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Schrödinger and Maxwell's Equations: on their similarities

Friday Talk, 16th April 2010

Oka KurniawanComputational Electronics and Photonics

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Electron and Photon

ħω

E2

E1

ħω

E2

E1

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The equations

i ℏ

t=−

ℏ2

2m

2V 00

2Et 2

=2E

00

2B

t 2=

2B

c2 2 f

t 2=

2 f

−i 2mℏ

t=

2

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Classical Case

z

EC1

EC2

T(E)

E

0 1

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Comparison with Classical Case

z

EC1

EC2

T(E)

E

0 1

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The case of a single barrier

z

EC1

EC2

ε1

ε2

z

r ,t =C z eik x x e

ik y ye−iEt/ħ E y r , t =C E y0 z ex xe

y ye−i t

d 2

dz2 2m

ħ2 E−EC z −k x2−k y

2=0d 2 E y0dz2

2n2

z

c2−x

2−y

2E y0=0

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The analogy: Quantum and Optics

■ Streams of electrons

■ At interface: Reflection Transmission

■ Interface → energy barrier

■ Streams of photons (EM wave)

■ At interface: Reflection Transmission

■ Interface → refractive index difference

R= n1−n2

n1n2

2

R= k 1−k 2

k 1k 2

2

T=2 v2

1 v1 2n1

n1n2

2

T=k 2

k 1 2 k1

k 1k 2

2

I=12 v E0

2

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Another simple case

ax

V(x)

a

d 2 E0

dx2 k 2 E0=0

E0=Aqsin k q x

E0=0, for x=0 and x=a

k q=qa

d 2

d x2 ħ2

2mE=0

=An sin k n x

=0 , for x=0 and x=a

k n=na

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Simple example: discrete frequencies

v = c/2a

ax

V(x)

a

E = π2Ћ2/2ma2

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Quantum and Optical Confinement

ħω

Taken from wikipedia

EC

EV

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Laser: Light Amplification by Stimulated Emission of Radiation

ħωE

2

E1

ħωħω Active medium

aActive medium

ħω

EC

EV

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Nano Laser

[1] M.T. Hill,et al., “Lasing in metallic-coated nanocavities,” Nat Photon, vol. 1, Oct. 2007, pp. 589-594.

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References■ Datta, “Quantum Phenomena,” Modular Series on Solid State Devices, Vol VIII, Addison-

Wesley (1989). page 12-28.■ Joannopoulus, et al., “Photonic Crystals: Molding the flow of light,” 2nd Ed, Princeton

(2008). page 22 and Appendix A. (E-book download from TWiki)

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Solid-State and Photonic Crystal

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Summary