Patent Publication Number: US-11048044-B2

Title: Collimation device

Description:
This application is a national stage filing under 35 U.S.C. 371 of International Patent Application Serial No. PCT/FR2017/053672, filed Dec. 18, 2017, which claims priority to French patent application FR16/63501, filed Dec. 29, 2016. The entire contents of these applications are incorporated herein by reference in their entireties. 
     BACKGROUND 
     The present invention generally concerns optoelectronic circuits made up of semiconductor materials and methods of manufacturing the same. The present invention more specifically relates to optoelectronic circuits comprising a collimation device. 
     DISCUSSION OF THE RELATED ART 
     An optoelectronic circuit is generally intended to be coupled to an external system, for example, an optical fiber or another optoelectronic circuit. The optoelectronic circuit should then emit a light beam which is received by the external system. For this purpose, the optoelectronic circuit generally comprises a collimation device which enables to provide a collimated light beam and which further enables to adapt the size of the emitted light beam according to the external system having the optoelectronic circuit coupled thereto. A collimated light beam is a beam having substantially parallel rays so that the beam only has a low divergence. 
     An example of a collimation device comprises a point-shaped silicon element housed in a cladding. Such a type of collimation device is described in the publication entitled “Cantilever couplers for intra-chip coupling to silicon photonic integrated circuits” by Peng Sun and Ronald M. Reano (Mar. 16, 2009/Vol. 17, No. 6/OPTICS EXPRESS 4565). 
     A disadvantage of such a collimation device is that it requires the manufacturing of a very fine point having dimensions which should be accurately obtained, given that a variation of the dimensions of the point may significantly disturb the operation of the collimation device. It may however be difficult to precisely and reproducibly manufacture a very fine point with specific dimensions at an industrial scale. Further, for certain applications, a collimation device comprising a point-shaped element may not enable to sufficiently collimate the light beam emitted by the optoelectronic circuit. 
     SUMMARY 
     An object of an embodiment is to overcome all or part of the disadvantages of the previously-described collimation devices. 
     Another object of an embodiment is for the collimation device to deliver a substantially collimated light beam. 
     Another object of an embodiment is for the collimation device to have a simple structure. 
     Another object of an embodiment is to be able to form the collimation device at an industrial scale with conventional integrated circuit manufacturing technologies. 
     Thus, an embodiment provide a light beam collimation device comprising a monomode waveguide, a first element of collimation of the light beam parallel to a first plane and a second element of collimation of the light beam parallel to a second plane, the first collimation element coupling the waveguide to the second collimation element. 
     According to an embodiment, the first collimation element comprises a body coupled at a first end to the waveguide and coupled at a second end to the second collimation element and having a dimension along a first direction perpendicular to the first plane increasing from the first end to the second end. 
     According to an embodiment, the second collimation element comprises a refraction index at the wavelength of the light beam which, along a second direction perpendicular to the second plane, increases from a first value n g  to a second value n c  and then decreases from the second value to the first value. 
     According to an embodiment, the refraction index at the wavelength of the light beam of the second collimation element varies along the second direction at least partly according to a parabolic law. 
     According to an embodiment, the refraction index at the wavelength of the light beam of the second collimation element, along the second direction, increases from the first value to the second value, comprises a plateau at the second value, and then decreases from the second value to the first value. 
     According to an embodiment, the refraction index at the wavelength of the light beam increases from the first value to the second value in a first portion slower than a parabolic law and then, in a second portion, faster than the parabolic law. 
     According to an embodiment, the second collimation element has a half-height H along the second direction and has a length L 4  along a third direction parallel to the first plane and to the second plane, length L 4  being provided by the following relation: 
     
       
         
           
             
               L 
               4 
             
             = 
             
               
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     According to an embodiment, the second collimation element comprises at least one alloy having its composition varying along the second direction. 
     According to an embodiment, the second collimation element comprises a stack along the second direction of a plurality of layers of materials having different refraction indexes at the wavelength of the light beam. 
     According to an embodiment, the second collimation element comprises an alternation of first and second layers, each first layer being made of a first material having a first refraction index at the wavelength of the light beam and each second layer being made of a second material having a second refraction index at the wavelength of the light beam. 
     According to an embodiment, the thicknesses of the first layers are not identical and the thicknesses of the second layers are not identical. 
     An embodiment also provides a method of manufacturing the collimation device such as previously defined, comprising the successive steps of: 
     forming a first portion of the second collimation element; 
     forming the waveguide and the first collimation element; and 
     forming a second portion of the second collimation element. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The foregoing and other features and advantages will be discussed in detail in the following non-limiting description of specific embodiments in connection with the accompanying drawings, in which: 
         FIGS. 1 and 2  respectively are a top view and a cross-section view, partial and simplified, of an embodiment of a collimation device comprising first and second collimation elements; 
         FIG. 3  shows a curve of the variation of the refraction index in an embodiment of the second element of the collimation device shown in  FIGS. 1 and 2 ; 
         FIGS. 4A, 5A, and 6A  show cross-section views similar to  FIG. 2  of collimation devices respectively used for first, second, and third simulations, and  FIGS. 4B, 5B, and 6B  show isoline maps of the component along direction (Oy) of the magnetic excitation field respectively obtained for the first, second, and third simulations; 
         FIGS. 7 and 8  illustrate two applications of the collimation device shown in  FIGS. 1 and 2 ; 
         FIGS. 9, 10, and 11  are cross-section views similar to  FIG. 2  illustrating three embodiments of the second collimation element of the collimation device; 
         FIGS. 12A to 12H  are partial simplified cross-section views of structures obtained at successive steps of another embodiment of a method of manufacturing the collimation device shown in  FIGS. 1 and 2 ; and 
         FIGS. 13A and 14A  show curves of the variation of the refraction index in an element of the collimation device respectively for fourth and fifth simulations and  FIGS. 13B and 14B  show isoline maps of the component along direction (Oy) of the magnetic excitation field respectively obtained for the fourth and fifth simulations. 
     
    
    
     DETAILED DESCRIPTION 
     The same elements have been designated with the same reference numerals in the different drawings. For clarity, only those steps and elements which are useful to the understanding of the described embodiments have been shown and are detailed. In particular, the possible structures of an optoelectronic circuit are well known by those skilled in the art and will not be described in detail hereafter. The terms “approximately”, “substantially”, and “in the order of” are used herein to designate a tolerance of plus or minus 10%, preferably of plus or minus 5%, of the value in question. 
     In the following description, an orthonormal reference frame (Oxyz) is considered. In an embodiment, the collimation device is formed by a stack of semiconductor layers along a stacking direction corresponding to direction (Oz). 
       FIGS. 1 and 2  respectively are a top view and a cross-section view, partial and simplified, of an embodiment of a collimation device  5 . Collimation device  5  forms part of an optoelectronic circuit  10  which is only partially shown in the drawings. Optoelectronic circuit  10  may in particular comprise at least one optical signal generation device, for example, a laser source, optical signal transmission devices, optical signal processing (modulation, amplification) devices, and/or optical signal conversion devices. A substrate  11  having collimation device  5  formed thereon has been schematically shown in  FIG. 2 . 
     Collimation device  5  enables to emit a collimated light beam  12  having rays substantially parallel to direction (Ox).  FIG. 2  shows rays  13  of beam  12 . Light beam  12  may be a monochromatic or polychromatic beam. Light beam  12  is preferably substantially monochromatic. When beam  12  is polychromatic, the wavelength of the beam may take different values over a wavelength range. The wavelength of the light beam collimated by collimation device  5  is called wavelength of interest hereafter. In the following description, unless otherwise indicated, when the refraction index of a material is mentioned, this designates the refraction index at the wavelength of interest. 
     According to an embodiment, collimation device  5  has a structure with a planar symmetry with respect to plane (Oxy) and with respect to plane (Oxz). 
     Collimation device  5  comprises three elements: 
     a monomode waveguide  14 ; 
     a first element  16  which enables to collimate the light rays parallel to a first plane P 1 , for example, plane (Oxz); and 
     a second element  18  which enables to collimate the light rays parallel to a second plane P 2 , for example, plane (Oxy), different from first plane P 1  and preferably substantially perpendicular to first plane P 1 . 
     Waveguide  14  is made of a first material, for example, a first semiconductor material, and is surrounded with a cladding  20  made of a second material, for example, a second semiconductor material. The refraction index of the first material is greater than the refraction index of the second material. The selection of the first and second materials depends, in particular, on the wavelength of the light beam to be collimated. 
     According to an example, for the collimation of a monochromatic beam having a wavelength which may vary from 2 μm to 12 μm, the first material may be an alloy of silicon and germanium (SiGe) and the second material may be silicon (Si). According to another example, for the collimation of a monochromatic beam used in telecommunications and having a wavelength in vacuum which may be in the order of 1.55 μm, the first and second materials may correspond to semiconductor materials mainly comprising a III-V compound, for example, a III-N compound, particularly when collimation device  5  is provided on an optoelectronic circuit  10  comprising a light beam generation device, for example, a laser diode. Examples of group-III elements comprise gallium (Ga), indium (In), or aluminum (Al). Examples of III-N compounds are GaN, AlN, InN, InGaN, AlGaN, or AlInGaN. Other group-V elements may also be used, for example, phosphorus (P) or arsenic (As). According to another example, for the collimation of a monochromatic beam used in telecommunications and having a wavelength in vacuum which may be in the order of 1.55 μm, the first material may be Si and the second material may be silicon dioxide (SiO 2 ), particularly when collimation device  5  is provided on an optoelectronic circuit  10  comprising no light beam generation device. For this case, silicon nitride (SiN) may also be envisaged as a first material and silicon dioxide (SiO 2 ) may be envisaged as a second material. 
     Due to its dimensions, waveguide  14  only allows the propagation of an electromagnetic radiation at the wavelength of interest along a propagation mode. According to an embodiment, waveguide  14  has a rectangular cross-section having a height T, measured along direction (Oz), and a length L 1 , measured along direction (Oy). 
     First collimation element  16  is made of the first material and is surrounded with a cladding  22  made of the second material. First collimation element  16  comprises a first end  24  connected to an end of waveguide  14  and a second end  26  connected to second collimation element  18  and a body  28  extending between first end  24  and second end  26 . Body  28  has a diverging shape from first end  24  to second end  26 . According to an embodiment, first collimation element  16  has a rectangular cross-section having the same height T, measured along direction (Oz), as the waveguide and having a width, measured along direction (Oy), increasing from width L 1  at first end  24  to a width L 2  to second end  26 . Call length L 3  of first collimation element  16  the distance measured along direction (Ox) between first end  24  and second end  26 . 
     According to an embodiment, first collimation element  16  has, in top view, the shape of a trapeze having its small base corresponding to first end  24  and having a large base corresponding to second end  26 . 
     Height T may be in the range from 0.1 μm to 10 μm. Width L 1  may be in the range from 0.1 μm to 10 μm. Width L 2  may be in the range from 1 μm to 100 μm. Length L 3  may be in the range from 10 μm to 1 mm. 
     Second collimation element  18  has a height 2*H measured along direction (Oz) and a length L 4 , measured along direction (Ox). 
     Second collimation element  18  comprises a refraction index which is substantially constant in any plane parallel to plane (Oxy) and which, along direction (Oz), increases from a minimum refraction index n g  to a maximum refraction index n c , and then decreases to the minimum refraction index n g . The refraction index is equal to the maximum refraction index n c  for z equal to 0 and is equal to the minimum refraction index n g  for z equal to ±H. 
     The variation law of the index gradient in second collimation element  18  is selected to enable to obtain a collimation of the beam emitted by second collimation element  18 . 
     According to an embodiment, in second collimation element  18 , refraction index n varies parabolically, for example, according to the following relation (I): 
                       n   2     ⁡     (   z   )       =       n   c   2     [     1   -       (         n   c   2     -     n   g   2         n   c   2       )     ⁢       (     z   H     )     2         ]             (   I   )               
for z in the range from −H to H.
 
     The gradient of the index, which results in curving the light, is linked to the derivative of relation (I). The gradient is zero for plane (Oxy) and increases as the distance to plane (Oxy) increases. The more a ray becomes distant from plane (Oxy), the more it tends to be curved to return to plane (Oxy). Since the index is smaller, it also propagates faster. 
       FIG. 3  shows the curve of variation of the squared refraction index n 2  in second collimation element  18  according to relation (I) along direction (Oz). 
     Collimation device  5  operates as follows. At the output of waveguide  14 , the light beam is widened in direction (Oy) so that, at second end  26 , the light beam is substantially collimated parallel to plane (Oxz). In second collimation element  18 , due to the index gradient along direction (Oz), the light rays follow curved paths in planes substantially parallel to plane (Oxz) periodically oscillating along propagation direction (Ox). 
     Length L 4  is then selected so that the light rays escape from second collimation element  18  while being substantially parallel to plane (Oxy). A collimation of the beam parallel to direction (Ox) is thus obtained. 
     Length L 4  is provided by the following relation (II): 
     
       
         
           
             
               
                 
                   
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     Preferably, in the case where the incident beam is polychromatic, refraction indexes n c  and n g  are substantially independent from the wavelength over the wavelength range of the beam so that a collimation of the beam is obtained over the entire wavelength range of the beam. 
     First, second, and this simulations have been performed by time domain finite difference calculation. 
       FIGS. 4A, 5A, and 6A  show cross-section views similar to  FIG. 2  of the collimation devices respectively used for the first, second, and third simulations. In the first simulation, the collimation device had the structure shown in  FIGS. 1 and 2 , with the difference that second collimation element  18  was not present. In the second simulation, the collimation device had the structure shown in  FIGS. 1 and 2 , with the difference that second collimation element  18  was replaced with an element  29  of same dimensions but having a constant refraction index. In the third simulation, collimation device  5  had the structure shown in  FIGS. 1 and 2 . 
     For the first, second, and third simulations, the wavelength of interest λ was 4.5 μm. For the three simulations, waveguide  14  and first collimation element  16  were made of SiGe with 40 wt. % of germanium, which has a refraction index at 4.5 μm of 3.6, and claddings  20  and  22  were made of Si, which has a refraction index at 4.5 μm of 3.4. Height T was 3 μm and height H was 10 μm. The radiation propagating in waveguide  14  had a TM polarization, that is, the magnetic field was oriented along (Oy). 
       FIGS. 4B, 5B, and 6B  show isoline maps of the component along direction (Oy) of the magnetic excitation field obtained with the first, second, and third simulations. 
       FIG. 4B  shows the diffraction of the light beam in the absence of the second collimation element.  FIG. 5B  shows a lack of collimation of the light beam in element  29 .  FIG. 6B  shows that the structure of collimation device  5  of  FIG. 6A  causes an alternation of collimation and of focusing of the light beam. To obtain the emission of a collimated light beam, one just has to select length L 4  so that second collimation element  18  stops at a location where the beam is collimated, that it, halfway between two nodes  30  where the light beam is concentrated in a small area.  FIG. 6B  shows that it is sufficient to select L 4  equal to 47 μm, which can further be deduced from relation (II). 
       FIG. 7  illustrates an application of collimation device  5  where optoelectronic circuit  10  is coupled to an optical fiber  31 , collimation device  5  being arranged opposite an end of optical fiber  31 . 
       FIG. 8  illustrates another application of collimation device  5  where optoelectronic circuit  10  is coupled to another optoelectronic circuit  32 , collimation device  5  being placed opposite another collimation device  34  of optoelectronic circuit  32 , which may have the same structure as collimation device  5 . 
     Advantageously, for the applications illustrated in  FIGS. 7 and 8 , the light beam emitted by collimation device  5  having a large size, a significant alignment tolerance may be obtained. 
       FIG. 9  illustrates an embodiment of second collimation element  18 . In this embodiment, second collimation element  18  is formed by deposition, for example, by epitaxy, of a semiconductor layer having its composition modified continuously during the deposition to obtain the desired variation of the refraction index, particularly according to relation (I). When the second collimation element  18  corresponds to an alloy of a first element and of a second element, for example, SiGe, the variation of the refraction index may be obtained by continuously varying the proportion of the first element with respect to the second element in the alloy during the deposition of the alloy. 
       FIG. 10  illustrates another embodiment of second collimation element  18 . In this embodiment, second collimation element  18  is formed by a stack of a plurality of semiconductor layers  40  having different compositions, each layer  40  having a homogeneous composition selected to obtain a determined refraction index. A stepped variation of the refraction index along direction (Oz), which may for example approximately follow relation (I), is then obtained. The method of manufacturing second collimation element  18  according to the embodiment illustrated in  FIG. 10  may be simpler than the method of manufacturing second collimation element  18  according to the embodiment illustrated in  FIG. 9 . 
       FIG. 11  illustrates another embodiment of second collimation element  18 . In this embodiment, second collimation element  18  is formed by a stack along direction (Oz) comprising an alternation of first layers  42  having a first composition, and thus a first refraction index n c , and of second layers  44  having a second composition, and thus a second refraction index n g . The thicknesses of layers  42  and  44  are smaller than at least one fifth of the wavelength of interest so that the light crossing second collimation element  18  locally sees a mean refraction index related to the ratio of the thicknesses of the closest first and second layers  42  and  44 . The thicknesses of the first and second layers are then selected so that the local mean refraction index varies as desired along direction (Oz), for example, approaching relation (I). The method of manufacturing second collimation element  18  according to the embodiment illustrated in  FIG. 11  may be simpler than the method of manufacturing second collimation element  18  according to the embodiment illustrated in  FIG. 10 . 
       FIGS. 12A to 12H  are partial simplified cross-section views of the structures obtained at successive steps of an embodiment of a method of manufacturing the collimation device  12  shown in  FIGS. 1 and 2  for which second collimation element  18  may have one of the structures shown in  FIG. 9, 10 , or  11 . In  FIGS. 12A to 12H , direction (Oz), not shown, corresponds to the vertical direction. 
     The method comprises the successive steps of: 
     (1) Forming, for example, by epitaxy, on a substrate  50  having at its top a refraction index equal to n g , a layer  52  having a refraction index varying along direction (Oz) from n g  at the base of layer  52  to n int  at the top of layer  52 , n int  being greater than n g  ( FIG. 12A ). Substrate  50  may correspond to a monoblock structure or to a layer covering a support made of another material. Substrate  50  may be a semiconductor substrate, for example, a substrate made of silicon, of germanium, of silicon carbide, of a III-V compound such as GaN or GaAs, or a ZnO substrate. Substrate  50  may correspond to a multilayer structure of silicon-on-insulator type, also called SOI. Layer  52  may have one of the structures shown in  FIG. 9, 10 , or  11 . 
     (2) Depositing a layer  54  of a material having a refraction index equal to n g  and a layer  56  of a semiconductor material having a refraction index equal to n c , which is greater than n int  ( FIG. 12B ). 
     (3) Etching a portion of layer  56  with a stop on layer  54  to delimit waveguide  14  and first collimation element  16  ( FIG. 12C ). 
     (4) Depositing a layer of the material having a refraction index equal to n g  over the entire structure and etching the layer, for example, by a chemical-mechanical planarization or CMP, to delimit a layer  58  of same height as waveguide  14  and first collimation element  16  around them ( FIG. 12D ); 
     (5) Depositing a layer  60  of the material having a refraction index equal to n g  over the entire structure ( FIG. 12E ). 
     (6) Etching the portions of layers  54 ,  58 , and  60  at the desired location of second collimation element  18  ( FIG. 12F ). 
     (7) Forming, for example, by epitaxy, over the entire structure, a layer having a refraction index increasing along direction (Oz) from n int  at the base of the layer to n c  in the middle of the layer and then decreasing to n int  at the top of the layer and etching the layer, for example, by chemical-mechanical planarization, with a stop on layer  60  to delimit a layer  62  resting on layer  52  ( FIG. 12G ). Layer  62  may have one of the structures shown in  FIG. 9, 10 , or  11 . 
     (8) Forming, for example, by epitaxy, over the entire structure, a semiconductor layer  64  having a refraction index increasing along direction (Oz) from n int  at the base of the layer to n g  at the top of the layer ( FIG. 12H ). Layer  64  may have one of the structures shown in  FIG. 9, 10 , or  11 . 
     Second collimation element  18  is formed by layers  52 ,  62 , and  64 . Cladding  20  of waveguide  14  and cladding  22  of first collimation element  16  are formed by layers  54 ,  58 , and  60 . 
     Another embodiment of a manufacturing method comprises all the previously-described steps, with the difference that steps (1), (7), and (8) are respectively replaced with the following steps (1)′, (7)′, and (8)′: 
     (1)′ identical to previously-described step (1), with the difference that the refraction index varies along direction (Oz) from value n g  at the base of layer  52  to value n c  at the top of the layer; 
     (7)′ identical to previously-described step (7), with the difference that the refraction index of layer  62  is constant and equal to n c ; 
     (8)′ identical to previously-described step (8), with the difference that the refraction index varies along direction (Oz) from value n c  at the base of layer  64  to value n g  at the top of layer  64 . 
     The present embodiment has the advantage that the forming of layer  62  is simpler, particularly at the etch step previously described at step (7)′. However, the structure of second collimation element  18  then does not enable to obtain a variation profile of the refraction index which follows the previously-described relation (I) since the refraction index in central layer  62  of second collimation element  18  is constant. 
     Further, the inventor has shown that when the refraction index is constant in central layer  62 , the index gradient in layers  52  and  64  cannot follow a parabolic law. Indeed, in this case, it is not possible to obtain a proper collimation of the beam emitted by second collimation element  18 . The inventor has shown that, to obtain a proper collimation, the refraction index should vary in second collimation element  18  according to the following relations (III), considering that central layer  62  extends from z equal to −z 0  to z equal to z 0 : 
     for z in the range from −z 0  to z 0 :
 
 n   2 ( z )= n   c   2   (III)
 
     for z in the range from −z 0  to H: 
     
       
         
           
             
               
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     for z in the range from −z 0  to −H: 
     
       
         
           
             
               
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     Fourth and fifth simulations have been performed by time domain finite difference calculation. 
       FIGS. 13A and 14A  show curves of the variation of the refraction index in second collimation element  18  of the collimation devices respectively used for the fourth and fifth simulations. For the fourth and fifth simulations, the collimation device had the structure shown in  FIGS. 1 and 2 . The fourth and fifth simulations have been performed in the same conditions as the first, second, and third previously-described simulations. 
     In the fourth simulation, the refraction index of second collimation element  18  was constant for z between −z 0  and +z 0  and was following a parabolic law between −H and −z 0  and +z 0  and H. 
     The refraction index was thus varying in second collimation element  18  according to the following relations (IV): 
     for z in the range from −z 0  to z 0 :
 
 n   2 ( z )= n   c   2   (IV)
 
     for z in the range from z 0  to H: 
     
       
         
           
             
               
                 n 
                 2 
               
               ⁡ 
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 n 
                 c 
                 2 
               
               [ 
               
                 1 
                 - 
                 
                   
                     ( 
                     
                       
                         
                           n 
                           c 
                           2 
                         
                         - 
                         
                           n 
                           g 
                           2 
                         
                       
                       
                         n 
                         c 
                         2 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           z 
                           - 
                           
                             z 
                             0 
                           
                         
                         
                           H 
                           - 
                           
                             z 
                             0 
                           
                         
                       
                       ) 
                     
                     2 
                   
                 
               
               ] 
             
           
         
       
     
     for z in the range from −z 0  to −H: 
     
       
         
           
             
               
                 n 
                 2 
               
               ⁡ 
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 n 
                 c 
                 2 
               
               [ 
               
                 1 
                 - 
                 
                   
                     ( 
                     
                       
                         
                           n 
                           c 
                           2 
                         
                         - 
                         
                           n 
                           g 
                           2 
                         
                       
                       
                         n 
                         c 
                         2 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           z 
                           + 
                           
                             z 
                             0 
                           
                         
                         
                           H 
                           - 
                           
                             z 
                             0 
                           
                         
                       
                       ) 
                     
                     2 
                   
                 
               
               ] 
             
           
         
       
     
     In the fifth simulation, the refraction index of second collimation element  18  was following a previously-described law (III). 
       FIGS. 13B and 14B  show isoline maps of the component along direction (Oy) of the magnetic excitation field obtained in second collimation element  18  with the fourth and fifth simulations. 
       FIG. 13B  shows that the collimation device  5  having a second collimation element  18  with the variation profile of the refraction index shown in  FIG. 13A  does not provide a good focusing of the light beam and thus does not provide a good collimation of the light beam at mid-distance between two focusings. 
       FIG. 14B  shows that the collimation device  5  having its second collimation element  18  with the variation profile of the refraction index shown in  FIG. 14A  enables to obtain a good focusing of the light beam and thus enables to obtain a good collimation of the light beam at mid-distance between two focusings. The previous relation (II) remains valid as a first approximation. 
     Specific embodiments have been described. Various alterations, modifications, and improvements will readily occur to those skilled in the art.