Solution 2. n n. n n R. n n. R n + = 1. n n = + n n. n n. n n. Insert R N = 0.95 and solve, + R n. 1 ln. =14.58 i.e. 15 pairs are needed.

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1 .9 Dieletri irror Cosider a dieletri irror that is ade up of quarter wave layers of GaAs with H.8 ad AlAs with L.9, both aroud 5. The GaAs-AlAs dieletri irror is iside a vertial avity surfae eittig laser diode operatig at.5 µ. The substrate is GaAs with substrate.8. The light is iidet o the irror fro aother seiodutor that is GaAlAs with a idex.4. Calulate how ay pairs of layers would be eeded to get a refletae above 95%. What would be the badwidth? Solutio Isert.95 ad solve, l l 4.58 i.e. 5 pairs are eeded arsi 4 π λ λ.9567 But, λ 5, λ 4. Coheree legth A partiular laser is operatig i sigle ode ad eittig a otiuous wave lasig eissio whose spetral width is MHz. What is the oheree tie ad oheree legth? Solutio The spetral width i frequey ad the oheree tie t are related by t Thus, the oheree tie is t / /( 6 Hz) -6 s or µs. The oheree legth is l t 8 ( s ) -6 s. Spetral widths ad oheree (a) Suppose that frequey spetru of a radiatio eitted fro a soure has a etral frequey o ad a spetral width. The spetru of this radiatio i ters of wavelegth will have a etral wavelegth λ o ad a spetral width λ. Clearly, λ o / o. Sie λ << λ o ad << o, usig λ /, show that the lie width λ ad hee the oheree legth l are

2 λo λo λ ad o l λo t λ (b) Calulate λ for a lasig eissio fro a He-e laser that has λ o 6.8 ad.5 GHz. Fid its oheree tie ad legth. Solutio (a) See Exaple.9. (b) Cosider the width i wavelegth, λ ( λ /) (.5 9 s - (6/8-9 ) /( 8 s - ).6-6. The oheree tie is t / /(.5 9 Hz) s The oheree legth is l t ( 8 s - )( s)..4 Coheree legths Fid the oheree legth of the followig light soures (a) (b) () (d) (e) A LED eittig at 55 with a spetral width 5 A seiodutor laser diode eittig at 55 with a spetral width A quatu well seiodutor laser diode eittig at 55 with a spetral with of. A ultiode Hee laser with a spetral frequey with of.5 GHz A speially desiged sigle ode ad stabilized Hee laser with a spetral width of MHz Solutio d (a) λ dλ λ so that λ(/λ ) (5-9 )( 8 s - )/(55-9 ).87 Hz Thus, the oheree tie is t / /(.87 Hz) s The oheree legth is l t.6-5 or 6 µ (b) λ(/λ ) ( -9 )( 8 s - )/(55-9 ).746 Hz The oheree tie is t / /(.746 Hz).67 - s or.67 fs The oheree legth is l t or.8 () Apply λ(/λ ), that is, λ(/λ ) (. -9 )( 8 s - )/(55-9 ).48 Hz The oheree tie is t / /(.48 Hz) 8. - s or 8. fs The oheree legth is l t.4 - or 4

3 (d) Apply t Thus, the oheree tie is t / /(.5 9 Hz) s or 666 fs. The oheree legth is 8 l t ( s ) s. (e) t Thus, the oheree tie is t / /( 6 Hz) -8 s The oheree legth is 8 l t ( s ) -8 s.5 Fabry-Perot optial avity Cosider a optial avity fored betwee two idetial irrors. The avity legth is 5 ad the refrative idex of the ediu is. The irror refletaes are.97 eah. What is the earest ode uber that orrespods to a radiatio of wavelegth 6.8? What is the atual wavelegth of the ode losest to 6.8? What is the ode separatio i frequey ad wavelegth? What are the fiesse F ad Q-fators for the avity? Solutio For λ λ o 6.8, the orrespodig ode uber o is, o L / λ o (.5 ) / (6.8-9 ) ad atual o has to be the losest iteger value to 5878., that is 5878 The atual wavelegth of the ode losest to 6.8 is λ o L / o (.5 ) / (5878) 6.85 The frequey separatio of two oseutive odes is λ λ L ( ) L L or L 8 8 Hz. (.5) The wavelegth separatio of two oseutive odes is λ λ L (6.8 9 ) or.4 p. (.5) Fiesse is / π F. Quality fator is Q F Fabry-Perot optial avity spetral width Cosider a optial avity of legth 4. Assue the refrative idex is, ad use Eq. (..) to plot the peak losest to 6.8 for 4 values of

) Solutio f ; L f /(L) Cavity resoat frequeies () I I o avity ( ) 4 si ( kl) Cavity itesity () I o I ax ; k L π Maxiu avity

4 .99,.9,.75 ad.6. For eah ase fid the spetral width δλ,, the fiesse F ad Q. How aurate is Eq.(..5) i preditig δλ? (You ay wat to use a graphig software for this proble.) Solutio f ; L f /(L) Cavity resoat frequeies () I I o avity ( ) 4 si ( kl) Cavity itesity () I o I ax ; k L π Maxiu avity itesity (4) ( ) / f π δ ; F Spetral width (5) F esoat frequey Quality fator, Q F (6) Spetral width δ Cavity fudaetal ode is f /(L).75 8 Hz. The Graph below shows that the peak losest to 6.8 is 6.85 whih orrespods to Hz.

5 AU: Arbitrary Uits Soure δλ,.5e-6.675e-5 4.6E-5 8.E-5 Fro Graph δν exp, MHz Fro Graph δν al, MHz f δ F F alulatio / π F F experiet Q 4.9E4.778E.76E 7.64E f F δ Q δ.8 Diffratio Suppose that a olliated bea of light of wavelegth 6 is iidet o a irular aperture of diaeter of µ. What is the divergee of the trasitted bea? What is the diaeter at a distae? What would be the divergee if the aperture were a sigle slit of width µ? Solutio (a) 9 λ 6 siθo D θ o.9 The divergee agle is θ o.48 If is the distae of the sree fro the aperture, the the radius of the Airy disk, approxiately b, a be alulated fro b/ taθ o θ o b taθo ta(.9 ).6.6 Thus, the diaeter is b.7 or 7. (b) Divergee fro a sigle slit of width a is 9 λ 6 6 θ θo 6 rad a π.4 Bragg diffratio Suppose that parallel grooves are ethed o the surfae of a seiodutor to at as a refletio gratig ad that the periodiity (separatio) of the grooves is iro. If light of wavelegth. µ is iidet at a agle 89 to the oral, fid the diffrated beas. Solutio

6 Whe the iidet bea is ot oral to the diffratio gratig, the the diffratio agle θ for the -th ode is give by, d(siθ siθ i ) λ ;, ±, ±, so that for first order ad ( µ)(siθ si(89 ) ()(. µ) ( µ)(siθ si(89 ) ()(. µ) Solvig these two equatios, we fid θ oplex uber for, ad θ 7.5 for. θ 7.5 for, i fat, is the oly solutio. Figure Q4- There is oly oe diffrated bea, whih orrespods to.

Planar dielectric waveguides

Planar dielectric waveguides Plaar dieletri waveguides Abstrat: A optial waveguide is a physial struture that guides eletroageti waves i the optial spetru. They are used as opoets i itegrated optial iruits, as the trasissio ediu i