1Wuhan National Laboratory for Optoelectronics, School of Optoelectronics Science and Engineering,Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan, Hubei 430074, China
2Department of Electrical Engineering and Computer Science, Northwestern University,2145 Sheridan Road, Evanston, Illinois 60208, USA
Arrayed waveguide gratings (AWGs) are key compo-nents in dense wavelength division multiplexing sys-tems. In recent decades, many methods have beenproposed to reduce the temperature sensitivity ofconventional AWGs [1,2]. Because of the demand forhigh-density photonic integration, ultrasmall AWGsbased on silicon (Si) photonic wire waveguides haveattracted much attention in recent years [3–5]. Thelarge refractive index contrast ratio between Si coreand silica (SiO2) cladding in photonic wire wave-guides makes it possible to confine light in the corewith a submicrometer dimension mode size. Thus,the typical size of AWGs can be reduced from the or-der of 10 cm2 for those based on SiO2 waveguides(SiO2 AWGs) to the order of 100 μm2 for those based
on Si wire waveguides (Si-AWGs). On the other hand,since most of the light energy is confined within theSi core, the central wavelength of Si-AWGs is moresensitive to thermal variations than that of SiO2AWGs, because the thermo-optic coefficient of Si isabout 20 times larger than that of SiO2.
To reduce the temperature sensitivity of Si wirewaveguides without the use of electrical control cir-cuits for heating or cooling, polymer upper claddingwith a negative thermo-optic coefficient (TOC) hasbeen proposed for use as a passive compensator tocompensate the thermal sensitivity of the Si corewith a positive TOC [6–8]. The same principle wasalso applied in optofluidic photonic crystal cavitiesusing liquid with an appropriate TOC [9]. The com-pensating effects of polymer can be enhanced when itis filled into the slot of a slot waveguide [10,11]. Atypical Si slot waveguide has a narrow slot with lowrefractive index sandwiched by two strips of Si withhigh refractive index [12]. As a result, most of the
optical field energy for a TE-polarized wave (with po-larization vector perpendicular to the slot’s Si wall) isconfined within the low-refractive-index slot region,which means that its thermal performance is alsomainly relying on the low-refractive-index materialin the slot, such as SiO2, air, and polymer. Hence,it is possible to tune the temperature coefficient ofslot waveguides from positive to negative by care-fully designing the structure parameters and select-ing proper low-refractive-index materials. Slotwaveguides with negative TOC have been proposedfor insertion into arrayed waveguides to constructtemperature-independent arrayed waveguide grat-ings (TI-AWGs) [13].
Recently, a TI-AWG utilizing the combination ofnarrow and wide Si wire waveguides (narrow–wide-wire) was presented [14]. The design scheme was ex-perimentally tested by fabricating prototype devicesof Mach–Zehnder interferometers [15], which arealso optical filters based on the optical path-lengthdifference. It was shown that the temperature sensi-tivity can be suppressed by adjusting the path-lengthdifference of narrow and wide waveguides. Thismethod avoids the potential instability of polymers,making the approach compatible with the comple-mentary metal oxide semiconductor (CMOS) fabrica-tion technique. However, the approach results in asignificantly reduced interference order, which is adisadvantage for constructing compact TI-AWGs.This method of the combination of two types ofwaveguide has also been used to reduce control pre-cision of fabrication [3] and to enhance the dispersionperformance [16] in the AWGs.
In this paper, a complete theory for the design ofTI-AWGs by a combined use of different types ofwaveguide is developed. The ratios ofΔLwj are intro-duced as tuning parameters, where ΔLwj is thepath-length difference between adjacent arrayedwaveguides for the wj type of waveguides. As an ex-ample, a TI-AWG is designed by the combination ofSi wire and slot waveguides (wire–slot-hybrid). TheSi slot waveguides are filled with SiO2 or air, which iscompatible with CMOS fabrication techniques. Then,the tuning effects on the temperature sensitivity, thedispersion enhancement, and the device size are stu-died. In Section 2, the layout of the wire–slot-hybridTI-AWG is given for illustration, and then the designtheory is developed. In Section 3, the design stepsare presented, the results of the tuning effects aregiven, and the design example is verified by thetwo-dimensional finite-difference time-domain (2D-FDTD) simulation. In Section 4, conclusions aresummarized.
2. Model and Theory
The layout of a design example, i.e., the wire–slot-hybrid TI-AWG, is shown in Fig. 1. The combi-nation of Si wire and slot waveguides is introducedto arrayed waveguides for adjusting the temperaturesensitivity.
A. General Theory
The transmission of a light beam with wavelength λis described as follows. It is launched from the pthinput waveguide and diffracts in the input slab wave-guide. The divergent beam is coupled into arrayedwaveguides and propagates through individualwaveguides with different path lengths. Then, themultiple beams interfere in the output slab wave-guide, and focus on the qth output waveguide. Thephase difference of λ from the pth input waveguide tothe qth output waveguide via two adjacent arrayedwaveguides is
ΔΦ ¼ k0nseΔLin þ k0nb
eΔLb þ k0nw1e ΔLw1
þ k0nw2e ΔLw2 þ k0ns
eΔLout; ð1Þ
where
ΔLini;iþ1 ¼ linp;iþ1 − linp;i;
ΔLbi;iþ1 ¼ 2Rðφiþ1 − φiÞ;
ΔLw1i;iþ1 ¼ 2½ðhw1
iþ1 þ lw1iþ1Þ − ðhw1
i þ lw1i Þ�;
ΔLw2i;iþ1 ¼ 2½ðhw2
iþ1 þ lw2iþ1Þ − ðhw2
i þ lw2i Þ�;
ΔLouti;iþ1 ¼ loutiþ1;q − louti;q :
k0 is the wave number in vacuum, and nse, nb
e , nw1e ,
and nw2e are the effective refractive indices of the slab
waveguides, the bent wire waveguides, the first typeof waveguides (type 1 waveguides), and the secondtype of waveguides (type 2 waveguides), respectively.ΔLin
i;iþ1, ΔLbi;iþ1, ΔLw1
i;iþ1, ΔLw2i;iþ1, and ΔLout
i;iþ1are the path-length differences in the input slabwaveguides, the bent wire waveguides that connectthe adiabatic tapers and straight waveguides, thetype 1 waveguides, the type 2 waveguides, and theoutput slab waveguides, respectively. linp;i and louti;qare the path lengths from the pth input waveguide orthe qth output waveguide to the ith arrayed wave-guide, respectively. R and φi are the radius and ra-dian of the bent waveguides, respectively. hj
i and ljiare the path lengths of the jth type of waveguides inthe Y direction and the X direction, respectively.Note that other waveguides, including the adiabatictapers between slab waveguides and arrayed wave-guides, the mode converters between the differenttypes of waveguide, and the 90° bent waveguideshave the same path length in each arrayed wave-guide. Here, we start the analysis from the phase dif-ference instead of the path-length difference. This isbecause ΔΦ is constant, but ΔLb, ΔLw1, and ΔLw2
are piecewise functions of i due to the layout design.In Fig. 1(a), the ΔLb is determined by the layout andcan be given as
N is the number of arrayed waveguides, da is thepitch of adjacent arrayed waveguides at the gratingcircle, and Lf is the diameter of the Rowland circle.Note that i ¼ 1; 2;…;N − 1. From the phase matchcondition, the grating equation can be expressed as
nseΔLin þ nb
eΔLb þ nw1e ΔLw1 þ nw2
e ΔLw2
þ nseΔLout ¼ mλ; ð3Þ
wherem is the nominal grating order. It is an integerfor the central wavelength. The effective refrac-
tive indices are functions of temperature and wave-length. Thus, the temperature coefficient of thetransmission wavelength ∂λ=∂T can be described as
∂λ∂T
¼ 1M
��∂ns
e
∂Tþ ns
eαs�ðΔLin þΔLoutÞ
þ�∂nb
e
∂Tþ nb
eαb�ΔLb þ
�∂nw1
e
∂Tþ nw1
e αw1
�ΔLw1
þ�∂nw2
e
∂Tþ nw2
e αw2
�ΔLw2
�; ð4Þ
where αs, αb, αw1, and αw2 are the thermal expansioncoefficients of the slab waveguides, the bent wirewaveguides, the type 1 waveguides, and the type 2waveguides, respectively. M is the modified gratingorder, which is defined by
Fig. 1. Design example: the temperature-independent arrayed waveguide gratings based on silicon wire and slot waveguides. (a) Theentire structure, (b) the schematic of the arrayed waveguides, (c) the cross section of Si wire waveguides, (d) the cross section of Si slotwaveguides, and (e) the mode converter between Si wire waveguides and slot waveguides [17].
g are the group refractiveindices of the slab waveguides, the bent wire wave-guides, the type 1 waveguides, and the type 2 wave-guides, respectively. By substituting for m fromEq. (5) into Eq. (3), the grating equation can berewritten as
nsgΔLin þ nb
gΔLb þ nw1g ΔLw1 þ nw2
g ΔLw2
þ nsgΔLout ¼ Mλ: ð6Þ
Considering that a multiplexed beam is launchedfrom an input waveguide and focused on differentoutput waveguides, the wavelength dispersion canbe derived from Eq. (6) as
∂ΔLout
∂λ ¼ Mnse: ð7Þ
If da ≪ Lf , then ΔLin ≈ da sin θin and ΔLout ≈
da sin θout, where θin and θout are the dispersion an-gles of corresponding input waveguides and outputwaveguides, respectively. Assuming θout ≈ 0, then
Lf ¼nsedadio
MΔλ ; ð8Þ
where dio is the pitch of the input/output waveguidesat the Rowland circle andΔλ is the wavelength chan-nel spacing. Using the first-order approximation inthe variation of effective refractive index, the wave-length free spectral range ΔλFSR can be derived as
ΔλFSR ¼ λM − 1
: ð9Þ
B. Design Equations for Central Wavelength
Equations (3)–(9) indicate that λ, ∂λ=∂T, M, and,thus, ∂ΔLout=∂λ, Lf , and ΔλFSR are dependent onthe input/output ports. For the central wavelengthλc, Eqs. (3)–(6) can be simplified by the conditionof ΔLin þΔLout ¼ 0 and are rewritten as
nbeΔLb þ nw1
e ΔLw1 þ nw2e ΔLw2 ¼ mλc; ð10Þ
M ¼ mnbgΔLb þ nw1
g ΔLw1 þ nw2g ΔLw2
nbeΔLb þ nw1
e ΔLw1 þ nw2e ΔLw2
; ð11Þ
nbgΔLb þ nw1
g ΔLw1 þ nw2g ΔLw2 ¼ Mλc; ð12Þ
∂λc∂T
¼ 1M
�∂nb
e
∂TΔLb þ ∂nw1
e
∂TΔLw1 þ ∂nw2
e
∂TΔLw2
�: ð13Þ
In Eq. (13), the thermal expansion effects areneglected, because they are much smaller thanthe thermo-optic effects [15]. The temperature-independent condition of the central wavelength,∂λc=∂T ¼ 0, can be derived from Eq. (13) as
∂nbe
∂TΔLb þ ∂nw1
e
∂TΔLw1 þ ∂nw2
e
∂TΔLw2 ¼ 0: ð14Þ
Generally, the wavelength free spectral range ΔλFSRis given at first as the basic parameters for the designof an AWG, and then the modified grating order thatsatisfied the given dispersion requirementMgiven canbe calculated from Eq. (9). Thus, ðΔLw1Þgiven andðΔLw2Þgiven can be derived from Eqs. (12) and (14) as
8>>>>>>>><>>>>>>>>:
ðΔLw1Þgiven ¼ −
∂nw2e
∂T Mgivenλc −�nbg∂nw2
e∂T −nw2
g∂nb
e∂T
�ΔLb
nw2g
∂nw1e
∂T −nw1g
∂nw2e
∂T
ðΔLw2Þgiven ¼∂nw1
e∂T Mgivenλc −
�nbg∂nw1
e∂T −nw1
g∂nb
e∂T
�ΔLb
nw2g
∂nw1e
∂T −nw1g
∂nw2e
∂T
:
ð15Þ
Note that the phase-match condition of Eq. (10)should be satisfied and the grating order m shouldbe an integer for the central wavelength λc. The in-teger value of m can be obtained by substitutingfor ΔLw1 and ΔLw2 from Eq. (15) into Eq. (10), andis given as
m¼round�nbeΔLbþnw1
e ðΔLw1Þgivenþnw2e ðΔLw2Þgiven
λc
�;
ð16Þ
where round (x) is the arithmetic operation to findthe closest integer of x. Then, ΔLw1 and ΔLw2 shouldbe modified according to the integer value of m andthe expressions can be derived from Eqs. (10) and(14) as
8>>>>>>><>>>>>>>:
ΔLw1 ¼ −
∂nw2e
∂T mλc −�nbe∂nw2
e∂T − nw2
e∂nb
e∂T
�ΔLb
nw2e
∂nw1e
∂T − nw1e
∂nw2e
∂T
ΔLw2 ¼∂nw1
e∂T mλc −
�nbe∂nw1
e∂T − nw1
e∂nb
e∂T
�ΔLb
nw2e
∂nw1e
∂T − nw1e
∂nw2e
∂T
: ð17Þ
The M that determined by the structure should alsobe modified according to the integer value of m andcan be calculated using Eq. (11). From Fig. 1(b),
the path length of the ith arrayed waveguide Li isgiven as
Li ¼ L0 þ 2Rφi þ Sw1i þ Sw2
i ; ð18Þwhere
Sji ¼ 2ðhj
i þ ljiÞ;
where j ¼ w1 or w2 for the type 1 or type 2 wave-guides, respectively. L0 is the total path length of theadiabatic tapers, the mode converters, and the 90°bent waveguides. Considering the path-length differ-ence between adjacent arrayed waveguides, the pathlength of straight waveguides in the ith arrayedwaveguide Sj
i can be written as
Sji ¼ Sj
1 þΔLj1;2 þ · · ·þΔLj
i−1;i: ð19Þ
From Eq. (17), ΔLw1 and ΔLw2 have opposite signs,because ∂ne=∂T is generally positive and ΔLb is verysmall. In the following analysis, we assume that Si
w2
increases when i increases, i.e., ΔLw1 < 0 andΔLw2 > 0, which is the case shown in Fig. 1. Thus,the path length in the longest arrayed waveguidesSN
j is
�Sw1N ¼ Sw1
1 − jΔLw11;2j − � � � − jΔLw1
N−1;N jSw2N ¼ Sw2
1 þ jΔLw21;2j þ � � � þ jΔLw2
N−1;N j: ð20Þ
Since the path length of each waveguide should notbe negative, then hj
j ≥ 0, ljj ≥ 0, and Sij ≥ 0. Applying
these inequalities to Eq. (20), one can obtain thatS1
w2 ≥ 0 and S1w1 ≥ ðjΔL1;2
w1j þ � � � þ jΔLN−1;Nw1jÞ.
To make the size compact, we can set S1w2 ¼ 0 and
S1w1 ¼ jΔL1;2
w1j þ � � � þ jΔLN−1;Nw1j. Then Si
j canbe expressed as
�Sw1i ¼ jΔLw1
i;iþ1j þ � � � þ jΔLw1N−1;N j
Sw2i ¼ ΔLw2
1;2 þ � � � þΔLw2i−1;i
: ð21Þ
If the differences of the effective refractive indices be-tween bent Si wire waveguides and the type 1 wave-guides are neglected, Eqs. (10)–(13) can be simplifiedby the condition of ne
b ¼ new1, ng
b ¼ ngw1, and
∂neb=∂T ¼ ∂ne
w1=∂T. Thus, Eqs. (17) and (21) canbe simplified as
8<:
ΔLw2 ¼ mλcnw2e − rAnw1
eΔLw1 ¼ −rAΔLw2 −ΔLb
; ð22Þ
�Sw1i ¼ rAðN − iÞΔLw2 þ 2RðφN − φiÞ
Sw2i ¼ ði − 1ÞΔLw2 ; ð23Þ
where
rA ¼ ∂nw2e
∂T=∂nw1
e
∂T: ð24Þ
The integer value of m can also be obtained by thesimplified form of Eqs. (15) and (16). Moreover, aftertheMgiven is determined fromΔλFSR using Eq. (9), theinteger value ofm can be obtained using Eq. (11) andis given as
m ¼ round�Mgiven
nw2e − rAnw1
e
nw2g − rAnw1
g
�: ð25Þ
Then, the M is modified according to the integervalue of m and is given as
M ¼ mnw2g − rAnw1
g
nw2e − rAnw1
e: ð26Þ
If we assume Siw1 increases when i increases,
i.e., ΔLw2 < 0 and ΔLw1 > 0, we can set S1w1 ¼ 0
and S1w2 ¼ jΔL1;2
w2j þ � � � þ jΔLN−1;Nw2j; then
Siw1 ¼ −rAði − 1ÞΔLw2 þ 2Rðφ1 − φiÞ and Si
w2 ¼−ðN − iÞΔLw2. Since LN > L1, this assumption leadsto r > 1, which cannot satisfy the temperature-inde-pendent condition of Eq. (24) for the example of thewire–slot-hybrid design.
C. Tuning Parameter
Taking hints from Eq. (22), we define the path-lengthdifference ratio as
r ¼ −ΔLw1 þΔLb
ΔLw2 : ð27Þ
The ∂λc=∂T can be simplified from Eq. (13) and berepresented as
∂λc∂T
¼ ΔLw2
M
�∂nw2
e
∂T− r
∂nw1e
∂T
�: ð28Þ
It can be seen that the rA in Eq. (24) is a special value,which satisfies the theoretically temperature-independent condition, of the r in Eq. (27). Thus,the rA in Eqs. (22), (23), (25), and (26) can be replacedby the r, when a temperature-independent conditionis not applied. Equations (22)–(28) show that the de-vice size and the transmission characteristics arecontrolled by the path-length difference ratio r. Thus,the r can be used as the tuning parameter to adjustthe thermal performance of AWGs.
D. Theory Extension
Actually, the theory can be generalized for a hy-brid combination of multiple types of waveguide inthe arrayed waveguides, which is useful in thedesign of a special TI-AWG that satisfies multiple-requirements. Define the path-length differenceratio as
rj ¼ −ΔLwj=ΔLwJ ; ð29Þwhere j ¼ 1; 2;…;J. Note that rJ ¼ −1. Then, thegrating order m, the path-length difference betweenadjacent arrayed waveguides for the Jth type of
waveguide ΔLwJ, the total path-length differenceΔLtotal, the path length of the jth type of waveguidesin the ith arrayed waveguide Si
wj, the total pathlength Li, the modified grating orderM, and the tem-perature coefficient of the central wavelength ∂λc=∂Tcan be derived as follows:
m ¼ round�Mgiven
PJj¼1 rjn
wjeP
Jj¼1 rjn
wjg
�; ð30Þ
ΔLwJ ¼ mλcPJj¼1ð−rjnwj
e Þ; ð31Þ
ΔLtotal ¼XJj¼1
ΔLwj; ð32Þ
Swji ¼
� ði − 1ÞΔLwj; ΔLwj > 0ðN − iÞjΔLwjj; ΔLwj < 0
; ð33Þ
Li ¼ L0 þXJj¼1
Swji ; ð34Þ
M ¼ m
PJj¼1 rjn
wjgP
Jj¼1 rjn
wje
; ð35Þ
∂λc∂T
¼ ΔLwJ
M
XJj¼1
�−rj
∂nwje
∂T
�: ð36Þ
Equations (29)–(36) can be used to design an AWGwith a combined use of JðJ ≥ 1Þ types of waveguidein the arrayed waveguides. The layout size and thedevice transmission characteristics are controlled bya number of parameters, such as the rj, the weightedeffective-refractive-indices difference
wjÞ, and the weighted temperature coeffi-cients of effective refractive indices
PJj¼1½−rj∂ne
wj
×ð∂TÞ−1�, which renders a high level of flexibilityfor optimizing the design.
3. Design Example
In this paper, we choose a TI-AWG design using nar-row Si wire waveguides with Wcore ¼ 400 nm andslot waveguides with WSi ¼ 150 nm and Wslot ¼100 nm as an example. The narrow Si wire wave-guide and the slot waveguide are selected as type1 and type 2 waveguides, respectively. Equations(2), (8), (9), and (22)–(26) are used to determinethe structure parameters of the wire–slot-hybridTI-AWG. Here, four assumptions are introduced.
First, the design is based on the parameters of theTE-like fundamental mode of the central wavelengthat room temperature. Second, the pitch between in-put/output/arrayed waveguides is far less than thediameter of the Rowland circle, i.e., da ≪ Lf anddio ≪ Lf . Third, the thermal expansion effects areneglected, i.e., α ≪ ð∂ne=∂TÞ=ne. Fourth, the differ-ences of effective refractive indices between Si bentand straight wire waveguides are neglected, i.e.,ne
b ¼ new1, ng
b ¼ ngw1, and ∂ne
b=∂T ¼ ∂new1=∂T.
A. Design Steps
The main design steps are outlined as follows. First,the basic parameters of AWG are set. In our design,λc ¼ 1:55 μm, Δλ ¼ 16 nm, and the numbers of inputwaveguides and output waveguides are one and six,respectively.
Second, the material parameters of the wave-guides are estimated. The silicon-on-insulator waferwith a 260 nm Si layer is selected as the design plat-form. The material parameters that are estimated bythe full-vector finite-element method are listed inTable 1.
Third, the parameter adjustments that are neededto make a compromise between performance and de-vice size are made. The compromised parameters arelisted in Table 2. The large gap between arrayedwaveguides is to make sure that the coupling be-tween adjacent slot waveguides can be neglected.The adiabatic tapers are inserted between the slabwaveguides and the arrayed input/output wave-guides to reduce the coupling loss.
Fourth, the structure parameters are determined.The nominal free spectral range is 128 nm, as calcu-lated byΔλFSR ¼ ΔλNmax. From Eqs. (2), (8), (9), (22),and (24)–(26), Mgiven ¼ 13:109, rA ¼ 0:184, m ¼ 11,M ¼ 13:146, Lf ¼ 45:673 μm, ΔLw2 ¼ 15:252 μm,ΔLw1 ¼ −2:807 μm −ΔLb, and ΔLb is −0:788 μmfor 1 ≤ i ≤ 14 or 0:788 μm for 15 ≤ i ≤ 28, respectively.Then, Si
jði ¼ 1; 2;…;N; j ¼ w1;w2Þ can be calculatedfrom Eq. (23). From Fig. 1, it can be observed that thewaveguide path length in the X direction liw1 þ liw2 isdetermined by other structural parameters, i.e., Lf ,ltaper,R, and φi. In this example, only slot waveguidesare used in the X direction, i.e., liw1 ¼ 0, and, thus,liw2 is determined. For the path length in the Y direc-tion, we have hi
w1 ¼ Siw1 and hi
w2 ¼ Siw2 − liw2. So
far, all the structural parameters are determined,
and the entire structure as shown in Fig. 1(a) can begenerated by a drawing program.
Finally, the parameters should be adjusted accord-ing to the results. Some errors are introduced fromthe design to fabrication, including the four assump-tions mentioned above and the fabrication errors,which means the measured temperature sensitivityof the fabricated AWG may not exactly satisfy∂λc=∂T ¼ 0. According to the results of the spectralthermal shift, a different r should be used in Eqs.(22), (23), and (25)–(28) to adjust the ratio of ΔLw1
and ΔLw2, and, thus, to fine tune the thermal perfor-mance of AWGs.
B. Results and Analysis
Following the design steps in Subsection 3.A, thetuning effects of the path-length difference ratio ron the temperature sensitivity, the dispersion perfor-mance, and the structural parameters of the wire–slot-hybrid example are studied. The temperaturecoefficient of the central wavelength ∂λc=∂T is calcu-lated using Eq. (28). The dispersion enhancementfactor is defined as
F ¼ M=m: ð37Þ
The nominal grating order m and the modified grat-ing order M are calculated from Eqs. (25) and (26),respectively. The path-length difference between ad-jacent arrayed waveguides for the type 2 waveguidesΔLw2 and the total path-length difference ΔLtotal ¼ΔLb þΔLw1 þΔLw2 are selected as the characteris-tic parameters for the device size. This is because thedevice size is mainly determined by the path lengthof the longest arrayed waveguide LN and the layout.The LN can be derived from Eqs. (18) and (23) as
LN ¼ L0 þ 2RφN þ ðN − 1ÞΔLw2: ð38Þ
It can be seen that LN is proportional to ΔLw2, whilethe latter is a basic parameter in the design and thenis selected as the characteristic parameter. The lay-out is mainly restricted by the ΔLtotal. For example,the ΔLtotal in Fig. 1(a) should be larger than 4 times
of the minimum gap between arrayed waveguides.Otherwise, another type of layout should be used,such as the serpentine design [14] for an ultrasmallΔLtotal.
The tuning effects of the r on ∂λc=∂T, 1=F, ΔLw2,and ΔLtotal for the wire–slot-hybrid design exampleare shown in Fig. 2. We choose 1=F instead of F forillustration, because these four parameters have thesame singularity rs ¼ ng
w2=ngw1 ¼ 0:496 and will be
infinity at that point. In the vicinity of rv ¼ne
w2=new1 ¼ 0:657, it can be seen that space intervals
exist in Figs. 2(a) and 2(b), while the lines with zerovalues exist in Figs. 2(c) and 2(d). This is because theinteger value of m is zero, and then M, ΔLw2, andΔLtotal are zeros; thus, the ∂λc=∂T and 1=F are math-ematical indeterminacy. A positive or negative rmeans that the path length of the two types of wave-guide, i.e., Si
w1 and Siw2, have different or the same
change trends when i increases, which was discussedin Subsection 2.B.
The dashed curves, solid curves, and dashed-dotted curves are results for λ and are 1.5, 1.55,and 1:6 μm, respectively. The corresponding theore-tical temperature-independent points are indicatedby circles, diamonds, and pentagrams. It is can beclearly seen that the lines in Figs. 2(c) and 2(d) arepiecewise other than continuous, because the gratingorderm is a discrete integer rather than a continuousnumber. For comparison, the results calculated fromthe m before the arithmetic operation round (x), i.e.,m0 ¼ Mðne
w2 − rnew1Þ=ðng
w2 − rngw1Þ, are also shown
in the insets.From Fig. 2(a), it can be observed that the AWGs
with only Si wire waveguides, i.e., r ¼ ∞, are moretemperature sensitive than those with only Si slotwaveguides, i.e., r ¼ 0. This is because the thermalperformance of the slot waveguide mainly relies onthe low-refractive-index material SiO2 with a smallTOC. The temperature coefficient can be tuned fromnegative to positive in an ultralarge range over thevicinity of the singularity. At that point, both ∂λ=λTand ΔLw2 [see Fig. 2(c)] are infinite, which indicatesthat the tuning range will be restricted by the devicesize. The temperature-independent point for the cen-tral wavelength rA ¼ 0:184 is far from the singular-ity, which means favorable fabrication tolerance.
It also shows the wavelength dispersion effect inthe range of ½1:5; 1:6� μm. From the inset in Fig. 2(a),the r that satisfied the temperature-independentcondition will increase when the wavelength de-creases. Thus, if an AWG is designed according to thetemperature-independent point for the central wave-length λc, the output ports with the channel wave-length λlong > λc will be overcompensated and beblueshifted, while those with the channel wave-length λshort > λc will be undercompensated and beredshifted. In Fig. 2(a), the temperature coefficientfor λlong ¼ 1:6 μm is about −4:321 pm=K, and thatfor λshort ¼ 1:5 μm is about 3:988 pm=K, while λc ¼1:55 μm is temperature independent.
Table 2. Compromised Parameters
Maximal number of channels Nmax 8Number of arrayed waveguides Na 29Minimum gap between arrayed waveguides dmin 1:85 μmPitch of arrayed waveguides at grating circle da 1:85 μmPitch of input/output waveguidesat Rowland circle
dio 1:85 μm
Width of tapers of arrayed waveguidesat grating circle
wataper 1:8 μm
Width of tapers of input/output waveguidesat Rowland circle
wiotaper 1:5 μm
Length of tapers between Si wire andslab waveguides
ltaper 15 μm
Length of mode converters betweenSi wire and slot waveguides
Figure 2(b) shows that the AWGs with only onetype of waveguide have the inherent dispersion en-hancement property, i.e., F > 1, because ng is largerthan ne for Si wire and slot waveguides. The F can betuned from 0 to ∞ by controlling the r, but it will alsobe restricted by the device size. At the temperature-independent point, F is about 1.195. The modified in-terference order is closed to the nominal interferenceorder and is a little enhanced, which is another ad-vantage of our proposed wire–slot-hybrid TI-AWGs.The wavelength dispersion effect is also illustrated.In the vicinity of the temperature-independent point,the (1=F) will decrease when the wavelength de-creases. In other words, the short-wavelength regimehas a larger dispersion enhancement factor than thelong-wavelength regime.
TheΔLw2andΔLtotal areshowninFigs.2(c)and2(d).Note thatΔLw2 decreases to zerowhen the r increasesto infinity. It canbeobserved that theΔLw2 andΔLtotal
at the temperature-independent point are larger thanthoseatr ¼ 0andr ¼ ∞,whichmeansthedevicesizeofa TI-AWG is larger than that of a conventional AWGwith only one type of waveguide. In the wire–slot-hybrid TI-AWG, ΔLw2 ¼ 15:252 μm and ΔLtotal ¼12:445 μm. The ΔLw2, and thus LN, is small, which
is suitable for the compact device. The ΔLtotal, whichshould be small but also be larger than 4 times theminimumgap between arrayedwaveguides, is appro-priate for the layoutshowninFig.1(a).Thedeviceareais about 192:846 μm× 262:954 μm when the channelspacing is Δλ ¼ 16 nm.
For comparison, a narrow–wide-wire TI-AWG isalso designed following the design steps in Subsection3.A. The width of the narrow and wide Si wire wave-guides are set to be 400 and 1000 nm, respectively.The parameters of the 1000 nm wide Si wire wave-guide are ne
w2 ¼ 2:867,ngw2 ¼ 3:725, and ∂ne
w2=∂T ¼1:936 × 10−4=K. Other parameters are the sameas in Subsection 3.A. Then, rA ¼ 0:934, ΔLw2 ¼−73:143 μm, and ΔLtotal ¼ −4:796 μm. The ΔLw2 ofthis narrow–wide-wire case is about 5 times largerthan that of the wire–slot-hybrid design. Consideringthe linear relation between the path length of thelongest arrayed waveguide LN and the ΔLw2, this in-dicates that the device area for the wire–slot-hybriddesign is about one-fifth that of the narrow–wide-wiredesign.However, the layout inFig. 1(a) cannot beusedin this case, because theΔLtotal is too small and, thus,the arrayedwaveguideswill cross each other. Instead,the serpentine layout [14] should be used.
Fig. 2. (Color online) Tuning effects of r on the performance of the wire–slot-hybrid example. (a) ∂λ=∂T, (b) 1=F, (c)ΔLw2, and (d) ΔLtotal.The dashed curves, solid curves, and dashed-dotted curves are results for λ is 1.5, 1.55, and 1:6 μm, respectively. The corresponding the-oretical temperature-independent points are indicated by circles, diamonds, and pentagrams. The insets show the details in the vicinity oftemperature-independent points. For comparison, the results calculated from the m before the arithmetic operation round (x), i.e.,m0 ¼ Mðne
The imbalance loss of the ith and the (iþ 1)th ar-rayed waveguide ΔB can be given from Eq. (32) as
ΔB ¼XJj¼1
BwjΔLwj; ð39Þ
where Bwj is the loss per unit path length of the wjthtype of waveguide. For the design example of thewire–slot-hybrid TI-AWG, ΔB is about 0:01 dB,while the loss of the longest arrayed waveguide (ex-cluding the loss of L0) is about 0:43 dB. Here, thelosses of the narrow Si wire waveguide (Wcore ¼400 nm) and the slot waveguide (WSi ¼ 150 nmand Wslot ¼ 100 nm) are set to be 2:6 dB=cm [18]and 10 dB=cm [19], respectively. This indicates thatthe lossy Si wire and slot waveguides are not seriousproblems in the compact TI-AWGs.
Three AWGs with different r are designed and si-mulated by our 2D-FDTD program. The 2D approx-imation may change the mode of the optical field andthen alter the loss and the cross-talk performance.However, it can still verify the tuning effects of ther. In the 2D planar waveguide structure, the effectiverefractive indices and their temperature coefficientsof the waveguide core layer and cladding layer arecalculated as ne
core ¼ 2:965, neclad ¼ 1:45, ∂ne
core=∂T ¼ 1:883 × 10−4=K, and ∂ne
clad=∂T ¼ 1 × 10−5=K.We assume that the temperature coefficients are con-
stants over the temperature variation of ΔT ¼100 K. The grid size is 25 nm × 25 nm, and the timestep is 4:17 × 10−17 s for a total of 8:5 × 10−12 s. Thetotal calculation time should be larger than thetransmission time, which is about 7 × 10−12 s, inthe whole structure. The small additional time of1:5 × 10−12 s means that the reflection power is ne-glected in the spectral distribution shown in Fig. 3.This will not affect the thermal performance of thesimulated devices.
Figure 3 shows the spectral shift of three AWGswith different r when the temperature varies. TheAWG in Fig. 3(a) with r ¼ 0:184, which is the theo-retical temperature-independent point in the three-dimensional waveguide structure at the designedcentral wavelength λc ¼ 1:55 μm, is selected as theinitial value for the 2D simulation. It is shown thatthe temperature coefficient is about 30 pm=K. This isbecause of the approximations mentioned at the be-ginning of this section, the 2D approximation, andthe limited grid size of the FDTD program. For thesame reasons, the calculated central wavelength ismodified and blueshifts from the designed values.From Fig. 2(a), the spectra thermal shift will bereduced by increasing appropriately the r. Then,another two AWGs with r ¼ 0:222 and 0.26 are de-signed and simulated. The spectral shifts are illu-strated in Figs. 3(b) and 3(c), respectively. It is
Fig. 3. (Color online) Spectra distributions (simulated by 2D-FDTD program) of three AWGs with r are (a) 0.184, (b) 0.222, and (c) 0.26.The solid curves and dashed-dotted curves are for ΔT ¼ 0 K and ΔT ¼ 100 K, respectively.
shown that the normal red spectral shift becomesblueshift when r increases from 0.184 to 0.26, whichmeans that an ideal temperature-independent AWGcan be obtained by fine adjusting r. When the r is0.222, ∂λ=∂T is suppressed to about −5 pm=K ataround 1:55 μm.
It is also shown that the compensated effect ofthe output ports with long-wavelength channels isstronger than those with short-wavelength channels,which is consistent with the results in Fig. 2. TakingFig. 3(b) as an example, the spectra is redshift ataround the wavelength of 1:48 μm while it is blue-shift at around 1:63 μm.
4. Conclusion
A complete theory for the design of a TI-AWG basedon the combination of two types of waveguide is de-veloped. This theory is also extended to the combineduse of multiple types of waveguide, which is usefulfor the design of a TI-AWG that satisfies multiple re-quirements. The ratio of the path-length differenceand its derived parameters are introduced as tuningparameters to control the device transmission char-acteristics and the layout size. The dependence on anumber of adjustable parameters renders a high le-vel of flexibility for optimizing the design of a TI-AWG. The detailed design steps for the TI-AWGsare presented, and then a TI-AWG realized by thecombination of Si wire and slot waveguides (wire–slot-hybrid) is given as an example. The tuningeffects of the path-length ratio on the temperaturesensitivity, the dispersion enhancement, and the de-vice size are studied. The results show that the tem-perature coefficient and the dispersion enhancementfactor can be tuned over a very large range, but willbe restricted by the device size. At the temperature-independent point, the modified grating order is a lit-tle enhanced compared to the nominal grating order.The device size is about one-fifth that of the narrow–
wide-wire design that uses a combination of narrowand wide silicon wire waveguides. The wavelengthdispersion effects are also studied. A TI-AWG de-signed according to the temperature-independentpoint of the central wavelength will be over-compensated and undercompensated for the longwavelength and short wavelength, respectively.Moreover, the short-wavelength regime has a largerdispersion enhancement factor than the long-wavelength regime. The prototype devices are simu-lated by a 2D-FDTD program and the results areconsistent.
The authors thank Haiying Hu for part of the si-mulation and Jiafu Wang for fruitful discussion.Huamao Huang thanks China Scholarship Councilfor financial support.
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