4. integrated photonics
TRANSCRIPT
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Mach-Zehnder modulator made from Indium Phosphide (InP) designed for 128 Gbs.
Are we experiencing a
similar transformation in Photonics ?
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βPhotonic Integrated Circuits are the next logical step in the world of optics!β, Infinera Corporation.
βWaveguide Integrated Optics involves the control of light analogous to integrated circuits in electronics. Processing and routing of data in the optical domain can offer advantages compared to electronic solutions, especially at increasing data ratesβ, Optical Society of America, 2015.
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lasers photodetectors
A Few Examples of Integrated Photonic Components
optical fibers planar waveguides
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M. Liu et al., Nature 474, 64 (2011)
A graphene-based electro-absorption modulator: In a device such as the one demonstrated by Liu et al. in 2011, electrically connected graphene is coupled to a SiO2
waveguide carrying a CW photon stream.
Driving Fundamental Research on Novel Materials and Devices
A Crucial Element: Light Guiding Geometries
2D (slab) and 3D (channel & optical fiber)
ππ > ππ
ππ > ππ
graded refractive index
step refractive index
π > π‘0
Requirements
Plane Waves
discrete set of modes
continuous set of modes
continuous set of modes
ππ > π > ππ
π > ππ > ππ
ππ > ππ > π
Maxwellβs Equations (isotropic, linear, lossless, non-magnetic)
π» Γ π¬ = βπ0 ππ―
ππ‘
π» Γ π― = π2 π0 ππ¬
ππ‘
Faradayβs law
Ampereβs law
π¬ β β π―
π― β π¬
Note:
π = π2 π0 β π0
π» Γ π» Γ π¬ = βπ0 ππ―
ππ‘
π» Γ π» Γ π― = π2 π0 ππ¬
ππ‘
π»2π¬ = π2
π2π2π¬
ππ‘2
π»2π― = π2
π2π2π―
ππ‘2
Wave Equations
A Propagating Wave along the Guide
π¬ π₯, π¦, π§, π‘ = πΈ π₯, π¦ ππ π π‘ β π½ π§
π― π₯, π¦, π§, π‘ = π» π₯, π¦ ππ π π‘ β π½ π§
π2
ππ‘2= βπ2
π»2 =π2
ππ₯2+π2
ππ¦2β π½2
π2πΈ π₯, π¦
ππ₯2+π2πΈ π₯, π¦
ππ¦2+π2π2
π2 β π½2 πΈ π₯, π¦ = 0
π2π» π₯, π¦
ππ₯2+π2π» π₯, π¦
ππ¦2+π2π2
π2 β π½2 π» π₯, π¦ = 0
2D Optical Waveguides
By considering the symmetry along y-axis: (slab case)
πΈ π₯, π¦ = πΈ π₯
π» π₯, π¦ = π» π₯
π2πΈ π₯
ππ₯2+π2π2
π2 β π½2 πΈ π₯ = 0
π2π» π₯
ππ₯2+π2π2
π2 β π½2 π» π₯ = 0
Transverse Electric (TE)
πΈ π₯ =0πΈπ¦ π₯
0
π» Γ π¬ π₯, π¦, π§, π‘ = βπ0 ππ― π₯, π¦, π§, π‘
ππ‘ π» π₯ =
βπ½ πΈπ¦ π₯
π π00
β 1
π π π0 ππΈπ¦ π₯
ππ₯
Faradayβs law
π½ β‘π
π π
π2πΈπ¦ π₯
ππ₯2+π2
π2π2 π₯ β π2 πΈπ¦ π₯ = 0
Guided TE Solution π2πΈπ¦ π₯
ππ₯2+π2
π2π2 π₯ β π2 πΈπ¦ π₯ = 0
π₯
π§
π =?
ππ
ππ
ππ
π₯ > 0 β π π₯ = ππ < π
βπ < π₯ < 0 β π π₯ = ππ > N
π₯ < βπ β π π₯ = ππ < N
πΈπ¦ π₯ = πΈπ πβπΎπ π₯
πΈπ¦ π₯ = πΈπ ππΎπ π₯+π
π
πΎπ =π
ππ2 β ππ
2
πΎπ =π
ππ2 β ππ
2
πΈπ¦ π₯ = πΈπ πππ ππ₯ π₯ + ππ
ππ₯ =π
πππ2 β π2
Boundary Condition at Cladding-Film Interface
πΈπ = πΈπ πππ ππ
π₯ = 0
πΈπ¦
π»π§ =β 1
π π π0 ππΈπ¦ π₯
ππ₯ πΎππΈπ = ππ₯ πΈπ sin ππ
tan ππ =πΎπππ₯
Boundary Condition at Substrate-Film Interface
πΈπ = πΈπ πππ βππ₯ π + ππ
π₯ = βπ
πΈπ¦
π»π§ =β 1
π π π0 ππΈπ¦ π₯
ππ₯ πΎπ πΈπ = βππ₯ πΈπ sin βππ₯ π + ππ
tan ππ₯ π β ππ =πΎπ ππ₯
Dispersion Relation for TE Modes
tan ππ =πΎπππ₯
tan ππ₯ π β ππ =πΎπ ππ₯
ππ₯ π = π‘ππβ1πΎπ ππ₯+ π‘ππβ1
πΎπππ₯+π π
&
2 π
ππ ππ
2 β π2 = π‘ππβ1π2 β ππ
2
ππ2 β π2
+ π‘ππβ1π2 β ππ
2
ππ2 β π2
+π π
b-V diagram
2 π
ππ ππ
2 β π2
π β‘2 π
ππ ππ
2 β ππ 2
ππΈ β‘π2 β ππ
2
ππ2 β ππ
2
ππΈ β‘ππ 2 β ππ
2
ππ2 β ππ
2
π 1 β ππΈ = π‘ππβ1
ππΈ1 β ππΈ
+ π‘ππβ1ππΈ + ππΈ1 β ππΈ
+π π
cut-off:
π ππ
0 ππΈ
ππ = π0 +π π
π0 β‘ π‘ππβ1 ππΈ
asymmetry factor
Transverse Magnetic (TM)
π» π₯ =0π»π¦ π₯
0
π» Γ π― π₯, π¦, π§, π‘ = π2 π0ππ¬ π₯, π¦, π§, π‘
ππ‘ πΈ π₯ =
π½ π»π¦ π₯
π π2 π00
1
ππ π2 π0 ππ»π¦ π₯
ππ₯
Ampereβs law
π2π»π¦ π₯
ππ₯2+π2
π2π2 π₯ β π2 π»π¦ π₯ = 0 π½ β‘
π
π π
Guided TM Solution π2π»π¦ π₯
ππ₯2+π2
π2π2 π₯ β π2 π»π¦ π₯ = 0
π₯
π§
π
ππ
ππ
ππ
π₯ > 0 β π π₯ = ππ < π
βπ < π₯ < 0 β π π₯ = ππ > N
π₯ < βπ β π π₯ = ππ < N
π»π¦ π₯ = π»π πβπΎπ π₯
π»π¦ π₯ = π»π ππΎπ π₯+π
π
πΎπ =π
ππ2 β ππ
2
πΎπ =π
ππ2 β ππ
2
π»π¦ π₯ = π»π πππ ππ₯ π₯ + ππ
ππ₯ =π
πππ2 β π2
Boundary Condition at Cladding-Film Interface
π»π = π»π πππ ππ
π₯ = 0
π»π¦
πΎπππ2π»π =ππ₯ππ2 π»π sin ππ
tan ππ =πΎπππ2
ππ2
ππ₯
πΈπ§ =1
π π π2 π0 ππ»π¦ π₯
ππ₯
Boundary Condition at Substrate-Film Interface
π»π = π»π πππ βππ₯ π + ππ
π₯ = βπ
π»π¦
πΈπ§ =1
π π π2 π0 ππ»π¦ π₯
ππ₯
πΎπ ππ 2π»π = β
ππ₯ππ2π»π sin βππ₯ π + ππ
tan ππ₯ π β ππ =πΎπ ππ 2
ππ2
ππ₯
Dispersion Relation for TM Modes
ππ₯ π = π‘ππβ1πΎπ ππ 2
ππ2
ππ₯+ π‘ππβ1
πΎπππ2
ππ2
ππ₯+π π
&
2 π
ππ ππ
2 β π2 = π‘ππβ1ππ2
ππ 2
π2 β ππ 2
ππ2 β π2
+ π‘ππβ1ππ2
ππ2
π2 β ππ2
ππ2 β π2
+π π
tan ππ =πΎπππ2
ππ2
ππ₯ tan ππ₯ π β ππ =
πΎπ ππ 2
ππ2
ππ₯
Overall Dispersion Relation 2 π
ππ ππ
2 β π2 = π‘ππβ1ππ
ππ
2ππ2 β ππ
2
ππ2 β π2
+ π‘ππβ1ππ
ππ
2ππ2 β ππ
2
ππ2 β π2
+π π
π = 0
π = 1
TE
TM
vππππ’π =ππ
ππ½π π
phase velocity:
group velocity:
vπβππ π =π
π½π π=π
ππ π
Different Types of Dispersion in a Waveguide
β’ modal dispersion β’ material dispersion β’ waveguide dispersion
Field Profile of Guided Modes Discrete Set of Solutions
evanescent field
oscillatory behavior
m = mode order
Propagating Power along the Waveguide
π = 1
2Re πΈ Γ π»β ππ§ =
1
2ππ§ ππ₯
β
ββ
Power/unit-width:
TE mode:
ππ§ = β1
2πΈπ¦ π»π₯
βππ₯β
ββ
π»π₯ =βπ½ πΈπ¦ π₯
π π0
ππ§ =π½
2 π π0 πΈπ¦
2 ππ₯
β
ββ
Poynting vector:
ππ§ =π½
2 π π0 πΈπ¦
2 ππ₯
β
ββ=π½
4 π π0πΈπ2 ππππ
ππππ β‘ π + π
2π π2 β ππ 2
+π
2π π2 β ππ2
effective thickness or mode size wavelength dependent
How much power can we put on each mode of a guide
from an incoherent blackbody source?
π =π
π2 β π π
πβ ππΎ π β 1
, π = π
π Γππππ
π = 19ππ
ππ Γππππ= β77
ππ΅π
ππ Γππππ
π = 550 ππ π = 3,000 πΎ π = 0.33
Intensity Profile propagating in Multimode Guides
pure excitation of mode 0
pure excitation of mode 1
mixed excitation of modes 0 & 1
Easier Route to Dispersion Relation:
Phase-change under total internal reflection
ππ = ππππ
ππ ππ
phase-change at film/substrate interface
phase-change at film/cladding interface
ππ = β2 π‘ππβ1ππ
ππ
2ππ2 β ππ
2
ππ2 β π2
ππ = ππππ
ππ = β2 π‘ππβ1ππ
ππ
2ππ2 β ππ
2
ππ2 β π2
π = ππ π πππ
Phase Change due to Propagation
πππ = ππ ππ π΄π΅ + π΅πΆ = ππ
ππ 2 π πππ π = 2 π ππ₯
ππ
ππ
ππ π π΄
πΆ
π΅
π
2 π
Resonant Condition:
πππ +ππ + ππ = 2 π π
2 ππ₯π β 2 π‘ππβ1ππ
ππ
2ππ2 β ππ
2
ππ2 β π2
β2 π‘ππβ1ππ
ππ
2ππ2 β ππ
2
ππ2 β π2
= 2 π π
Waveguide Couplers: injecting light into waveguides
β’ End couplers (usually used for channel and optical fibers)
β’ Transverse couplers (prism-coupler or grating-coupler) (typically used for slab waveguides)
End Coupler
πΌππ π₯, π¦ = ππΌ
πΌ
π¬πΌ π₯, π¦
ππΌ = πΌππ . π¬πΌ
βππ΄2
πΌππ2ππ΄ π¬πΌ
2ππ΄
πΌππ
π¬πΌ
input field decomposed into modes of the guide
Overlap Integral: fraction of coupled power into each mode