The Negative Index of Refraction Meta- · PDF fileThe Negative Index of Refraction Meta-String A. Camacho, ... As the system oscillates above the ... we Fourier transform this equation

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WJP, PHY 381 (2014) Wabash Journal of Physics v1.3, p.1

The Negative Index of Refraction Meta-String

A. Camacho, R. C. Dennis, D. T. Tran, and M. J. Madsen.

Department of Physics, Wabash College, Crawfordsville, IN 47933

(Dated: October 19, 2014)

As in acoustic and optical mediums, we explore the possibity of creating a one

dimensional negative index metamaterial in a simple mechanical elastic system with

dispersion. Using these models, it is possible to more easily understand the propa-

gation of waves in negative index materials.


I. INTRODUCTION

The term ”metamaterial” characterizes a synthetic material which displays properties

unseen in nature. Such a material can exist because its design dictates that it be comprised

of many small structural units. The original research for this phenomenon was introduced by

Victor Veselago [7] demonstrating that it is theoretically possible to create a negative index

of refraction for electromagnetic waves. As research in this field accumulated, it was shown

that this effect can also be seen in other types of waves such as sound waves [5]. Based on

an acoustic metamaterial model [1], [2], [3], [4], [6], we can see that in a certain frequency

range, the effective Young’s modulus of an elastic solid can become negative since it exhibits

an unusual frequency dependent spring constant. Also, one can design a component with

an effective negative mass which is dependent on frequency as well. From that, we aim to

produce a ”meta-string” that exhibits both negative effective tension and mass.

A note should be made here that in one dimension, negative refraction only refers to

negative phase velocity as there are not enough dimensions for refraction to occur. The

phase velocity is the velocity of the wave crests in accordance to the formula:

vphase ≡ω

k(1)

where ω is the angular frequency and k is the wavenumber. Notice here that the frequency

is always taken to be positive and the sign of the wavenumber determines the direction in

which waves are travelling. An index of refraction is given by n where

n ≡ v0vphase

. (2)

Here, v0 is simply a constant index velocity; for electromagnetism, this is the speed of light,

and for acoustics, this is the speed of sound. In a way, this is arbitrary and since no such

natural speed exists in a string, we may define this as whatever we please. A negative index

of refraction characterizes a medium in which the phase velocity of its waves is antiparallel

to the direction of the system’s energy propagation. To create such a system, we first look

at the energy flux on a string in the parameterized x direction:

Ix = −T(∂ψ(x, t)

∂x

)(∂ψ(x, t)

∂t

)(3)

where T refers to the tension of the system and ψ(x, t) is the equation of motion. We can

simply substitute the harmonic wave equation ψ(x, t) = ei(kxx−ωt) to see that

Ix = −Tkxωψ2(x, t). (4)


From this equation it is clear that the direction of energy propagation (Ix) is antiparallel

to the direction the waves are travelling (kx) only if the tension of the system is negative.

Therefore, a negative tension material is necessary to achieve a negative refractive index.

If we consider the formula for the phase speed of a string, we see that

v2ph ≡T

µ(5)

where µ describes the mass per unit length of the string and T is again the tension. If µ is

positive and T is negative, then the phase speed is imaginary and the result is an evanescent

wave. If, on the other hand, µ is negative, a negative phase velocity (and by extension a

negative index of refraction) is achieved.

II. MODELS

To properly create a string which mimics negative tension and mass/length, we have to

consider what it means for these quantities to be negative and that is where we introduce our

models. The ordinary model for a string is a series of equally spaced beads of identical mass

each of which is connected by a spring (See FIG 1). Here, our model has two components,

FIG. 1. This is the normal model of a beaded string. Each sphere represents a point mass and they

are connected by springs. The idea is that in the continuum limit, this body would act exactly like

a one dimensional string. This model is useful for our purposes because the springs and masses

independently give rise to the tension and mass/length respectively.

the masses and the springs. The masses are only permitted to move in the parameterized y


direction which means that compression of the springs is not possible. This model works very

well given that the separation between the point masses is much smaller than the wavelength

of the waves it is supporting. A metamaterial is a material that modifies these components

to achieve novel behavior. Conveniently, the masses are responsible for determining the

mass/length and the springs are responsible for determining the tension, so if we replace

these with more complicated components, the desired effect can be seen. By modifying the

models proposed for acoustic systems by Huang and Sun [3], it is theoretically possible to

create a meta-string. One way to find the equations for the tension and mass/length of these

systems is to consider how they behave differently to harmonic forces as compared to the

ordinary components. This is acceptable because a harmonic force is experienced by each

spring and mass in the ordinary string model.

FIG. 2. This is the negative mass component, where m1 is the mass of the loop, m2 is the mass

of the sphere inside the loop, K1/2 is the spring constant, and ψ′i − ψi is the displacement of the

inner sphere from its equilibrium position due to a harmonic force F. As the inner mass is driven

above resonance, it applies a force on m1 that opposes the harmonic force F and creates a behavior

unseen in point masses.

FIG 2 shows the altered bead model which consists of a mass m2 that is allowed to

oscillate inside an outer ring of mass m1. As the inner mass begins oscillating above the

resonance frequency, the entire system responds to the harmonic force F in the opposite


way a point mass would. As the system oscillates above the resonance frequency of the

inner mass m2, this mass is overwhelmed and begins moving in the direction opposite the

force being applied. If this effect is large enough, the entire component will move in the

opposite direction of the applied force. More specifically, an ordinary point mass that is

being forced in this fashion would follow the equation:

F (t) = m∂2ψ(x, t)

∂t2(6)

according to Newton’s second law. Because we are interested in how frequency affects this

system, we Fourier transform this equation according to the following wave convention:

f(ω) =

∫ ∞−∞

f(t)e−iωtdω (7)

to get

F (ω) = −ω2ψ(x, ω)m, (8)

where the tilde represents the Fourier transform of the given variable. For the system in

FIG 2, the equations of motion reveal that

F (ω) = −ωψ(x, ω)

[m1 +m2 +m2

(ω2(Ω2

µ − ω2)− ω2γ2µ(Ω2

µ − ω2)2 + ω2γ2µ

)+ im2

(ωγµΩ2

µ

(Ω2µ − ω2)2 + ω2γ2µ

)](9)

where Ωµ is the resonance frequency of m2 and γµ is the attenuation factor of the inner

oscillating mass. The basic idea behind this exercise is that equation (9) is the same as

equation (8) if the mass is described as being dynamic and following the equation:

mdyn = m1 +m2 +m2

(ω2(Ω2

µ − ω2)− ω2γ2µ(Ω2

µ − ω2)2 + ω2γ2µ

)+ im2

(ωγµΩ2

µ

(Ω2µ − ω2)2 + ω2γ2µ

). (10)

Because the separation distance between all of these components is constant and much less

than the entire length of the string, the dynamic mass/length can be found to take the form:

µdyn = µ1 + µ2 + µ2

(ω2(Ω2

µ − ω2)− ω2γ2µ(Ω2

µ − ω2)2 + ω2γ2µ

)+ iµ2

(ωγµΩ2

µ

(Ω2µ − ω2)2 + ω2γ2µ

). (11)

A very similar argument can be applied to the tension component seen in FIG 3. where

again we take advantage of the phenomenon of resonance. In this model, m3 will pull

the trusses causing the component to compress. We said earlier that the springs in the


FIG. 3. This is the dynamic negative tension component, where F is the harmonic force exerted

by the spring-mass systems with spring constants K2 and masses m3. D and L refer to the vertical

and horizontal rest-lengths of the trusses respectively. v1 is the displacement of the truss from its

equilibruim point, v2 is the displacement of m3 from its equilibrium, and u is the displacement

the trusses have been stretched from equilibrium. As the masses m3 are driven slightly below

resonance, they apply a force to the trusses which opposes the harmonic force. This effect makes

the component act as though it is a spring with a negative spring constant, that is, expanding

when it is compressed and compressing when it is expanded.

ordinary beaded string model cannot be compressed. However, when the amplitude of m3

is in phase with the force and very large (i.e. slightly under its resonance frequency), the

system displays compression. In the case of an ordinary spring where the displacement from

equilibrium position is u(t), we expect the equation of motion to take the form:

F (t) = −Ku(t) (12)

and for the Fourier transformed equation to be

F (ω) = −Ku(ω). (13)

This time, F (ω) itself describes the tension component. When we find the equations of

motion for our altered tension component, we see that for small displacements,

F (ω) = −1

4

(L

D

)2

K2

1 +

[ω2(ω2 − Ω2

T )

(ω2 − Ω2T )2 + ω2γ2T

]− i[

ω3γT(ω2 − Ω2

T )2 + ω2γ2T

]u(ω) (14)


where ΩT is the resonance frequency of m3 and γT is the attenuation factor of m3. From

equation (14) it is clear that the dynamic tension is

Tdyn = −1

4

(L

D

)2

K2

1 +

[ω2(ω2 − Ω2

T )

(ω2 − Ω2T )2 + ω2γ2T

]− i[

ω3γT(ω2 − Ω2

T )2 + ω2γ2T

]u(ω) (15)

and we can see from this equation that the dynamic tension is negative for frequencies

slightly below the resonance frequency. Further, in the limit that m3 → 0, ΩT →∞ and

limm3→0

Tdyn = −1

4

(L

D

)2

K2u(ω) (16)

which is not frequency dependent. In fact, the 12

(LD

)2portion is a direct result of the trusses

and K2

2is the reduced spring constant of the pair of springs attached to m3. Unsurprisingly,

equation (16) is also the low frequency limit of Tdyn. For the high frequency limit, we expect

m3 to stay still and for the end equation to not depend on the spring that is attached to the

wall. This is what we see because in this limit

limω→∞

Tdyn = −1

2

(L

D

)2

K2u. (17)

This is also equivalent to the limit as m3 →∞ which is reasonable in that the upper spring

would have no effect.

III. EXPERIMENTAL DATA

One of the primary challenges of creating a system like this is determining appropriate

parameters for the large number of variables in question. After obtaining appropriate com-

binations of masses and springs, we resonated a series of spring-mass systems to determine

their natural frequencies. The effects of damping in our system were also taken into con-

sideration. By attaching one end of our spring-mass system to the wall and the other end

to a force sensor (See FIG 4), we pulled the mass from its equilibrium position and let it

oscillate. The force sensor then measured the oscillating force as a function of time, and

from a graph of force vs. time (See FIG 5), we were able to obtain our damping parameter.

The formula used to obtain attenuation was

Ae−γt sin (ωt+ φ) +D (18)

where γ is our damping factor.


FIG. 4. This figures shows our resonance and damping experiment with mass m and spring constant

K. The mass was displaced from its equilibrium position and was free to resonate while the force

sensor measured its attenuation. By definition, the mass would oscillate at its natural or resonance

frequency and attenuate according to its attenuation factor.

FIG. 5. This graph depicts the attenuation and resonance in our resonance and damping experi-

ment. The periodic nature of the graph shows the natural frequency and the exponential decrease

in amplitude shows the attenuation. From our fit, the angular frequency, ω, was measured to be

420.500± 0.018 rad/s (95% CI), and damping coefficient, γ, was measured to be 1.896± 0.018 s−1

(95% CI).

After doing this procedure on a series of resonators, we chose a resonator for our mass


component with a resonance frequency

Ωµ = 420.500± 0.035 rad/s (95% CI, Gaussian PDF) (19)

and damping factor

γµ = 1.896± 0.036 rad/s (95% CI, Gaussian PDF). (20)

The tension component we chose had a resonance frequency

ΩT = 539.80± 0.25 rad/s (95% CI, Gaussian PDF) (21)

and damping factor

γT = 4.37± 0.25 rad/s (95% CI, Gaussian PDF). (22)

With these values and their uncertainties, we are ready to begin testing the effect using our

models. However, it is wise to individually test the components beforehand.

IV. TESTING NEGATIVE MASS

Before putting the entire system together, we decided to test this negative mass effect

directly. Like in our derivations, we attempted to apply a harmonic force to the mass

component to see if it indeed behaves as the formula predicts. Our experiment involved

connecting this component with a set of springs, one of which was fixed and the other

attached to a mechanical driver (See FIG 6). Using this setup, we created and numerically

solved the following differential equations:

m1ψ1(t) =− 2K3ψ1(t) +K3u(t) + 2K4

[ψ2(t)− ψ1(t)

]−m1γ1ψ1(t) (23)

m2ψ2(t) =− 2K4

[ψ2(t)− ψ1(t)

]−m2γ2

[ψ2(t)− ψ1(t)

](24)

in order to predict the behavior we would see before peforming the experiment. Here, ψ1(t)

refers to the motion of the outer shell mass, ψ2(t) refers to the motion of the inner mass,

u(t) is the motion of the driver, γ1 is the damping factor of the system associated with K3,

and γ2 is associated with K4. We were able to create such a setup using a 1200 fps camera.

The mechanical driver can be described as displacing the end of the spring according to the

equation

u(t) = a0 cos(ω0t) (25)


FIG. 6. This is the dynamic mass setup. The inner springs have spring constant K4 and the outer

springs have spring constant K3. The loop has mass m1 and the inner mass has mass m2. We were

able to qualitatively predict the motion of m1, m2, and the driver.

FIG. 7. The system was driven at 10 Hz. At this frequency, the driver, the outer ring, and the

inner mass move in phase with one another. The left side shows the different frames of our setup,

and the right side shows our numerical fit.

where ω0 is the frequency at which the driver pin oscillates and a0 is its amplitude.

Because our system has damping, there is a phase difference between the driver and the

mass that depends on the driving frequency. If we consider the equation of motion of an

ordinary mass in this system, we would discover that it can be calculated by measuring the

amplitude of the driver, a0, the amplitude of the mass, A, the spring constant of the outer

springs, K, and the phase difference between the mass and the driver, φ. The equation for


FIG. 8. The system was driven at 100 Hz. At this frequency, the inner mass is out of phase with

the driver and out of phase with the outer ring. The left side shows the different frames of our

setup, and the right side shows our numerical fit.

FIG. 9. The system was driven at 122 Hz. At this frequency, the inner mass, outer ring, and driver

were equally out of phase. The left side shows the different frames of our setup, and the right side

shows our numerical fit.


FIG. 10. The system was driven at 200 Hz. At this frequancy, the driver and the inner mass are in

phase, while the outer ring is out of phase. The left side shows the different frames of our setup,

and the right side shows our numerical fit.

the mass is

m =K[2− (a0/A) cosφ

]ω20

. (26)

In this model, for small frequencies, the mass’s amplitude is about half that of the driver

and they are both nearly in phase. For larger amplitudes, these become completely out of

phase (a π phase shift) and the amplitude of the mass becomes increasingly small.

This formula can also be directly applied to a setup like that of FIG 6 and we see that for

certain frequencies, we achieve a negative result. Qualitatively, there are four frequencies of

interest in such a setup. The first is at low frequencies when the inner mass, outer mass, and

driving pin are all in phase with each other (FIG 7). As the frequency increases, there is a

point where the inner mass is completely out of phase with the outer mass and driver (FIG

8). Increasing the frequency still, we hit a point where the masses and driver are shifted

2π/3 rad from each other (FIG 9). Finally, there is a frequency where the inner mass and

driving pin are both completely out of phase with the outer mass (FIG 10).

That is to say that even when utilizing the high speed camera, it is not obvious that this

component is behaving as though it has negative mass. In our case, the negative mass effect

can be clearly seen in the case where the inner mass is completely out of phase with the

driver. As one can imagine, the inner mass is opposing the force provided by the driver and


causing the outer mass to behave in a way that an ordinary mass would not. Numerically

solving the set of differential equations for each mass, we were able to describe the phase of

each mass in relation to the driver. The model fit qualitatively well when compared to the

results observed by the high speed footage.

To ultimately answer whether or not this created negative mass, we decided to test

the model described by equation (26). We did this by first attaching an ordinary mass,

m = 2.489 ± 0.00048 g (95% CI, rect pdf) to the springs of spring constant K3 = 750 ±

12 N/m (95% CI). After tracking the motion of the mass and driver, we were able to find

values for the amplitude of each and the phase difference between them; using this formula,

the measured mass was mpos = 2.904 ± 0.050g (95% CI). From this measurement, we did

not find the result to be within uncertainty of the predicted value and in future work will

strive to improve our methods. Using the same setup and methodology, but attaching our

negative mass component, we found its mass to be mneg = −27.0± 1.9 g (95% CI). This is

enough evidence in support of the idea to proceed. This should be tested more rigorously

as observing negative mass in this manner is subtle.

V. FUTURE WORK

A. Testing Negative Tension

The setup to test negative tension is very much the same, but does not necessitate being

connected to springs on both ends. Rather, we need only look at how one end of the system is

moving with respect to the mechanical driver (See FIG 11). The negative tension component

can be thought of as a spring with a frequency dependent spring constant. So to prove that

the effect is achieved, we require that this ”spring” expands rather than contracts when the

mechanical driver moves toward it and contracts rather than expands when the driver moves

away. Once this component is built, we can test that this effect is achieved.

B. Testing Negative Phase Velocity

Once both of these components have been tested, we may proceed to put them together

in series in accordance to FIG 1 and drive one end up and down. We have utilized our

formulas to find a range of frequencies that capture this effect, so using a high frame rate


FIG. 11. This is the dynamic tension setup. L and D are the rest lengths of the trusses, v1 and

v2 are displacements from equilibrium of the trusses and mass m3 respectively. K6 is the spring

constant of the spring attached to the driver and K5 the spring constant of the spring-mass system.

At certain frequencies, the driver will compress the negative tension component rather than stretch

it which demonstrates a negative spring constant.

camera, we should be able to see if the effect is achieved. One thing to keep in mind is that

we need the wavelength to be much larger than the separation distance. This calls for a

very large phase velocity as we are using frequencies in the 70-80 Hz range. In fact, the low

frequency limit of each combination of springs and masses we investigated was too small to

produce such a wavelength at these frequencies. Luckily, the models predict that everything

should work smoothly using all of the parameters we have established.

VI. CONCLUSIONS

The negative index of refraction defies intuition and may appear mysterious at first, but

in reality it is well within reach for the average undergraduate physics student to under-

stand. These simple mechanical systems show that negative phase velocity is not limited to


electromagnetic waves.

[1] Huang X., Schaich W.L.,Wave Packet Propagation into a Negative Index Medium,

American Journal of Physics 72, 1232 (2004); doi: 10.1119/1.1764560

[2] Huang H. H. Dynamic characteristics of an acoustic metamaterial with locally reso-

nant microstructures. Purdue University Graduate School Thesis/Dissertation. (2007).

[3] Huang H. H., Sun C.T. Anomalous wave propagation in a one-dimensional acoustic

metamaterial having simultaneously negative mass density and Youngs modulus.

Acoustical Society of America. (2012). http://dx.doi.org/10.1121/1.4744977.

[4] Huang H. H., Sun C.T. Theoretical investigation of the behavior of an acoustic meta-

material with extreme Young’s modulus. Journal of Mechanics and Physics of Solids V

59 2070-2081. (2011).

[5] Kadic M, Buckmann T, Schittny R, Wegener M. Metamaterials Beyond Electromag-

netism. Rep. Prog. Phys. 76 (2013). http://dx.doi.org/10.1088/0034-4885/76/12/126501

[6] Sun C.T. Development and Applications of Acoustic metamaterials with locally res-

onant microstructures. Purdue University School of Aeronautics and Astronautics. (2012).

[7] Veselago V, Braginsky L, Shklover V, Hafner C. Negative Refractive Index Materials.

American Scientific Publishers. Journal of Computational Theoretical Nanoscience Volume 3,

Number 2, pp 189-218 [2006].

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The Negative Index of Refraction Meta- · PDF fileThe Negative Index of Refraction Meta-String A. Camacho, ... As the system oscillates above the ... we Fourier transform this equation