Splitting in Multimode Waveguides Based on

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 31, NO. 21, NOVEMBER 1, 2013

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Mode Conversion/Splitting in Multimode Waveguides Based on Invariant Engineering Kai-Hsun Chien, Chi-Shung Yeih, and Shuo-Yen Tseng, Member, IEEE

Abstract—We discuss mode conversion and splitting in multimode waveguides based on Lewis–Riesenfeld invariant theory. The design of mode converters/splitters using the dynamical invariant is discussed. Closed form expressions for the modal power at the output for converter/splitter design are derived. Various design approaches are discussed and assessed by beam propagation simulations. We show that the invariant-based engineering scheme reduces device length as compared to the common adiabatic scheme. Index Terms—Coupled mode analysis, gratings, multimode waveguide, optical beam splitting, optical planar waveguides.

I. INTRODUCTION PTICAL communication systems have long been based on single-mode waveguides. However, recent studies show that the data capacity over single-mode optical waveguides is approaching its fundamental limit [1], [2]. It has been suggested that space-division multiplexing (SDM) over coupled or uncoupled channels can be used to overcome this limit [3], [4]. Mode-division multiplexing (MDM) is a promising SDM technique where multiple optical modes are used as independent data channels to transmit optical data [5]. While MDM was primarily targeted for short distance SDM in optical interconnects due to strong mode coupling and large modal differential group delay [4], recent efforts in fiber design and signal processing have made possible the use of MDM for long-haul transmissions [6]. So far, most of the efforts in MDM systems are focused on fiber based systems, but there is also interest in realizing integrated multimode systems [7]. Several basic building blocks for MDM in integrated optics, such as the mode add/drop multiplexer [8] and the mode generator [9], have been proposed. Recently, analogies between quantum mechanics and wave optics in integrated optics devices have been exploited to

O

Manuscript received June 22, 2013; revised September 2, 2013; accepted September 17, 2013. Date of publication September 19, 2013; date of current version October 11, 2013. This work was supported by the National Science Council of Taiwan under Contracts NSC100-2221-E-006-176-MY3 and NSC 102-3113-P-110-004. K.-H. Chien is with the Department of Photonics, National Cheng Kung University, Tainan 701, Taiwan. C.-S. Yeih is with the Department of Photonics, National Cheng Kung University, Tainan 701, Taiwan. S.-Y. Tseng is with the Department of Photonics, National Cheng Kung University, Tainan 701, Taiwan, and also with the Advanced Optoelectronics Technology Center, National Cheng Kung University, Tainan 701, Taiwan (email: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2013.2283032

realize many interesting devices [10]–[12]. Of particular interest is the analogy between light passage in engineered photonics systems and coherent population transfer among quantum states of atoms and molecules by laser pulses. We have previously proposed mode conversion/splitting devices [13], [14] using computer-generated planar holograms (CGPHs) [15] in multimode waveguides based on optical analogy of stimulated Raman adiabatic passage (STIRAP) [16]. In these devices, longperiod waveguide gratings with spatially varying coupling coefficients are used to couple the guided modes in the multimode waveguide; the coupled mode equations describing modal amplitude evolution is analogous to the Schrödinger equation describing population evolution in an n-level system coupled by laser pulses. The devices are based on the adiabatic theorem, so they do not require a precise definition of coupling length, but they need to be sufficiently long to satisfy the adiabatic condition. However, long device length reduces device density and induces more transmission losses. Moreover, for mode splitting, previous approach using optical analogies of the multistate STIRAP [14] requires coupling of auxiliary modes to generate the required adiabatic mode for mode splitting. The additional coupling terms complicate the design and fabrication of the mode converter/splitter. On the other hand, the fastest process to complete population transfer in atomic and molecular physics is to use a resonant pulse with area π for the Rabi frequency. In photonics systems, this corresponds to resonant mode coupling by designing the coupling region of a resonant mode converter to be half the beat length, however, like its counterpart in quantum mechanics being nonrobust against variations of the pulse area; while resonant mode converter can be made very short, it is nonrobust against device parameter variations from fabrication. Efforts have been made to speed up quantum adiabatic processes. These “shortcuts to adiabaticity” protocols are alternative fast processes which reproduce the same final populations, or the same final state, as the adiabatic process in a shorter time [17]–[22]. A shortcut protocol that has received a lot of attention is the inverse engineering by Lewis–Riesenfeld invariants [22], which provides adiabatic-like shortcuts to quantum adiabatic processes. Previously, we applied the invariant engineering to the design of fast mode converters in engineered multimode waveguides [23]. In this paper, we show that the invariant-based engineering provides an intuitive approach for the design of mode converter/splitter by describing the system evolution using the eigenstates of the invariant. Also, the approach does not require coupling of auxiliary modes for mode splitting. We derive closed form expressions for the modal power at the output for converter/splitter design and discuss the engineering

0733-8724 © 2013 IEEE

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Fig. 1. (a) Level scheme of a three-level system. (b) Schematic of a CGPH loaded multimode waveguide. (c) Amplitude profiles of the three guided modes.

of the invariant parameters and their relations to the device performances.

where |Ψ1 = [1, 0, 0]T , |Ψ2 = [0, 1, 0]T , and |Ψ3 = [0, 0, 1]T . An adiabatic mode of H0 with a zero eigenvalue can be found as va = (κ23 |Ψ1 − κ12 |Ψ3 )/( κ212 + κ223 ). When va is excited initially, the system will remain in it if the coupling coefficients are varied slowly. In STIRAP, if the two variable coupling coefficients κ12 and κ23 are applied in a counterintuitive scheme, where κ23 precedes κ12 , then va approaches |Ψ1 at the input and adiabatically evolves to |Ψ3 at the output. The design goal is to minimize coupling among the eigenmodes of H0 during the evolution through the engineering of κ12 and κ23 [25]. Using the WKB approximation, a closed form solution of (1) can be obtained [26]; however, the design is only applicable under the adiabatic approximation. B. Lewis–Riesenfeld Invariant Theory

II. THEORETICAL ANALYSIS A. Coupled Mode Equations In the step-index multimode waveguide shown schematically in Fig. 1, we consider three forward-propagating modes |Ψ1 , |Ψ2 , and |Ψ3 , coupled by a CGPH. The CGPH consists of multiplexed long-period gratings which couple the guided modes depending on the grating shape and periodicity [14]. We design the CGPH to couple modes |Ψ1 , |Ψ2 and modes |Ψ2 , |Ψ3 by gratings Λ12 and Λ23 with spatially varying coupling coefficients κ12 and κ23 . Fig. 1(b) shows the schematic of a CGPH loaded multimode waveguide, and amplitude profiles of the three guided modes are shown in Fig. 1(c). The electric field in the waveguide can be expressed as a superposition of the waveguide modes as E(x, y, z) = ΣAn (z) exp(−iβn z)|Ψn (n = 1, 2, 3), where An is the mode amplitude and βn is the propagation constant corresponding to the nth mode. When the CGPH is on resonance with the input wavelength, the evolution of mode amplitude An obeys the coupled mode equations [24] ⎤ ⎤ ⎡ ⎤⎡ ⎡ 0 A1 A1 0 κ12 (z) d ⎢ ⎥ ⎥ ⎥⎢ ⎢ 0 κ23 (z) ⎦ ⎣ A2 ⎦ i ⎣ A2 ⎦ = ⎣ κ21 (z) dz 0 κ32 (z) A3 0 A3 ⎤ ⎡ A1 ⎥ ⎢ = H0 (z) ⎣ A2 ⎦ . (1) A3 The term κ13 vanishes because we assume that the CGPH composed by gratings Λ12 and Λ23 does not couple modes |Ψ1 and |Ψ3 . When the spatial variation z is replaced with the temporal variation t, (1) is used to describe the probability amplitudes of a three-level atomic system driven by two laser pulses shown in Fig. 1(a) using the Schrödinger equation ( = 1) under the rotating-wave approximation, in which An represents the probability amplitudes of the state |Ψn being populated, κm n is the Rabi frequency of the pulse coupling states |Ψm and |Ψn . The system state can be expressed as a linear combination of the waveguide modes |Ψ(z) = An (z)e−iβ n z |Ψn (2) n

A quantum system driven by a time-dependent Hamiltonian H(t) can be described by the Schrödinger equation i∂t |Ψ(t) = H(t)|Ψ(t). An invariant I(t) of H(t) satisfies i∂t (I(t)|Ψ(t)) = H(t)(I(t)|Ψ(t)) [27]. In other words, the invariant produces another solution of the Schrödinger equation when acting on a solution of the Schrödinger equation. According to Lewis–Riesenfeld theory, we can thus write the system state as a linear combination of the eigenstates of the invariant |Ψ(t) =

cn eiα n φn (t)

(3)

n

where φn (t) is the eigenvector of I(t), cn is the amplitude, and αn is the Lewis–Riesenfeld phase [27], 1 αn (t) =

0

t

φn (t )|i

∂ − H(t )|φn (t )dt . ∂t

(4)

Since the system evolution can be described by the superposition of φn (t)s, and their amplitude cn is time-independent, we can engineer the evolution of Ψ(t) using φn (t) without worrying about the coupling among them. System engineering is now achieved by controlling the evolution of I(t) and its corresponding eigenvectors. C. Invariant of the CGPH-Loaded Multimode Waveguide By replacing t with z, the discussions in Section II-B. can be applied to the CGPH-loaded multimode waveguide described by (1). The Hamiltonian H0 possesses SU(2) symmetry; by solving i∂z (I(z)|Ψ(z)) = H0 (z)(I(z)|Ψ(z)), we obtain its invariant I(z) [22] ⎡ ⎤ 0 cos γ sin β −i sin γ ⎢ ⎥ I(z) = ⎣ cos γ sin β 0 cos γ cos β ⎦ (5) 2 i sin γ cos γ cos β 0 and its parameters are related to the coupling coefficients by κ12 = 2(β˙ cot γ sin β + γ˙ cos β)

(6)

= 2(β˙ cot γ cos β − γ˙ sin β).

(7)

κ23

CHIEN et al.: MODE CONVERSION/SPLITTING IN MULTIMODE WAVEGUIDES BASED ON INVARIANT ENGINEERING

The Lewis–Riesenfeld phase are obtained using (4) (where t is replaced by z):

The eigenmodes of I(z) are ⎡ ⎤ cos γ cos β |φ0 = ⎣ −i sin γ ⎦ − cos γ sin β ⎡

(8)

sin γ cos β ± i sin β 1 ⎦. |φ± = √ ⎣ i cos γ 2 − sin γ sin β ± i cos β

(9)

α0 (z) = 0

(10)

z 1 α± (z) = ∓ β˙ sin γ + (κ12 sin β + κ23 cos β) cos γ dz . 2 0 (11) By designing the evolution of γ and β and implementing it with κ12 and κ23 using (6) and (7) in a CGPH, the eigenmodes of the invariant I(z) in (8) and (9) can now be used to engineer mode evolution in the waveguide. III. MODE CONVERTER/SPLITTER DESIGN USING THE INVARIANT In this section, we illustrate the engineering of mode conversion/splitting from |Ψ1 to combinations of |Ψ1 , |Ψ2 , and |Ψ3 by designing the evolution of γ and β. This is achieved by setting the boundary conditions for γ and β at z = 0 and z = L to obtain the desired input state and output state. At the output z = L, we set γ(L) =

(12)

β(L) = η.

(13)

The modal power of |Ψ1 , |Ψ2 , and |Ψ3 at the output can be calculated as

|Ψ(L) =

2

Cn eiα n (L ) |φn (L)

(14)

(15)

n

with

⎤ cos cos η |φ0 (L) = ⎣ −i sin ⎦ − cos sin η

⎡

⎤ sin cos η ± i sin η 1 ⎦ |φ± (L) = √ ⎣ i cos 2 − sin sin η ± i cos η

(20)

α± (L) = ∓

Substituting (8) and (9) into (4), we can obtain their corresponding Lewis–Riesenfeld phases as

where

α0 (L) = 0

⎤

Pn (L) = |Ψn |Ψ(L)|

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(16)

⎡

π . 2 sin

(21)

Substituting (15)–(17) into (14), we obtain the expressions for the modal power at the output

η

P1 (L) =

cos η cos2 + sin2 cos sin

2 η

+ sin η sin sin (22) sin

η 2

P2 (L) = cos sin cos (23) −1 sin

η

P3 (L) =

sin η cos2 + sin2 cos sin η

2

. (24) − cos η sin sin sin Mode conversion and arbitrary split ratios can be obtained by solving (22)–(24) to obtain the required γ and β at the output. On the other hand, the boundary conditions at z = 0 are not critical because the input can always be decomposed into a linear combination of |φ0 and |φ± , no matter what choices are made on γ(0) and β(0). In order to obtain fast adiabatic-like evolution [22], we set the additional boundary conditions γ(0) ˙ = γ(L) ˙ =0

(25)

and for γ and β at z = 0, γ(0) =

(26)

β(0) = 0.

(27)

As a result of the boundary conditions in (25), H0 (z) and I(z) commute at z = 0 and z = L, suggesting that the matrices share the same eigenmodes at the input and output. So, the input and output modes can transform into the eigenmodes of I(z) smoothly. Along with (12) and (13), (25)–(27) constitute the boundary conditions for γ and β. We note that once the boundary conditions are set, the evolution of γ and β can be freely chosen to satisfy any additional constraints demanded by the system. It is beyond the scope of this study to categorize or evaluate the various design approaches that are possible; rather, in the following, we use specific designs to demonstrate mode conversion/splitting, and to illustrate the effect of different evolutions of γ and β on device characteristics.

(17) A. Path A

and the time-independent amplitude Cn s are obtained by Cn = φn (0)|Ψ1 as C0 = cos

(18)

1 C± = √ sin . 2

(19)

In path A, we consider the basic boundary conditions described in (12), (13), and (25)–(27), which are shown again here γ(0) = γ(L) = , β(0) = 0,

γ(0) ˙ = γ(L) ˙ =0

β(L) = η.

(28) (29)

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Consistent with the boundary conditions, we obtain the simplest evolutions of γ and β as γA (z) = η βA (z) = z. L

(30) (31)

B. Path B Next, we impose additional boundary conditions for β, ˙ ˙ β(0) = β(L) =0

(32)

such that the change of |Ψn will be smoother (more adiabaticlike) at the input and output of the waveguide. Interpreting β with a third degree polynomial, we obtain γB (z) = βB (z) =

3η 2 2η 3 z − 3z . L2 L

(33) (34)

C. Path C Finally, in addition to (32), we impose the following boundary conditions for β, ¨ ¨ β(0) = β(L) =0

(35)

such that the change of |Ψn will be even more adiabatic-like at the input and output of the waveguide. Interpreting β with a fifth degree polynomial, we obtain γC (z) =

(36)

10η 15η 6η βC (z) = 3 z 3 − 4 z 4 + 5 z 5 . (37) L L L Substituting the γs and βs into (6) and (7), we can obtain the coupling coefficients for CGPH design. Next, we use numerical examples to illustrate mode conversion/splitting with invariant engineering described previously and illustrate the differences among the three paths discussed previously. IV. DESIGN EXAMPLES AND BEAM PROPAGATION SIMULATIONS In this section, we design CGPHs on a five-moded waveguide to implement mode conversion/splitting using invariant engineering. A polymer ridge waveguide structure is chosen with the following design parameters [14]: 3 μm thick SiO2 on a Si wafer for the bottom cladding layer, 2.4 μm layer of Cyclotene (BCB) for the core, the width of the waveguide is 3 μm, and the length of the waveguide is 6 mm. The device is designed at a wavelength of 1.55-μm input wavelength and the TE polarization. The maximum effective index modulation of the CGPH is assumed to be Δn = 0.003. We use a finite-difference mode solver [28] to verify that this waveguide geometry indeed supports five guided modes. Subsequent analysis is performed on the two-dimensional (2-D) structure obtained using the effective index method. The wide-angle beam propagation method (WA-BPM) [29] is used for device simulations. The first three modes of the waveguide are used in the simulation and CGPH design.

Fig. 2. Left column: Evolution of the coupling coefficients κ 1 2 and κ 2 3 along z for the mode converters. Right column: The corresponding CGPH pattern. Path A: (a) and (b). Path B: (c) and (d). Path C: (e) and (f). Dashed white lines indicate the waveguide core.

A. Mode Converter We consider a mode converter to convert |Ψ1 at the input to |Ψ3 at the output. The modal power at the output is thus P1 (L) = 0,

P2 (L) = 0,

P3 (L) = 1.

(38)

Using (22)–(24), we obtain π . (39) 2 Substituting into (30), (31), (33), (34), (36), and (37), we obtain the evolution of γs and βs for the three paths as follows. 1) Path A: = 0.2527,

η=

γA (z) = 0.2527 π z. βA (z) = 2L

(40) (41)

2) Path B: γB (z) = 0.2527 βB (z) =

3π 2 π z − 3 z3 . 2L2 L

(42) (43)

3) Path C: γC (z) = 0.1253

(44)

5π 3 15π 4 3π 5 z − z + 5z . (45) L3 2L4 L The γs and βs are substituted into (6) and (7) to obtain the coupling coefficients for CGPH design. The resulting κ12 and κ23 for the three paths and their corresponding CGPHs are shown in Fig. 2. The results of BPM simulations for the three design paths using |Ψ1 as input are shown in Fig. 3. The beam evolution corresponding to paths A, B, and C are shown in Fig. 3(a), (c), and (d), respectively. Their corresponding modal power evolution is shown in Fig. 3(b), (d), and (f). It is clear that for all three designs, |Ψ1 is converted into |Ψ3 in 6 mm βC (z) =


Fig. 3. BPM simulations of beam evolution in the mode converter and their corresponding modal power evolution. Path A: (a) and (b). Path B: (c) and (d). Path C: (e) and (f). Dashed white lines indicate the waveguide core.

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Fig. 4. Left column: Evolution of the coupling coefficients κ 1 2 and κ 2 3 along z for the one-to-two mode splitters. Right column: The corresponding CGPH pattern. Path A: (a) and (b). Path B: (c) and (d). Path C: (e) and (f). Dashed white lines indicate the waveguide core.

length, which violates the adiabatic criterion [14]. Due to the additional boundary conditions (32) and (35), paths B and C are more adiabatic-like in such a way that the conversion is more robust to device length variations. The conversion is completed in a shorter length within the device for paths B and C, with path C shorter than path B. B. One-to-Two Splitter Next, we consider a mode splitter to split |Ψ1 at the input to equal amount of |Ψ1 and |Ψ3 at the output. The modal power at the output is P1 (L) =

1 , 2

P2 (L) = 0,

P3 (L) =

1 . 2

(46)

Using (22)–(24), we obtain = 0.1253,

η=

π . 4

(47)

Again, we obtain the evolution of γs and βs for the three paths as follows. 1) Path A: γA (z) = 0.1253 π z. βA (z) = 4L

(48) (49)

2) Path B: γB (z) = 0.1253

(50)

3π 2 π 3 βB (z) = z − z . 2 4L 2L3

(51)

Fig. 5. BPM simulations of beam evolution in the one-to-two mode splitter and their corresponding modal power evolution. Path A: (a) and (b). Path B: (c) and (d). Path C: (e) and (f). Dashed white lines indicate the waveguide core.

The resulting κ12 and κ23 for the three paths and their corresponding CGPHs are shown in Fig. 4. The results of BPM simulations for the three paths are shown in Fig. 5. The beam evolution corresponding to paths A, B, and C are shown in Fig. 5(a), (c), and (d), respectively. Their corresponding modal power evolution is shown in Fig. 5(b), (d), and (f). The same observations on path differences can be made, that is, path C is more adiabatic-like than path B, which is more adiabatic-like than path A.

3) Path C: γC (z) = 0.1253 βC (z) =

5π 3 15π 4 3π 5 z − z + z . 2L3 4L4 2L5

(52)

C. One-to-Three Splitter

(53)

Finally, we consider a mode splitter to split |Ψ1 at the input to equal amount of |Ψ1 , |Ψ2 , and |Ψ3 at the output. The

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Fig. 6. Left column: Evolution of the coupling coefficients κ 1 2 and κ 2 3 along z for the one-to-three mode splitters. Right column: The corresponding CGPH pattern. Path A: (a) and (b). Path B: (c) and (d). Path C: (e) and (f). Dashed white lines indicate the waveguide core.

Fig. 7. BPM simulations of beam evolution in the one-to-three mode splitter and their corresponding modal power evolution. Path A: (a) and (b). Path B: (c) and (d). Path C: (e) and (f). Dashed white lines indicate the waveguide core.

modal power at the output is 1 1 , P2 (L) = , 3 3 Using (22)–(24), we obtain P1 (L) =

= 0.3144,

P3 (L) =

η = 0.8874.

1 . 3

(54)

(55)

The evolutions of γs and βs for the three paths are as follows. 1) Path A: γA (z) = 0.3144

(56)

0.8874 z. βA (z) = L

(57)

2) Path B: γB (z) = 0.3144 βB (z) =

2.6622 2 1.7748 3 z − z . L2 L3

(58) (59)

3) Path C: γC (z) = 0.3144

(60)

8.874 3 13.311 4 5.3244 5 z − z + z . (61) L3 L4 L5 The resulting κ12 and κ23 for the three paths and their corresponding CGPHs are shown in Fig. 6. The results of BPM simulations for the three paths are shown in Fig. 7. The beam evolution corresponding to paths A, B, and C are shown in Fig. 7 (a), (c), and (d), respectively. Their corresponding modal power evolution is shown in Fig. 7 (b), (d), and (f). Again, the same path dependent characteristics can be observed. βC (z) =

V. DISCUSSIONS AND CONCLUSION Through engineering of the evolution of the invariants, we have designed mode converters/splitters with three different

paths. Paths B and C show more adiabatic-like evolution because the additional boundary conditions (33) and (35) require the evolution of the invariants to be slower at the input and the output. For all three paths, the area under κ12 and κ23 are constants given by

L κ12 (z)dz = 2 cot (1 − cos η) (62) 0

L

κ23 (z)dz = 2 cot sin η.

(63)

0

In terms of population transfer/splitting by laser pulses, the results indicate that the total pulse energy required is identical for all three paths. For path C, the transfer is completed within the shortest period of time (distance), indicating the largest maximum value of the coupling coefficients κ12 and κ23 as seen in Figs. 2, 4, and 6. In terms of CGPH, this corresponds to larger refractive index modulation needed to implement path C than path B, which also requires larger refractive index modulation than path A. We note that the maximum value of coupling coefficient corresponding to Δn = 0.003 is 2.022 mm−1 for κ12 and 1.755 mm−1 for κ23 in this polymer waveguide platform in our numerical example. Using the mode converter as an example, we calculate the shortest mode converter length achievable with the three paths. We numerically solve the coupled mode equation (1) for the three paths and plot the normalized modal power of |Ψ3 at the output of the converter as a function of the device length in Fig. 8. We cap the maxima of κ12 and κ23 at 2.022 mm−1 and 1.755 mm−1 , respectively, in the calculation to account for physical realizability in fabrication and to avoid additional scattering loss resulting from large effective index modulation. Although the mode conversion in path C takes place much faster than path A as shown in Fig. 3, it requires a larger maximum value of the coupling coefficient as shown in Fig. 2. When the device length is reduced, the path C design reaches the cap we put on the maxima of κ12 and κ23 earlier than


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coupling [16]. By engineering the device characteristics to be adiabatic-like using the dynamical invariants, the resulting device combines the advantages of short resonant devices and robust adiabatic devices. In conclusion, mode converter/splitter design based on invariant engineering is discussed. The method provides an intuitive method for device design. By fixing the input and output boundary conditions of the dynamical invariants, system mode evolution follows the eigenmodes of the invariant from the input mode to the desired output mode/modes. Several design examples are discussed analytically and verified with BPM simulations. The device characteristics and their relations to the invariant parameters are discussed. The design approach is applicable to a wide range of coupled-wave devices. Fig. 8. Normalized modal power in |Ψ 3 at the mode converter output with different device lengths for the three design paths.

ACKNOWLEDGMENT We are grateful to J. G. Muga and X. Chen for discussions. REFERENCES

Fig. 9. Normalized modal power at the mode converter output with Δn variations.

the other two paths, resulting in lower conversion efficiency. To obtain a conversion efficiency ≥99%, the device length can be reduced to 3.2 mm for path A, 4.1 mm for path B, and 4.8 mm for path C. The minimum device lengths for the three paths are all shorter than the 11.4 mm reported for adiabatic mode converter designed using optical analogy of STIRAP [20]. The results indicate that the device characteristics can be engineered to be more adiabatic-like, but at a cost of longer final device. Using the path C mode converter design introduced in Section IV-A, we analyze the robustness of the proposed devices against fabrication errors. From the modal power evolution plot shown in Fig. 3, it is clear that the device is robust against device length variations resulting from typical fabrications errors. Errors in the CGPH etch depth is directly related to the refractive index modulation, which in turn affects the coupling coefficients [14]. In Fig. 9, using |Ψ1 as the input to the mode converter, we calculate the power in |Ψ1 , |Ψ2 , and |Ψ3 at the output as a function of Δn variations . For a wide range of Δn variations, the device retains its mode conversion property very well. For large negative Δn variations, the mode conversion efficiency drops significantly due to insufficient coupling; this is similar to resonantly coupled devices. For large positive Δn variations, the device behaves like an adiabatic device, because the adiabatic condition is satisfied due to the increased

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Kai-Hsun Chien received the B.S. degree in optoelectronic engineering from National Chi Nan University, Nantou County, Taiwan, in 2011 and the M.S. degree in photonics from National Cheng Kung University, Tainan City, Taiwan, in 2013. He is currently serving in the Taiwan Armed Forces.

Chi-Shung Yeih received the B.S. degree in physics from National Chiayi University, Chiayi City, Taiwan, in 2010 and the M.S. degree in photonics from National Cheng Kung University, Tainan City, Taiwan, in 2013. He is currently with the EPISTAR corporation.

Shuo-Yen Tseng received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, in 1999 and the M.S. and Ph.D. degrees in electrical engineering from the University of Maryland, College Park, MD USA, in 2002 and 2006, respectively. Since 2008, he has been with the Department of Photonics, National Cheng Kung University, as an Assistant and then Associate Professor. His research interests include optical communications, numerical simulation, nanotechnology, and integrated photonics.