4. Integrated Photonics. (or optoelectronics on a flatland)

Transcription

1 4. Integrated Photonics (or optoelectronics on a flatland) 1

2 x Benefits of integration in Electronics:

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4 Are we experiencing a similar transformation in Photonics? Mach-Zehnder modulator made from Indium Phosphide (InP) designed for 18 Gbs. 4

5 Waveguide 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 rates, Optical Society of America, 015. Photonic Integrated Circuits are the next logical step in the world of optics!, Infinera Corporation. 5

6 A Few Examples of Integrated Photonic Components lasers photodetectors optical fibers planar waveguides 6

7 modulators optical amplifiers add/drop filters 7

8 Bragg gratings wavelength division multiplexing (WDM) couplers optical isolator 8

9 Driving Fundamental Research on Novel Materials and Devices M. Liu et al., Nature 474, 64 (011) A graphene-based electroabsorption modulator: In a device such as the one demonstrated by Liu et al. in 011, electrically connected graphene is coupled to a SiO waveguide carrying a CW photon stream. 9

10 Early Days

11 A Somewhat Recent (008) Retrospect

12 A Crucial Element: Light Guiding Geometries Requirements D (slab) and 3D (channel & optical fiber) n f > n c n f > n s T > t 0 step refractive index graded refractive index

13 Plane Waves discrete set of modes θ > θ s > θ c continuous set of modes θ s > θ > θ c continuous set of modes θ s > θ c > θ

14 Maxwell s Equations (isotropic, linear, lossless, non-magnetic) E = μ 0 H t H = n ε 0 E t Faraday s law Ampere s law Note: E H H E ε = n ε 0 μ 0 Wave Equations E = μ 0 H t E = n E c t H = n ε 0 E t H = n H c t

15 A Propagating Wave along the Guide E x, y, z, t H x, y, z, t j ω t β z = E x, y e = H x, y ej ω t β z t = ω = x + y β E x, y x + E x, y y + n ω c β E x, y = 0 H x, y x + H x, y y + n ω c β H x, y = 0

16 D Optical Waveguides By considering the symmetry along y-axis: (slab case) E x, y H x, y = E x = H x d E x dx + n ω c β E x = 0 d H x dx + n ω c β H x = 0

17 Transverse Electric (TE) E x = 0 E y x 0 d E y x dx + ω c n x N E y x = 0 β ω c N H x, y, z, t E x, y, z, t = μ 0 t Faraday s law H x = β E y x ω μ de y x j ω μ 0 dx

18 Guided TE Solution d E y x dx + ω c n x N E y x = 0 x > 0 n x = n c < N E y x = E c e γ c x x n c N =? γ c = ω c N n c z T < x < 0 n x = n f > N T n f E y x = E f cos k x x + φ c k x = ω c n f N n s x < T n x = n s < N E y x = E s e γ s x+t γ s = ω c N n s

19 Boundary Condition at Cladding-Film Interface x = 0 E y H z = 1 j ω μ 0 de y x dx E c = E f cos φ c γ c E c = k x E f sin φ c tan φ c = γ c k x

20 Boundary Condition at Substrate-Film Interface x = T E y E s = E f cos k x T + φ c H z = 1 j ω μ 0 de y x dx γ s E s = k x E f sin k x T + φ c tan k x T φ c = γ s k x

21 Dispersion Relation for TE Modes tan φ c = γ c tan k x T φ c = γ s & k x k x k x T = tan 1 γ s k x + tan 1 γ c k x + m π π λ T n f N = tan 1 N n s n f N + tan 1 N n c n f N + m π

22 b-v diagram V 1 b E = tan 1 b E 1 b E + tan 1 a E + b E 1 b E + m π V π λ T n f n s π λ T n f N cut-off: b E N n s n f n s N b E n s 0 V m = V 0 + m π a E n s n c n f n s V 0 tan 1 a E asymmetry factor

23 Transverse Magnetic (TM) H x = 0 H y x 0 d H y x dx + ω c n x N H y x = 0 β ω c N H x, y, z, t = n ε 0 E x, y, z, t t Ampere s law E x = β H y x ω n ε dh y x jω n ε 0 dx

24 Guided TM Solution d H y x dx + ω c n x N H y x = 0 x > 0 n x = n c < N H y x = H c e γ c x x n c N γ c = ω c N n c z T < x < 0 n x = n f > N T n f H y x = H f cos k x x + φ c k x = ω c n f N n s x < T n x = n s < N H y x = H s e γ s x+t γ s = ω c N n s

25 Boundary Condition at Cladding-Film Interface x = 0 H y H c = H f cos φ c E z = 1 j ω n ε 0 dh y x dx γ c n c H c = k x n f H f sin φ c tan φ c = γ c n c n f k x

26 Boundary Condition at Substrate-Film Interface x = T H y H s = H f cos k x T + φ c E z = 1 j ω n ε 0 dh y x dx γ s n s H s = k x n f H f sin k x T + φ c tan k x T φ c = γ s n s n f k x

27 Dispersion Relation for TM Modes tan φ c = γ c n c n f & k x tan k x T φ c = γ s n s n f k x k x T = tan 1 γ s n s n f k x + tan 1 γ c n c n f k x + m π π λ T n f N = tan 1 n f n s N n s n f N + tan 1 n f n c N n c n f N + m π

28 Overall Dispersion Relation π λ T n f N = tan 1 n f n s ρ N n s n f N + tan 1 n f n c ρ N n c n f N + m π TE TM ρ = 0 ρ = 1 phase velocity: v phase = ω β m ω = c N m ω group velocity: v group = dω dβ m ω

29 Different Types of Dispersion in a Waveguide modal dispersion material dispersion waveguide dispersion

30 Field Profile of Guided Modes Discrete Set of Solutions m = mode order evanescent field oscillatory behavior

31 Propagating Power along the Waveguide Poynting vector: S = 1 Re E H P z = TE mode: Power/unit-width: 1 S z dx P z = 1 E y H x dx P z = β E ω μ y dx 0 H x = β E y x ω μ 0 P z = β E ω μ y dx 0 = β 4 ω μ 0 E f T eff T eff T + λ π N n s + λ π N n c effective thickness or mode size wavelength dependent

32 How much power can we put on each mode of a guide from an incoherent blackbody source? S = c λ h ν ε e h ν K T 1, S = W m mode λ = 550 nm T = 3,000 K ε = 0.33 S = 19 pw nm mode = 77 dbm nm mode

33 Intensity Profile propagating in Multimode Guides pure excitation of mode 0 pure excitation of mode 1 mixed excitation of modes 0 & 1

34 Easier Route to Dispersion Relation: r c r s N = n f sinθ Phase-change under total internal reflection r c = e jφ c r s = e jφ s φ c = tan 1 n f n c ρ N n c n f N φ s = tan 1 n f n s ρ N n s n f N phase-change at film/cladding interface phase-change at film/substrate interface

35 Phase Change due to Propagation C n c A θ T n f B T n s φ pr = n f ω c AB + BC = n f ω c T cosθ = T k x

36 Resonant Condition: φ pr + φ s + φ c = π m k x T tan 1 n f n s ρ N n s n f N tan 1 n f n c ρ N n c n f N = π m

37 Waveguide Couplers: injecting light into waveguides End couplers (usually used for channel and optical fibers) Transverse couplers (prism-coupler or grating-coupler) (typically used for slab waveguides)

38 End Coupler E in E α E in x, y = a α Overlap Integral: fraction of coupled power into each mode α E α x, y η α = input field decomposed into modes of the guide E in. E α da E in da Eα da

39 Prism Coupler n p sin θ p = N

40 Grating Coupler n 0 sin θ 0 + q λ Λ = N