Ch 4: Integrated Optic Waveguides Integrated optics (optoelectronics, photonics) is the technology of constructing optic devices & circuits on substrates Integrated optics (optoelectronics, photonics) is the technology of constructing optic devices & circuits on substrates Within integrated structure, light is transformed through waveguides Within integrated structure, light is transformed through waveguides Dielectric slab waveguides Dielectric slab waveguides Light propagates in the film Light propagates in the film

Dielectric slab waveguide Critical angles at lower and upper boundaries Critical angles at lower and upper boundaries  1 >largest of both  1 >largest of both Surfaces must be smooth (specular) Surfaces must be smooth (specular) Material (film) homogenous Material (film) homogenous Absorption small Absorption small Typical materials Typical materials Symmetrical and asymmetrical waveguides Symmetrical and asymmetrical waveguides

Dielectric slab waveguide Propagation factors Propagation factors Fields Fields Phase velocity Phase velocity Effective index of refraction Effective index of refraction Evanescent fields Evanescent fields

Modes in the Symmetric WG n 2 ≤n eff ≤n 1 n 2 ≤n eff ≤n 1 Mode condition,  =m2  Mode condition,  =m2  Cavity like condition Cavity like condition The waves supported by the waveguide are the modes The waves supported by the waveguide are the modes TE & TM polarizations, Reflections TE & TM polarizations, Reflections

TE Mode Chart For even modes, the solution is tan(hd/2)=(1/n 1 cos  )(n 2 1 sin 2  -n 2 2 ) 1/2 For even modes, the solution is tan(hd/2)=(1/n 1 cos  )(n 2 1 sin 2  -n 2 2 ) 1/2 For odd modes, hd/2 ► ( hd/2)-  /2 For odd modes, hd/2 ► ( hd/2)-  /2 Example of AlGaAs WG, n 1 =3.6, n 2 =3.55, 3.55≤n eff ≤3.6, 80.4≤  ≤90 Example of AlGaAs WG, n 1 =3.6, n 2 =3.55, 3.55≤n eff ≤3.6, 80.4≤  ≤90 hd=dkcos  =dn 1 k o cos  =2  dn 1 cos  / hd=dkcos  =dn 1 k o cos  =2  dn 1 cos  / d/  hd  2  n 1 cos  d/  hd  2  n 1 cos 

TE Mode chart Note d/ ~ evanescent wave

Higher order modes Tangent function periodicity, multiple solutions Tangent function periodicity, multiple solutions Smallest solution is (d/ ) o and higher order ones are (d/ ) m Smallest solution is (d/ ) o and higher order ones are (d/ ) m (d/ ) m = (d/ ) o +m/2n 1 cos  (d/ ) m = (d/ ) o +m/2n 1 cos   (d/ )= 1/2n 1 cos   (d/ )= 1/2n 1 cos 

Multi modes Example 4-1, using mode chart Example 4-1, using mode chart In the example 3 modes exist, higher modes are cut off In the example 3 modes exist, higher modes are cut off Cut off occurs when propagation angle for a given mode is just the critical angle Cut off occurs when propagation angle for a given mode is just the critical angle Condition for cut off for mth TE mode, (d/ ) m,c = m/2 (n 2 1 -n 2 2 ) 1/2 Condition for cut off for mth TE mode, (d/ ) m,c = m/2 (n 2 1 -n 2 2 ) 1/2 If d/ < this value, mth mode will not propagate If d/ < this value, mth mode will not propagate

Multi modes Highest mode of a WG: m=2d (n 2 1 -n 2 2 ) 1/2 / Highest mode of a WG: m=2d (n 2 1 -n 2 2 ) 1/2 / # of modes for a WG: N=1+2d (n 2 1 -n 2 2 ) 1/2 / # of modes for a WG: N=1+2d (n 2 1 -n 2 2 ) 1/2 / For a single mode, TE o : d/ <1/2 (n 2 1 -n 2 2 ) 1/2 For a single mode, TE o : d/ <1/2 (n 2 1 -n 2 2 ) 1/2 Multi mode WG Multi mode WG

TM mode chart Solution for phase condition for TM polarization: tan(hd/2)=(n 1 /n 2 2 cos  )(n 2 1 sin 2  -n 2 2 ) 1/2 Solution for phase condition for TM polarization: tan(hd/2)=(n 1 /n 2 2 cos  )(n 2 1 sin 2  -n 2 2 ) 1/2 For odd modes, hd/2 ► ( hd/2)-  /2 For odd modes, hd/2 ► ( hd/2)-  /2 For n1 close to n2, difference between TE and TM solutions is negligible For n1 close to n2, difference between TE and TM solutions is negligible Two modes having the same propagation factor are said to be degenerate Two modes having the same propagation factor are said to be degenerate Even when n1 is not close to n2, cut off values for TE and TM modes are the same Even when n1 is not close to n2, cut off values for TE and TM modes are the same It follows that # of modes is same It follows that # of modes is same Total # of modes is twice the value Total # of modes is twice the value Single mode operation? Single mode operation?

Mode pattern Variation of light in transverse plane Variation of light in transverse plane Fields outside film Fields outside film m: # of zero crossings m: # of zero crossings High order modes: High order modes: Steeper angles Steeper angles Travel longer Travel longer Suffer greater absorption Suffer greater absorption Scattering might deflect them below critical angle Scattering might deflect them below critical angle Higher order modes attenuate more quickly than lower order modes Higher order modes attenuate more quickly than lower order modes

Modes in Asymmetric WG The practical choice The practical choice Let n 1 =2.29 (ZincSulfide) Let n 1 =2.29 (ZincSulfide) n 2 =1.5 (glass), n 3 =1 (air) n 2 =1.5 (glass), n 3 =1 (air)  C1 =25.9 o,  C2 =41 o  C1 =25.9 o,  C2 =41 o n 2 ≤n eff ≤n 1 n 2 ≤n eff ≤n 1 Following similar solutions as before, mode chart results Following similar solutions as before, mode chart results

Mode Chart and Pattern TE and TM are not degenerate TE and TM are not degenerate Truly single mode operation is possible Truly single mode operation is possible OIC are single mode, asymmetrical structures OIC are single mode, asymmetrical structures Mode patterns Mode patterns Unequal amplitude at two boundaries Unequal amplitude at two boundaries

Waveguide Coupling Edge (Butt) coupling Edge (Butt) coupling Different sizes, loss of power Different sizes, loss of power Mismatching between radiation & mode patterns Mismatching between radiation & mode patterns NA approximates coupling efficiency for large WGs NA approximates coupling efficiency for large WGs

Edge Coupling In single mode, pattern matching is critical in determining the coupling efficiency In single mode, pattern matching is critical in determining the coupling efficiency  =(n 1 -n 2 )/n 1 ► NA=n 1 (2  ) 1/2 when indices close to each other  =(n 1 -n 2 )/n 1 ► NA=n 1 (2  ) 1/2 when indices close to each other Transmission plane boundaries (15%) Transmission plane boundaries (15%) Advantages: Compactness, simplicity Advantages: Compactness, simplicity When WG is small, lens used to reduce beam ► creates an alignment problem When WG is small, lens used to reduce beam ► creates an alignment problem

Prism Coupling When n 3 =1 When n 3 =1 Frustrated total internal reflection Frustrated total internal reflection Field added must be in phase with field inside or supported by WG Field added must be in phase with field inside or supported by WG n p sin  p = n 1 sin  n p sin  p = n 1 sin   p is adjusted to make matching (synchronous)  p is adjusted to make matching (synchronous)

Prism Coupling For n 1 ~n 2,  c ~90 o, sin  ~1 For n 1 ~n 2,  c ~90 o, sin  ~1 We conclude n p >n 1 We conclude n p >n 1 Materials problem: Rutile and flint glass Materials problem: Rutile and flint glass Position of beam in prism Position of beam in prism Optimum coupling is 81% Optimum coupling is 81% Back coupling Back coupling

Prism Coupling: Out coupling Reciprocity, synchronous Reciprocity, synchronous # of beams indicate # of modes # of beams indicate # of modes Angles represent specific modes Angles represent specific modes If base is long enough, all power is extracted If base is long enough, all power is extracted If projected out, beam is not gaussian If projected out, beam is not gaussian Max coupling is 81% Max coupling is 81% Disadvantages Disadvantages High index materials High index materials Alignment Alignment Not integrated Not integrated

Grating Coupling Amplitude or phase periodic structure Amplitude or phase periodic structure Longitudinal propagation factor matching Longitudinal propagation factor matching Position of beam to grating (to prevent back coupling) Position of beam to grating (to prevent back coupling) Gaussian beam max efficiency is 81% Gaussian beam max efficiency is 81%

Waveguide dispersion Mode chart shows dependence of n eff on just similar to n( ) Mode chart shows dependence of n eff on just similar to n( ) This n eff (  is called waveguide dispersion This n eff (  is called waveguide dispersion It follows same eq. as material dispersion:  (  /L)=-M g , where  is source linewidth It follows same eq. as material dispersion:  (  /L)=-M g , where  is source linewidth

Modal Distortion Not wavelength dependant Not wavelength dependant If  =0, modal distortion does exist If  =0, modal distortion does exist Single mode, no modal distortion Single mode, no modal distortion Consider shortest (along axis) and longest (along qc) paths, and find their travel time difference Consider shortest (along axis) and longest (along qc) paths, and find their travel time difference Axial ray, t a =n 1 L/C Axial ray, t a =n 1 L/C Critical angle ray, t c =n 2 1 L/Cn 2 Critical angle ray, t c =n 2 1 L/Cn 2  /L)=n 1 (n 1 – n 2 )/Cn 2 =n 1  /C  /L)=n 1 (n 1 – n 2 )/Cn 2 =n 1  /C