Optical Airy beams and bullets

Demetri Christodoulides CREOL-College of and Photonics

Medical applications

Lasers

Military

2 CREOL-College of Optics and Photonics

Nanolasers

Integrated optics

Optical fibers 3 Light travels in straight lines

Euclid of Alexandria 325-265 BC

Optica Yet, can light bend ?

5 Can light bend? Are curved light trajectories possible?

arts science fiction Bending light

Refraction Negative refraction Cloaking

Gravitational lensing Optical

9 Diffraction

Diffraction is defined as “any deviation of light rays from rectilinear paths that can not be interpreted as Francesco Grimaldi reflection or refraction”. 1613-1663

Diffraction of a Gaussian beam Diffraction

A beam from a green laser pointer, 1 mm in diameter, will have a diameter of 1500 km when it reaches the moon.

That will be the size of Texas !

Why optical beams diffract?

Intuitively one may say that all plane waves components comprising the beam tend to walk-off from each other and a as a result the beam diffracts. Non-diffracting beams & waves

13 Diffraction-free patterns? X

θ Z θ

E = E0 exp[i(kx x + kz z)]+ E0 exp[i(− kx x + kz z)]

E = 2E0 cos(kx x)exp[ikz z] 2 2 2 I = E = 4E0 cos (kx x) 14 Non-diffracting beams - conical plane wave superposition z

4-waves

21 waves 15 Non-Diffracting Beams

A non-diffracting beam remains intensity invariant during propagation.

Bessel Mathieu

Non-diffracting beams share two common characteristics:

Non-diffracting beams (like plane All the known non-diffracting beams can be waves) are known to convey generated through conical superposition of infinite power or energy. plane waves.

On the other hand, finite energy beams/ z pulses are known to eventually diffract k 2 + k 2 = const. or disperse. x,i y,i

J. Durnin, J. J. Miceli, and J. H. Eberly, PRL 58, 1499 (1987). J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, Opt. Lett. 25, 1493-1495 (2000) 16 One-dimensional non-diffracting accelerating beams:

Airy-beams

17 Non-spreading Airy wavepackets

2 2 ∂ψ  ∂ ψ Free particle i + = 0 1D Schrödinger equation ∂t 2m ∂x2 The Airy wavepacket is a unique, non-spreading and accelerating solution!

⎡ B ⎛ B3t 2 ⎞⎤ x,t Ai ⎜ x ⎟ exp iB 3t / 2m x B3t 2 / 6m2 ψ ( ) = ⎢ 2/3 ⎜ − 2 ⎟⎥ {( )[ − ( )]} ⎣ ⎝ 4m ⎠⎦

Airy function Acceleration of a acceleration term nonspreading Airy wave Ai 2 (x)

Airy wavefunction is … not square integrable

Michael Berry and Nandor Balazs, 18 Am. J. Phys. 47, 264 (1979) The Airy function

d 2 y = xy dx2

George Biddell Airy 1801-1892

19 Rainbow Why do Airy packets freely accelerate?

V=mgx - g

Free-falling observer

g

Equivalence principle & quantum mechanics D. M. Greenberger, Am. J. Phys. 48, 256 (1980) 20 Uniqueness

It can be formally shown that the Airy state is the only non-dispersing wavepacket in 1D.

K. Unnikrishnan and A. R. P. Rau, Am. J. Phys. 64, 1034 (1996)

21 Finite Energy Optical Airy Beams

Schrödinger Equation Paraxial Diffraction Equation ∂ψ  ∂ 2ψ ∂ϕ 1 ∂ 2ϕ i + = 0 i + = 0 ∂t 2m ∂x2 ∂z 2k ∂x2 Finite power optical Airy beams have recently been suggested*:

ϕ(x, z = 0) = Ai(x / x0 )exp[x / w]

x0 : transverse scale

w: aperture width

* G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007) 22 Acceleration dynamics of Airy Beams

Finite energy Airy stripe beams propagate according to* :

2 2 ⎡ 1 1 ⎤ ⎛ 1 ⎞ I x, z Ai x / x z 2 i z exp 2x / w exp⎜ z 2 ⎟ = ϕ( ) = ⎢ 0 − 2 4 + ⎥ ( ) ⎜ − 2 3 ⎟ ⎣ 4k x0 k wx0 ⎦ ⎝ k wx0 ⎠

z = 0cm

z = 10cm

z = 20cm

* G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007) 23 Optical analog of projectile ballistics  z g The Airy beam moves on a parabolic trajectory x very much like a body under the action of gravity!

Flat Optical Wavefront Tilted Optical Wavefront

x x

z z 24 Experimental Set-up

Angular 2 & i 3 # Phase Mask for 1D Φ0 (k) = exp(− ak )exp$ k ! Spectrum % 3 "

We can synthesize a truncated Airy wave by imposing a cubic phase on a Gaussian beam and then taking its using a lens

488 nm Phase Mask for 2D

25 Other possibilities-2D cubic phase masks

U. of Arizona

Papazoglou et al, PRA A 81, 061807(R) (2010)

26 Ballistic dynamics of Airy beams

An Airy beam can move on parabolic trajectories very much like a cannonball under the action of gravity!

 z g x

2 2 3 Parabolic deflection of the beam: xd = θ z + z /(4k x0 )

* G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Optics Letters 33, 207 (2008) 27 Ballistic dynamics of Airy beams Experiment

Beam’s center of gravity moves on straight lines

∞ 2 x = ∫ dxx E −∞

Theory

In agreement with Ehrenfest’s theorem

Transverse electromagnetic momentum is conserved ! 28 2-D Airy Beams

Similarly, we can introduce 2D finite energy Airy beams:

φ(x, y, z = 0) = Ai(x / x0 ) Ai(y / y0 )exp[(x / w1 )+ (y / w2 )]

z = 0 cm z = 15 cm

Example: x0 = y0 = 50µm w1 = w2 = 0.5mm 29 Diffraction of the main lobe

Input z = 30cm

Experiment

Theory

The main lobe launched in isolation has experienced a 5-fold increase in the beam width, while the peak intensity has dropped to 5% of its initial value.

30 Possibilities

Airy beams can “heal” themselves during propagation.

Airy beams can circumvent opaque obstacles.

Spatio-temporal optical bullets resisting both diffraction and dispersion effects.

31 Self-healing of Airy Beams

Input Self-healed Beam Beam

Input Self-healed Beam Beam

Airy beams are robust and self-reconstruct even under severe perturbations

Broky, Siviloglou, Dogariu, and Christodoulides, "Self-healing properties of optical Airy beams," Opt. Express 16, 12891 (2008)

Self-healing

Regeneration

33 Airy Beams in adverse environments: I. Scattering media

Z = 0 cm Z = 5 cm Z = 10 cm

Diameter = 0.5 µm Moderate Scattering

Silica microspheres (n=1.47) Diameter = 1.5 µm suspended in water (n=1.33) Severe Scattering Concentration: 0.2 % w/w

34 Optical nano-particle manipulation using curved Airy beams

Baumgartl, Mazilu & Dholakia Nature Photonics 2, 675 - 678 (2008) Airy Beams in adverse environments: Turbulent media

Airy beam Gaussian beam

Airy beams are resilient under turbulence while a comparable Gaussian beam is badly deformed.

36 Scintillation dynamics of Airy beams under turbulence

Optics Letters vol. 35, pp. 3456, (2010) 37 Airy beams in optical filamentation studies

38 Curved plasma channel generation using ultra-intense Airy beams in air

U. of Arizona /CREOL

Science, 324, 5923 (2009) 39 Curved plasma channel generation using ultra-intense Airy beams in air

7 f=1mf=1m data 1.2 6 f=1mf=1m fit A B 5 f=75cmf=75cm data 1.0 f=75cm fit 4 f=75cm fit 0.8

3 0.6 2 0.4 1 0.2 0 density, a.u. Plasma Beam displacement, mm displacement, Beam -1 0.0 -80 -60 -40 -20 0 20 40 60 80 100 -40 -20 0 20 40 Distance from Fourier plane, cm Distance from Fourier plane, cm

3 C 3 D

2 2

2.0 2.0

1 3

1 3 1.5 1.5 1/cm 1.0 1.0 1/cm Beammmdisplacement, Beammmdisplacement, 16 0 0 16 0.5 0.5 x 10 x 10 0.0 0.0 -1 -40 -20 0 20 40 -1 -40 -20 0 20 40 Distance from Fourier plane, cm Distance from Fourier plane, cm 40 Curved plasma channel generation using ultra-intense Airy beams in air

Experiment Simulation A B C D E

5mm 5mm 5mm 5mm

0.5 Intensity 0.4 F distribution of the forward emission along the filament, for different values of pulse energy. 0.3 At energies above 5 mJ, the distribution develops two peaks 0.2 consistent with the experimentally observed emission patterns 0.1

0.0

Energy in forward emission, mJ emission, forward in Energy -40 -20 0 20 Distance from Fourier plane, cm

41 Abruptly auto-focusing waves

Efremidis, Christodoulides, Opt. Lett. 35, 4045-4047 (2010)

Autofocusing waves versus Gaussian beams

140

120 Autofocusing

100 Gaussian 80

60

40

20

0 20 40 60 80 100 120

43 Experimental observation of auto-focusing waves

Fused silica ablation

Time-domain enhancement Airyx3 using Airy bullets PRL 2010

Experimental results: Tzortzakis,FORTH Crete, OL vol. 36, p. 1842 (2011)

Optical Airy Bullets

45 Optical Bullets

An spatio-temporal optical wave propagating under the influence of diffraction and dispersion obeys:

∂E 1 ⎛ ∂ 2 E ∂ 2 E ⎞ k '' ∂ 2 E i ⎜ ⎟ 0 0 + ⎜ 2 + 2 ⎟ − 2 = ∂z 2k ⎝ ∂x ∂y ⎠ 2 ∂τ

Diffraction Dispersion

'' k0 > 0 : Normal dispersion '' k0 < 0 : Anomalous dispersion

Broadening in both space and time occurs

46 Nonlinear optical bullets

In the presence of nonlinearity one finds that:

2 2 '' 2 ∂E 1 ⎛ ∂ E ∂ E ⎞ k ∂ E 2 i ⎜ ⎟ 0 k n E E 0 + ⎜ 2 + 2 ⎟ − 2 + 0 2 = ∂z 2k ⎝ ∂x ∂y ⎠ 2 ∂τ

If the dispersion and diffraction lengths are equal and if the dispersion is anomalous: Spherical optical bullet τ 2 w 2 L L 0 0 X dispersion = diffraction = '' = k0 k

2 2 2 ∂ψ 1 ⎛ ∂ ψ ∂ ψ ⎞ 1 ∂ ψ 2 i ⎜ ⎟ 0 T + ⎜ 2 + 2 ⎟ + 2 + ψ ψ = ∂Z 2 ⎝ ∂X ∂Y ⎠ 2 ∂T

Y Y. Silberberg, Optics Letters 15, 1282 (1990) 47 Nonlinear optical bullets

• They demand equalization of dispersion and diffraction lengths (they are spherical)

• They only exist under anomalous dispersive conditions

• They need high power levels

• Nonlinear optical bullets are highly unstable- they implode/explode during propagation ! X

Never observed experimentally except in phase matched T chi-2 crystals and only in 2+1 D (Frank Wise group-Cornell)

Y 48 Are linear optical bullets possible?

∂ψ 1 ⎛ ∂ 2ψ ∂ 2ψ ⎞ 1 ∂ 2ψ i + ⎜ + ⎟ + = 0 Z 2 ⎜ X 2 Y 2 ⎟ 2 T 2 Anomalous dispersion ∂ ⎝ ∂ ∂ ⎠ ∂

Yes, as long as the diffraction and dispersion lengths are again equal! X sin X 2 + Y 2 + T 2 More complicated bullets ψ = exp(iµZ ) are possible as well. X 2 + Y 2 + T 2 T

Y

• Very difficult to synthesize • Complex spectra • Need dispersion +diffraction equalization • Possible under anomalous dispersion 49 Are linear optical bullets possible?

∂ψ 1 ⎛ ∂ 2ψ ∂ 2ψ ⎞ 1 ∂ 2ψ i ⎜ ⎟ 0 + ⎜ 2 + 2 ⎟ − 2 = Normal dispersion ∂Z 2 ⎝ ∂X ∂Y ⎠ 2 ∂T

Again, as long as the diffraction and dispersion lengths are equal!

1 ψ = X 2 + Y 2 + (iT + c)2

X-waves

• Very difficult to synthesize • Complex spectra Lu and Greenleaf, Ultrasonics 39 (1992). Di Trapani et al, PRL 1994. • Need diffraction +dispersion equalization Christodoulides, Efremidis, Optics Letters 29, 1446 (2004). • Possible under normal dispersion 50 Optical bullets

To overcome these problems one has to disengage space and time- e.g. use separation of variables!

Given that 3 can be written only as:

2+1=3 1+1+1=3

A 1D non-dispersing packet is absolutely necessary as a building block.

51 Spatio-Temporal Airy Bullets

∂ψ 1 ⎛ ∂ 2ψ ∂ 2ψ ⎞ 1 ∂ 2ψ i ⎜ ⎟ 0 L ≠ L + ⎜ 2 + 2 ⎟ + ε 2 = dispersion diffraction ∂Z 2 ⎝ ∂X ∂Y ⎠ 2 ∂T

ψ = Ai(α T ) J 0 ( β r )

Airy-Bessel Bullet

• Easy to synthesize • Does not need diffraction +dispersion equalization • Possible under any dispersion conditions

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007) 52 Airy-Bessel Optical bullets

Cornell-CREOL

Frank Wise’s group

Self-healing in time after 6 Ld 53 Airy+Bessel optical bullets

12 diffraction + 6 dispersion lengths 90 micros-90fs 1020 nm

Cornell / CREOL 54 Airy-Bessel optical bullets

Two-photon fluorescence

Bessel-Gauss (90 fs)

Bessel-Airy (90 fs main pulse lobe)

55 Spatio-Temporal Airy Bullets

Airy is the only non-dispersing wavepacket in 1-D

(a) Airy-Bessel (b) 3D Airy and (c) Airy-X optical non-dispersing bullets.

56 Airy Bullets

Papazoglou et al, PRL 105, 253901 (2010) 57 Airy plasmons

2 2 ∇ Ey + k0 εEy = 0 y dielectric ikz z −αy ⎧Ey (x, y, z) = A(x, z)e e ⎪ 2 2 2 ⎨αd = kz − k0 ε d ⎪ z k k ⎩⎪ z = 0 ε dε m (ε d +ε m )

2 ∂ A ∂A metal + 2ik = 0 ∂x2 z ∂z Airy plasmon propagation.

2 3 2 ⎡ x ⎛ z ⎞ az ⎤ ⎡ ⎛ x + a2 x z 1 ⎛ z ⎞ ⎞⎤ ⎡ x a ⎛ z ⎞ ⎤ A(x, z) = Ai⎢ − ⎜ ⎟ + i ⎥exp⎢i⎜ 0 − ⎜ ⎟ ⎟⎥exp⎢a − ⎜ ⎟ ⎥ x ⎜ 2k x2 ⎟ k x2 ⎢ ⎜ 2x k x2 12 ⎜ k x2 ⎟ ⎟⎥ x 2 ⎜ k x2 ⎟ ⎣⎢ 0 ⎝ z 0 ⎠ z 0 ⎦⎥ ⎣ ⎝ 0 z 0 ⎝ z 0 ⎠ ⎠⎦ ⎣⎢ 0 ⎝ z 0 ⎠ ⎦⎥

A. Salandrino and D. Christodoulides, Opt. Lett. 35, 2082-2084 (2010) Airy plasmons

 1DSelf-healing diffraction behavior free propagation

Gaussian Plasmon Airy Plasmon Experimental observation of Airy plasmons

Kivshar’s group-Australia: PRL 107, 116802 (2011). Zhang’s group, Opt. Lett. 36, 3191 (2011). Zhu’s group-Nanjing University PRL (2011). Super-continuum generation using Airy pulses

Polynkin et al. U. of Arizona, Phys. Rev. Lett. 107, 243901 (2011) New directions

PRL 108, 163901 (2012)

Technion Segev’s group

Sending femtosecond pulses along curved trajectories

John Dudley’s group Can we identify similar accelerating beams which are non-paraxial?

To do so we must use a non-paraxial formulation based on Helmholtz equation New directions

Non-paraxial accelerating Bessel wave-packets, solutions to Maxwell’s equations

 E = yˆ Jν (kr)exp(iνθ )exp(−iωt)

 E(x,0) = yˆ Jν (kx)

Kaminer, Bekenstein, Nemirovsky, and Segev, PRL 108, 163901 (2012). Non-paraxial Airy and Bessel beams

------Airy------

For a FWHM of 500 nm, the main lobe can bent 500 aer 10 µm

------Bessel------• For a FWHM of 500 nm, the main lobe can bent 700 aer 10 µm Are there any other accelerating vectorial solutions of Maxwell’s equations ? Elliptic Helmholtz equation

Even Radial/Angular Mathieu Funcons Odd Radial/Angular Mathieu Funcons

* P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012) * P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012) Future directions

experimental results

Aleahmad, Miri, Mills, Kaminer, Segev, and Christodoulides PRL 109, pp. 203902 (2012) Prolate Spheroidal Coordinates

Radial Prolate-Spheroid funcon Angular Prolate-Spheroid funcon Diametric drive acceleration

- m

+ m

warp drive ?? diametric drive Diametric drive acceleraon: Newton’s third law

Experimental observation of diametric drive acceleration

Max Planck Erlangen-CREOL, Nature Physics, pp. 780-784, 2013 In principle Airy beams can be used in:

• Optics

• Microwaves

• Acoustics-Ultrasonics Airy beams and pulses: applications

Biophotonics Filamentation Plasmonics

Berkeley/ANU/Nanjing St. Andrews Arizona/UCF

Super-continuum Micromachining generation

J. Europ. Opt. Soc. Rap. Public. 8, 13019 (2013)

Franche-Comté U. Of Arizona Airy beams and pulses:applications

Light-sheet using Stochastic optical reconstruction Airy beams microscopy (STORM) using Airy

CONVENTIONAL Airy STORM

Higher contrast and resolution 10X FOV

Harvard St. Andrews Nature Photonics, April 2014 Nat. Methods, April 2014

Tel-Aviv Electron Airy beams Nature, 331 (2013) Caustics are everywhere