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One Dimensional Metallo-Dielectric Structures and their Applications I. Introduction It is known that both electrons and electromagnetic waves have both particle and wave nature. From solid state physics, it is also known that when an electron wave travels in a periodic potential of a crystal, they are arranged into discrete energy bands separated by energy bands called Band Gaps. Analogues to this, EM waves travelling in a periodic structure experience frequency band gaps, and the waves which fall in this gap do not propagate. These frequency gaps are called as Photonic Band Gaps (PBG). (Soukoulis, 1996) Nearly all early applications of PBGs have been in optical domain and numerous techniques and structures for the application of PBGs have been proposed in literature. Many of these devices have electronic analogs. Some of these are (Scalora, 1998): 1

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Page 1: Layered Medium

One Dimensional Metallo-Dielectric Structures and their Applications

I. Introduction

It is known that both electrons and electromagnetic waves have both particle and

wave nature. From solid state physics, it is also known that when an electron wave travels

in a periodic potential of a crystal, they are arranged into discrete energy bands separated

by energy bands called Band Gaps. Analogues to this, EM waves travelling in a periodic

structure experience frequency band gaps, and the waves which fall in this gap do not

propagate. These frequency gaps are called as Photonic Band Gaps (PBG). (Soukoulis,

1996)

Nearly all early applications of PBGs have been in optical domain and numerous

techniques and structures for the application of PBGs have been proposed in literature.

Many of these devices have electronic analogs. Some of these are (Scalora, 1998):

1. Optical Transistor or Switch: This is an optical limiter that allows the propagation of a

low intensity beam of light, while it reflects a high intensity beam. When used in

combination with a second reference beam, it has been shown that the limiter can

operate as an optical transistor or switch.

2. Optical Diode: a beam can either be reflected or transmitted through a device

depending on the direction of approach: right propagating waves may be reflected,

while a left propagating signal may be transmitted.

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3. Frequency Up Converter: These are second harmonic generator that utilizes band-edge

and nonlinear effects to provide the phase matching needed for efficient frequency up-

conversion.

All these early band gap structures have a generic structure as shown in Figure 1, which

are composed of alternating high and low index layers. Each layer can be chosen such that

its width is a fraction of the size of a reference wavelength, usually one quarter of the

reference wavelength. This forms a quarter wave stacks. As a consequence of this

arrangement of the dielectric layers, interference effects cause some wavelengths to be

transmitted, while a range of wavelengths centered about the reference wavelength, often

referred to as ‘‘band-gap’’ wavelengths, are completely reflected (Soukoulis, 1996).

Figure 1. Transmittance vs frequency for the generic PBG structure shown in the inset (Scalora, 1998)

Typically, the materials used in the fabrication of PBG structures are dielectric or

semiconductor substances, due to their low absorption characteristics. The main concern

over the material choice is, however, that the materials used should not absorb light to any

significant extent, so as not to compromise device operation. For this reason, metallic

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substances are almost exclusively used to enhance the reflective properties of dielectric or

semiconductor materials by designing and incorporating within particular device thick

metallic films, such as silver, nickel, copper, aluminum, or gold.

Nevertheless, owing to the interesting properties metallic inclusions inside dielectrics

exhibit a number of 2-D and 3-D PBG structures have been studied and applied to various

applications in both Optical (Scalora, 1998) and lower RF frequency domains. At the RF

frequency they are however categorized into Electromagnetic Band Gap Structures (EBG),

which have been widely used previously in Antenna applications for miniaturization and

surface wave reduction. But in all these cases only the reflective property of metals is used.

In this report the focus is on the applications of transmissive properties of Metals by

using alternate layers of metals and dielectrics. At this point it is important to notice that,

we know from skin depth theory that externally incident waves will propagate

approximately these respective distances inside the metal, depending on the incident

wavelength, before most of the part is reflected back. However, the concept of skin depth is

applicable only when the wave is incident on uniform, thick, highly reflective, metal films.

However, we find that the concept of skin depth looses its meaning in the case of a periodic

structure, where the presence of closely spaced boundaries, i.e., spatial discontinuities of

the index of refraction, alters the physical properties of the structure as a (Scalora, 1998).

The important modifications include;

1. Effective group velocity near the band edge

2. Transmission and Reflection Coefficient

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3. Absorption Coefficient inside the metal.

It is worth noticing that each of this modification of properties from conventional thick

metal could be used separately in different applications. In the following sections firstly a

brief theory and results reported in (Scalora, 1998) on One Dimensional Metallo-Dielectric

Photonic Band Gap Material is presented, followed by some of its applications in Optical

Frequency Domain (section II), low frequency power line applications (section III) and

some limited applications at Microwave frequencies (section IV).

II. 1 D – Meallo-Dielectric Material at Optical Frequencies

1. Structure

Similar to the generic PBG Structure shown in figure 1, the 1 D Meallo-Dielectric

structure has alternating layers of Metals and dielectric. The usual metals used are gold,

silver and copper. For the material separating the metal films a low loss dielectric or

semiconducting materials may be used (Soukoulis, 1996). The individual metal layers are

required to be as small as possible (of the order of skin depth), however, the net thickness

across the entire structure may be a hundreds of skin depth. The thickness of the dielectric

layers may be greater than skin depth.

2. Examples and Theory

Figure 2 shows an example of the 1 D Meallo-Dielectric structure and its Transitivity

compared to a single layer of metal, in this case silver (Ag). These calculations are

performed using Transmission Matrix Method.

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It is shown here that this sample transmits 2.5% of the incident red light, 8% of green

light, and about 15% of blue light for the case of single solid Ag layer. Thus, this film is

fairly opaque to visible light. However, if original 40 nm film is sliced into four films each

about 10 nm in thickness, and space each Ag layer with approximately 110 nm of MgF2

then the total transmission of visible light increases to an average of 70%. Another similar

example is shown in figure 3, for longer periodicity and compared to 200nm thick silver

layer. Similar to the generic materials involving only dielectric materials the periodicity

determines the number of peaks/ valleys in the transmission coefficient. Also, increasing

the overall thickness reduces the transitivity of the material.

Figure 2. Transmission vs wavelength for a four-period PBG sample (solid line) and a solid silver film 40 nm thick (dotted line). Silver layers are 10 nm thick, while the MgF2 layers are 110 nm thick. (Scalora, 1998)

Inherently, these structures could be used for shielding or other transparent circuits.

Figure 4 shows a plot for three layers of 30nm thick Ag layers staked between 140nm thick

MgF2 layers using Transmission Matrix Method (TMM) method and in Figure 5 using

drude model approximation, which hold the best for metals. It can be seen here though the

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transmission properties are found to accurate the drude model fails to predict the

transmission occurring due to the plasma resonance of metals.

It was also shown in (Scalora, 1998)that it is also possible to use a combination of two

or more metals, or two or more types of dielectric or semiconductor materials within the

same structure, without any significant departure from the basic characteristics that we have

described. The frequency range where light is transmitted can be changed by either

increasing or decreasing the thickness of the magnesium fluoride layers. Increasing the

thickness of the dielectric material cause a shift of the band structure toward longer

wavelengths.

Figure 3. Same as Fig. 2, except that for the PBG sample MgF2 layers are 140nm thick, and the solid silver film is 200 nm thick. (Scalora, 1998)

Figure 4 TMM- Transmission vs wavelength for a Ag/MgF2 PBG (solid line) and the continuous silver film (dotted line) (Scalora, 1998)

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Figure 5 Drude model - Transmission vs wavelength for a Ag/MgF2 PBG (solid line) and the continuous silver film (dotted line) (Scalora, 1998)

The physical interpretation of this high transmission through these structures may

be explained using resonant enhanced tunnelling of Electromagnetic waves in periodic

structures similar to the electron tunneling effect through crystal lattice. The optical path of

the 40 nm silver film is only approximately where is the wavelength of light in

silver. The introduction of a second metal layer, and hence additional boundary conditions,

can create the right set of circumstances that lead to a kind of induced transparency such

that the effective absorption coefficient inside the metal is also suppressed. This suggests

that boundary conditions cause a significant redefinition of skin depth for metals. (Scalora,

1998)

In (C Sibilia, 1999) it has been shown that the transitivity and the operational

frequency can further be optimized by using quasi periodic structures (Figure 6), for the

results on transitivity see (C Sibilia, 1999). Proposed applications for these structures

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include sensors, UV blocking films, transparent electrodes for light-emitting polymer

stacks, and conductive displays just to name a few.

Figure 6 (a) Periodical multilayer, (b) 3-stage Cantor-like multilayer, (c) Fibonacci multilayer, (d) ‘chirped’

set (C Sibilia, 1999)

III. Laminated Conductors at Low Frequency Power Transmission

We have seen in the previous section that 1-D layered metal-dielectric structure

could be used to make an electromagnetic wave penetrate(transmitted) more into metallic

medium before they are absorbed (lost) in the material due to its high conductivity. In this

section implementation of similar technique to power transmission lines is discussed based

on the literature given in (Clogston, 1951), (Morgan S. P., 1952), (Morgan S. P., 1952),

(King, 1959). It is to be however noted that these techniques were proposed in the 1950s

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and have well been in use even before they were applied to the high optical frequency

domain.

Unlike in optical frequencies where transmission through the material was of chief

importance, here the main focus is the improvement of the efficiency of the transmission

line itself. The loss in a conducting object is mainly governed by skin depth, which gives us

the measure of to what extent the fields penetrate into the conducting body and is given by

the equation (Clogston, 1951):

Where the symbols indicate their usual meanings. It can be seen here the skin depth gets

worse as the frequency of operation goes high. In it was discovered that the skin depth of a

particular conductor could be increased to certain extent and the loss occurring due to skin

depth could be reduced.

In order to make a wave penetrate more into the medium the technique used by the

authors (Clogston, 1951) is to fabricate conductors of many insulated laminae or filaments

of conducting material parallel to the flow of current, if the transverse dimension ie the

thickness of the conducting filaments are smaller than that of skin depth, at the frequency of

operation the waves will penetrate more into the conducting medium. Physically , the

lateral change of the wave through the conducting regions is cancelled by the insulating

region. Figure 7 shows an example of this structure applied to a coaxial transmission line.

The effective dielectric constant for this structure is found to be:

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Figure 7 Coaxial transmission line with laminated inner conductor (Clogston, 1951)

and when the phase velocity with which the wave travels inside the transmission line

becomes:

Maximum penetration of wave into the conductor is obtained. This velocity can be

achieved by having the dielectric constant is the coax to be:

The new skin depth thus obtained is larger and is given by:

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Figure 8 shows the attenuation constant of transmission lines with laminated inner

conductor and for single thick conductor. It is clear here that at the frequency of interest the

attenuation constant of the proposed conductor is linear and less than that of conventional

solid conductor.

Figure 8 Attenuation constant for laminated transmission line and solid conductor (Clogston, 1951)

Another example for transmission line with laminated conductors is shown in figure

9. These are parallel plate wave guides with solid walls and laminated walls respectively.

The comparison of attenuation for these two structures is shown in figure 10. Also the

effect of the dielectric constant of the space between the upper and lower walls on the

attenuation is shown in figure 11. It is clear here that the maximum penetration occurs only

at one specific dielectric constant which is obtained from the equation shown previously.

Further studies including mathematical theory (Morgan S. P., 1952), equivalent

transmission line equations and experimental demonstrations have been carried out in

(King, 1959). Figure 11 shows the experimental structure used and the corresponding

improvement in attenuation.

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Figure 10 Parallel plate wave guide with solid and laminated conducting walls (Clogston, 1951)

Figure 9 Attenuation in Solid walled parallel plate waveguide and laminated waveguide (Clogston, 1951)

Figure 10 Variation of attenuation ratio with dielectric constant of the waveguide (Clogston, 1951)

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Figure 11 Experimental results and fabricated structure (King, 1959)

IV. Loss Reduction due to skin depth using Laminated Conductors at Microwave

frequencies.

Recently the concept discussed in the previous section has been applied to

microwave components like filters and resonators for their improvement in quality factor

by ohmic loss reduction. Some of these examples are explained in this section.

In (Anders Eriksson, 2012) a circular disc microstrip resonator is constructed using

laminated conductors of sub-micron thickness. This is shown in figure 12.

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Figure 12 Parallel Plate Resonator with laminated conductor walls (Anders Eriksson, 2012)

Figure 13 Skin Depth Enhancement using laminated conductors (Anders Eriksson, 2012)

It was also shown in this paper that, in the calculation of resonant frequency the

boundary condition has to be modified. In figure 13 it can be seen that the wave penetrates

deeper into material only at resonant condition. Further the thicknesses of these structures

were also optimized to obtain the maximum possible Q by the authors. The modified

structure is shown in figure 14. In figure 15 it can be seen that with the increase in the

number of laminated conductors the surface resistance of the material decreased, and thus

improving the Q factor of the resonator. Resonant tunneling condition particularly follows

from the phase distribution in the system shown in Fig.14, where the inductive response of

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the metal layers (-90’ phase shift) is compensated by capacitive response (+90° phase shift)

of the dielectric layers. Phase matching - between different layers is observed for optimized

thickness of the sublayers. It is also clear here that there tunneling phenomenon is highly

resonant in nature.

Figure 14 optimized thickness for laminated conductors (Anders Eriksson, 2012)

Figure 15 Surface resistance and Q factor variation with frequency and number of laminated conductors (Anders Eriksson, 2012)

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In (Jun Hattori, 1999) similar technique was applied to a filter design using cavity

resonators at 1.9 GHz. This construction of this filter is shown in figure 16.

Figure 16 Construction of band elimination filter (Jun Hattori, 1999)

Figure 17 Surface resistance and Q improvement of Band Elimination filter using laminated conductors (Jun Hattori, 1999)

Further in (Latif, 2013) an attempt has been made in the application of this technique to

Antennas and microstrip line through simulations. Figure 18 shows the antenna presented

in it and the low gain due to material losses. In table 1 it can be seen that the efficiency of

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the antenna can be improved by using multiple layers of the order of skin depth. Figure 19

shows the effect of laminating microstrip lines and there by improvement in its efficiency.

V. Conclusion

In this report it one dimensional metallo dielectric materials was presented. It was

shown that both at optical and lower frequencies the electromagnetic waves can be made to

penetrate deeper into metal than in conventional solid conductor case. Applications of this

technique were also discussed showing superior performance in terms of transmission at

optical frequencies and ohmic loss at lower RF and microwave frequencies. With the

advancement of micro fabrication techniques one could extend the described method of

micron scale laminated conductors easily to lossy antennas and circuits where most of the

loss comes from the skin effect.

Figure 18 Loaded small antenna with high loss (Latif, 2013)

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Table 1 Gain improvement of antenna shown in figure 18 using laminated conductors (Latif, 2013)

Figure 19 Loss reduction of microstrip line using laminated conductors (Latif, 2013)

BibliographyAnders Eriksson, A. D. (2012). Resonant Tunneling of Microwave Energy in Thin Film Multilayer.

IEEE MTT-S Digest, 3, 2009-2012.

C Sibilia, M. S. (1999). Electromagnetic properties of periodic and quasi-periodic one-dimensional, metallo-dielectric photonic band gap structures. Journal of Optics A: Pure and Applied Optics.

Clogston, A. (1951). Reduction of skin effect losses by the use of laminated conductors. Proceedings of the IRE, 39, 767-782.

Jun Hattori, S. H. (1999). Low Profile Dielectric Band Elimination Filter using Thin Film Layered Electrode for 2 GHz Band Cellular Base Station. MTT-S 99, (pp. 1025-1028).

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King, R. A. (1959). An Experimental Clogston 2 Transmission Line. Bell System Technical Journal., 517-536.

Latif, S. I. (2013). An Engineered Conductor for Gain and Efficiency Improvement of Miniaturized Microstrip Antennas. Antennas and Propagation Magazine, 77-90.

Morgan, S. P. (1952). Mathematical Theory of Laminated Transmission Lines—Part I. Bell System Technical Journal, 883-949.

Morgan, S. P. (1952). Mathematical Theory of Laminated Transmission Lines—Part II. Bell System Technical Journal, 1121-1206.

Scalora, M. e. (1998). Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures. Journal of Applied Physics, 2377-2383.

Soukoulis, C. M. (1996). Photonic band gap materials: the "semiconductors" of the future. Physica Scripta , T66(146).

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