NanophotonicsAtilla Ozgur Cakmak, PhD

Unit 3Lecture 36: Metamaterials-Part3

Outline

• Transmission Line Approach

• Chiral Metamaterials

A couple of words…

In this lecture we start by looking into non-resonant approaches for metamaterials. Chiral metamaterials have been at the heart of this discussion. Later on, as the second part of the discussion we will examine hyperbolic metamaterials and zero index metamaterials. We will finish the regarding discussions with a very intriguing application, namely cloaking with metamaterials.

Suggested readings: “Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications” by Christophe Caloz and Tatsuo Itoh Ch. 1, “Fundamentals and Applications of Nanophotonics” by D. de Ceglia, J. W. Haus, N. M. Litchinister, A. Sarangan, M. Scalora, J. Sun, M. A. Vincenti Ch. 9, Bahaa E. A. Saleh and Malvin Carl Teich “Fundamentals of Photonics” 2nd Edition Ch. 6

Transmission Line ApproachDue to the problems regarding the resonance based approaches, other methods are saught. Transmission line approach is developed, which treats the propagation of the waves in metamaterials as the waves in transmission lines with a small twist.

A transmission line includes distributed inductance and capacitance (rather than lumped elements as in the case of last lecture’s topic). Conventional TL is shown in a). For the infinitesimal length (∆x), we can write Krichhoff’s voltage and current laws for the per unit length capacitance and inductance:(V is voltage, i is current)

( , ) ( , )( , ) ( , ), ( , ) ( , )

( , ) ( , ) ( , ) ( , ) ( )( ) ( ) For Harmonic voltages

( , ) ( , ) ( , ) ( , ) ( )

s

i x t V x x tV x t L x V x x t i x t C x i x x t

t t

V x t V x x t V x t i x t dV xL j Li x Z i x

x x t dx

i x t i x x t i x t V x t di xC j

x x t dx

( ) ( )For Harmonic currentspCV x Y V x

Transmission Line ApproachDo you start seeing any resemblance to the wave equations that we derived before?

22

2

22

2

( ) ( )( ) ( ) For Harmonic voltages ( )

( ) ( )( ) ( )For Harmonic currents ( )

where k is the propagation constant for this transmission line. These equat

s

p

dV x d V xj Li x Z i x k V x

dx dx

di x d i xj CV x Y V x k i x

dx dx

22 2 2 2

2

0

0

ions are known as Telegrapher's

equations.

instead of our

Then, voltages and currents can be written as:

( )

1( ) ( ), Z is the characteristic impedance

jkx jkx

jkx jkx

k LC k nc

V x V e V e

i x V e V eZ

of the transmission line. There will be reflection

due to the end points of the TL. The phase velocity and group velocity can be defined as:

1 10, 0p g pv v v

k kLC LC

These are all valid and very well known for the conventional transmission line.

Transmission Line ApproachNow, interchange the places of L and C in the transmission line approach:

2 2

You will see that the propagation constant will become:

1, hence

0, 0p g

kLC

v LC v LCk k

Accordingly, we start witnessing negative phase velocity as we would expect in LH medium. This treatment tells us that we can treat the atoms as distributed L and C elements in the bulk metamaterial and opens up new ways to treat metamaterials in a similar way that frequency selective surfaces were treated in 20th century. Let us investigate a concrete example.

TL Approach (problem)

1) Show that we get a negative phase velocity for the TL in b) compared to the conventional one in a).

TL Approach (solution)

1) Show that we get a negative phase velocity for the TL in b) compared to the conventional one in a).

1

2

1 2

If you think of them as impedances for harmonic waves propagating

( ) ( ) ( )

( ( ) ( )) ( ) ( )

Here Z is the series impedance, Z is the parallel impedance, if one defines the

per u

Z i x V x V x x

Z i x i x x V x x V x

1 2

2 22

2 2 2

nit lenght inductance and capacitance as

1 1 1 1, inverse capacitance/inductance per unit length

,Z

( ) ( )1

,

I I

I

I I I

I I

I I

C x C L x L

Lx dV i di VZ j

j C dx j C x dx j L

i i

j C j Ld i d Vi V k

dx j L dx j C L

I IC

TL Approach (solution)

1) Show that we get a negative phase velocity for the TL in b) compared to the conventional one in a).

2

Compare with what we had for the traditional TL:

1

1 ?But actually it merely shows that

and are antiparallel shows the flow direction of energy so it must be

RH g

LH g

I I

g p g

k LC vLC

dk v LC

dkL C

v v v

positive

phase velocity becomes negative.

Transmission Line ApproachA famous fishnet type structure is shown below. A prism is built out of the meta-atoms depicted in (a). These meta-atoms can be treated as resonance structures as in the case of (c), where we have L and C elements. Based on this modelling effort, an equivalent circuit shown together with the transmission spectrum when waves are sent at normal incidence.

The resonance circuit model well coincides with the transmission spectrum where we see two peaks. Their characteristics are labelled as LH and RH. Around 14 GHz we have LH band and RH band for 18 GHz.

RH band

LH band

Transmission Line ApproachNext, we can apply a retrieval procedure to retrieve the effective parameters around the LH band (14 GHz). Finally, just as in the case of photonic crystals, transmission line approach can be applied to retrieve the dispersion diagrams of the wave propagating at normal incidence for various unit lattice sizes (the dimensions are shown).

There are two transmission bands as expected. a) Band1 around 14 GHz: Has a

negative slope! Pointing out the negative phase velocity or the antiparallel Poynting vector and phase velocities.

b) Band2 around 18 GHz: An ordinary RH band with positive slope.

Chiral Metamaterials

https://en.wikipedia.org/wiki/Chirality_(electromagnetism)#/media/File:Chirality_with_hands.svg

The fabrication difficulties of bulk metamaterials with cascaded several meta-atomic layers and scaling problems due to the resonance conditions forced the researchers to look for alternatives. Chiral metamaterials emerged as one of the main solutions. This has also thinned the metamaterials significantly since the light interacted along the direction of propagation with the metamaterial.

The picture depicts a design that would illustrate chiral optical activity while the geometry cannot coincide with its mirror image as in the case of our hands.

J. K. Gansel et al. Opt. Express vol. 18 pp. 1059 (2010).

Chiral MetamaterialsNow, let us investigate the light propagation in a chiral metamaterial by first defining the polarization of light.

The polarization of light is defined based on the phase differences of the fields along x and y directions while the light propagates along z in this example. The amplitudes of the fields along x and y are also critical.

z

x

y

k x

y

E

E

Chiral Metamaterials

E E

E E

Circular Elliptical Random Linear

Based on this information, we have these polarizations available as depicted above. The linear polarization checks if the fields are in phase while the magnitudes might differ as in the case above where E is only along one direction. Circular polarization means that these two components have 90 degree (+ or -) phase difference between each other while the magnitudes are the same. Elliptical polarization shows a more general situation where the magnitudes and the phases do not match.

Chiral MetamaterialsThese fields are shown as below in what is called Jones matrix form.

Linearly polarized Linearly polarized

Right circularly polarized Left circularly polarized

Here, the amplitudes of Ex and Ey and the phase difference determine the orientation and shape of the polarization. If Ex=1 and Ey=0 as in 1st example it is linearly polarized. Let us say we have both Ex and Ey are in phase so it is still linearly polarized and amplitudes are different along x and y related by an angle of θ.

Chiral Metamaterials

0

When we take snapshots of the wave as it is propagating in air we will only see the real part

Re{ exp( ( )} cos( ( ) ) cos( ( ) )

Fix t, eg. t=0, observe how the field is evolving

x x y y

z z zE E j t a t a t

c c c

r r

while you are changing/evolving z

Evolving like a right handed helix

Evolving like a left handed helix

Fixed time snapshots (easiest is t=0)

Chiral MetamaterialsShortly,

= , - = for RHCP and LHCP, respectively.2

Now, it should be clear why the Jones matrices are shown as depicted.

x y y xa a

We can express CP waves in terms of LP waves and vice versa.

Chiral Metamaterials (problem)

2) How would you express LP wave as a combination of (linear series of) RHCP and LHCP waves if you know that the system supports RHCP and LHCP waves as basis functions (Hint: Remember our quantum mechanics discussions)? LP wave is shown below.

1

1LP

Chiral Metamaterials (solution)

2) How would you express LP wave as a combination of (linear series of) RHCP and LHCP waves if you know that the system supports RHCP and LHCP waves as basis functions (Hint: Remember our quantum mechanics discussions)? LP wave is shown below.

1 1 01 1

1 0 12 2

1 1 1( ' '') ( ' '') ( ' ') ( '' '')' ' '' '' 2, 0

0 2 2 2

'' '', '' '' 0 '' '' 0,

2' ' 2, ' ' ' '

2

0 1( ' '')

1 2

LP a b

a ja b jb a b j a ba b ja jb

j j

a b a b a b

a b a b a b

x jx

j

1( ' '')' ' 0, '' '' 2 2

2

y jyx y x y

j

Chiral Metamaterials (solution)

2) How would you express LP wave as a combination of (linear series of) RHCP and LHCP waves if you know that the system supports RHCP and LHCP waves as basis functions (Hint: Remember our quantum mechanics discussions)? LP wave is shown below.

1 1 01 1

1 0 12 2

1 1 1 1( ' '') ( ' '') ( ' '') ( ' '')( ) ( )

2 2 2 2 2 2

1 1 1 1 12 2 2 2( ) ( )

2 2 2 2 2 2 2 2 2 2 2 2

LP a b

a a ja b jb b x jx y jy

j j j j

a b j j a bj a b

j j j j j

1 1

12 2

1, 1, I solved it in a long way you could directly expand it in RHCP and LHCP2 2

1 1,

2 2 2 2 2 2 2 2

j

j

a b

a bj j a bj j

Chiral MetamaterialsHence, polarizers can be built optically. Now, they are shown as a black box but we can built them out of our chiral metamaterials as you will see.

Chiral metamaterials are not sought to give negative refractive index but also because of their unique electromagnetic characters of circular dichroism and optical activity.

Chiral Metamaterials

0

0

Magnetic and electric flux densities are written in the following way for a medium with chiral bianisotropy:

. where is the chiral parameter, which determines the j cD E

B j c H

0 0 0 0 0 0

0 0 0

coupling between E and H

We can write for plane waves (go back to Maxwell's equations)

( ) ( ) ( ) ( )

( )

E jk E j B k E B

H jk H j D k H D

k k E k B k j E H j k E k H

j k E

2

0 0 0

2 2

0 0 0 0 0 0 0 0 0

2

0 0 0 0 0 0

0 0 0 0 0

0

( ) ( )

( ) ( ) ( )

It makes:

( ) ( )

We can write H in terms of E again

1

k H j k E D

j k E D j k E E j H

k k E j k E E j H

k E k Ej E H B H j

2

0 0 0 0 0 0 0 0

0

1( ) ( )

E

k Ek k E j k E E j j E

Chiral Metamaterials2

0 0 0 0 0 0 0 0

0

2 2 2 2

0 0 0 0 0 0 0 0 0 0

0 0

2 2 2

0 0 0 0 0 0 0 0

1( ) ( )

1 1( )

This further simplifies to

( )(

k Ek k E j k E E j j E

k Ek k E j k E E j E

k k E E E j j k E

2 2 2 2

0 0 0 0 0 0

)

( ) 2 ( ) ( ) 2 ( )

This equation is the master equation (eigen value equation) for the chiral metamaterials.

Let us plug in our plane wave ( ) exp( )x y

k k E E j k E k E jk k E

E E x E y jkz

) )

2

2 2 2 2

0 0 0 0

propagating inside the chiral medium

You can drop exp( ) from each term (using k =kz):

( ( )) ( )

( )( ) 2 ( ( )) ( )( ) 2

z

z z x y x y

x y z x y x y x

jkz

k k E x E y k E x E y

k E x E y jk k E x E y k E x E y jk kE y

)

) ) ) )

) ) ) ) ) ) )0

2 2 2

0 0

2 2 2

0 0

2

So, it makes

( ) 2

( ) 2

y

x x y

y y x

jk kE x

k E k E jk kE

k E k E jk kE

)

Chiral Metamaterials2 2 2

0 0

2 2 2

0 0

2 2 2 2 2 2

0 0

0 0

( ) 2

( ) 2

These two equations should be satisfied by various and pairs, under these conditions

( ) ( ) and

2 2

y

x

x

y

x y

y yx

x y

Ek k jk k

E

Ek k jk k

E

E E

E Ek k E k k

E jk k E jk k

x

x y

Ej

E E

Mystery? Not really, the solution says that for the chiral medium, circularly polarized wave are the basis functions!

0 ( ) exp( )E E x jy jkz ) )

Now, plug the circularly polarized wave into the eigenvalue equation.

Chiral Metamaterials2

0 0

2 2 2 2

0 0 0 0 0 0 0 0

2 2 2

0 0

2

2 2 2 2 2

0 0 0

( ( ( ))) ( )

( )( ( )) 2 ( ( ( ))) ( )( ( )) 2 ( ( )))

( ) 2

Remember that

2

Let us define two w

z z

z

k k E x jy k E x jy

k E x jy jk k E x jy k E x jy jk k E y jx

k k k k

n

k n k k k k

) ) ) )

) ) ) ) ) ) ) )m

m

m

2 2 2 2 2 2

0 0 0 0

2 2 2 2 2 2

0 0 0 0

0

avevectors for RHCP and LHCP as k

2 ( )

2 ( )

( )

k n k k k k k k

k n k k k k k k

k k n

Isn’t it amazing? The wave sees different refractive index for RHCP and LHCP waves. As we have seen in the HW problem, any incoming wave can be expressed in terms of the basis RHCP and LHCP waves, hence we will start seeing optical activity where there will be phase retardation due to this seen effective refractive indices for different polarizations. Chiral metamaterials would be used as polarizers. Moreover, we will have circular dichroism for which RHCP and LHCP waves will be absorbed differently when losses are present. On top of it, we would have our negative refractive index for one of the polarizations even though permittivity and permeability resonances are not met and they do not have to be negative.

Chiral MetamaterialsWe will not derive here (quite messy algebra but I can send the derivation to those interested) but

azimuth rotation angle and ellipticity of the incoming wave will be determined by how well it

transmits LHCP (T ) and RHCP (T ) waves.

https://www.intechopen.com/books/advances-in-geoscience-and-remote-sensing/polarimetric-responses-and-scattering-mechanisms-of-tropical-forests-in-the-brazilian-amazon

1

arg( ) arg( ) 2 : rotation angle

where T are transmission coefficients for RHCP and LHCP

tan ( )

T T

T T

T T

Arg: phase of the complex transmission coefficient.

Chiral Metamaterials

http://xploreandxpress.blogspot.com/2011/10/fun-with-mathematicssymmetry-in-art-and.html

Most of the chiral metamaterial structures proposed had C4 rotational symmetry.

Z. Li et al. J. Optics vol. 15, 023001, 2013

Chiral MetamaterialsSuch symmetric systems cannot show asymmetric transmission because:

for C4 symmetry,x xx xy x x xx yx x

xy yx

y yx yy y y xy yy yf b

T T T I T T T IT T

T T T I T T T I

Symmetries are broken to attain asymmetric transmission for linearly polarized light

M. Mutlu et al. vol. 19, 14290-9 (2011)

Forward T-matrix Backward T-matrix

C. Menzel et al. Phys. Rev. Lett. vol. 104, 253902 (2010)

Chiral Metamaterials

Tunable chirality with fabrication parameters

Optical broadband circular polarizersY. Zhao et al. vol. 3, 870 (2012)

A. S. Karimullah et al. Adv. Matter. vol. 27 5610-6 (2015)

Chiral Metamaterials

DNA origami self assembly of chirality

Chirality without chirality

E. Plum et al. Phys. Rev. Lett. vol. 102, 113902 (2009)

A. Kuzyk et al. Nature vol. 483, 311-14 (2012)

Chiral Metamaterialsa, Schematic diagram. Two gold nanorods (AuNRs) are

hosted on a switchable DNA origami template

consisting of two connected bundles, which subtends a

tunable angle. The relative angle between the AuNRs

and therefore the handedness of the 3D chiral

nanostructure can be actively controlled by two DNA

locks, which are extended from the sides of the DNA

origami template. Specifically designed DNA strands

work as fuel to drive the plasmonic nanostructure to

desired states with distinct 3D conformations by altering

the relative angle between the two DNA bundles and

hence the AuNRs. Unreactive waste is produced during

a cycle. The red and blue beams indicate the incident

left-and right-handed circularly polarized light,

respectively. b, Switching mechanism. The four arms of

the two DNA locks are labelled a, b, c and d. Through

toehold-mediated strand displacement reactions, the

plasmonic nanostructure can be driven to either the left-

or right-handed state by adding removal strands R1 or

R2.A. Kuzyk et al. Nature Materials vol. 13 862-66 (2014)

Chiral Metamaterials

A. Kuzyk et al. Nature Materials vol. 13 862-66 (2014)

a, The plasmonic metamolecules can be driven to either the left- or right-handed state by

adding R1 or R2. The CD signal was monitored over time at a fixed wavelength of 725

nm. b, CD spectra measured in the first cycle of a. The red, black and blue curves

correspond to the CD spectra of the left-handed, relaxed and right-handed states,

respectively. The right-handed state exhibits much larger CD response than the relaxed

state.