Waveguide Coupler II. Free-Space Mach-Zehnder Interferometer

Transcription

1 Waveguide Couper II Cass: Integrated Photoni Devies Time: Fri. 8:00am ~ 11:00am. Cassroom: 資電 06 Leturer: Prof. 李明昌 (Ming-Chang Lee) Free-Spae Mah-Zehnder Interferometer Wikipedia L1 L φ π λ Detetor 1 L1 L φ π + π λ Detetor ~os L L 1 λ ~sin L L 1 λ

2 Guided-Wave Mah-Zehnder Interferometer L π if κ 4 A1 A0os( κ) B1 ja0sin( k) π A0 A1 A0os( κ) A0os( ) 4 π A0 B1 ja0sin( k) ja0sin( ) j 4 (3-dB Couper) Guided-Wave Mah-Zehnder Interferometer L Then A0 A A1exp( jβl) exp( jβl) A B B j L+ j j j L+ j 0 1exp( β φ) exp( β φ) A B φ φ A3 + j ja0sin( )exp( jβ L+ j ) B A φ φ B3 j ja0os( )exp( jβ L j ) + +

3 Guided-Wave Mah-Zehnder Interferometer L φ A3 A0 sin φ B3 A0 os If f is sighty moduated with δφ δφ A3 A0 B3 A0 It is not inear moduation! Guided-Wave Mah-Zehnder Interferometer L To make the moduation is more inear, π φ + δφ π δφ 1 1 A3 A0 sin + A0 [1 + sin( δφ)] A0 [1 + δφ] 4

4 Muti-Mode Interferene (MMI) Couper Waveguide I Waveguide II Waveguide I Waveguide II MMI (3dB) Couper The MMI region an be seen as a muti-mode waveguide. The fundamenta of the power transfer omes from mutipe modes beating. Muti-Mode Interferene (MMI) Couper h + β k n v 0,1,, yν ν 0 r where π k0 λ 0 and h yν ( ν + 1) π W eν Effetive width Suppose the MMI is we-onfined σ 1 λ0 n ev e0 M + ( r ) π nr W W W n n σ 1 for TM,0 for TE β kn 0 0 r

5 Muti-Mode Interferene (MMI) Couper β ( ν + 1) πλ 0 v kn 0 r ν ( ν + ) πλ0 and ( β0 βv) 4nW r e 4nW r e Define the L π as the beating ength of the two owest-order modes then L π π 4nW r β β 3λ e ν ( ν + ) π ( β0 βv) 3 L π Muti-Mode Interferene (MMI) Couper Z Suppose the input fied profie ψ ( y,0) imposed at z 0 ψ( y,0) ϕ( y) ϕ( y) is the eigenmode (inude radiation mode) where ν ν v ψ( yo, ) ϕ ( ydy ) ϕ v v ( ydy ) Fied-orthogonaity reations

6 Muti-Mode Interferene (MMI) Couper If the spatia spetrum of the input fied Ψ is narrow enough not to exite unguided modes, it may be deomposed into the guided modes m 1 ψ( y,0) vϕv( y) ν 0 even mode ( v is even) ϕv( y): odd mode ( v is odd) symmetri antisymmetri The fied profie at a distane z m 1 ψ( yz, ) vϕv( y)exp( jβvz) ν 0 The reative phase orrespondent to fundamenta mode m 1 ψ( yz, ) ϕ ( y)exp[ j( β β ) z] v v 0 v ν 0 m 1 νν ( + ) π ψ( yl, ) vϕv( y)exp[ j L] 3L ν 0 π Muti-Mode Interferene (MMI) Couper (A) Singe Images If L p(3 L π ) p 0,,4, (even) Then ν ( ν + ) π exp[ j L] 1 3 L π ψ( yl, ) ψ( y,0) (Image is reprodued) If ϕv ( y) ϕv( y) ϕv( y) L p(3 L π ) p 1,3,5, (odd) ν ( ν + ) π Then exp[ j L] 1 ( v is even) or 1 ( v is odd) 3L π ψ( yl, ) ϕ ( y) + ϕ ( y) for v even (os) for v odd (sin) v v v v v: even v: odd ϕ ( y) + ϕ ( y) v v v v v: even v: odd ψ ( y,0) (Image is mirrored at y 0)

7 Muti-Mode Interferene (MMI) Couper (B) Mutipe Images Then p If L (3 L π ) p 1,3,5, (odd) m 1 p π ψ( y, 3 Lπ ) vϕv( y)exp[ jνν ( + ) p ] ν 0 veven : v: odd ψ( y,0) + ψ( y,0) vϕv( y) ψ( y,0) ψ( y,0) vϕv( y) p ψ( y, 3 Lπ ) ϕ ( y) + ( j) ϕ ( y) p v v v v v: even v: odd p p 1 + ( j) 1 ( j) ψ( y,0) + ψ( y,0) Two Images Muti-Mode Interferene (MMI) Couper 3 db ouper 100% ouper Advantage of MMI: The ength oud be shorter It oud be extend to a N N ouper/spitter

8 Arrayed Waveguide for Star Couper Prinipe of Arrayed-Waveguide Grating (AWG) AWG an be used for waveength de-mutipexer Katsunari Okamoto Input/Ouput Waveguides Two fous sab regions A phase array of mutipe hanne waveguides

9 Prinipe of Arrayed-Waveguide Grating (AWG) For the phase array of waveguides, the path-ength differene ΔL between neighboring waveguides resuts in phase differene by L / λ In the first sab, the input waveguide separation is D 1, the array waveguide separation is d 1, and the radius of urvature is f 1. Simiar definition is shown in the seond sab. The ight input at the x 1 position (x 1 is measured ounterokwise from the enter of the input waveguide) is radiated to the first sab and then exites the arrayed waveguides. The ampitude profie of a i, eetri fied at eah arrayed waveguide, is usuay a Gaussian distribution. After traveing through the arrayed waveguides, the ight beams onstrutivey interfere into one foa point x in the seond sab. Prinipe of Arrayed-Waveguide Grating (AWG) Output Waveguides Input sab Arrayed waveguide Output sab Seond sab region dx dx βs( λ0) f1 + β( λ0) [ L + ( i 1) L] + βs( λ0) f + f1 f dx dx βs( λ0) f1+ + β( λ0) [ L + i L] + βs( λ0) f mπ f1 f where β s and β denote the propagation onstants in the sab region and array waveguide λ 0 is the enter waveength of the WDM system, and L is the minimum array waveguide ength β ( λ dx dx ) β ( λ ) + β ( λ ) L m π s 0 s 0 0 f1 f (a) Arrayed Waveguides dx f1 + f 1 d dx f1 f 1 m: Integer, diffrative order

10 Prinipe of Arrayed-Waveguide Grating (AWG) Output Waveguides To satisfy the equation, if β λ π ( 0) L m Seond sab region and dx f1 dx f, the ight input position x 1 wi fous on the output position x We define the effetive index of the array waveguide β βs n and n s k k and group index N n dn λ d λ Arrayed Waveguides Prinipe of Arrayed-Waveguide Grating (AWG) Output Waveguides The dispersion of the foa position x with respet to waveength λfor the fixed ight input position x 1 is given by x N f L (differentiate (a)) λ ndλ s 0 Seond sab region The dispersion of the input-side position x 1 with respet to waveength λfor the fixed ight output position x is given by x1 λ N f1 L ndλ s 1 0 (differentiate (a)) The input and output waveguide separations are x D and x D Arrayed Waveguides

11 Prinipe of Arrayed-Waveguide Grating (AWG) Output Waveguides The waveength spaing in the output side for the fixed ight input position x 1 ndd s λ0 λout N f L Seond sab region The waveength spaing in the input side for the fixed ight output position x nddλ λ in s 0 N f1 L Generay, the waveguide parameters in the first and seond sab regions are the same. Then the hanne spaings are the same λ λ λ in out Arrayed Waveguides WDM hanne spaing Prinipe of Arrayed-Waveguide Grating (AWG) Output Waveguides The path-ength differene ΔL is obtained as ndd s λ0 L N f λ Seond sab region Arrayed Waveguides

12 Free Spatia Range of AWG (m0,1,, ) m1 m0 The spatia separation of the mth and (m+1)th fous beams for the same waveength is given λ0 f XFSR xm xm+ 1 nd X FSR represents for free spatia range of AWG. The number of avaiabe waveength hannes N h is given by s N X λ f nd FSR 0 h D s Katsunari Okamoto