Using optical eigenmodes for single photon description

Using optical eigenmodes for

single photon description

Kyle Ballantine and Michael Mazilu

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Outline

•

Optical eigenmode decomposition

–

Motivation, definition and properties

–

Application to optical forces

•

_{Stress tensor equivalence to momentum collision}

representation

•

_{Photon quantum eigenstates}

–

Single and multi photons (natural Schmidt decomposition)

•

_{Example applications}

–

Beam splitter, waveplates, finite scatters (Mie scattering)

•

_Conclusion

Characterising momentum interference for 2 beams

4

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2 ⇣

(✏0E⇤·E+µ0H⇤ ·H) ˜I ✏0E⇤⌦E µ0H⇤ ⌦H ✏0E⌦E⇤ µ0H⌦H⇤⌘

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˜12 =

1 2

⇣

(✏0E⇤1 ·E2 +µ0H⇤1 ·H2) ˜I ✏0E⇤1 ⌦E2 µ0H⇤1 ⌦H2 ✏0E2 ⌦E⇤1 µ0H2 ⌦H⇤1

⌘

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Interference term:

Any superposition:

Momentum transfer

PW1+PW2: p(x)_{= 2.7}

PW1: p(x)_{=2.5 PW2: p(x)= 2.4 PW1-PW2: p}(x)_{= 7.1}

PW1

PW2

General definition

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⌘

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Torque: Momentum:

with:

Properties of optical eigenmodes

6

Mazilu, J. Opt. A, 11, 094005, (2009) & Mazilu et al. Opt Express 19. 933 (2011)

Eigenvalue sorting -> Optimisation

Optical eigemodes are:

•

independent of the numerical method used

•

characteristic to each optical device/object and measure

•

some measures commute and some do not (i.e. it is not possible to

define optical eigenmodes simultaneously orthogonal with respect to

both measures.

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Orthogonality with respect to measure

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Hermitian

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Plane wave collision representation

7 far field in/out:

n n

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Single surface

Ei_{1 /} kˆ _^

Z

S ⇣

n_^ E+pµ0/✏0kˆ ^ (n^ H) exp( ik ·r)

⌘

ds

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Ef_{1 /} kˆ _^ Z

S

⇣

n _^E pµ₀/✏₀kˆ _^ (n _^H) exp(ik _· r)⌘ds

<latexit sha1_base64="6ZTHl4eLsltRVKVH4SW9zqN5jPI=">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</latexit>

H<latexit sha1_base64="jiWKh/GqvDi/IltAFGiscHOZnnk=">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</latexit> ₁ = p✏0/µ0kˆ ^ E1

p(x) = u_{x ·} ˜ _· n

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p

(x)

_/

(

_|

a

f₁

_|

2

+

_|

a

₂f

_|

2

_|

a

i₁

_|

2

_|

a

i₂

_|

2

)

<latexit sha1_base64="x81hSpLTuaO8I8IVm6upzJ03CJ8=">AAACJnicbZDLSsNAFIYn9VbrrerSzWARWsSSREE3QtGNywr2Am1aJtNJO3SSDDMTsaR9Gje+ihsXFRF3PoqTtAtt/WHg4z/ncOb8LmdUKtP8MjIrq2vrG9nN3Nb2zu5efv+gLsNIYFLDIQtF00WSMBqQmqKKkSYXBPkuIw13eJvUG49ESBoGD2rEieOjfkA9ipHSVjd/HfNOXHwqTSYQtrkIuQphcYw6Xtcad+zTlGxN8EwjTc0ZJWapmy+YZTMVXAZrDgUwV7Wbn7Z7IY58EijMkJQty+TKiZFQFDMyybUjSTjCQ9QnLY0B8ol04vTMCTzRTg96odAvUDB1f0/EyJdy5Lu600dqIBdriflfrRUp78qJacAjRQI8W+RFDOosksxgjwqCFRtpQFhQ/VeIB0ggrHSyOR2CtXjyMtTtsnVetu8vCpWbeRxZcASOQRFY4BJUwB2oghrA4Bm8gil4N16MN+PD+Jy1Zoz5zCH4I+P7B4EWo+o=</latexit>

Directional Stratton-Chu far-field integrals (1/r terms for momentum)

• Field is conceptually

split between incident and “scattered field”

• Both fields are free

space fields

• Can be associated to

before and after the “collision”

Equivalence between representations

8

• Optical eigenmodes are identical the same but in different representations

• the plane wave collision representation introduces a conceptual

sequence between incident and scattered field

E = i

✓

~!

2"0V

◆1/2 _X

k,s

ek,sak,seir·k+i!t

<latexit sha1_base64="hsaHL83FSFisLG1wsDSfhqYpM1M=">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</latexit>

p

(x)

=

Z

s

u

x

·

˜(

F

s

)

·

n

ds

=

Z

1

u

_x

_·

˜(

F

₁

)

_·

n

ds

=

X

k,s

k

x

|

a

f_k,s

|

2

k

x

|

a

i_k,s

|

2

<latexit sha1_base64="3Wkv0McfUtX0876BHnO3TCjp4FE=">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</latexit>

Note: No momentum is lost in a closed volume by a free space plane waves.

Single photons

9

ak ! aˆk

a⇤_{k !} aˆ†_k

<latexit sha1_base64="1NG8xAbRFbWm6nF/+3cqQPE26BY=">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</latexit>

K. Ballantine, and M. Mazilu “Optical eigenmode description of single-photon light-matter interactions.” SPIE Proceedings, Complex Light and Optical Forces XIII, 10935, 1B (2019).

Quantum optics, coefficients become operators

ˆ

E = X

i

ˆ

aiE~i, [ˆai,ˆa†_j] = ij (1)

Observable quantities correspond to Hermitian operators,

ˆ

m = X

ij

ˆ

a†_iMijˆaj (2)

Eigenvalues of M correspond to possible outcomes of measurement, after mea-surement state is in corresponding eigenvector.

Change of basis

ˆ

ai ! Aˆi = Uijˆaj (3)

Diagonalise matrix (no summation over (i))

U M U† _ij = (i) ij (4)

ˆ

m = X

i

iAˆ†_iAˆi (5)

Quantum states corresponding to classical optical eigenmodes.

Quarter-wave plate torques

General quarter-wave plate

I

_{Eigenvectors are elliptically}

polarised at

_±

45 to fast axis.

I

_{Average value of}

_~

_{comes because}

input diagonal polarisation is more

aligned with one ellipse

Ex

Ey

Consider general wave-plate:

_|

H

_{i !}

_|

H

_i

,

_|

V

_{i !}

e

i

_|

V

_i

.

I

_{Find ˆ}

_M

_{= 2}

_~

_{sin (}

/

2)(ˆ

a

₊†

a

ˆ

+

a

ˆ

†

a

ˆ

)

I

_{Torque per photon can vary from 0 to 2}

_~

I

_{Eigenmodes go from linear (}

_!

_{0) to circular (half-wave}

plate,

=

⇡

)

Proof by contradiction

11 :Fast axis :Slow axis :Torque :Electric field

:First quarter wave plate :Second quarter wave plate :Right circular polarisation :Left circular polarisation :Linear polarisation horizontal :Linear polarisation vertical :Linear polarisation 45o :Linear polarisation 135o

L0 L45 RCP L45 L135 LCP

QWP1 QWP2

L90

L135

L0 L0

L90 L90 QWP₁

QWP2 RCP LCP L0 L90 L45 L135 same rotated

by 45o

Polarisation 1st _{QWP Torque 2}nd _{QWP Torque 3}rd _{QWP Torque 4}th _{QWP Torque}

L0 0 0 -1 +1

L45 +1 -1 0 0

L90 0 0 +1 -1

Optical eigenmodes and

multiphoton eigenstates

12

[1] J.C. Garcia-Escartin et al. Opt. Comm. 430, 434 (2019).

M

(_pqx)

=

V

†

D

(x)

V

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ˆ

_b

_p

₌

X

q

V

pq

ˆ

a

i_q

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Using the effective Hamiltonian approach [1] we can look at the

photon eigenstates of p(x) _{for the multiphoton case}

ˆ

p

(x)

=

X

k

k

x

(ˆ

a

f_k†

a

ˆ

_kf

ˆ

a

i_k†

ˆ

a

i_k

)

=

X

pq

M

_pq(x)

a

ˆ

i_p†

a

ˆ

i_q

=

X

pq

D

_pq(x)

ˆ

b

†_p

ˆ

b

_q

<latexit sha1_base64="ED5LREd6G8WeVRVxcGi/pfZHhI8=">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</latexit>

where

V

are the optical

eigenmodes of

M

|

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n

₁

n

₂

...n

_m

>

_E

= ˆ

U

_|

n

₁

n

₂

...n

_m

>

_k

U

ˆ

= exp(

i

H

ˆ

U

)

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i

H

ˆ

U

=

i

X

pq

H

S pq

a

ˆ

i_p†

ˆ

a

i_q

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H

=

i

log(

V

)

Momentum eigenmodes 50/50 BS

13 a1 a4 a3 a2

S

=

_p

1

2

✓

1

i

i

1

◆

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✓

a

3

a

4

◆

=

S

✓

a

1

a

2

◆

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M

(?)

=

_p

1

2

✓

1

i

i

1

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v2 =

8 < :

ip2 p2 2 ,

1

q

2 2 p2

9 = ;

<latexit sha1_base64="P0uRs1CXuu6fYmLBnua317krjLA=">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</latexit>

v1 =

8 < :

ip2 +p2 2 ,

1

q

2 2 +p2

9 = ;

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Momentum matrix in the orthogonal direction

Scattering matrix

Eigenvalues and eigenvectors

Single photon case

Two photon case

Collapsing criteria:

?

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k

<latexit sha1_base64="QgoegamjyYsaKIJo1iOUrTZ/H80=">AAAB8HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ie0oWy2k3bpbhJ2N0IJ/RVePCji1Z/jzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSNsGm4EdhKFVAYC28H4dua3n1BpHkcPZpKgL+kw4iFn1FjpsZdQRYVA0S9X3Ko7B1klXk4qkKPRL3/1BjFLJUaGCap113MT42dUGc4ETku9VGNC2ZgOsWtpRCVqP5sfPCVnVhmQMFa2IkPm6u+JjEqtJzKwnZKakV72ZuJ/Xjc14bWf8ShJDUZssShMBTExmX1PBlwhM2JiCWWK21sJG9kImLEZlWwI3vLLq6RVq3oX1dr9ZaV+k8dRhBM4hXPw4ArqcAcNaAIDCc/wCm+Ocl6cd+dj0Vpw8plj+APn8wfz8pCB</latexit>

input |11> |10>

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k

<latexit sha1_base64="QgoegamjyYsaKIJo1iOUrTZ/H80=">AAAB8HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ie0oWy2k3bpbhJ2N0IJ/RVePCji1Z/jzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSNsGm4EdhKFVAYC28H4dua3n1BpHkcPZpKgL+kw4iFn1FjpsZdQRYVA0S9X3Ko7B1klXk4qkKPRL3/1BjFLJUaGCap113MT42dUGc4ETku9VGNC2ZgOsWtpRCVqP5sfPCVnVhmQMFa2IkPm6u+JjEqtJzKwnZKakV72ZuJ/Xjc14bWf8ShJDUZssShMBTExmX1PBlwhM2JiCWWK21sJG9kImLEZlWwI3vLLq6RVq3oX1dr9ZaV+k8dRhBM4hXPw4ArqcAcNaAIDCc/wCm+Ocl6cd+dj0Vpw8plj+APn8wfz8pCB</latexit>

1

=

~

k

<latexit sha1_base64="6xl+Hr9Ssltsu0JOgAwsjP2gEoc=">AAAB+3icbVDLSsNAFJ34rPUV69LNYBHcWJIq6EYounFZwT6gCeFmMm2HTiZhZiKW0F9x40IRt/6IO//GaZuFth4YOJxzLvfOCVPOlHacb2tldW19Y7O0Vd7e2d3btw8qbZVkktAWSXgiuyEoypmgLc00p91UUohDTjvh6Hbqdx6pVCwRD3qcUj+GgWB9RkAbKbArHjfhCAL3+swbhiDxKLCrTs2ZAS8TtyBVVKAZ2F9elJAspkITDkr1XCfVfg5SM8LppOxliqZARjCgPUMFxFT5+ez2CT4xSoT7iTRPaDxTf0/kECs1jkOTjEEP1aI3Ff/zepnuX/k5E2mmqSDzRf2MY53gaRE4YpISzceGAJHM3IrJECQQbeoqmxLcxS8vk3a95p7X6vcX1cZNUUcJHaFjdIpcdIka6A41UQsR9ISe0St6sybWi/VufcyjK1Yxc4j+wPr8AcbUk54=</latexit>

2

=

~

k

<latexit sha1_base64="4qgNZC5NOrVSkEDR0legkj7uJpc=">AAAB+nicbVDLSsNAFJ34rPWV6tLNYBFclaQKuhGKblxWsA9oQriZTNqhk0mYmSil9lPcuFDErV/izr9x2mahrQcGDuecy71zwowzpR3n21pZXVvf2Cxtlbd3dvf27cpBW6W5JLRFUp7KbgiKciZoSzPNaTeTFJKQ0044vJn6nQcqFUvFvR5l1E+gL1jMCGgjBXbF4yYcQVC/8gYhSDwM7KpTc2bAy8QtSBUVaAb2lxelJE+o0ISDUj3XybQ/BqkZ4XRS9nJFMyBD6NOeoQISqvzx7PQJPjFKhONUmic0nqm/J8aQKDVKQpNMQA/UojcV//N6uY4v/TETWa6pIPNFcc6xTvG0BxwxSYnmI0OASGZuxWQAEog2bZVNCe7il5dJu15zz2r1u/Nq47qoo4SO0DE6RS66QA10i5qohQh6RM/oFb1ZT9aL9W59zKMrVjFziP7A+vwBWkSTaA==</latexit>

input

85% prob. eigenstate 1 à 0.92i|10> - 0.38|01>

15% prob. eigenstate 2 à 0.38i|10> +0.92|01>

output state

|n3,n4>

25% (state l=-2hk) à -0.14 |20>-0.5i|11>+0.85|02>

50% (state l=0) à -0.5 |20>- 0.71|11>-0.5|02>

25% (state l=2hk) à -0.85 |20>+0.5i|11>+0.14|02>

0% (state l=0) à 0.5 |20>+0.71i|11>-0.5|02>

100% (state l=0) à -0.71i |20> -0.71i|02>

0% (state l=0) à -0.5 |20>+0.71i|11>+0.5|02>

Dynamic response of scatterer (eg. recoil, deformation) which

Optical forces: Vector spherical representation (Mie scattering)

beam shape coefficients: incident (A,B) and scattered (a,b)

Momentum matrix (z-direction)

M_kj = 1 2

mj mk pkpj

0

@ nj+1

nk

s

(1 n_k)2_(n2

j m2k) n2_k(4n2_k 1)

nj nk+1

s

(1 nj)2(n2_k m2_k) n2_j(4n2_j 1)

1 A

1

pk(bk + b⇤j 2bkb⇤j) + pk2 (ak + a⇤j 2aka⇤j)

+i 2

nj nk mkmj

mk nk(nk + 1)

1

pk pj

2 (bk + a⇤k 2bka⇤k) pk2 pj

1 (ak + b⇤k 2akb⇤k)

F

_z

=

g

k

M

_k

j

g

_j

Maxwell’s stress tensor

Momentum transferred = Force

e

=

1

2

⇣

n

2₀

✏

₀

E

_·

E

+

µ

₀

H

_·

H

I

e

2

n

2₀

✏

₀

E

_⌦

E

2

µ

₀

H

_⌦

H

⌘

Hermitian with respect to the

beam shape coefficients

g

k

=

g

k⇤

with

F

z

=

<

I

S

e

z

e

>

Use translation and rotation matrices for other positions and directions.

Force (z-direction) matrix

a

a

b

m->

b m->

a

b

g

k

=

g

_k⇤

Hermitian

Angular intensity dependence (&OAM)

F

_z

F

_z

=0

F

_z

=0

F

_z

OAM

Equivalent representation

P

_z

P

_z

=0

F

_z

=0

F

_z

Size dependence

F

_z

F

_z

=0

F

_z

=0

F

_z

r=5 r=3

Number of momentum bound

levels increases with size

Birefringence effect on Fz modes

Fundamental Fz momentum level change with birefingence

Mode change

Slow axis index of refraction

Slow axis Fast a

xis

Beam

Fundamental mode Mode splitting

Conclusion

•

Introduce the basics of optical eigenmode decomposition

•

Optical eigenmodes can be used as a natural representation of the

electromagnetic fields

•

Hypothesized that at single photon level the optical eigenmodes act

as photon eigenstates. Photon collapse depends on the system, its

degrees of freedom and inertial mass (bulk, soft matter,

mesoscopic).

Next steps:

•

_{Far field plane wave collision equivalence for angular momentum}

•

Multiphoton cases

•

_{Normalization using intensity -> frequency encoding}

21

f(t) = _Q(f0) = p.v.

t

Z

1

f0(⌧)

i 1

p

hp⌧ td⌧

Bibliography

Acknowledgement

Prof Kishan Dholakia and his group

[1] M. Mazilu, “Spin and angular momentum operators and their conservation,” J. of Opt., 11, 094005, 2009.

[2] J. Baumgartl, et. al. , “Far field subwavelength focusing using optical eigenmodes,”Applied Physics Letters, vol. 98, p. 181109, 2011.

[3] M. Ploschner, et. al., “Numerical investigation of passive optical sorting of plasmon nanoparticles,” Optics Express, vol. 19, p. 13922, 2011.

[4] M. Mazilu, et. al., “Optical Eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,”Optics Express, vol. 19, pp. 933–945, 2011.

[5] A. C. De Luca, et. al. , “Optical eigenmode imaging,” Phys. Rev. A, 84, p. 021803, 2011.

[6] M. Mazilu, “Optical eigenmodes; spin and angular momentum,” Journal of Optics, 13, p. 064009. 2011.

[7] M. Mazilu, et. al. , “Orbital-angular-momentum transfer to optically levitated microparticles in vacuum,” Phys. Rev. A. 94, p. 053821, 2016.

[8] D. Craig, et.al., “Enhanced Optical Manipulation of Cells Using Antireflection Coated Microparticles,” ACS Photonics, 2, 1403–1409, 2015.

[9] Y. Arita, et.al., “Rotation of two trapped microparticles in vacuum: Observation of optically mediated parametric resonances,” Opt. Lett., 40, 4751–4754, 2015.

[10] E. De Tommasi, L. Lavanga, S. Watson, and M. Mazilu, “Encoding complex valued fields using intensity,” Opt. Express, vol. 24, no. 20, pp. 23186–23197, Oct. 2016.

[11] M. Chen, K. Dholakia, and M. Mazilu, “Is there an optimal basis to maximise optical information transfer?,” Sci. Rep., vol. 6, p. 22821, Mar. 2016.

[12] G. Tkachenko, M. Chen, K. Dholakia, and M. Mazilu, “Is it possible to create a perfect fractional vortex beam?,”

Optica, OPTICA, vol. 4, no. 3, p. 330, 2017.

[13] M. Mazilu, Tom Vettenburg, Martin Ploschner, Ewan M. Wright and Kishan Dholakia, “Modal beam splitter: determination of the transversal components of an electromagnetic light field,” Scientific Reports, 7: 9139, 2017.

Further applications of optical eigenmodes

Optical degrees of freedom

Hyper-spectral Raman and fluorescence imaging Compressive imaging

Optimized micromanipulation:

Trapping, tractor beams and sorting Coherent control

Beating the diffraction limit Aberration

correction

Generalized spatial mode

beam splitter Whispering _{gallery mode}

coupling Chirality

decomposition

(E,H)

(iµ0cH, i✏0cE)

( µ0c@tH,✏0c@tE)

Enhanced

nonlinear effects (photo-poration)

Torque matrix (birefringent)

a

a

b

m->

b m->

a

b

Fz for

comparison isotropic

birefringent

Tz modes for birefringent spheres

Mode shape

Slow axis Fast a

xis

Beam

Tmatrix

OAM (m) Tz

Torque eigenmodes

Quarter-wave plate torques

Scattering from finite object

I

_{Consider numerical calculation of scattering from an arbitrary}

finite object

I

Basis of Gaussian beams incoming from range of angles

0

<

✓

<

2

⇡

I

_{Use angular spectrum to determine momentum output at}

each angle for each basis input, gives transfer matrix between

Quarter-wave plate torques

Results: eigenvalues and eigenmodes

Finding eigenmodes of force in e.g.

x

direction

The first input eigenmode of

F

_x

, i.e. this input field distribution for

a single photon is such that it imparts a precise quanta of force in

the

x

direction.

17 / 20

momentum in e.g. x direction

Decreasing |F

_x

| value ->

Quarter-wave plate torques

Scattering eigenmodes and Force

Before&After; probability: 7% _{Before&After; probability: 6}% _{Before&After; probability: 5}%

0.504535 _1.50494 0.589885

The input momentum (red) minus output momentum (blue) of the

optical eigenmodes imparts a force (gray) on the particle. The

x

component (black) is a quantum eigenvalue of this measurement.