   t  = λ o .sup.(i), but we usually find this to be unnecessary as is illustrated by the example below. 
     Example 
     Let Λ = 3500A and m = 3 for the DH structure shown in FIG. 3 with t = 2μm, n 1  = n 3  = 3.4 and n 2  = 3.6. 
     1. Assume λ o   t  = 8500A. Note that 2Λn 2  /m ≈ 8400A. 
     2. The calculation gives B.sup.(1) = 26.57381 μm as listed in Table II with only 26.574 retained. 
     3. λ g .sup.(1) = 2π/26.57381 = 2364.4277 A. 
     4. Since 2Λ/m = 2333.3333 A. ##EQU3## 
     5. Set λ o   t  = 8388.2A 
     2&#39;. The calculation gives B.sup.(1) = 26.9288μm 
     3&#39;. λ g .sup.(1) = 2333.2516 A ##EQU4## which is not a significant change in the calculated value of λ o .sup.(1). 
     Referring to Table III, we note that for Λ = 3623 A the modes adjacent to the ones resonant at λ o  = 8500 A are 8584 A and 8393 A which are shifted so far from the main resonant frequency that they are outside of the spectral width of the gain and thus experience substantially reduced net gain. The same is true for the modes adjacent λ o  = 8500 A for Λ = 3669 A. Since the higher modes resonant at greater frequency separation than the lower order modes, and the adjacent modes are outside the spectral width of the gain, single mode operation is more easily obtained. 
     
                       TABLE III                                                   
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Mode     λ.sub.o (A)                                               
                     λ.sub.o (A)                                   
                                 λ.sub.o (A)                       
Number   Λ=3547A                                                   
                     Λ=3623A                                       
                                 Λ=3669A                           
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1        8500        8682        8792                                     
2        8464        8645        8755                                     
3        8405        8584        8694                                     
4        8322        8500        8608                                     
5        8218        8393        8500                                     
6        8097        8270        8375                                     
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 FREE-SPACE WAVELENGTHS FOR VARIOUS GRATING PERIODS IN A DH STRUCTURE.    
 
    
     Regarding single mode operation, reference is had to FIG. 4 which is a plot of K (the coupling coefficient) vs. grating height for propagated modes of a double heterojunction diode geometry with n 1  =n 2  =3.4, n 2  =3.6, τ= 2mm, W=875A, and λ = 3500A. The coupling constant K is calculated for a rectangular grating using the equation ##EQU5## Generally K increases with mode number, which reflects the fact that the higher modes have larger relative amplitudes in the vicinity of the grating and hence interact more with the grating. Also K increases with the grating height, g; however, when g approximates the zero of a particular mode d K/d g = O. This occurs for the 6th mode at g ≈ 2500A; K does increase for that mode with further increases in g  and in fact K 6  exceeds K 5  for g≈4500A. Clearly, K 4 , K 5 , and K 6  are substantially larger than K 1  or K 2  for small value of g. For example, for g = 1500A, K 4  and K 5  are over an order of magnitude greater than K 1 . Thus, referring to Table II, a double heterojunction diode laser having t = 2μm and λ o  = 8500 A will have a substantially lower threshold with Λ = 3623 A or 3669 A then with Λ = 3547A since the coupling coefficients are greater for the latter spacings than for the former spacing. Thus, it is shown for TE modes that a particular mode will oscillate at a lower pumping threshold if the spacing of the grating is chosen to optimize oscillations and the coupling coefficient of that mode. 
     Identical calculations have been carried out for TM modes with very similar results. Generally, K for TM modes is slightly smaller than that for corresponding TE modes, but the differences are not significant. It should also be noted that the foregoing calculations are based on perturbations rather than an exact solution of the boundary value problem (with grating present). 
     In conclusion it has been shown that higher order transverse modes in guided wave structures often have much larger coupling coefficients than do lower modes. Grating spacings required to resonate the higher order transverse modes have been calculated and these modes were shown to have large separations in frequency, thus facilitating single mode operation. The results indicate that it is often desirable to fabricate the DFB grating of a guided wave laser at a period which differs substantially from that required to resonate the lowest order mode, and calculated in accordance with the standard formula Λ = mλ/2n.