IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 6, NOVEMBER/DECEMBER 2002
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A Three-Focal-Point Method for the Optimal Design of a Flat-Top Planar Waveguide Demultiplexer Zhimin Shi and Sailing He, Senior Member, IEEE
Abstract—A three-focal-point method is introduced for an optimal design of a planar waveguide demultiplexer with a flattened spectral response. The grating in an etched diffraction grating demultiplexer is divided into three interleaved subgratings with some appropriate ratio of the facet numbers of the subgratings. With a Gaussian approximation, an analytical formula is derived for the optimal condition for minimal ripple in the flat-top spectral response. A required passband width determines the separation between the peaks of the two outmost subimages, which then determines the optimal ratio of the facet numbers of the subgratings according to the derived analytical formula. The present method is illustrated and verified with a design example. The ripple for the designed flat-top demultiplexer is suppressed greatly by controlling appropriately the facet numbers of the three subgratings according to the present analytical formula. Index Terms—Demultiplexer, flat-top, Gaussian approximation, passband flattening, ripple, three-focal-point, wavelength-division multiplexing (WDM).
I. INTRODUCTION
W
AVELENGTH-DIVISION-MULTPLEXING (WDM) technology has become an essential part of an optical communications system today. The performance of a WDM optical network depends greatly on the spectral characteristics of the components it uses. Multiplexers/demultiplexers/routers based on the integrated planar waveguide technologies (e.g., arrayed waveguide gratings (AWGs) [1] and etched diffraction gratings (EDGs) [2]) have shown a great potential due to their low insertion loss, low crosstalk, high compactness, and high spectral finesse. However, a conventional design of such devices gives a Gaussian-type spectral response, which seriously restrains the concatenation of the devices and requires a strict wavelength control in the system. To overcome this problem, various techniques [3]–[8] have been proposed to broaden or flatten the spectral response of a multiplexer/demultiplexer/router. In this paper, we introduce a three-focal-point method for an optimal design of a flat-top EDG demultiplexer. By introducing multiple focal points for the grating, the passband width increases significantly due to a broadened imaging field. To
Manuscript received August 13, 2002; revised September 30, 2002. This work was supported by the Government of Zhejiang Province (China) under a major research Grant (001101027). Z. Shi is with the Centre for Optical and Electromagnetic Research, State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou, 310027, R.O.C. (e-mail:
[email protected]). S. He is with the Centre for Optical and Electromagnetic Research, State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Yu-Quan, Hangzhou, 310027, R.O.C. and also with the Division of Electromagnetic Theory, Alfven Laboratory, Royal Institute of Technology, S-100 44 Stockholm, Sweden (e-mail:
[email protected]). Digital Object Identifier 10.1109/JSTQE.2002.805967
achieve a broadened spectral response, two focal points can be easily made by two subgratings (see [6] and [7] for a similar treatment for the case of AWG demultiplexers). Since the two subimages should have the same amplitude in order to obtain a symmetric spectral response, the case of two focal points does not require any formula similar to those derived in Section III of the present paper for the case of three focal points. The passband is broadened when the two focal points depart from each other. However, the ripples of the spectral response near the central wavelength cannot be controlled using only two focal points. When a large passband is required, one has to increase the separation between the two focal points and consequently the ripple problem becomes serious. A large ripple in the spectral response requires a strict wavelength control in a network. In this paper, we introduce a three-focal-point method to reduce the ripple. By using a Gaussian approximation, an analytical formula is derived for the optimal ratio of the peak amplitudes of the subimages (or the optimal ratio of the facet numbers of the interleaved subgratings). The etched grating is divided into three interleaved subgratings with the optimal ratio of facet numbers. Each subgrating forms a subimage near its focal center. The three subimages overlap with each other partly and give a flat-top spectral response with minimal ripple. In this paper, a required passband width determines the separation between the peaks of the two outmost subimages, which then determines the optimal ratio of the facet numbers of the subgratings according to the derived analytical formula. The design method is illustrated and verified with a numerical example of a typical SiO EDG demultiplexer. II. PRINCIPLE FOR AN EDG DEMULTIPLEXER SCALAR DIFFRACTION THEORY
AND A
In this paper, we consider a planar waveguide demultiplexer consisting of an EDG (see Fig. 1), which is usually based on a Rowland circle construction [9] (as illustrated in Fig. 2). The field launched from the ending facet of the input waveguide to the free propagation region (FPR) is diffracted by each grating facet. It is then refocused on the imaging curve and guided into different output waveguides according to the wavelength. In a conventional design of an EDG demultiplexer, the fields “reflected” by all grating facets should add in phase so that they are “focused” at one focal point for a designed wavelength. The center point of the th facet on the grating is thus determined by (1) is the center of the ending facet of the input where is the focal point, and , , and denote waveguide,
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integral over the cross-sectional line where the ending facet of the output waveguide is positioned (4)
Fig. 1.
and represent the imaging field distribution where and the fundamental mode profile of the output waveguide, respectively. Here the superscript is for the complex conjugate. Alternatively, the passband can also be broadened by using a multimode output waveguide which is then tapered down to a single-mode waveguide [3], [12]. However, the obtained spectrum is not good since the transition “skirt” becomes worse (i.e., the adjacent crosstalk will become worse as the width of the multimode waveguide increases). Since an EDG demultiplexer is a linearly dispersive element, the imaging field shifts linearly when the frequency changes, , where is the dispersive coefficient deteri.e., mined by the grating structure. Therefore, (4) can be written in the following convolution form:
Schematic diagram for an EDG demultiplexer.
Fig. 2. Configuration for a Rowland circle construction.
the diffraction order of the grating, the designed wavelength in vacuum, and the effective refractive index in the FPR, respectively. The demultiplexing of the grating can be analyzed with a scalar diffraction theory [10], [11]. Using Kirchhoff– Huygens’ diffraction formula, the incident electric field at a point on the grating can be calculated by the following integral over the cross-sectional line where the ending facet of the input waveguide is positioned:
(2) denotes the incident electric field, is the wave where is the distance between a point number in the FPR, on the cross-sectional line and point , and is the diffraction angle with respect to the normal of the ending facet of the input waveguide (see Fig. 2). Similarly, we can obtain the following formula for the electric field at a point on the imaging plane:
(5) where denotes the spatial convolution. In a conventional design, an EDG demultiplexer has only one focal point, and the image formed near the focal center has a shape that is almost the same as the input field. Thus, (5) results in a spectral response of Gaussian type, which is not good for use in a practical dense WDM (DWDM) system. To achieve a broadened spectral response, two focal points can be easily made by two interleaved subgratings with equal number of facets. To obtain a symmetric spectral response, the two subimages should have the same amplitude and thus no need for any analytical formula as those derived in the next section for the case with three focal points. The passband is broadened when the two focal points depart from each other. However, this two-focal-point method cannot control the ripples of the spectral response near the central frequency. The problem of ripples becomes serious when the separation between the two focal points is large (i.e., when a large passband is required; cf. Fig. 5 below). Large ripples require a strict wavelength control in a network. In this paper, we introduce a three-focal-point method to reduce the ripples. should also be considThe dispersion characteristic ered in the design of a demultiplexer. It can be calculated by [13] (6)
(3) where is the reflection coefficient of the grating, and are the incident and diffraction angles with respect to the normal of the grating facet, respectively (see Fig. 2). In most cases, can be considered as a constant. When the output waveguide is of single-mode, the spectral response of a channel can be obtained from the following overlap
where
is the group delay given by (7)
and where is the velocity of the light in vacuum, and is the phase response determined by the overlap integral .
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approximations are used for the three field distributions of the subimages. The total imaging field distribution with respect to the imaging center at the designed wavelength ( and are omitted hereafter for simplicity of notation) can be approximated as
(9)
Fig. 3. Geometrical configuration for the three-focal-point method.
III. ANALYTICAL FORMULAS FOR THE THREE-FOCAL-POINT METHOD In this paper, we divide the grating into three interleaved subgratings, each of which forms a subimage near its focal center. The three subimages overlap with each other partly, and give a broadened total imaging field (see Fig. 3). From Fig. 3, one sees that the peak amplitude of the central subimage 2 should be different from (smaller than) the peak amplitude of the two outmost subimages 1 and 3 (the amplitudes of the two outmost subimages 1 and 3 should be the same in order to obtain a symmetric spectral response). Therefore, the three subgratings should be interleaved appropriately so that they give appropriate ratio of the peak amplitudes. To obtain a grating structure which gives three focal points, (1) should be replaced by the following formula: (8) is the focal point for the th facets and is one of where the three focal points. When the subgratings are interleaved as shown in Fig. 3 [cf. (14) below], the ratio of the peak amplitudes of the subimages is approximately equal to the ratio of the facet numbers of the corresponding subgratings (e.g., when the ratio of the facet numbers of two subgratings is 2 : 1, the ratio of the peak amplitudes of the corresponding subimages is approximately equal to 2 : 1). Furthermore, the subimages are usually in phase (as the numerical example considered below). In fact, the present three-focal-point method is realized by choosing appropriate focal points near the imaging center, which then determines the position of the center of each facet uniquely by fulfilling the condition of the light path difference [cf. (8)]. The normal to the grating facet is typically along the direction in the middle of the incident ray and the diffracted ray (and the facet angle has a good fabrication tolerance). Thus, incident field with a wavelength different from the designed wavelength would also have three focal points with the same separation near its corresponding imaging centers (i.e., the noncentral channel also has good spectral response with a flat top, cf. Fig. 9), and the present three-focal-point method is effective over a wide spectrum range. To obtain an analytical formula for the optimal ratio of the facet numbers of the three interleaved subgratings, Gaussian
denotes the effective width of each Gaussian where subimaging field (the effective width is defined such that the of the peak value when ; the effective widths field is are approximately the same for the three subimages since the subgratings are interleaved), denotes the separation between the 2 outmost focal points (i.e., between the peaks of subimages , and denote the peak amplitudes of 1 and 3), and and are chosen to the three subimages, respectively. be identical in order to obtain a symmetric spectral response. For simplicity of analysis, the input and output waveguides are assumed to have the same width in the present paper. Then the effective width of each subimage (which is approximately equal to the effective width of the fundamental mode profile for the input waveguide according to the approximate 1 : 1 imaging theory of the demultiplexer) is approximately equal to the effective width of the fundamental mode profile for the output waveguide. The spectral response (5) can thus be written as shown in (10) at the bottom of the page. Its first-order derivative has the expression shown in (11) at the bottom of the page. , (corresponding to the central Since frequency for the channel) is always an extremum point of the spectral response. In order to make the spectral response flat at this central frequency, the curvature of the spectral response at . From (11), this point should be zero, i.e., one can obtain the following formula:
(12) The right-hand side of (12) becomes zero when at least one of the two terms in brackets is zero. The first term can be zero only is negative (i.e., the two subimages have when the ratio a phase difference of ), which indicates a destructive interference and large energy loss. To minimize the excess loss of the EDG chip, the subgratings should be designed in such a way and always have the same phase. Therefore, we can that only let the second term be zero, which gives (13) As discussed before, the above equation for the ratio of the peak amplitudes of the subimages also gives the optimal ratio of the facet numbers of the corresponding subgratings for reducing the ripple. From the analytical formula (13), we obtain the following two conclusions.
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1) To minimize the central ripple, a third (the central) focal . When , (13) point is necessary when is negative (which is cannot be satisfied unless not preferable). , a broadened spectral response with no 2) When ripple at the center can be obtained if the subgratings are interleaved according to the ratio given by the analytical formula (13).
IV. A DESIGN EXAMPLE AND NUMERICAL VERIFICATION A numerical example of a typical SiO EDG demultiplexer based on a Rowland circle construction is used to illustrate the present three-focal-point method. We choose the following pafor the effective refractive index of rameters: for the effective refractive the FPR and the core, m for the width of the input index of the cladding, m, and output waveguides, the central wavelength nm (i.e., GHz in the channel spacing m for the radius of the the frequency domain), (see Fig. 2 and [9]), Rowland circle, the grating angle , the total number of the facets the diffraction order is 701 [the facets are coated with a (reflective) metallic thin film], and the separation between two adjacent output wavegm (thus the dispersive coefficient has uides is m/GHz). The etched grating is coated a value of with metal at the backside, and thus the reflection coefficient . To verify our design, the scalar diffraction formulas (2) and (3) are used to simulate the imaging of the designed EDG demultiplexer, and (5) is used to calculate the spectral response. Note that these formulas do not adopt Gaussian approximations, which lead to the analytical formula (13) for designing the interleaved subgratings. The effective width of the fundamental mode profile of the m [1]. The 5 m waveguide is calculated to be
Fig. 4. Relation between the ratio E =E (normalized with the effective half-width w ).
and the half-separation a
relation (13) between the ratio of the peak magnitudes and the half-separation is plotted in Fig. 4. The ratio of the , facet numbers of subgratings are chosen to be close to and the three subgratings are interleaved. The ripple is defined as the maximal difference among the extremum points within the 3-dB passband. Fig. 5 shows the ripple level of the spectral response as the half-separation varies for both cases of two and three focal points. When the EDG demultiplexer has only two focal points, the ripple rises quickly when the half-separation increases (see the dashed line in Fig. 5). Using the present three-focal-point method, the ripple is always kept at a very low level (see the solid line in Fig. 5). It is well known that flattening of a spectral response will introduce some additional excess loss for the DWDM chip. Therefore, an appropriate passband width should be specified first in a design. In the present design method, a required passband width
(10)
(11)
SHI AND HE: A THREE-FOCAL-POINT METHOD FOR THE OPTIMAL DESIGN OF A FLAT-TOP PLANAR WAVEGUIDE DEMULTIPLEXER
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0
Fig. 5. Relation between the ripple and the half-separation a (normalized with the effective half-width w ).
Fig. 7. Relation between 1-dB bandwidth and the half-separation a (normalized with the effective half-width w ).
Fig. 6. Relation between the excess loss and the half-separation a (normalized with the effective half-width w ).
Fig. 8. The flattened spectral response for the designed EDG demultiplexer.
determines the half-separation , which then determines the optimal ratio of the facet numbers of the subgratings according to the analytical formula (13). Like all the other methods for passband flattening, the excess loss of an EDG demultiplexer designed with the present three-focal-point method increases as the passband is more flattened (i.e., as the half separation increases). Fig. 6 shows the excess loss level as increases (the solid line is the excess loss of a designed EDG demultiplexer using the present method and the dashed line is that for an EDG demultiplexer designed with 2 focal points). One sees that the present three-focal-point method does not introduce extra ex(the corresponding relative 1-dB cess loss when passband width can already reach up to 65% of the channel spacing) as compared to those methods with two focal points (in the numerical example shown below we have and the excess loss is a little bit lower than that for a corresponding EDG demultiplexer designed with a method with two focal points; see Fig. 8). The relative 1-dB bandwidth (divided
by the channel spacing) as varies is shown in Fig. 7. As a design example, we require the 1-dB bandwidth to be 60% of the channel spacing. From the solid line in Fig. 7, we obtain the m. From the corresponding half-separation analytical formula (13), we obtain the corresponding ratio of the (the ratio peak amplitudes of the subimages as of the facet numbers of the corresponding interleaved subgratin (8) can be ings is thus chosen to be 3 : 4). Therefore, expressed as
(14) is the imaging center of the designed wavelength. where The spectral response of the central channel for the designed EDG demultiplexer is shown by the solid line in Fig. 8. The 1-dB bandwidth, the ripple, and the excess loss for the flat-top EDG demultiplexer designed with the present three-focal-point
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Fig. 9. The flattened spectral responses of different channels for the designed EDG demultiplexer.
Fig. 10. The dispersion characteristic for the designed EDG demultiplexer.
method are 62.00 GHz, 0.013 dB, and 5.73 dB, respectively. As expected, the ripple for this flat-top EDG demultiplexer is indeed very small. A flattened spectral response for an EDG demultiplexer obtained by using two equally interleaved subgratings (i.e., two focal points) is shown by the dashed line in the same figure for comparison. The spectral response of the EDG demultiplexer obtained by using only two focal points has a ripple of 1.1869 dB (see the dashed line in Fig. 8; its 1-dB bandwidth and excess loss are 60.95 GHz and 5.32 dB, respectively). A demultiplexer with such a large ripple cannot be used in a practical DWDM system. Fig. 9 shows the spectral responses of the other seven channels with larger frequencies and a channel spacing of 100 GHz besides the central channel for the designed EDG demultiplexer. The spectral response for the seventh channel is still quite good, with a 1-dB passband of 61.95 GHz and a ripple of only 0.019 dB. The dispersion characteristic for the EDG demultiplexer designed with the present three-focal-point method is calculated [by using (6) and (7)] and shown in Fig. 10. From this figure, one sees that the designed EDG demultiplexer has a nearly perfect dispersion characteristic (less than 1 ps/nm through all the passband). (the separation of the peaks When the half-separation of the two outmost subimages is too large), the crosstalk will then become a main issue (instead of the ripple). The present three-focal-point method is thus effective in the range of , which corresponds to a 1-dB bandwidth from 45% to 85% of the channel spacing (this range is large enough to cover the passband of any practical flat-top demultiplexer).
The three subimages overlap with each other partly, and give a broadened imaging field. By using a Gaussian approximation and the 1 : 1 imaging method, an analytical formula has been derived for the optimal ratio of the peak amplitudes of the subimages (or the optimal ratio of the facet numbers of the interleaved subgratings). In the present optimal design, a required passband width determines the separation between the outmost peaks of the subimages, which then determines the optimal ratio of the facet numbers of the subgratings according to the derived analytical formula. The design method has been illustrated with a numerical example of a SiO EDG demultiplexer. The designed EDG demultiplexer has a nearly perfect dispersion characteristic besides the flat-top spectral response with virtually no ripple. Note that the present scalar diffraction theory is not capable of calculating the polarization-dependent loss (PDL) and an EDG may have a large PDL (particularly if the etched grating has no metal coating at the backside). Progress has been made toward reducing PDL in EDGs [14]. In the present paper, we assume that the backside of the etched grating is coated with . metal and thus we choose the reflection coefficient The present method can be generalized to the case of more than three focal points. The present method and explicit formulas are also applicable to other planar waveguide multiplexers/demultiplexers/routers, such as AWGs. For the case of a flat-top AWG, the ratio of the peak amplitudes of the subimages gives approximately the ratio of the waveguide numbers of the interleaved subarrays.
V. CONCLUSION In this paper, a three-focal-point method has been introduced for an optimal design of a flat-top EDG demultiplexer with minimal ripple. The etched grating has been divided into three interleaved subgratings with some appropriate ratio of facet numbers. Each subgrating forms a subimage near its focal center.
REFERENCES [1] M. K. Smit, “Phasar-based WDM-devices: Principles, design and applications,” IEEE J. Select. Topics Quantum Electron., vol. 2, pp. 236–250, 1996. [2] J.-J. He et al., “Integrated polarization compensator for WDM waveguide demultiplexers,” IEEE Photon. Technol. Lett., vol. 11, pp. 224–226, Feb. 1999. [3] K. Okamoto and A. Sugita, “Flat spectral response arrayed-waveguide grating multiplexer with parabolic waveguide horns,” Electron. Lett., vol. 32, no. 18, pp. 1661–1662, 1996.
SHI AND HE: A THREE-FOCAL-POINT METHOD FOR THE OPTIMAL DESIGN OF A FLAT-TOP PLANAR WAVEGUIDE DEMULTIPLEXER
[4] J. B. D. Soole et al., “Use of multimode interference couplers to broaden the passband of wavelength-dispersive integrated WDM filters,” IEEE Photon. Technol. Lett., vol. 8, pp. 1340–1342, Oct. 1996. [5] C. Dragone and L. Siliver, “Frequency routing devices having a wide and substantially flat passband,” U.S. Patent 5 706 377, May 2, 1995. [6] A. Rigny, A. Bruno, and H. Sik, “Multigrating method for flattened spectral response wavelength multi/demultiplexer,” Electron. Lett., vol. 33, no. 20, pp. 1701–1702, 1997. [7] Y. P. Ho, H. Li, and Y. J. Chen, “Flat channel-passband-wavelength multiplexing and demultiplexing devices by multiple-Rowland-circle design,” IEEE Photon. Technol. Lett., vol. 9, pp. 342–344, Mar. 1997. [8] J.-J. He, E. S. Koteles, and B. Humphreys, “Passband flattening in waveguide grating devices using phase-dithering,” Integrated Photonics Research, Vancouver, Canada, paper IFE2, July 17–19, 2002. [9] R. Marz and C. Cremer, “On the theory of planar spectrographs,” J. Lightwave Technol., vol. 10, pp. 2017–2022, Dec. 1992. [10] A. C. McGreer, “Diffraction from concave gratings in planar waveguides,” IEEE Photon. Technol. Lett., vol. 7, pp. 324–326, Mar. 1995. [11] J.-J. He, B. Lamontagne, A. Delage, L. Erickson, M. Davies, and E. S. Koteles, “Monolithic integrated wavelength demultiplexer based on a waveguide Rowland circle grating in InGaAsP/InP,” J. Lightwave Technol., vol. 16, pp. 631–638, 1998. [12] D.-K. Han et al., “Low loss AWG demultiplexer with flat spectral response,” U.S. Patent, 6 188 818 B1, Feb. 13, 2001. [13] M. C. Parker and S. D. Walker, “Adaptive chromatic dispersion controller based on an electro-optically chirped arrayed-waveguide grating,” in Proc. Optical Fiber Communication Conf., vol. 2, 2000, pp. 257–259.
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[14] S. Janz et al., “The scalable planar waveguide component technology: 40 and 256-channel echelle grating demultiplexers,” in Proc. Integrated Photonics Research, Vancouver, Canada, July 17–19, 2002, paper IFE1.
Zhimin Shi was born in June 1979. He received the B.Sc. degree in optical engineering from Zhejiang University, China, in 2001. He is currently working toward the M.S. degree at the same university. He is with the Centre for Optical and Electromagnetic Research, Zhejiang University. His research interests are in the areas of design, optimization, and fabrication of planar waveguide devices for optical communication systems.
Sailing He (M’92–SM’98) received the Licentiate of Technology and Ph.D. degree in electromagnetic theory from the Royal Institute of Technology, Stockholm, Sweden, in 1991 and 1992, respectively. He has worked at the Royal Institute of Technology since 1992. He has also been with the Centre for Optical and Electromagnetic Research of Zhejiang University, China, since 1999 as a Special Professor appointed by the Ministry of Education of China. He has authored one monograph and over 140 papers in refereed international journals. He is also the Chief Scientist for the Joint Laboratory of Optical Communications of Zhejiang University.
