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Visualization method of modal interference in multimode interference structures
Optics Communications 214 (2002) 47–53 www.elsevier.com/locate/optcom
Visualization method of modal interference in multimode interference structures Marek Błahut *, Paweł Karasi nski, Damian Kasprzak, Roman Rogozi nski Institute of Physics, Silesian University of Technology, ul. Krzywoustego 2, 44-100 Gliwice, Poland Received 29 July 2002; received in revised form 7 September 2002; accepted 21 October 2002
Abstract A method of modal field interference examination in multimode interference structures (MMI) is proposed, using fluorescence of the substance covering the MMI section. The method enables direct observation of modal fields interference and allows to determine the propagation lengths of N-fold images for the given window width of MMI section and technological process parameters. Testing investigations are carried out for MMI structures made by Kþ –Naþ ion exchange process in glass. The investigations concern the self-imaging phenomena for symmetrical and paired interference. Ó 2002 Published by Elsevier Science B.V.
1. Introduction During the excitation of multimode waveguide the matching effects of the input field to modal fields of multimode waveguide are observed and then the interference of excited waves. The socalled self-imaging effects of the input field accompany the intermode interference. As a result of these effects, the input field coming from singlemode waveguide or a group of single-mode waveguides is reproduced in simple, reflected and multiple images. This phenomenon is a basis for the operation of multimode interference structures (MMI) [1].
*
Corresponding author. E-mail address: [email protected] (M. Blahut).
The development of integrated optics circuits using MMI structures based on step-index optical waveguides, and in particular systems based on semiconductor structures, has been observed since early 1990s [2–6]. In the work [7,8] it has been affirmed that the self-imaging effects can occur in gradient waveguides produced by ion exchange method. Properties of MMI structures depend on modal properties of multimode waveguide – the number of guided modes, the geometrical mismatch of modal fields, and the propagation constants dependence on the mode number. All these parameters can be determined only approximately, particularly in the case of gradient-index waveguides, and measurements of the wave propagation characteristics in MMI devices are of potential interest. In this paper we proposed a method of examination of MMI structures which enables the direct
0030-4018/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 2 1 7 5 - 2
48
M. Błahut et al. / Optics Communications 214 (2002) 47–53
observation of modal fields interference and allows to determine the propagation lengths of N-fold images for the given window width of MMI section and technological process parameters. We adopt the observation system proposed in the paper [9] for the wave propagating in a Y branch waveguide.
2. Self-imaging in MMI sections MMI structure consists of a group of singlemode waveguides (a) of the window width w, defining the input field, a multimode section (b) of the width W, where the mode fields interference effects are observed and single-mode output waveguides (c) of the input geometry – Fig. 1. It is assumed for the interference section that it is multimode for the direction consistent with the section width (X) and single mode for the perpendicular direction (Y). In connection with it, only one-dimensional effects of modal fields interference appear. Single-mode waveguides give stable input field distribution Eðx; y; 0Þ. This field introduced to multimode section is decomposed into modal fields ul0 ðx; yÞ of multimode waveguide with propagation constants bl0 (in further formulas the not important index 0 is omitted) X Eðx; y; 0Þ ¼ cl ul ðx; yÞ: ð1Þ l
The field at the distance z is a superposition of modal fields propagating with different phase velocities and hence, in a different way shifted in phase
Eðx; y; zÞ ¼
cl ul ðx; yÞ expðj bl zÞ:
ð2Þ
l
The interference pattern observed in MMI section depends on modal properties of the multimode waveguide. Particularly important is the propagation constants dependence on the mode number. For step-index waveguide it is nearly quadratic and the self-imaging phenomena appear. Similar quadratic dependence show gradient index waveguides made by Kþ –Naþ and Agþ –Naþ ion exchange [8]. According to the theory presented in paper [2] for step-index waveguides, for propagation lengths z satisfying conditions: n p z ¼ ð3 Lz Þ for n ¼ 0; 1; . . . ; Lz ¼ ; N b0 b1 ð3Þ N-fold images of the input field are formed: Eðx; y; zÞ ¼
N 1 1 X Eðx xq ; y; 0Þ expðj /q Þ; C q¼0
ð4Þ
situated round the points: W for q ¼ 1; . . . ; N 1; ð5Þ N where W is the MMI section width, and qp /q ¼ n ðN qÞ ; ð6Þ N determines the phase shifts which are an inherent feature of N-fold interference images. The dependences presented characterize the so-called general interference. It can be also shown [3] that for the input field position xin satisfying conditions: xq ¼ ð2 q N Þ
W for i ¼ 1; 0; . . . ð7Þ N some of the input field images overlap giving, in general, non-uniform field distribution at the output. Only for selected input field positions
xin ¼ i
W W 2W ; xin ¼ ; xin ¼ ð8Þ 2 3 3 self-imaging effects with uniform distribution of the energy of the field are observed. It is the case of so-called restricted interference – symmetric and paired, when mode fields are selectively excited. Symmetric interference occurs when the MMI section is fed by a single central waveguide. In that xin ¼
Fig. 1. The scheme of MMI structure.
X
M. Błahut et al. / Optics Communications 214 (2002) 47–53
case only symmetric modes l ¼ 0; 2; 4; . . . are excited. For the step-index MMI structures, the input field image appears at the propagation length of 3=4Lz and multiple images can be observed at the distances equal 3=4 Lz =N , where N is the number of multiplications. Paired interference occurs when only the l ¼ 0; 1; 3; 4; 6; 7; . . . modes are excited. For the step-index structure, a possible way of attaining such selective excitation is by launching the singlemode input waveguide at 1/3 or 2/3 of the MMI width. In that case the input field image is at the distance 2Lz and its mirror image occurs at Lz . At the distance Lz =N , multiple images appear. Propagation lengths L for N-fold input field images for the described cases of interference obey the relations: LðgeneralÞ ¼ 3 LðpairedÞ ¼ 4 LðsymmetricÞ:
ð9Þ
3. Experimental stand Experimental investigations of MMI structures are carried out in the arrangement shown in Fig. 2. The MMI section in which the mode fields interference is observed excited from laser by the light of wavelength k ¼ 0:63 lm from single-mode fiber is covered with fluorescent substance by spincoating. The covering layer has the thickness of about 1 lm. This substance is the Nile Blue ‘‘A’’ perchlorate suspension in PMMA, which absorbs the light at 0.63 lm. Emission occurs for the wavelength at k ¼ 0:69 lm [9]. Fluorescence is forced by evanescent waves of modal fields which penetrating the layer of fluorescent substance produce lightening proportional
Fig. 2. Experimental arrangement.
49
Fig. 3. The geometry of the designed mask and micro-image of the fragment of right side of the mask.
to the energy of the excited mode. Wave field distribution is recorded by sensitive CCD camera and registered and analyzed in a computer. Recording of successive sequences of mode field interference patterns makes it possible to determine the dependence of propagation lengths for N-fold images on the window width, time and temperature of diffusion for ion-exchange process. Testing investigations are made for gradient index MMI structures produced by ion exchange in glass. The investigated MMI sections are fabricated in the process of Kþ –Naþ ion exchange in the time of 1 h and temperature of 400 °C in BK-7 glass. Diffusion is performed through the mask opening, in which, depending on the direction of excitation, symmetric or paired interference can be observed. The geometry of the designed single mask is presented in Fig. 3. Next to it, the exemplary micro-image of the mask is presented. Widths WM of the masks of MMI sections are 30, 39, 51, 60, 69 and 81 lm. The multimode waveguides obtained after diffusion support, respectively, 5, 7, 9, 11, 13 and 15 guided modes.
4. Numerical simulation of the field propagation in investigated MMI structures For the convenience of experimental results analysis, numerical simulations of TE and TM wave propagation in the structures investigated are carried out by BPM method for symmetrical excitation and asymmetrical excitation at 1/3rd of the section width. The distribution profile of refractive index Dnðx; yÞ of MMI section obtained in the diffusion process through the window opening is calculated numerically from the non-linear diffusion equation [10]. In Fig. 4 are presented exemplary results of simulations of the TE wave field
50
M. Błahut et al. / Optics Communications 214 (2002) 47–53
propagation for MMI section of the window width of 51 lm. The presented results preserve the real geometry of interference patterns. The propagation length for twofold images L3dB amounts, respectively, to 3300 lm for symmetrical images and 4400 lm for asymmetrical. N-fold images appear at propagation distances 2 L3dB =N . This property makes it possible to establish the coupling length L3dB on the base of the picture of mode field interference images. It is important for further analysis to determine the influence; of dielectric cover on modal fields interference pictures. The dielectric cover changes external wave propagation conditions and there-
fore influences propagation constants values. It comes out, however, that decisive about modal interference, the difference of propagation constants of modes b0 bl weakly depends on refractive index of the cover. In Fig. 5 are presented the dependences of relative propagation constants difference ½bl ðnÞ bl ð1:0Þ=bl ð1:0Þ on the mode number l in the multimode waveguide of the window width 81 lm, covered by dielectric medium of the refractive index n ¼ 1:45, 1.5 and surrounded by air (n ¼ 1:0). For both coverings, the relative propagation constants difference practically does not depend on the mode number. Therefore, one can expect that the modal field interference pic-
Fig. 4. Numerical simulations of the TE wave field propagation for MMI section of the window width of 51 lm.
M. Błahut et al. / Optics Communications 214 (2002) 47–53
51
5. Experimental results
Fig. 5. The dependence of relative propagation constants differences on the mode number in MMI sections covered by dielectric medium of the refractive index n, and surrounded by air (n ¼ 1:0).
tures in MMI section covered with fluorescent substance and the section uncovered will not differ. It is illustrated by the next Fig. 6. The characteristics present the numerically calculated TE field evolution in MMI section asymmetrically excited at 1/3rd of the section width surrounded by air (n ¼ 1) and covered by dielectric medium of the refractive index n ¼ 1:45. Despite of essential differences in propagation conditions, both images of mode field interferences are practically the same.
The proposed method of investigation of MMI structures enables complete analysis of their properties. In Figs. 7 and 8 are presented, for example, successive sequences of recorded mode field interference images for gradient index MMI section made by Kþ –Naþ ion exchange for the window width 39 and 60 lm. The first MMI section guides 7 modes, and the second one 11 modes. Sections are symmetrically excited by the field from single-mode fiber. The input field coming out from single-mode waveguide of the window width 5 lm extends as a result of diffraction, achieving waveguide boundaries. Then, during successive reflections, the field matches to the mode structure and next an interference patterns evolution along the propagation length is observed. For the determination of selfimaging lengths of twofold images L3dB one should recognize N-fold images which appear in interference images. It can be easily done by comparison with the results of numerical simulations. The quantity of N-fold input field images depends on the input field spreading and MMI section width. For narrower section distinctly visible are 1 4, 1 3, 1 2 and 1 1 self-images marked in the interference pattern. In wider section, observation
Fig. 6. Numerically calculated TE field evolution in MMI section of the width 81 lm, asymmetrically excited at 1/3rd of the section width surrounded by air (n ¼ 1) and covered by dielectric medium of the refractive index n ¼ 1:45.
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M. Błahut et al. / Optics Communications 214 (2002) 47–53
Fig. 7. The successive sequences of recorded mode field interference images for MMI section of the width 39 lm symmetrically excited.
Fig. 8. The successive sequences of recorded mode field interference images for MMI section of the width 60 lm symmetrically excited.
of 1 7, 1 6, 1 5 N-fold images is also possible. On the base of it, using dependence: z ¼ 2 L3dB =N ; where z is the propagation length for N-fold images, the average value of coupling length L3dB can be determined. Interference images shown in Figs. 9 and 10 for MMI sections of the width 39 and 60 lm excited at 1/3rd of the section width, are the examples of paired interference. The images are not as clear as in the case of symmetrical interference. This can be explained by the effects of polarization, particularly important for asymmetrical excitation.
Fig. 9. The successive sequences of recorded mode fields interference images for MMI section of the width 39 lm excited at 1/3rd of the section width.
Fig. 10. The successive sequences of recorded mode fields interference images for MMI section of the width 60 lm excited at 1/3rd of the section width.
The TE and TM waveguide modes which appear in MMI section for the excitation by the un-polarized field from single-mode fiber have different modal characteristics depending on propagation conditions at the substrate surface. Additionally, the material birefringence is observed for waveguides obtained by Kþ –Naþ ion exchange, caused by stresses produced in the technological process – maximum of the refractive index change is equal Dn ¼ 0:0084 for TE modes and 0.0106 for TM modes in BK-7 glass. For these reasons, the differences in interference patterns for both polarizations should appear. The interference images observed are the averaging images for TE and TM polarization.
M. Błahut et al. / Optics Communications 214 (2002) 47–53
53
6. Conclusions
Acknowledgements
In the paper we proposed a method of visualization of the optical field propagation in multimode waveguides. This method enables the direct observation of modal fields interference in MMI structures and allows to determine the propagation lengths of N-fold images for the given window width of MMI section and technological process parameters. On the base of it, investigations of gradient index MMI structures made by Kþ –Naþ ion exchange were performed to be used in passive elements technology of integrated optics. Investigations concerned self-imaging effects of the input field for symmetrical and paired interference. The knowledge of the propagation length of twofold images for fixed window width and technological process parameters of ion-exchange makes it possible to design basic elements of integrated optics – gradient index splitters and couplers N M and 1 N having very good optical properties in which the splitting of the input field is effected within a small area.
This work was carried out under Research Project No. 8 T11B 052 18 of State Committee for Scientific Research, Poland.
References [1] R. Ulrich, G. Ankele, Appl. Phys. Lett. 27 (1975) 337. [2] M. Bachman, P.A. Besse, H. Melchior, Appl. Opt. 33 (1994) 3905. [3] L.B. Soldano, E.C.M. Pennings, J. Lightwave Technol., LT 13 (1995) 615. [4] J.M. Heaton, R.M. Jenkins, D.R. Wright, J.T. Parker, J.C.H. Birbeck, K.P. Hilton, Appl. Phys. Lett. 61 (1992) 1754. [5] Q. Lai, M. Bachman, H. Melchior, Electron. Lett. 33 (1997) 1699. [6] T. Saida, A. Himeno, M. Okuna, A. Sugita, K. Okamoto, Electron. Lett. 35 (1999) 2031. [7] S. Das et al., in: Righini, Honkanen (Eds.), Proc. SPIE, vol. 3936, Integrated Optics Devices IV, 2000, p. 239. [8] M. Blahut, Opt. Appl. 30 (2000) 401. [9] Y. Okamura, A. Miki, S. Yamamoto, Appl. Opt. (1986) 3405. [10] M. Blahut, A. Opilski, R. Rogozi nski, Opt. Appl. 22 (3–4) (1992) 161.
Visualization method of modal interference in multimode interference structures Marek Błahut *, Paweł Karasi nski, Damian Kasprzak, Roman Rogozi nski Institute of Physics, Silesian University of Technology, ul. Krzywoustego 2, 44-100 Gliwice, Poland Received 29 July 2002; received in revised form 7 September 2002; accepted 21 October 2002
Abstract A method of modal field interference examination in multimode interference structures (MMI) is proposed, using fluorescence of the substance covering the MMI section. The method enables direct observation of modal fields interference and allows to determine the propagation lengths of N-fold images for the given window width of MMI section and technological process parameters. Testing investigations are carried out for MMI structures made by Kþ –Naþ ion exchange process in glass. The investigations concern the self-imaging phenomena for symmetrical and paired interference. Ó 2002 Published by Elsevier Science B.V.
1. Introduction During the excitation of multimode waveguide the matching effects of the input field to modal fields of multimode waveguide are observed and then the interference of excited waves. The socalled self-imaging effects of the input field accompany the intermode interference. As a result of these effects, the input field coming from singlemode waveguide or a group of single-mode waveguides is reproduced in simple, reflected and multiple images. This phenomenon is a basis for the operation of multimode interference structures (MMI) [1].
*
Corresponding author. E-mail address: [email protected] (M. Blahut).
The development of integrated optics circuits using MMI structures based on step-index optical waveguides, and in particular systems based on semiconductor structures, has been observed since early 1990s [2–6]. In the work [7,8] it has been affirmed that the self-imaging effects can occur in gradient waveguides produced by ion exchange method. Properties of MMI structures depend on modal properties of multimode waveguide – the number of guided modes, the geometrical mismatch of modal fields, and the propagation constants dependence on the mode number. All these parameters can be determined only approximately, particularly in the case of gradient-index waveguides, and measurements of the wave propagation characteristics in MMI devices are of potential interest. In this paper we proposed a method of examination of MMI structures which enables the direct
0030-4018/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 2 1 7 5 - 2
48
M. Błahut et al. / Optics Communications 214 (2002) 47–53
observation of modal fields interference and allows to determine the propagation lengths of N-fold images for the given window width of MMI section and technological process parameters. We adopt the observation system proposed in the paper [9] for the wave propagating in a Y branch waveguide.
2. Self-imaging in MMI sections MMI structure consists of a group of singlemode waveguides (a) of the window width w, defining the input field, a multimode section (b) of the width W, where the mode fields interference effects are observed and single-mode output waveguides (c) of the input geometry – Fig. 1. It is assumed for the interference section that it is multimode for the direction consistent with the section width (X) and single mode for the perpendicular direction (Y). In connection with it, only one-dimensional effects of modal fields interference appear. Single-mode waveguides give stable input field distribution Eðx; y; 0Þ. This field introduced to multimode section is decomposed into modal fields ul0 ðx; yÞ of multimode waveguide with propagation constants bl0 (in further formulas the not important index 0 is omitted) X Eðx; y; 0Þ ¼ cl ul ðx; yÞ: ð1Þ l
The field at the distance z is a superposition of modal fields propagating with different phase velocities and hence, in a different way shifted in phase
Eðx; y; zÞ ¼
cl ul ðx; yÞ expðj bl zÞ:
ð2Þ
l
The interference pattern observed in MMI section depends on modal properties of the multimode waveguide. Particularly important is the propagation constants dependence on the mode number. For step-index waveguide it is nearly quadratic and the self-imaging phenomena appear. Similar quadratic dependence show gradient index waveguides made by Kþ –Naþ and Agþ –Naþ ion exchange [8]. According to the theory presented in paper [2] for step-index waveguides, for propagation lengths z satisfying conditions: n p z ¼ ð3 Lz Þ for n ¼ 0; 1; . . . ; Lz ¼ ; N b0 b1 ð3Þ N-fold images of the input field are formed: Eðx; y; zÞ ¼
N 1 1 X Eðx xq ; y; 0Þ expðj /q Þ; C q¼0
ð4Þ
situated round the points: W for q ¼ 1; . . . ; N 1; ð5Þ N where W is the MMI section width, and qp /q ¼ n ðN qÞ ; ð6Þ N determines the phase shifts which are an inherent feature of N-fold interference images. The dependences presented characterize the so-called general interference. It can be also shown [3] that for the input field position xin satisfying conditions: xq ¼ ð2 q N Þ
W for i ¼ 1; 0; . . . ð7Þ N some of the input field images overlap giving, in general, non-uniform field distribution at the output. Only for selected input field positions
xin ¼ i
W W 2W ; xin ¼ ; xin ¼ ð8Þ 2 3 3 self-imaging effects with uniform distribution of the energy of the field are observed. It is the case of so-called restricted interference – symmetric and paired, when mode fields are selectively excited. Symmetric interference occurs when the MMI section is fed by a single central waveguide. In that xin ¼
Fig. 1. The scheme of MMI structure.
X
M. Błahut et al. / Optics Communications 214 (2002) 47–53
case only symmetric modes l ¼ 0; 2; 4; . . . are excited. For the step-index MMI structures, the input field image appears at the propagation length of 3=4Lz and multiple images can be observed at the distances equal 3=4 Lz =N , where N is the number of multiplications. Paired interference occurs when only the l ¼ 0; 1; 3; 4; 6; 7; . . . modes are excited. For the step-index structure, a possible way of attaining such selective excitation is by launching the singlemode input waveguide at 1/3 or 2/3 of the MMI width. In that case the input field image is at the distance 2Lz and its mirror image occurs at Lz . At the distance Lz =N , multiple images appear. Propagation lengths L for N-fold input field images for the described cases of interference obey the relations: LðgeneralÞ ¼ 3 LðpairedÞ ¼ 4 LðsymmetricÞ:
ð9Þ
3. Experimental stand Experimental investigations of MMI structures are carried out in the arrangement shown in Fig. 2. The MMI section in which the mode fields interference is observed excited from laser by the light of wavelength k ¼ 0:63 lm from single-mode fiber is covered with fluorescent substance by spincoating. The covering layer has the thickness of about 1 lm. This substance is the Nile Blue ‘‘A’’ perchlorate suspension in PMMA, which absorbs the light at 0.63 lm. Emission occurs for the wavelength at k ¼ 0:69 lm [9]. Fluorescence is forced by evanescent waves of modal fields which penetrating the layer of fluorescent substance produce lightening proportional
Fig. 2. Experimental arrangement.
49
Fig. 3. The geometry of the designed mask and micro-image of the fragment of right side of the mask.
to the energy of the excited mode. Wave field distribution is recorded by sensitive CCD camera and registered and analyzed in a computer. Recording of successive sequences of mode field interference patterns makes it possible to determine the dependence of propagation lengths for N-fold images on the window width, time and temperature of diffusion for ion-exchange process. Testing investigations are made for gradient index MMI structures produced by ion exchange in glass. The investigated MMI sections are fabricated in the process of Kþ –Naþ ion exchange in the time of 1 h and temperature of 400 °C in BK-7 glass. Diffusion is performed through the mask opening, in which, depending on the direction of excitation, symmetric or paired interference can be observed. The geometry of the designed single mask is presented in Fig. 3. Next to it, the exemplary micro-image of the mask is presented. Widths WM of the masks of MMI sections are 30, 39, 51, 60, 69 and 81 lm. The multimode waveguides obtained after diffusion support, respectively, 5, 7, 9, 11, 13 and 15 guided modes.
4. Numerical simulation of the field propagation in investigated MMI structures For the convenience of experimental results analysis, numerical simulations of TE and TM wave propagation in the structures investigated are carried out by BPM method for symmetrical excitation and asymmetrical excitation at 1/3rd of the section width. The distribution profile of refractive index Dnðx; yÞ of MMI section obtained in the diffusion process through the window opening is calculated numerically from the non-linear diffusion equation [10]. In Fig. 4 are presented exemplary results of simulations of the TE wave field
50
M. Błahut et al. / Optics Communications 214 (2002) 47–53
propagation for MMI section of the window width of 51 lm. The presented results preserve the real geometry of interference patterns. The propagation length for twofold images L3dB amounts, respectively, to 3300 lm for symmetrical images and 4400 lm for asymmetrical. N-fold images appear at propagation distances 2 L3dB =N . This property makes it possible to establish the coupling length L3dB on the base of the picture of mode field interference images. It is important for further analysis to determine the influence; of dielectric cover on modal fields interference pictures. The dielectric cover changes external wave propagation conditions and there-
fore influences propagation constants values. It comes out, however, that decisive about modal interference, the difference of propagation constants of modes b0 bl weakly depends on refractive index of the cover. In Fig. 5 are presented the dependences of relative propagation constants difference ½bl ðnÞ bl ð1:0Þ=bl ð1:0Þ on the mode number l in the multimode waveguide of the window width 81 lm, covered by dielectric medium of the refractive index n ¼ 1:45, 1.5 and surrounded by air (n ¼ 1:0). For both coverings, the relative propagation constants difference practically does not depend on the mode number. Therefore, one can expect that the modal field interference pic-
Fig. 4. Numerical simulations of the TE wave field propagation for MMI section of the window width of 51 lm.
M. Błahut et al. / Optics Communications 214 (2002) 47–53
51
5. Experimental results
Fig. 5. The dependence of relative propagation constants differences on the mode number in MMI sections covered by dielectric medium of the refractive index n, and surrounded by air (n ¼ 1:0).
tures in MMI section covered with fluorescent substance and the section uncovered will not differ. It is illustrated by the next Fig. 6. The characteristics present the numerically calculated TE field evolution in MMI section asymmetrically excited at 1/3rd of the section width surrounded by air (n ¼ 1) and covered by dielectric medium of the refractive index n ¼ 1:45. Despite of essential differences in propagation conditions, both images of mode field interferences are practically the same.
The proposed method of investigation of MMI structures enables complete analysis of their properties. In Figs. 7 and 8 are presented, for example, successive sequences of recorded mode field interference images for gradient index MMI section made by Kþ –Naþ ion exchange for the window width 39 and 60 lm. The first MMI section guides 7 modes, and the second one 11 modes. Sections are symmetrically excited by the field from single-mode fiber. The input field coming out from single-mode waveguide of the window width 5 lm extends as a result of diffraction, achieving waveguide boundaries. Then, during successive reflections, the field matches to the mode structure and next an interference patterns evolution along the propagation length is observed. For the determination of selfimaging lengths of twofold images L3dB one should recognize N-fold images which appear in interference images. It can be easily done by comparison with the results of numerical simulations. The quantity of N-fold input field images depends on the input field spreading and MMI section width. For narrower section distinctly visible are 1 4, 1 3, 1 2 and 1 1 self-images marked in the interference pattern. In wider section, observation
Fig. 6. Numerically calculated TE field evolution in MMI section of the width 81 lm, asymmetrically excited at 1/3rd of the section width surrounded by air (n ¼ 1) and covered by dielectric medium of the refractive index n ¼ 1:45.
52
M. Błahut et al. / Optics Communications 214 (2002) 47–53
Fig. 7. The successive sequences of recorded mode field interference images for MMI section of the width 39 lm symmetrically excited.
Fig. 8. The successive sequences of recorded mode field interference images for MMI section of the width 60 lm symmetrically excited.
of 1 7, 1 6, 1 5 N-fold images is also possible. On the base of it, using dependence: z ¼ 2 L3dB =N ; where z is the propagation length for N-fold images, the average value of coupling length L3dB can be determined. Interference images shown in Figs. 9 and 10 for MMI sections of the width 39 and 60 lm excited at 1/3rd of the section width, are the examples of paired interference. The images are not as clear as in the case of symmetrical interference. This can be explained by the effects of polarization, particularly important for asymmetrical excitation.
Fig. 9. The successive sequences of recorded mode fields interference images for MMI section of the width 39 lm excited at 1/3rd of the section width.
Fig. 10. The successive sequences of recorded mode fields interference images for MMI section of the width 60 lm excited at 1/3rd of the section width.
The TE and TM waveguide modes which appear in MMI section for the excitation by the un-polarized field from single-mode fiber have different modal characteristics depending on propagation conditions at the substrate surface. Additionally, the material birefringence is observed for waveguides obtained by Kþ –Naþ ion exchange, caused by stresses produced in the technological process – maximum of the refractive index change is equal Dn ¼ 0:0084 for TE modes and 0.0106 for TM modes in BK-7 glass. For these reasons, the differences in interference patterns for both polarizations should appear. The interference images observed are the averaging images for TE and TM polarization.
M. Błahut et al. / Optics Communications 214 (2002) 47–53
53
6. Conclusions
Acknowledgements
In the paper we proposed a method of visualization of the optical field propagation in multimode waveguides. This method enables the direct observation of modal fields interference in MMI structures and allows to determine the propagation lengths of N-fold images for the given window width of MMI section and technological process parameters. On the base of it, investigations of gradient index MMI structures made by Kþ –Naþ ion exchange were performed to be used in passive elements technology of integrated optics. Investigations concerned self-imaging effects of the input field for symmetrical and paired interference. The knowledge of the propagation length of twofold images for fixed window width and technological process parameters of ion-exchange makes it possible to design basic elements of integrated optics – gradient index splitters and couplers N M and 1 N having very good optical properties in which the splitting of the input field is effected within a small area.
This work was carried out under Research Project No. 8 T11B 052 18 of State Committee for Scientific Research, Poland.
References [1] R. Ulrich, G. Ankele, Appl. Phys. Lett. 27 (1975) 337. [2] M. Bachman, P.A. Besse, H. Melchior, Appl. Opt. 33 (1994) 3905. [3] L.B. Soldano, E.C.M. Pennings, J. Lightwave Technol., LT 13 (1995) 615. [4] J.M. Heaton, R.M. Jenkins, D.R. Wright, J.T. Parker, J.C.H. Birbeck, K.P. Hilton, Appl. Phys. Lett. 61 (1992) 1754. [5] Q. Lai, M. Bachman, H. Melchior, Electron. Lett. 33 (1997) 1699. [6] T. Saida, A. Himeno, M. Okuna, A. Sugita, K. Okamoto, Electron. Lett. 35 (1999) 2031. [7] S. Das et al., in: Righini, Honkanen (Eds.), Proc. SPIE, vol. 3936, Integrated Optics Devices IV, 2000, p. 239. [8] M. Blahut, Opt. Appl. 30 (2000) 401. [9] Y. Okamura, A. Miki, S. Yamamoto, Appl. Opt. (1986) 3405. [10] M. Blahut, A. Opilski, R. Rogozi nski, Opt. Appl. 22 (3–4) (1992) 161.