OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 19 — Sep. 18, 2006
  • pp: 8753–8764
« Show journal navigation

Principles and application of reduced beat length in MMI couplers

Lung-Wei Chung, San-Liang Lee, and Yen-Juei Lin  »View Author Affiliations


Optics Express, Vol. 14, Issue 19, pp. 8753-8764 (2006)
http://dx.doi.org/10.1364/OE.14.008753


View Full Text Article

Acrobat PDF (481 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

A unified method was proposed to reduce the beat length of a multimode interference (MMI) coupler. By properly adjusting the phase difference of the N-fold images, the mode evolution is changed to generate self-images at a much shorter distance. The effect of adjusting the phase difference can be regarded as dividing the original MMI coupler into multiple sub-MMI couplers. Such an effect can be applied for both symmetric- and paired-interference cases. We applied the principle to design compact optical splitters operating at dual wavelength bands. The simulation shows that excellent performance with reduced coupler length can be obtained for splitters operating at both 1.3 and 1.55 µm bands.

© 2006 Optical Society of America

1. Introduction

For many applications, the IOCs need to work for multiple wavelength bands. Due to the wavelength dependency, the components based on MMI couplers usually require the device length to be a common multiple of the beat lengths for the wavelength bands. This results in difficulties for designing the MMI-based circuits when the wavelength bands are far apart. For instance, due to the weak wavelength dependency for MMI couplers, it takes a large common multiple and needs a long device for constructing a multi-band power splitter or multiplexers [9

9. K.-C. Lin and W.-Y. Lee, “Guided-wave 1.3/1.55µm wavelength division multiplexer based on multimode interference,” Electron. Lett. 32, 1259–1261 (1996). [CrossRef]

].

The effect of adjusting the phase difference can also be regarded as dividing the original MMI coupler into multiple sub-MMI couplers. The effective width of the sub-MMI is reduced to a fraction of the original MMI width. The principle of phase adjustment and the effect of reduced beat length will be discussed for both symmetric- and paired-interference cases. Moreover, the effect of reduced beat length is applied to design MMI splitters operating at dual wavelength bands.

2. Operation principles

2.1 Field representation in MMI

For easy reference, some of the basic formulas for MMI couplers are summarized below. The operation of MMIs device is based on the self-imaging effect of multimode interference [13

13. O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Amer. 63, 416–419 (1973). [CrossRef]

]. From the superposition of the guided modes of a multimode waveguide, as shown in fig. 1, the local field distribution at any position along a MMI device can be given by:

Φ(x,z)=mcmφm(x)exp(jβmz),
(1)

where cm denotes the expansion coefficient for the m-th eigenmode of which the field distribution is φm and the propagation constant is βm. The coordinates of x and z stand for the transverse and longitudinal directions, respectively. βm can be expressed as:

βm=2πnrλ0(m+1)2πλ04nrWe2,
(2)

where λ 0 and nr denote the free-space wavelength and the refractive index of the core waveguide, respectively. We is the effective width that includes the Goos-Hanschen shifts at the waveguide boundaries. The beat length Lπ is defined as [8

8. L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995). [CrossRef]

]:

Lπ=πβ0β14nrWe23λ0.
(3)

φm(x)={sin[(m+1)πxWe]form=oddcos[(m+1)πxWe]form=even,
(4)

Therefore, the field profile given by Eq. (1) can be rewritten by replacing φm with Eq. (4).

Fig. 1. The modal field distribution of a MMI coupler [8]. Symbol m represents the number of mode. The squares and circles mark the zero-crossings of the non-excited modes for the paired- and symmetric-type excitation, respectively.

2.2 Principles of reduced beat length

Since the beat length is squarely proportional to the MMI width, it can be reduced by separating the MMI into multiple isolated regions along the transverse plane. With the approximate mode fields given in Eq. (4), a MMI coupler can be treated as if it is bounded by perfect conductors. Thus, inserting a perfect conductor at the middle of the MMI will turn it into two isolated half-width MMIs. With this approach, the self imaging occurs in a quarter of period than the original one.

The effect of inserting a perfect conductor plane at the middle of a MMI can be obtained instead by changing the phase of the images to support only the anti-symmetric (odd) modes. This can be performed at the longitudinal positions (z=0) where two self images are formed. Assume that one of the two images is represented as Φ(x,0), as given in Eq. (5). The two images can be treated as the point sources for generating the following wave propagation. Therefore, the field evolution can be regarded as the superposition of responses from the two point sources. The relative phases of the N-fold images can be derived for the general or symmetric-interference cases [16

16. M. Bachmann, P. A. Besse, and H. Melchior, “General self-imaging properties in N×N multimode interference couplers including phase relations,” Appl. Opt. 33, 3905–3911 (1994). [CrossRef] [PubMed]

, 17

17. E. R. Thoen, L. A. Molter, and J. P. Donnelly, “Exact modal analysis and optimization of N×N×1 cascaded waveguide structures with multimode guiding sections,” IEEE J. Quantum Electron. 33, 1299–1307 (1997). [CrossRef]

]. By applying a phase change to one image such that the two images are out-of-phase, the field distribution can be written as:

Ψ(x)=Φ(x,0)+Φ(x,0)ejπ=m:oddcmφm(x)=m:oddcmsin((m+1)πWex),
(5)

Ψ(x)=m:evendmcos((m+1)πWe2x)+m:odddmsin((m+1)πWe2x),
(6)

where the expansion coefficient is given by

dm={cmcos[(m+1)π2]form=oddcmsin[(m+1)π2]form=even,
(7)

βm=2πnrλ0(m+1)2πλ04nr(We2)2,
(8)

Therefore, the propagation constants are separated by multiples of πλ 0/4nr(We′)2 with We′=We/2. Eqs (6) and (8) prove that the net effect of applying the π-phase shift is to divide the original MMI into two isolated regions. Since the effective width is halved, the beat length becomes Lπ=L π/4. The above principles can be applied to general types or restricted-interference types of MMI couplers. The phase-shift scheme can also be applied to the longitudinal positions where the N-fold (N>2) images are formed.

In the case of N=3, if phase changes are applied to the local images so that the adjacent images are out-of-phase, the resultant wave evolution can be treated as if the MMI is divided into three isolated regions with an effective width of We′=We/3. This can be easily verified for the case with three equally-spaced images. Assume that the images appear at x=-We/3, 0, We/3, respectively. When phase changes are applied to the three images to provide relative phase shifts of π, 0, and, π, respectively, the resultant field can be expressed as:

Ψ(x)=Φ(x,0)+Φ((xWe3),0)ejπ+Φ((x+We3),0)ejπ
(9)

By substituting Φ(x,0) with Eq. (1) and (4) for z=0, it is easy to show that the expansion coefficient is not vanishing only if m=2, 5, 8, … Let m=3m′+2, Eq. (9) can be rewritten as:

Ψ(x)=m=0,2,4,3cmcos((m+1)πWe3x)+m=1,3,5,3cmsin((m+1)πWe3x),
(10)

It can also be easily shown that, with m=3m′+2, the propagation constant can be written as:

βm=2πnrλ0(m+1)2πλ04nr(We3)2.
(11)

Similar derivations can be applied to the cases of N=4 or above to show the same effects from the phase shifts. Therefore, with appropriate phase shifts on the N-fold images, the effective width and beat length can be reduced to We/N and Lπ/N2, respectively.

The phase change can be applied by varying the refractive index of the waveguide around the imaging spots [18

18. S. Nagai, G. Morishima, H. Inayoshi, and K. Utaka, “Multimode interference photonic switches (MIPS),” J. Lightwave Technol. 20, 675–681 (2002). [CrossRef]

]. For a phase shift region, the phase change can be written as

Δθ=2πλ0·Δnr·Lm.
(12)

where Lm is the length of the phase shift region and Δnr is the change in the refractive index. The size and location of the phase shift region could affect the wave evolution in a MMI. In our design, the phase shift region is placed on the self-imaging spots. Therefore, the design is not sensitive to the size variation as long as the region covers the spot and does not overlap with other self-imaging spots [18

18. S. Nagai, G. Morishima, H. Inayoshi, and K. Utaka, “Multimode interference photonic switches (MIPS),” J. Lightwave Technol. 20, 675–681 (2002). [CrossRef]

]. The tolerance on the location depends on the beat length of the MMI coupler.

In the following analysis, the effect of phase shifts will be applied to both paired- and symmetric-interference types of MMI couplers [8

8. L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995). [CrossRef]

]. The beam propagation method (BPM) developed by RSoft, Inc. is used to simulate the wave evolution in a MMI coupler. Compared to other numerical methods, the BPM method is very robust for different device structure and can be calculated with less computation time. All the simulations are under the transparent boundary condition. The designs are based on the typical silica waveguide structure. The waveguide cross-section is 6×6µm2 for the single mode access waveguide and the refractive index difference between the core and the cladding is 0.75%.

3. Restricted interference cases

3.1. Modified symmetric interference

Fig. 2. Simulated light evolution along the MMI coupler for the symmetric-interference type. The dotted squares mark the phase shift regions. From left to right, the plots are for the conventional MMI and the ones with phase shifts on the two- to five-fold images.

Table 1. Phase differences before and after applying phase shifts for symmetric interference cases

table-icon
View This Table
| View All Tables

3.2. Modified paired-interference

For the paired-interference case, the expansion coefficients cm=0 for m=2, 5, 8…; and the periodic distance is Lc=Lπ. By applying a π/2-phase shift to one of the two-fold images and make them to have π phase difference, the modified paired-interference can also be described by Eq. (6). The net effect looks as if the MMI is divided into two sub-MMI’s. The wave evolution in each sub-MMI possesses the property of paired-interference. When the same principle is applied to the N-fold images, L′c/Lc/N2.

Figure 3 depicts the light propagation for the original and the modified paired-interference cases. Again, the net effect of applying the phase shifts is like adding mirrors to separate the MMI to N sub-MMI’s. The above schemes of applying phase shifts result in point sources with alternating 0 and π phases. The out-of-phase sources generate anti-symmetric wave evolution patterns that vanish at the intermediate planes. Another interesting arrangement of the phases is to make the N-fold images to have symmetric phases such that the paired-interference property is turned into the symmetric-interference case. This changes the self-imaging patterns and the periodic distance. A simulation example is shown in Fig. 3(e) by using the phase shifts listed in Table 2. The periodic distance is reduced to Lc/4.

Several combinations of the restricted interference types can be realized by properly adjusting the phase shifts. For example, the phases of the N-fold images can also be adjusted to include both symmetric- and paired-interference cases. Figure 3(f) depicts the wave evolution when the 4-fold images are adjusted to have two pairs of phases that have a phase difference of π, as listed in Table 2. The results reveal that the periodic interval can be reduced to Lc/16 though the effective width is only reduced to one half. From the above results, there are a variety of phase shift combinations to alter the periodic distance of a MMI coupler. Since the beat lengths for the symmetric-type and paired-type excitations are different, the proposed method offers flexibility in choosing the periodic distance for a MMI coupler.

Fig. 3. Wave evolution along a MMI for paired-interference type excitation without phase shift (a), with out-of–phase phase shifts on 2-fold (b), 3-fold (c), and 4-fold images (d), with in-phase phase shifts on the 2-fold images (e), and with a pair of 0 and π phase shifts on the 4-fold images (f). The squares indicate the phase adjustment spots.

Table 2. Phase differences before and after applying phase shifts for paired-interference type excitation

table-icon
View This Table
| View All Tables

4. Applications

The method of reducing the beat length of a MMI is applied to design dual-band optical splitters. Figure 4 shows the device schematic. In principle, 1×N optical splitters can easily be realized with MMI couplers. From Eq. (3), the beat length of a MMI coupler depends on the wavelength. To operate simultaneously at two wavelengths (λ 1 and λ2), the total length of the MMI coupler should satisfy:

LMMI=q1·(Lc,λ1N)q2·(Lc,λ2N)
(13)

where Lc,λ1 and Lc,λ2 are the beat lengths for the λ1 and λ2 wavelengths, respectively. q1 and q2 are integers with no common divisor with N. Using a conventional MMI structure to realize a dual-band 1×N power splitter would typically require a long device, because the beat length is only weakly wavelength dependent. The difference between Lc,λ1 and Lc,λ2 is usually small, so the values q1 and q2 are large.

Fig. 4. Schematic diagram of the 1×N dual-band optical power splitter.

By applying the scheme of reduced beat length, the periodic distance can be shortened. There may be several options to achieve this goal. Here, we proposed to adjust the phase shift for one wavelength band but keep the phase of the other wavelength unchanged. By doing so, it is possible to obtain a smaller common multiple of the coupler lengths such that uniform power splitting can be achieved for both wavelengths with a shorter MMI.

As shown in fig. 4, the MMI coupler is divided into two sections and LMMI=Ld1+Ld2. When the length of the first region Ld1 satisfies:

Ld1=s1·(Lc,λ1N)s2·Lc,λ2,
(14)

the N-fold images of λ1 light and the single image of λ2 light will appear at the same z position. s1 and s2 are integers that have no common divisor with N. Under such a condition, applying the phase shift on the N-fold images for λ1 except the single-image spot for λ2 can affect only the phase difference of λ1 light. Therefore, the N-fold images of λ1 light will repeat with a periodic distance of Lc,λ1/N, while the periodic distance of λ2 light remains unchanged. When the length of the second part Ld2 obeys this relation:

Ld2=s1·(Lc,λ1N2)s2·(Lc,λ2N),
(15)

the N-fold images of λ1 light and the N-fold images of λ2 light will appear at the same z position. Under such a condition, 1×N power splitting can be achieved for both wavelengths.

Similar equations as Eqs. (14) and (15) should work when λ1 and λ2 are interchanged. Thus, the alternative expressions for Ld1 and Ld2 are:

Ld1=s3·(Lc,λ2N)s4·(Lc,λ1)
Ld2=s3·(Lc,λ2N2)=s4·(Lc,λ1N).
(16)

where s3 and s4 are integers that have no common divisor with N.

After searching the solutions of Eqs. (14) to (16) and obtaining the smallest set of the s-integers, a compact dual-band 1×N optical power splitter can be realized. The uniformity of power splitting can be evaluated by using the power imbalance parameter:

Pim=10·log(POmaxPOmin)
(17)

where POmax and POmin are the maximum and minimum powers at the output waveguides.

4.1 1×2 dual-band splitter

Without loss of generality, let λ1 and λ2 be 1.3 µm and 1.55 µm, respectively. For obtaining compact devices, the s-integers and the MMI width should be optimized. Though a smaller MMI width is desired to shorten the beat length, the width should be large enough to support as many modes as required for self-imaging. From numerical simulation, we limit the MMI width to WM≧38µm. Moreover, the solution for the shortest MMI does not necessarily occur at the smallest WM. After calculation, the optimal solution for each case is listed in Table 3.

Table 3. Optimal solutions for 1×2 dual-band splitters

table-icon
View This Table
| View All Tables

The 1×2 splitter is designed with the paired-interference type of couplers. The symmetric-interference type can also be used. For the standard design, the best values of q2 and q1 are 43 and 37, respectively, when WM=38µm. It requires a long MMI section (LMMI=45045 µm). By using the phase-shift scheme listed in Table 2, the shortest device is found while (s3, s4)=(7, 3) and WMMI=45 µm. The MMI coupler is 15057 µm long under this condition. Figure 5(a) and (b) shows the light propagating pattern along the MMI coupler for the shortest solution. Power splitting can be achieved for the two wavelengths. Figure 6(a) illustrates the simulated imbalance and excess loss as a function of the deviation in device length (ΔLMMI). The solid and dashed lines show the performance for the TE and TM polarization, respectively. The imbalance is below 0.15 and 0.25 dB for the 1.3- and 1.55-µm wavelengths, respectively. The excess loss is below 0.1 dB for both wavelengths. Among the solutions listed in Table 3, the one with the shortest device has better performance in terms of the excess loss and imbalance.

The three-dimensional BPM simulation indicates very low polarization-dependent loss (PDL) for the design at the optimal length. The low PDL is due to our design in the waveguide cross-section and needs to be further verified experimentally. The same simulation method has been applied to verify the performance of a polarization converter [20

20. J. Z. Huang, R. Scarmozzio, G. Nagy, M. J. Steel, and R. M. Osgood Jr., “Realization of a compact and single-mode optical passive polarization converter,” IEEE Photon. Technol. Lett. 12, 317–319 (2000). [CrossRef]

]. The results indicate that the splitting function has a large tolerance on device length. Figure 6(b) shows that the wavelength tolerance is about 20 nm for an excess loss <0.5dB. The design is more sensitive to the variation of MMI width [19

19. M. Rajarajan, B. M. A. Rahman, T. Wongcharoen, and K. T. V. Grattan, “Accurate analysis of MMI devices with two-dimensional confinement,” J. Lightwave Technol. 14, 2078–2084 (1996). [CrossRef]

]. For ±0.2 µm of width fluctuation, the imbalance and excess loss rise to 0.8 dB and 1.8 dB, respectively.

Fig. 5. Light intensity along the MMI coupler at 1.3-µm (a) and 1.55-µm (b). The phase shift region is indicated with a rectangulat mark. The phase shift region locates at (x, z)=(11.25, 9965.5) µm with an area of 22.5×145 µm2.
Fig. 6. Imbalance and excess loss versus the deviation in device length (a) and deviation in wavelength (b).

4.2. 1×3 dual-band splitter

For the design of 1×3 power splitters, the symmetric-interference type of MMI coupler is used as the example. The optimal parameters for the conventional and modified device designs are calculated and summarized in Table 4.

Table 4. Optimal design parameters for 1×3 dual-band splitter

table-icon
View This Table
| View All Tables

For the conventional design, the shortest device occurs when (q2, q1)=(43, 37) and WMMI=38 µm, which gives a MMI length of 23100 µm for constructing a dual band splitter. By using the phase-shift scheme listed in Table 1, The best solution is obtained when (s3, s4)=(7, 2) and WMMI=52 µm, which gives the shortest MMI coupler (LMMI=8802µm). The MMI with phase shifts can be much shorter than the conventional one. Figure 7 shows the power evolution along the MMI coupler for the two wavelength bands. The 1×3 power splitting function can be obtained for both wavelengths. Figure 8 illustrates the simulated imbalance and excess loss as a function of the device length (ΔLMMI) and wavelength. The imbalance is below 0.01 and 0.04 dB for the 1.3- and 1.55-µm wavelengths, respectively. The excess loss is below 0.06 dB for both wavelengths. The wavelength tolerance for less than 0.5dB of excess loss is larger than 30 nm. Again, from Table 4, the solution with the shortest device has better performance in terms of the excess loss and imbalance.

5. Conclusions

A novel scheme of reducing the beat length of MMI couplers is demonstrated in this work. With proper phase shifts to the self-imaging spots, the original MMI waveguide becomes one having the properties of multiple sub-MMI’s. The effect is like inserting mirrors in the MMI couplers. It can be applied for both symmetric-interference and paired-interference types of couplers. Such concept is applied to design the 1×N optical power splitters operated at dual bands. It helps to shorten the device length. The simulation of 1×2 and 1×3 power splitters shows very good performance with much shorter device. The devices have also very large tolerance on the deviation of device length tolerance and operation wavelength. More splitting ports can be designed with the same principle or by cascading the designed 1×2 and 1×3 power splitters.

Fig. 7. Light intensity along a 1×3 MMI coupler of symmetric-interference type at 1.3- (a) and 1.55-µm (b). The phase shift regions are indicated with rectangulat marks. The phase shift regions locate at (x, z)=(17.3, 6492.6) µm and (x, z)=(-17.3, 6492.6) µm, respectively, with an area of 17.3×198 µm2
Fig. 8. Imbalance and excess loss versus the deviation in device length (a) and deviation in wavelength (b).for the 1×3 MMI coupler. The solid and dashed lines show the performance for the TE and TM polarization, respectively.

Acknowledgments

This work was supported in part by Ministry of Economic Affairs, R.O.C. under contract No. 91-EC-17-A-07-S1-0011 and by the National Science Council under contract no NSC94-2213-E011-016.

References and links

1.

H. Sasaki, E. Shki, and N. Mikoshiba, “Propagation characteristics of optical guided wave in asymmetric branching waveguides,” IEEE J. Quantum Electron. QE-17, 1051–1058 (1981). [CrossRef]

2.

M. Belanger, G. L. Yip, and M. Haruna, “Passive planar multibranch optical power divider: Some design considerations,” Appl. Opt. 22, 2283–2289 (1983). [CrossRef]

3.

M. Haruna and J. Koyama, “Electrooptic branching waveguide-switch and the application to 1×4 optical switching network,” J. Lightwave Technol. LT1-1, 233–247 (1983).

4.

R. Baets and P. E. Lagasse, “Calculation of radiation loss in integrated-optic tapers and Y-junctions,” Appl. Opt. 21, 1972–1978 (1982). [CrossRef] [PubMed]

5.

O. Mikami and S. Zembutsu, “Coupling-length adjustment for an optical direction coupler as a 2×2 switch,” Appl. Phys. Lett. 35, 38–40 (1979). [CrossRef]

6.

H. A. Haus and C. G. Fonstad, “Three waveguide couplers for improved sampling and filtering,” IEEE J. Quantum Electron. QE-17, 2321–2325 (1981). [CrossRef]

7.

M. Rajarajan, B. M. A. Rahman, and K. T. V. Grattan, “A rigorous comparison of the performance of directional couplers with multimode interference devices,” J. Lightwave Technol. 17, 243–248 (1999). [CrossRef]

8.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995). [CrossRef]

9.

K.-C. Lin and W.-Y. Lee, “Guided-wave 1.3/1.55µm wavelength division multiplexer based on multimode interference,” Electron. Lett. 32, 1259–1261 (1996). [CrossRef]

10.

Y.-J. Lin and S.-L. Lee, “InP-based 1.3/1.55µm wavelength demultiplexer with multimode interference and chirped grating,” Opt. and Quantum Electron. 34, 1201–1212 (2002). [CrossRef]

11.

Y. Ma, S. Park, L. Wang, and S. T. Ho, “Ultracompact multimode interference 3-dB coupler with strong lateral confinement by deep dry etching,” IEEE Photon. Technol. Lett. 12, 492–494 (2000). [CrossRef]

12.

Y. Gottesman, E. V. K. Rao, and B. Dagens, “A novel design proposal to minimize reflections in deep-ridge multimode interference couplers,” IEEE Photon. Technol. Lett. 12, 1662–1664 (2000). [CrossRef]

13.

O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Amer. 63, 416–419 (1973). [CrossRef]

14.

J. M. Heaton and R. M. Jenkins, “General matrix theory of self-imaging in Multimode Interference (MMI) couplers,” IEEE Photon. Technol. Lett. 11, 212–214 (1999). [CrossRef]

15.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, and K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992). [CrossRef]

16.

M. Bachmann, P. A. Besse, and H. Melchior, “General self-imaging properties in N×N multimode interference couplers including phase relations,” Appl. Opt. 33, 3905–3911 (1994). [CrossRef] [PubMed]

17.

E. R. Thoen, L. A. Molter, and J. P. Donnelly, “Exact modal analysis and optimization of N×N×1 cascaded waveguide structures with multimode guiding sections,” IEEE J. Quantum Electron. 33, 1299–1307 (1997). [CrossRef]

18.

S. Nagai, G. Morishima, H. Inayoshi, and K. Utaka, “Multimode interference photonic switches (MIPS),” J. Lightwave Technol. 20, 675–681 (2002). [CrossRef]

19.

M. Rajarajan, B. M. A. Rahman, T. Wongcharoen, and K. T. V. Grattan, “Accurate analysis of MMI devices with two-dimensional confinement,” J. Lightwave Technol. 14, 2078–2084 (1996). [CrossRef]

20.

J. Z. Huang, R. Scarmozzio, G. Nagy, M. J. Steel, and R. M. Osgood Jr., “Realization of a compact and single-mode optical passive polarization converter,” IEEE Photon. Technol. Lett. 12, 317–319 (2000). [CrossRef]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(230.1360) Optical devices : Beam splitters
(230.7370) Optical devices : Waveguides

ToC Category:
Optical Devices

History
Original Manuscript: June 22, 2006
Revised Manuscript: September 7, 2006
Manuscript Accepted: September 10, 2006
Published: September 18, 2006

Citation
Lung-Wei Chung, San-Liang Lee, and Yen-Juei Lin, "Principles and application of reduced beat length in MMI couplers," Opt. Express 14, 8753-8764 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-19-8753


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. H. Sasaki, E. Shki, and N. Mikoshiba, "Propagation characteristics of optical guided wave in asymmetric branching waveguides," IEEE J. Quantum Electron. 17, 1051-1058 (1981). [CrossRef]
  2. M. Belanger, G. L. Yip, and M. Haruna, "Passive planar multibranch optical power divider: Some design considerations," Appl. Opt. 22, 2283-2289 (1983). [CrossRef]
  3. M. Haruna and J. Koyama, "Electro-optic branching waveguide-switch and the application to 1 × 4 optical switching network," J. Lightwave Technol. 1, 233-247 (1983).
  4. R. Baets and P. E. Lagasse, "Calculation of radiation loss in integrated-optic tapers and Y-junctions," Appl. Opt. 21, 1972-1978 (1982). [CrossRef] [PubMed]
  5. O. Mikami and S. Zembutsu, "Coupling-length adjustment for an optical direction coupler as a 2 × 2 switch," Appl. Phys. Lett. 35, 38-40 (1979). [CrossRef]
  6. H. A. Haus and C. G. Fonstad, "Three waveguide couplers for improved sampling and filtering," IEEE J. Quantum Electron. 17, 2321-2325 (1981). [CrossRef]
  7. M. Rajarajan, B. M. A. Rahman, and K. T. V. Grattan, "A rigorous comparison of the performance of directional couplers with multimode interference devices," J. Lightwave Technol. 17, 243-248 (1999). [CrossRef]
  8. L. B. Soldano and E. C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995). [CrossRef]
  9. K.-C. Lin and W.-Y. Lee, "Guided-wave 1.3/1.55μm wavelength division multiplexer based on multimode interference," Electron. Lett. 32, 1259-1261 (1996). [CrossRef]
  10. Y.-J. Lin and S.-L. Lee, "InP-based 1.3/1.55μm wavelength demultiplexer with multimode interference and chirped grating," Opt. and Quantum Electron. 34, 1201-1212 (2002). [CrossRef]
  11. Y. Ma, S. Park, L. Wang, and S. T. Ho, "Ultracompact multimode interference 3-dB coupler with strong lateral confinement by deep dry etching," IEEE Photon. Technol. Lett. 12, 492-494 (2000). [CrossRef]
  12. Y. Gottesman, E. V. K. Rao, and B. Dagens, "A novel design proposal to minimize reflections in deep-ridge multimode interference couplers," IEEE Photon. Technol. Lett. 12, 1662-1664 (2000). [CrossRef]
  13. O. Bryngdahl, "Image formation using self-imaging techniques," J. Opt. Soc. Amer. 63, 416-419 (1973). [CrossRef]
  14. J. M. Heaton and R. M. Jenkins, "General matrix theory of self-imaging in Multimode Interference (MMI) couplers," IEEE Photon. Technol. Lett. 11, 212-214 (1999). [CrossRef]
  15. J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, and K. P. Hilton, "Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides," Appl. Phys. Lett. 61, 1754-1756 (1992). [CrossRef]
  16. M. Bachmann, P. A. Besse, and H. Melchior, "General self-imaging properties in N×N multimode interference couplers including phase relations," Appl. Opt. 33, 3905-3911 (1994). [CrossRef] [PubMed]
  17. E. R. Thoen, L. A. Molter, and J. P. Donnelly, "Exact modal analysis and optimization of N×N×1 cascaded waveguide structures with multimode guiding sections," IEEE J. Quantum Electron. 33, 1299-1307 (1997). [CrossRef]
  18. S. Nagai, G. Morishima, H. Inayoshi, and K. Utaka, "Multimode interference photonic switches (MIPS)," J. Lightwave Technol. 20, 675-681 (2002). [CrossRef]
  19. M. Rajarajan, B. M. A. Rahman, T. Wongcharoen, and K. T. V. Grattan, "Accurate analysis of MMI devices with two-dimensional confinement," J. Lightwave Technol. 14, 2078-2084 (1996). [CrossRef]
  20. J. Z. Huang, R. Scarmozzio, G. Nagy, M. J. Steel, R. M. Osgood, and Jr., "Realization of a compact and single-mode optical passive polarization converter," IEEE Photonics Technol. Lett. 12, 317-319 (2000). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited