particle, wave, and fluid properties of light · particle, wave, and fluid properties of light...

Particle, Wave, and Fluid Properties of Light

Mankei Tsang1,2, Martin Centurion1,3, and Demetri Psaltis1,4

[email protected]

1Department of Electrical Engineering, Caltech, USA2Center for Extreme Quantum Information Theory, MIT, USA

3Max Planck Institute of Quantum Optics, Germany4EPFL, Switzerland

Particle, Wave, and Fluid Properties of Light – p.1/24

Nature of Light


Fluid Picture


Ray Optics

Maxwell’s equations:

∇ · (ǫE) = 0, ∇ · (µH) = 0, ∇× E = iωµH, ∇× H = −iωǫE (1)

E = e(r) exph

ik0S(r)i

, H =1

η0h(r) exp

h

ik0S(r)i

(2)

S(r) is the Optical Path.

In the zeroth-order approximation of small λ,

∇S · ∇S = n2(r), n2 = ǫµ. (3)

called the Eikonal Equation.

satisfies Fermat’s Least Action (path) Principle.

Born and Wolf


Newton’s Stream of Particles

Ray optics = a stream of non-interacting particles.


Euler Fluid Description

Assume that rays are almost in the z direction (Paraxial Approximation):

S(x, y, z) = z + θ(x, y, z),∂θ

∂z≪ 1. (4)

Define u = ∇⊥θ as the transverse components of rays, and n ≈ n0 + ∆n,

∂u

∂z+ u · ∇⊥u = n0∇⊥(∆n) (5)


Euler Fluid Equations

Zeroth-order approximation of small λ+ Paraxial:

∂u

∂z+ u · ∇⊥u = ∇⊥(∆n), (6)

First-order approximation of small λ + Paraxial Approximation:

∂ρ

∂z+ ∇⊥ · (ρu) = 0, (7)

where ρ = |e × h∗| is the intensity.

If light propagates in a Kerr nonlinear medium, ∆n depends on the intensity ρ

For self-defocusing nonlinearity (∆n ∼ −|n2|ρ), these equations become Euler fluidequations that describe 2D inviscid fluids:

u: fluid velocity

ρ: fluid density

∆n(ρ): fluid pressure

z: time


Optics Experiment

Thermal self-defocusing nonlinearity: silver nanoparticles in chloroform.


Quantized Vortices

u = ∇θ, ∇× u = ∇×∇θ = 0 (8)

Vorticity (∇× u) is zero for optics?

But θ is a multi-valued function:

I

u · dr⊥ = θ′(r) − θ(r) = 2πm, m = 1, 2, ... (9)

Z

∇× u · dA =

I

u · dr⊥ = 2πm, (10)

∇× u ∼ 2πmδ2(r⊥) (11)

We must have quantized vortices! (m =”topological charge”)


Optical Vortex

E(x, y, z) ∼ (x+ iy)m at the center, continuous, differentiable

ρ = 0 at the center. phase of E is undefined there

J. F. Nye and M. V. Berry, Proc. R. Soc. Lond. A 336, 165 (1974).

P. Coullet, L. Gil, and F. Rocca Opt. Commun. 73, 403 (1989).


Breakdown of Ray Optics

Second-order approximation of small λ:

∂u

∂z+ u · ∇⊥u = ∇⊥(∆n) +

„

λ

2π

«

2

∇⊥

„

1

2√ρ∇2

⊥

√ρ

«

, (12)

This term is called quantum pressure

responsible for diffraction

When ρ = 0, this term blows up.


Gross-Pitaevskii Equation

Paraxial approximation of Maxwell’s equations [E(x, y, z) ∼ eA(x, y, z) exp(ik0z)]:

i∂A

∂z= − 1

2n0k0∇2

⊥A− k0∆n(|A|2)A (13)

In the study of superfluids this is called the Gross-Pitaevskii equation, while in optics itis more well known as the nonlinear Schrödinger equation.

It admits a stable vortex solution for self-defocusing nonlinearity:

A(r, φ) = fm (r) exp(imφ) exp(iβmz) (14)

First experimentally observed by Swartzlander and Law, PRL 69, 2503 (1992).


Optical Vortex Solitons


Interactions of Optical Vortices

dri

dz∝

X

j

Z

drj

(ri − rj) × Γj

4π|ri − rj |3(15)

Exactly the same as vortex interactions in Euler fluid dynamics!

A large number of discrete vortices can approximate continuous vorticity in fluids(Chorin, J. Fluid. Mech. 57, 785 (1973)).

may be useful as a fluid simulator.


Small-Angle Scattering off an Obstacle


Quantized Vortex Street

With self-defocusing, light fills in any dark region and the fringes become quantizedvortices:

Frisch, Pomeau, and Rica, PRL 69, 1644 (1992) in the context of superfluids

Pomeau and Rica, C. R. Acad. Sc. Paris 317, 1287 (1993).

Fluid experiment:

http://www-arctic.nipr.ac.jp/members/kaoru/Doc/karman/stillf/indexe.html


Numerical Simulations

Fluids:

http://www-arctic.nipr.ac.jp/members/kaoru/Doc/karman/stillf/inde


Superflow

Beyond a certain threshold of intensity, vortices cease to emit, and light perfectly flowsaround the object

reduces scattering, might be useful for cloaking

doesn’t work for images, self-defocusing of beam, “acoustic” waves give away thepresence of object


Quantized Gross-Pitaevskii Equation

Envelope becomes operators:

i∂A

∂z= − 1

2n0k0∇2

⊥A−γ∆n(A†A)A. (16)

Schrödinger picture:

i∂

∂zψ(x1, . . . ,xN , z) =

2

4− 1

2n0k0

X

j

∇2

j−γX

i<j

h(xi − xj)

3

5ψ(x1, . . . ,xN , z)

(17)

Same as a simple model of superfluids, or Bose-Einstein condensates.

May be used as a quantum simulator of quantum fluids.

Use the quantum dynamics to generate nonclassical beam for quantum metrology:

Tsang, PRL 97, 023902 (2006)

Tsang, PRA 75, 063809 (2007)

Tsang, e-print quant-ph/0604132.


Conclusion

Publications:Demetri Psaltis, Invited Talk, OSA Annual Meeting, Oct 2006, paper FWB4.

Martin Centurion, Invited Talk, LEOS Annual Meeting, Oct 2005, paper TuEE2.

Mankei Tsang, Oral Presentation, CLEO/QELS, May 2005, paper QML6.

Mankei Tsang and Demetri Psaltis, Invited Paper, Proceedings of SPIE, 5735, 1 (Apr 2005).

Mankei Tsang and Demetri Psaltis, "Metaphoric optical computing of fluid dynamics," e-print physics/0604149.

Demetri’s group in Switzerland will continue working on the experiments.

Fundng by DARPA and NSF Center for Science and Engineering of Materials.

[email protected], http://mankei.tsang.googlepages.comParticle, Wave, and Fluid Properties of Light – p.20/24

http://mankei.tsang.googlepages.com/

Quantum OpticsTsang , “Decoherence of quantum-enhanced timing accuracy,” PRA 75, 063809 (2007).

Tsang , “Relationship between resolution enhancement and multiphoton absorption rate in quantum lithography,” PRA 75, 043813(2007).

Tsang , “Quantum temporal correlations and entanglement via adiabatic control of vector solitons,” PRL 97, 023902 (2006).

Tsang and Psaltis, “Propagation of temporal entanglement,” PRA 73, 013822 (2006).

Tsang and Psaltis, “Spontaneous spectral phase conjugation for coincident frequency entanglement,” PRA 71, 043806 (2005).

Nano-OpticsTsang and Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” PRB, in press, e-printarXiv:0708.0262.

Tsang and Psaltis, “Theory of resonantly enhanced near-field imaging,” Opt. Express 15, 11959 (2007).

Tsang and Psaltis, “Reflectionless evanescent wave amplification via two dielectric planar waveguides,” Opt. Lett. 31, 2741 (2006).

Nonlinear OpticsPu, Wu, Tsang , and Psaltis, “Optical parametric generation in periodically poled KTiOPO4 via extended phase matching,” APL 91,131120 (2007).

Tsang , “Spectral phase conjugation via extended phase matching,” JOSA B 23, 861 (2006).

Centurion, Pu, Tsang , and Psaltis, “Dynamics of filament formation in a Kerr medium,” PRA 71, 063811 (2005).

Tsang and Psaltis, “Spectral phase conjugation by quasi-phase-matched three-wave mixing,” Opt. Commun. 242, 659 (2004).

Tsang and Psaltis, “Spectral phase conjugation with cross-phase modulation compensation,” Opt. Express 12, 2207 (2004).

Tsang , Psaltis, and Omenetto, “Reverse propagation of femtosecond pulses in optical fibers,” Opt. Lett. 28, 1873 (2003).

Tsang and Psaltis, “Dispersion and nonlinearity compensation by spectral phase conjugation,” Opt. Lett. 28, 1558 (2003).Particle, Wave, and Fluid Properties of Light – p.21/24

Euler Fluid Dynamics

Ray Optics + Paraxial Approximation + Self-Defocusing = Euler Fluid Dynamics

Both describe a stream of repulsive particles in the continuum limit.

(3+1)D dynamics if we have a pulse and anomalous group-velocity dispersionParticle, Wave, and Fluid Properties of Light – p.22/24

Linear Scattering

Linear propagation:


Quantum Metrology

Define a “center-of-mass” coordinate and relative coordinates,

X ≡ 1

N(x1 + x2 + · · · + xN ), ξj ≡ xj − X, (18)

i∂ψ

∂z=

2

4− 1

2k

0

@∇2

X+

X

j

∇2

ξj

1

A − γX

i<j

δ(ξi − ξj)

3

5ψ (19)

Nonlinearity does not affect the center of mass, which governs the beam positionaccuracy.

Use nonlinear propagation to beat the spatial Standard Quantum Limit:

Tsang, PRL 97, 023902 (2006)

Tsang, PRA 75, 063809 (2007)

Tsang, e-print quant-ph/0604132.


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