Twisted Photons and Photoexcitation of High

Angular Momentum StatesCarl Carlson

College of William and MaryJLab Cake Seminar

27 March 2013

work with Andrei Afanasev and Asmita Mukherjee(ArXiv placement coming soon)

30 transparencies

2

Spinning Photons

• Plane wave photons famously have helicity, spin in direction of motion, ± h/2𝞹

• Experimentally verified

• Plane wave writable as superposition of spherical waves, so can speak of orbital angular momentum (OAM) of photon components.

• But OAM projection along direction of motion is zero.

• So:

3

Jz = Sz = ±h

�L = �r��kLz = k ·�L = 0

target

r

k

Imagine the surprise

• in 1992 [Allen et al., PRA], when we learned we could make photon wave fronts that had Jz = m𝜸 = very large, like a

few (h/2𝞹) to 100 (h/2𝞹). “Twisted photons”

• Centered on some axis

• Also experimentally verified. Optical frequency twisted photons have been used to spin up dust and paramecia.

• Can we use them to spin up smaller things like atoms and/or nuclei or nucleons?

• For the latter, we need energetic photons.

• Can backscatter optical photons from energetic electrons. High OAM property is maintained [Serbo & Jentschura, 2011]

4

Scattering

• Thinkably useful for selectively producing high spin states.

• Scattering properties of the twisted photon are simple when it hits a target on its axis.

• E.g., in photoexcitation, the final state has the full angular momentum of the photon.

• Hitting small objects exactly on axis may be difficult

• Off axis results harder to see: selection rules and reaction rates not previously calculated

• To be shown: Twisted photoproduction on random target has many properties of plane wave photoproducton, but residuum of high OAM initial state does remain.

5

Description of twisted photons

• Serbo-Jenschura (2011) simpler than Allen et al. (1992).

• Wave front moves in z-direction, written as superposition of plane wave states,

6

|�m� kz�� =�

�

2�

� d�k2�

(�i)m� eim��k |�k, ��

• Momenta of plane wave states lie on periphery of cone in momentum space, • All have same kz

• All have same | k⊥ | = 𝜿

• All have same “pitch angle” 𝜽k

• Varying 𝝓k , with phase as above

• 𝚲 = helicity

Coordinate space

• A𝝁(x) = photon field operator

• Plane wave states in coordinate space

7

�0|Aµ(x)|�k, �� = �µ�k,�

e

�ikx

• Polarizations 𝜀𝝁 suitably rotated versions of

�µ±1 =

1�2

(0,�1,�i, 0) , �µ0 = (0, 0, 0, 1)

• The coordinate space potential for the twisted photon is

Aµ�m�k

z

�(x) = �0|Aµ(x)|�m�k

z

��

a. k. a.

• Coordinate space with cylindrical coordinates (𝝆, 𝝓𝝆, z)

• From here can calculate E, B, and Poynting vector

• Note speed of wavefront!

• (Ordinary photon got in limit 𝜽k → 0 and m𝜸 = 𝚲.)

8

Aµ�m�k

z

�(x) = e

�i(�t�k

z

z)�

�

2�i

��

�e

i(m���)��cos

2

�k

2

J

m���(��) �µ�

+i�2

e

im���sin �

k

J

m�(��) �µ0

� e

i(m�+�)��sin

2

�k

2

J

m�+�(��) �µ��

�

Poynting vector

• Points in direction of momentum flow

• Plane wave has

• Here,

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S� = 0 ,

S� =��2

4�sin �k Jm(��)

��

cos

2

�k2

Jm�1

(��) + sin

2

�k2

Jm+1

(��)�

,

Sz =��2

4�

�cos

4

�k2

J2

m�1

(��)� sin

4

�k2

J2

m+1

(��)�

�S = �E� �B

�S || z

☞

Poynting vector in x-y plane

10

-4 -2 0 2 4

-4

-2

0

2

4

x HmmL

yHmmL

� = 0.5 µmm� = 4�k = 0.2 radian

Poynting magnitude (S𝝓)

11

� = 0.5 µmm� = 4�k = 0.2 radian

Makeable in laboratory

12

• Optical wavelength: spiral phase plateFigure 5

A spiral phase plate can generate a helically phased beam from a Gaussian. Inthis case ` = 0 ! ` = 2.

see that upon exiting the helical face the ray is refracted in the azimuthaldirection. Thus the linear momentum of the light acquires an azimuthalcomponent that, when expressed with respect to a radius vector, givesan angular momentum along the beam axis. At a radius r, the azimuthalangle of the helical face is simply `�/2⇡r(n � 1). The application ofSnell’s law then gives the angular deviation in the transmitted ray as(n � 1)`�/2⇡r(n � 1) = `/k0r. Multiplying this by the linear momentumper photon, hk0, and the radius vector gives an angular momentum perphoton of `h.

2.2. Laguerre–Gaussian Modes

Helically phased light carries an OAM irrespective of the radialdistribution of the beam. However, it is useful to express most beamsin a complete basis set of orthogonal modes. For OAM carrying beamsthis is most usually the Laguerre–Gaussian (LG) mode set; these modeshave amplitude distributions, LGp`, given by [3,41,42]

LGp` =s

2p!⇡ (p + |`|)!

1w(z)

"rp

2w(z)

#|`|exp

�r2

w2(z)

�L|`|

p

✓2r2

w2(z)

◆exp[i`�]

exp

"ik0r2z

2(z2 + z2R)

#

exp�i(2p + |`| + 1)tan�1

✓z

zR

◆�(1)

where the 1/e radius of the Gaussian term is given by w(z) = w(0)[(z2 +z2

R)/z2R]1/2 with w(0) being the beam waist, zR the Rayleigh range, and

(2p + |`| + 1)tan�1(z/zR) the Gouy phase. L|`|p (x) is an associated Laguerre

Advances in Optics and Photonics 3, 161–204 (2011) doi:10.1364/AOP.3.000161 169

Yao & PadgettAdvances in Optics and Photonics (2011)

Angular momentum

13

• Field theoretically (e.g., Bjorken and Drell, ch. 14) for photons,

J

k =12

�ijk

�d

3x : �

A · (x

i�j � x

j�i)�A� (A

i

A

j � A

j

A

i) :

• After some effort,

��m�kz�|J3|�m�kz����m�kz�|�m�kz�� = m�

• I.e., total angular momentum projection in the direction of motion is m𝜸

Photoproduction rates

• Study atomic case. Available photon energies are atomic scale and wave functions are known.

• Some known calculations:

• Photoionization on axis Picon et al. [OE,2009]

• Forward Compton scattering on-axis, Davis et al. [JO, 2013]

• Backscattering from energetic electrons [Jenschura & Serbo, PRL & EPJC, 2011]

• Selection rules contrast to ordinary ones

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Photoproduction rates

• Some known results:

• Energetic backscattered photons retain high angular momentum [Jenschura & Serbo] Potentially relevant to FEL.

• Selection rules

• From Picon et al.: “violation of the standard dipolar selection rules during interaction with matter, yielding, in general, an exchange of angular momentum larger than h-bar per absorbed photon”

• Here: do off-axis photoproduction and treat on-axis as limit.

• Will study selection rules

• Rates

• Special twisted photon effects

15

Ordinary selection rules

• 𝛄 + ground state → (nf, lf, mf)

• mf = 𝚲 = photon helicity

• If needed, plane wave photon state is eikz 𝜺(k,𝚲) and

16

eikz = ��

i��

4�(2� + 1) j�(kr)Y�0(�, �)

• another way to see morbital = 0

• lf ≥ 1

• suppression factor (ka0)� f�1

• Interaction hamiltonian (omitting electron spin-flips)

H1 = � eme

�A ·�p

Twisted photon transitions

• Same interaction hamiltonian

17

b

y

xb

| |e−

Nucleus

• Setup: nucleus of target atom at origin

• Axis of twisted photon offset by impact parameter b

• Interacts with electron, so potential A evaluated at 𝝓𝝆’ instead of 𝝓𝝆 and

Bessel fcns at |𝝆 − b|, not 𝝆

Matrix elements

• or

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S f i = �2�i�(Ef � Ei � �) �n f l f m f | � eme

�A ·�p |nilimi; �m�kz��

S f i = 2�i�(E1

� E0

� �)e

me

��

2�

��

r2dr d(cos �r) d�� Rn f l f (r)Y�l f m f

(�r, 0)e�im f �� i��

��

cos

2

�k2

ei(m���)��� Jm���(�|�� ��b|)��� + rest

�

� eikzz ·�p R10

(r)Y00

Evaluate

• Good theorems:

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and

��·�p R10(r) = �i��·r R�10(r) = �i

�4�

3Y1�(�r, ��) R�

10(r)

ein��� Jn(�|�� ��b|) =

�

�N1=��

eiN1�� e�i(N1�n)�b JN1(��)JN1�n(�b)

Amplitudes

• The atomic factors are

20

gn1

l1

m1

� =� �

0

r2dr Rn f l f (r) R10

(r)

��

1

�1

d(cos �r) Jm f��(��) Yl f m f (�r, 0) Y1�(�r, 0)eikzz

• Get,

(for 𝝓b = 0)

S f idef= 2��(Ef � Ei ��)M(m�)

n f l f m f �(b)

M(m�)n f l f m f �(b) =

emea

0

�2��

3

Jm f�m�(�b)i�

�cos

2

�k2

gn f l f m f � + rest

�

On-axis selection rules

• Aiming well: axis of the twisted photon centered on atom

• Impact parameter b = 0, and

21

b

y

xb

| |e−

Nucleus• Selection rules,

• Contrast to plane wave: no excitation for lower lf

m f = m�,

l f � |m�| ,

gn f l f ,m f =m� ,� � (�a0

cos �k)l f�1 (tan �k)

|m���|

Jm f�m�(�b)� Jm f�m�(0) = �m f m�

if help needed for suppression rule

• for estimating suppression factors

22

Jm f��(��) = Jm f��(� sin �k r sin �r)

� (�r sin �k)|m f��|Y|m f��|,|m f��|(�r, 0)

eikz = ��

i��

4�(2� + 1) j�(kr)Y�0(�, �)

gn1l1m1� �� �

0r2dr Rn f l f (r) R10(r) (�r sin �k)|m f��|

��

d� Y�l f m f(�r, ��) Y|m f��|,|m f��|(�r, ��) Y1�(�r, ��)eikzz

• If exponential ≈ 1 insufficient, use

Off-axis selection rules

• mf = m𝜸 not forced

• Photon has z-component angular momentum m𝜸 about its axis. Need

to expand exp(im𝜸𝝓𝝆’) about origin, where atom is located. Need all exp(iN𝝓𝝆). All mf allowed.

• Off-axis selection rules,

23

y

xNucleus

m f = any,

l f ����m f

��� ,

gn f l f ,m f =m� ,� � (�a0

cos �k)l f�1 (tan �k)

|m f��|

Size of off-axis amplitudes

• Play between Bessel function Jmf-m𝜸(𝜿b) and size of reduced atomic amplitudes shown in selecton rule

• Test case: Pick some lf = m𝜸 with now many mf allowed.

(Will do nf = 4, lf = m𝜸 = 3, 𝚲 = 1.)

• mf = m𝜸 for on-axis twisted photon, mf = 1for plane wave photon

24

Off-axis amplitudes

• Observe:

• mf = m𝜸 only amplitude for b = 0

• other amplitudes important even for just half-wavelength offset

• biggest amplitude has same mf = 1 as for plane wave case.

25

Scattering in large targets

• if a photon passes through a target containing many atoms, one cannot expect that the location of the photon axis can be controlled at the level of the atomic spacing

• give cross sections averaged over transverse separations

• for a given b (f = incoming flux)

26

�n f l f m f � = 2��(Ef � Ei ��)

�|M(m�)

n f l f m f �(b)|2

f

-b

R

Averaging

• Average over the disk

27

�n f l f m f � = 2��(Ef�Ei��)f

� 1�R2

�d2b |M(m�)

n f l f m f �(b)|2

• Useful theorem,

limR��

� R

0b db J2

m1�m�(�b) =

R��

• Obtain

�n f l f m f � = 2��(Ef � Ei ��) 8�3�3

3kz

���� cos

2

�k2

gn f l f m f � + rest

����2

Twisted photon ratio

• Unusual mf are unique to photoproduction by twisted photons

• Consider ratio of cross sections for unusual mf (i.e., mf not possible with plane wave photons) to cross sections for all mf ,

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ftwisted =

�m f =l fm f =�l f ,

m f �=�

�n f l f m f �

�m f =l fm f =�l f

�n f l f m f �

• For specific case nf = 4, lf = m𝜸 = 3,

ftwisted [gnd. state� (4, 3)] = 20.3%

• Residue of twisted photon not small, even in averaged situation

Summary and discussion

• developed a formalism for atomic photoexcitation by twisted photons

• standard electric-dipole angular momentum selection rules in the limit of plane-wave photons

• if photon beam axis coincides with the atomic center, transitions between atomic levels have striking angular momentum selection rules, similar to earlier conclusions

• extended calculation more general case of the atoms located away off the photon beam axis

• the magnetic quantum number of the photoexcited state no longer matches m𝜸, and a range of final-state quantum numbers is generated

29

more

• Relative magnitudes of various transitions were studied both analytically and numerically

• Found that relative probability of transitions to "forbidden" states can reach tens of per cent in the examples we considered

• can be verified experimentally. E. g.,

• identification of excited states with high orbital angular momentum by analyzing angular distributions of photons from de-excitation

• comparison of photoabsorption with left vs right circular polarization (dichroism or asymmetry) at finite impact parameters.

30

Beyond the end

Results with other helicity

32

another m𝜸

33

first one shown

34