Diffraction engineering of multimode waveguides using computer-generated planar holograms

Shuo-Yen Tseng

doi:10.1364/OE.17.021465

1. Introduction

Integrated optical devices with self-imaging (SI) properties are widely used in optical communications for switching and routing applications [1]. It is widely known that graded index waveguides can produce periodic images of an object since the propagation constants of the propagating modes are equally spaced [2]. SI in uniform index planar optical waveguides was first suggested by Bryngdahl [3] and was demonstrated by Ulrich [4]. Since then, integrated optical components based on multimode interference (MMI) effects [5] have found wide use in power splitter applications because of their large fabrication tolerance, large bandwidth, and compact size [6]. Recently, it was shown that SI could also occur in evanescently coupled waveguide arrays with properly engineered coupling coefficients [7–10]. These advances are made possible by the recent progress in microfabrication technology, which allows the fabrication of waveguide arrays with precision-controlled spacing and geometry. The advances in microfabrication technology have also made possible the recent efforts in creating refractive index perturbations on planar waveguides as holograms to perform optical signal processing [11] and multiplexing/demultiplexing functionalities [12].

Traditionally, multimode waveguides can only have limited split ratios at specific lengths. Geometrical variations are required to obtain a free choice of split ratios [13–15]. In the past, we have presented a matrix analysis based on the coupled-mode theory [16] to show that arbitrary unitary transformations can be generated with computer-genearted planar holograms (CGPHs) on multimode waveguides [17]. In these devices, multiplexed long-period gratings are fabricated on a multimode waveguide to mix and transform the optical modes in the multimode guiding structure to generate the desired transformations. Based on the analysis, we have demonstrated variable-ratio power splitters with fixed dimensions using CGPHs on MMI couplers at intermediate imaging lengths [18,19]. These devices are designed for specific functionalities as determined by the CGPHs, however, the SI properties no longer applies to these multimode waveguides perturbed by the CGPHs.

The ability to control the SI lengths of multimode waveguides would provide greater flexibility in device design. However, the SI length of a multimode waveguide is uniquely determined by its diffraction property and is generally related to the waveguide width [5]. For a fixed width device, the linear relationship between the SI length and the refractive index indicates that only modest variations in the SI length can be obtained by a uniform refractive index change. In this work, we consider the effects of CGPHs on the diffraction property of multimode waveguides, particularly on the SI property. Using eigenmode analysis of the coupled-mode equations, we show that the SI properties of multimode waveguides can be engineered by the application of CGPHs. In particular, we show that the SI length can be varied by a greater extent using the CGPHs while maintaining the width of the multimode waveguide.

2. Coupled-mode theory

We briefly review the properties of multimode waveguides. With the usual assumptions that the multimode waveguide is single-moded in the transverse direction, and the effective index method can be used to separate the transverse and lateral components of the electric field [5], the electric field in a multimode waveguide with M guided modes can be written as:

E (x, z) = \sum_{1}^{M} A_{m} (z) ϕ_{m} (x) \exp (- j β_{m} z),

where A_m represents the mode amplitude, ϕ_m represents the lateral mode profile, and β_m is the propagation constant in the z direction corresponding to the mth mode. The modes are normalized with the following condition [16]:

\int ϕ_{m} ϕ_{n} d x = \frac{2 ω μ}{β_{n}} δ_{m n} .

In an unperturbed multimode waveguide, the propagation constants can be approximated by:

β_{m} ≅ k_{0} n_{r} - A m^{2} = k_{0} n_{r} - \frac{m^{2} π λ}{4 n_{r} W_{e}^{2}} (m = 1 \dots M),

where λ is the free-space wavelength, k₀ = 2π/λ, n_r is the core effective index, and W_e is the effective width [5]. The quadratic dependence of the propagation constants with respect to mode number m results in the SI properties of multimode waveguides. A discussion on the image formation properties and possible split ratios can be found in [20]. We now consider the effect of CGPHs on the SI properties of the multimode waveguides, particularly on the propagation constants of the eigenmodes of the perturbed multimode waveguide.

Similar to [17], we express the CGPH as an index perturbation on the multimode waveguide. It can be written as multiplexed long-period gratings by the following expression

Δ n (x, z) = \sum_{m \neq n} f_{m n} (x) \exp (- j 2 π z / Λ_{m n}),

with the constraint that f_mn = (f_nm)*. The grating period is related to the propagation constants of the unperturbed modes by Λ_mn = 2π/(β_m-β_n), which couples modes m and n. Using the coupled- mode theory (CMT), the evolution of the field of the mth mode E_m(z) = A_m(z)exp(-jβ_mz) satisfies

\frac{d E_{m}}{d z} = - j β_{m} E_{m} - j \sum κ_{m n} E_{m n},

where the coupling coefficients are given by

κ_{m n} = \frac{ω ε_{0} n_{r}}{2} \int f_{m n} (x) ϕ_{m} ϕ_{n} d x (m \neq n) .

A matrix form can be used for (5)

\frac{d E}{d z} = - j K E,

where the mnth element of the M × M matrix K is κ_mn, the diagonal entries of K is [β₁, β₂,…, β_M], E = [E₁(z), E₂(z),…, E_M(z)]^T, and T denotes the transpose. Notice that the matrix K in (7) is Hermitian.

We are interested in finding the eigenmodes of (7). The eigenmmodes can be written as

U_{m} (z) = U_{m} (0) \exp (- j γ_{m} z) .

The resulting eigenvalue problem is

K U_{m} = γ_{m} U_{m} .

The mth eigenmode of K consists of a superposition of the unperturbed waveguide modes ϕ_m(x)’s and propagates with a corrected phase factor γ_m, which is the eigenvalue corresponding to the mth eigenmode. An input field can be decomposed into the orthogonal set formed by these eigenmodes. When the perturbed system with the corrected phase factors γ_m have a quadratic spectrum as the original system, SI can be observed. With the CGPH in (4), we have the ability to engineer the spectrum of the multimode waveguide, thus changing its SI properties. In particular, we use the example below to show that the SI lengths can be engineered while the waveguide width remains fixed.

3. Device design example

For SI to occur, the γ’s need to have a quadratic relation as the β’s in (3):

γ_{m} = β^{'} - A^{'} m^{2} .

From [5], we know that the beat length of the unperturbed multimode waveguide can be calculated using (3) as

L_{π} = \frac{π}{β_{1} - β_{2}} ≅ \frac{4 n_{r} W_{e}^{2}}{3 λ} = \frac{π}{3 A},

and that single images and mirror images of the input electric field can be observed at lengths equal multiples of 3L_π [5]. Since the γ’s of the CGPH-loaded multimode waveguide also has a quadratic relation, SI can also be observed at lengths equal multiples of 3L’_π, where

{L^{'}}_{π} = \frac{π}{γ_{1} - γ_{2}} ≅ \frac{π}{3 A^{'}} .

Since the magnitude of A’ is related to the entries of matrix K, the SI lengths of multimode waveguides can thus be engineered by appropriately choosing the coupling coefficients κ’s.

As an example, we consider a four-moded waveguide, with gratings coupling the 1st and the 4th modes, and the 2nd and the 3rd modes. The matrix describing this system is

K = [\begin{matrix} β_{1} & 0 & 0 & κ_{1} \\ 0 & β_{2} & κ_{2} & 0 \\ 0 & κ_{2}^{*} & β_{3} & 0 \\ κ_{1}^{*} & 0 & 0 & β_{4} \end{matrix}],

where κ₁ is the coupling coefficient between modes 1 and 4, and κ₂ is the coupling coefficient between modes 2 and 3; their magnitudes are on the order of (2π/λ)Δn. Its four eigenvalues are found to be

\begin{matrix} γ_{1} = \frac{β_{1} + β_{4}}{2} + \frac{1}{2} {[{(β_{1} - β_{4})}^{2} + 4 {| κ_{1} |}^{2}]}^{1 / 2} \\ γ_{2} = \frac{β_{2} + β_{3}}{2} + \frac{1}{2} {[{(β_{2} - β_{3})}^{2} + 4 {| κ_{2} |}^{2}]}^{1 / 2} \\ γ_{3} = \frac{β_{2} + β_{3}}{2} - \frac{1}{2} {[{(β_{2} - β_{3})}^{2} + 4 {| κ_{2} |}^{2}]}^{1 / 2} \\ γ_{4} = \frac{β_{1} + β_{4}}{2} - \frac{1}{2} {[{(β_{1} - β_{4})}^{2} + 4 {| κ_{1} |}^{2}]}^{1 / 2} . \end{matrix}

After some algebraic calculations, we find that when κ₁ = 3κ₂, γ’s are related by the quadratic relation in (10). We now show that the SI lengths can be engineered by properly choosing κ₁ and κ₂.

We choose a silicon-on-insulator (SOI) rib waveguide structure to illustrate this numerically. The design parameters are chosen as follows: the waveguide height H is 3μm, the slab height h is 2μm, the buried oxide layer thickness d is 1μm, and the width W of the waveguide is 10μm. The device is optimized for 1.55μm input wavelength and the TE polarization. Using a finite difference modesolver [21], we verified that this waveguide geometry supports 4 guided modes. Subsequent analysis is performed on the 2D structure obtained from the 3D structure using the effective index method [22]. Setting κ₁ = 3κ₂ = (2π/λ)Δn, we use the CMT to calculate the A’ parameter in (10) by solving the eigenvalue Eq. (9) for different index modulation depths Δn. The results of the corresponding first mirror image lengths 3L’_π are shown in Fig. 1 . It can be seen that the SI lengths are reduced by properly designing the CGPHs.

Fig. 1 Self-imaging (SI) lengths of multimode waveguides with different index modulation depths. Coupled-mode theory (CMT) calculations are obtained by solving (9) numerically. WA-BPM simulations are obtained using the finite difference algorithm in [23]. Uniform index change indicates the SI lengths when Δn is applied uniformly throughout the multimode waveguide.

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For a comparison, we consider uniform refractive index change throughout the multimode waveguide. From (11), the SI lengths reduce linearly with the index modulation depths Δn. The calculated SI lengths with uniform index modulations are shown in Fig. 1. From (14), we can observe that the perturbations to A’ in (10) is quadratic in nature to the first degree. As a result, the CGPH approach provides a greater control over the SI lengths then simply modifying the refractive index of the multimode waveguide uniformly.

4. Device simulation

We use the wide-angle beam propagation method (WA-BPM) [23] to simulate the device outlined in the previous section. From (13), the CGPH consists of gratings coupling the 1st/4th mode and the 2nd/3rd mode. Using the design procedure outlined in [17], the corresponding CGPH pattern is calculated, and the grating strengths are adjusted such that κ₁ = 3κ₂ is satisfied. Figure 2 shows the calculated CGPH for SI length engineering. In device fabrication, the calculated pattern can be converted to a binary mask and transferred to the surface of the multimode section [17,19]; the etching depth d_etch of the surface relief hologram pattern is directly related to the magnitude of effective index modulation Δn. We find by the effective index method that Δn and d_etch follow an almost linear relation with a slope of Δn/d_etch = −7 × 10⁻⁶/nm [18].

Fig. 2 Calculated CGPH for self-imaging (SI) length engineering. The pattern is obtained using the WA-BPM based code and the design procedure outlined in [17].

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The WA-BPM based code is used to verify the matrix theory analysis in sections 2 and 3. In the simulations presented below, the CGPH in Fig. 2 is used as a perturbation to the core effective index. The hologram features are discretized into 1μm × 1μm pixels in the simulation. Considering a Gaussian input field with a FWHM of 2μm and is offset by 4.0μm with respect to the center of the multimode waveguide, the intensity patterns of the light in a waveguide without (a) and with (b) the CGPH are shown in Fig. 3 . As expected, the first mirror image length of the CGPH-loaded waveguide is reduced. We note that the intensity patterns appear different in these two figures, which is expected since the waveguides in these two figures have different eigenmodes, although they both have quadratically related eigenvalues. We also monitor the SI lengths for the 1st mirror image to occur for different index modulation depths using the WA-BPM code, and the results are shown together with results from the coupled-mode theory calculations in Fig. 1. It can be seen that the WA-BPM simulation results agrees quite well with the numerical calculations using the coupled-mode theory.

Fig. 3 The intensity patterns of the light in multimode waveguides without and with the CGPH. (a) no CGPH: SI occurs at 1379μm. (b) with CGPH (index modulation depth Δn = 0.002): SI occurs at a shorter length of 1250μm.

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5. Discussions and summary

We presented an example using a four-moded waveguide to illustrate the ability to engineer the SI lengths of multimode waveguides by CGPHs. The particular CGPH consists of grating elements contributing only to the anti-diagonal of the matrix K in (9). As the mode number is increased in the multimode waveguide, a matrix K with perturbation entries not only on its anti-diagonal can always be found for its eigenvalues to be quadratically related [24]. It is beyond the scope of this work to evaluate all possible designs; rather, we focus on a specific design to illustrate the concept of diffraction engineering in multimode waveguides.

The proposed CGPH approach provides a greater control over the SI lengths of multimode waveguides then simply uniformly modifying the refractive index. By a modest refractive index modulation of 0.06%, a 9% change in the SI lengths is observed in both the CMT calculation and the BPM simulation. The SI length reduction would be practically limited by the obtainable refractive index modulation by selective etching and the increased scattering loss.

In summary, diffraction engineering of multimode waveguides using CGPHs is considered using the coupled-mode theory. The concept of designing fixed width multimode waveguides with variable SI lengths is presented. Based on the design concept, reduced SI lengths multimode waveguides are designed using SOI rib waveguide structures. The same principle can be applied to devices based on multimode interference.

Acknowledgments

This material is based upon work supported in part by the National Science Council of Taiwan under contract NSC 98-2221-E-006-016-.

References and links

1. K. Okamoto, Fundamentals of Optical Waveguides. (Academic, Burlington, MA, 2006).

2. D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10(9), 1875–1881 ( 1993). [CrossRef]

3. O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Am. 63(4), 416–419 ( 1973). [CrossRef]

4. R. Ulrich, “Image formation by phase coincidence in optical waveguides,” Opt. Commun. 13(3), 259–264 ( 1975). [CrossRef]

5. L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 ( 1995). [CrossRef]

6. P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12(6), 1004–1009 ( 1994). [CrossRef]

7. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 ( 1999). [CrossRef]

8. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 ( 2003). [CrossRef] [PubMed]

9. R. Gordon, “Harmonic oscillation in a spatially finite array waveguide,” Opt. Lett. 29(23), 2752–2754 ( 2004). [CrossRef] [PubMed]

10. N. K. Efremidis and D. N. Christodoulides, “Revivals in engineered waveguide arrays,” Opt. Commun. 246(4-6), 345–356 ( 2005). [CrossRef]

11. T. W. Mossberg, “Planar holographic optical processing devices,” Opt. Lett. 26(7), 414–416 ( 2001). [CrossRef] [PubMed]

12. T. Hashimoto, T. Saida, I. Ogawa, M. Kohtoku, T. Shibata, and H. Takahashi, “Optical circuit design based on a wavefront-matching method,” Opt. Lett. 30(19), 2620–2622 ( 2005). [CrossRef] [PubMed]

13. P. A. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightwave Technol. 14(10), 2286–2293 ( 1996). [CrossRef]

14. D. S. Levy, Y. M. Li, R. Scarmozzino, and R. M. Osgood, Jr., “A multimode interference-based variable power splitter in GaAs-AlGaAs,” IEEE Photon. Technol. Lett. 9(10), 1373–1375 ( 1997). [CrossRef]

15. D. J. Y. Feng and T. S. Lay, “Compact multimode interference couplers with arbitrary power splitting ratio,” Opt. Express 16(10), 7175–7180 ( 2008), http://www.opticsinfobase.org/abstract.cfm?uri=oe-16-10-7175. [CrossRef] [PubMed]

16. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 ( 1973). [CrossRef]

17. S.-Y. Tseng, Y. Kim, C. J. K. Richardson, and J. Goldhar, “Implementation of discrete unitary transformations by multimode waveguide holograms,” Appl. Opt. 45(20), 4864–4872 ( 2006). [CrossRef] [PubMed]

18. S.-Y. Tseng, C. Fuentes-Hernandez, D. Owens, and B. Kippelen, “Variable splitting ratio 2 x 2 MMI couplers using multimode waveguide holograms,” Opt. Express 15(14), 9015–9021 ( 2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-14-9015. [CrossRef] [PubMed]

19. S.-Y. Tseng, S. Choi, and B. Kippelen, “Variable-ratio power splitters using computer-generated planar holograms on multimode interference couplers,” Opt. Lett. 34(4), 512–514 ( 2009). [CrossRef] [PubMed]

20. M. Bachmann, P. A. Besse, and H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Appl. Opt. 34(30), 6898–6910 ( 1995). [CrossRef] [PubMed]

21. A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightwave Technol. 26(11), 1423–1431 ( 2008). [CrossRef]

22. K. Kawano, and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, New York, 2001).

23. G. R. Hadley, “Wide-angle beam propagation using Pade approximant operators,” Opt. Lett. 17(20), 1426–1428 ( 1992). [CrossRef] [PubMed]

24. M. T. Chu, and G. H. Golub, Inverse Eigenvalue Problems: Theory, Algorithms, and Applications, (Oxford, 2005).

Diffraction engineering of multimode waveguides using computer-generated planar holograms

Abstract

1. Introduction

2. Coupled-mode theory

3. Device design example

4. Device simulation

5. Discussions and summary

Acknowledgments

References and links

Cited By

Figures (3)

Equations (14)

Optics Express