Chapter 6 Summary and

Conclusion

Chapter 6

Summary and Conclusion

Chapter 1 deals with the general introduction of the subject matter of

photonic crystals and photonic band gap materials. Various aspects of

photonic band gap materials related to applications have also been discussed.

In the last four chapters (from chapter 2 to chapter5) of this thesis, the

researcher has presented the theoretical study of some photonic crystal

structures. The purpose of the work is to study the propagation of

electromagnetic waves through the photonic band gap structures having

alternate layers of different types of materials. This chapter summarizes the

outline of the work and the main conclusion drawn from the results obtained.

In this chapter, it has been presented what the relevance of the present work

to technological applications in general and the scopes for further research

are. The work of the thesis is divided into four parts. In the first part (chapter

2), we have suggested through theoretical modeling the fabrication of optical

filter in different ranges of electromagnetic spectrum using the photonic

band gap materials (PBGs). The second part (chapter 3) is concerned with

the study of optical properties of dielectric-plasma photonic band gap

material for different frequencies. In the third part (chapter 4), we have

shown that it is possible to achieve negative refraction without meta-

materials in one dimensional PBG. Also, we have theoretically shown that

omnidirectional band gap can be enhanced by cascading such structures. In

the fourth part (chapter 5), we have shown that omnidirectional reflection

(ODR) can be enlarged by using photonic crystal quantum well structures.

Chapter 6: Summary and Conclusion

- 144 -

Further ODR range can also enlarged by using the gradual stacked PBG

structures.

From the analytical study of the structures considered in chapter 2, it

is clear that by selecting the desired value of the lattice period and angle of

incidence, we can achieve any desired ranges of wavelength in the filter

pattern. This type of filter may be used in many optical devices and in other

optical systems. By choosing appropriate values of controlling parameters,

we can design a frequency selector or rejecter. Also, by cascading two, three

or more filters we can design a perfect mirror or monochromator. We could

design a monochromator having a transmission window of 9Å for EM

radiation in ultraviolet region by cascading the three PBG structures.

In chapter 3, we studied the optical properties of the dielectric-plasma

PBG materials. This study showed that the thickness of plasma layers in

PPCs controls the forbidden band gaps. So we can use a PPC as a broad

band reflector or as a frequency selector by choosing appropriate values of

plasma thickness and suitable material for dielectric layers. For PPC1, the

mid point of a band gap shifts towards the higher normalized frequency

range, whereas for PPC2 the mid point of a band gap shifts towards the

lower normalized frequency range by increasing the thickness of the plasma

layers. For certain normalized frequency range, the group velocity becomes

negative. Because of this abnormal behaviour (Vg<0) superluminal flow of

photon occurs in the PPCs. Such structures in principle may be considered as

a flip flop as there is positive and negative symmetry of effective group

velocity. From the above discussion it is clear that the PPCs can be used as a

flip-flop for both cases ( < P and > P). But it gives better results for case I

because in this case, we achieve perfect symmetries and the band gap lies in

the same region as we increase the value of d. Also, the broadening of band

Chapter 6: Summary and Conclusion

- 145 -

gaps with d is more for case I. This symmetry is exhibited by PPC1 for a

wide normalized frequency range, and hence such a structure may

considered for the design of a flip flop working over a wide range of

frequency. On the other hand, the structure PPC2 may be considered for the

design of a flip flop working over narrower range of normalized frequency.

Since PPC2 has two critical values of normalized frequency where change of

states takes place, it may be used in the design of composite flip-flops. It

also exhibits superluminal propagation both for the smaller and larger values

of normalized frequency. Such a flip flop may be used to making logic

gates, optical switches in optical computing.

In chapter 4, it is shown that it is possible to achieve negative

refraction in one dimensional photonic band gap material without meta-

materials (materials having simultaneously negative permittivity and

permeability). We demonstrated theoretically that negative refraction may

occur near the low frequency edge of the second and fourth band gaps in

one-dimensional photonic crystals for an oblique incidence of

electromagnetic wave. These unique properties of refracting Bloch photons

have the potential to perfect the design of integrated photonic systems. Also,

the total reflection frequency range is substantially enlarged for all incident

angles and for both TM and TE polarizations. Subsequently, the

omnidirectional total reflection frequency range for any polarization is

enlarged. It is obvious that one can use more 1D PCs to form multiple

photonic heterostructures to get a very wide omnidirectional total reflection

frequency range as desired, provided that the directional PBGs of the

adjacent 1D PCs at any incident angle cascade each other in tandem.

In chapter 5, a simple design of omni-directional reflector using one-

dimensional photonic crystal quantum well structures in the form of a stack

Chapter 6: Summary and Conclusion

- 146 -

of periodic multi-layers has been suggested. By properly choosing the

geometry and dielectric parameters of the constituent materials, the

realization of a large wavelength range of omnidirectional reflection of

electromagnetic waves is possible, which may have potential applications

such as broadband micro-mirrors. Also, we have investigated theoretically

the ODR range of GSPC structure. It is found that the ODR range of GSPC

structure can be enhanced by increasing the value of gradual constant

(without increasing the number of the layers as considered in first case). It is

found that the ODR range for GSPC structure is more than that of

conventional PC and simple graded structure. Hence, a GSPC structure can

be used as a broadband optical reflector and the range of reflection can be

tuned to any wavelength region by varying the value of gradual constant and

also by choosing proper thickness of the period of first stack and relative

thicknesses of individual layers of each period. In the first case, the ODR

range enlarges towards the lower wavelength region whereas in the second

case, the ODR range enlarges towards the higher wavelength region. These

types of optical reflectors are compact in size and may have potential

applications in the field of optical technology and optoelectronics.

General remarks and scope for future work: The work presented in this thesis is purely theoretical and it is mainly

concentrated on the one-dimensional photonic band gap materials. The

purpose of the work is to introduce the new structure in the field of photonic

devices. The researcher is well known to the limitation of these theoretical

findings. In this work, some approximations have been done keeping in the

view that they do not violate the laws of physics. Although the 2-

dimensional and 3-dimensional photonic crystal has wide range of

Chapter 6: Summary and Conclusion

- 147 -

applications, a 1-dimensional photonic crystal gives basic understanding of

3-dimensional photonic crystals. Also the fabrication of 1-dimensional

photonic crystal is relatively simple and cheap, and one-dimensional

photonic crystals provide an effective alternative possibility to control the

propagation light. The future work of this thesis may be employed to study

photonic crystals of higher dimensions. Moreover, investigations of the

photonic band gap in the materials such as chiral, superconductors, metals,

polymers and liquid crystals in the various ranges of the electromagnetic

spectrum will be of immense importance. Also, negative refractive index

material can be added to multilayed system with chiral, superconductors,

metals, polymers and liquid crystals. The new combinations may give many

exciting results.

The studies presented in the present thesis, however, would be

complete if various results reported here are subject to experimental

confirmations. Therefore, it is expected that such experimental work will be

taken up by other researchers or the investigator himself once the necessary

resources and/or facilities can be managed in the near future to confirm the

predictions and findings obtained in this thesis.

OPTOELECTRONICS AND ADVANCED MATERIALS � RAPID COMMUNICATIONS Vol. 4, No. 1, January 2010, p. 19 - 22

Design of an omni directional reflector using one dimensional photonic crystal with a single defect V. KUMAR*, KH. S. SINGH, S. K. SINGHa, S. P. OJHAb

Department of Physics, Digamber Jain (P.G.) College, Baraut-250611, India a Chaudhary Charan Singh University, Meerut, India bDepartment of Physics, T. D. (P. G.) College, Jaunpur (U.P.), India

An analytical study of a one-dimensional photonic crystal with a defect has been presented. It is found that omni-directional reflection (with cent percent reflectivity) range of a dielectric multilayered structure can be enhanced by introducing a defectin the conventional photonic crystal (PC). In the present communication, we study the omnidirectional reflection in visible and infrared region. We choose the Si/SiO2 multilayer system for our study. It is found that introduction of a single defect in the structure considered is sufficient to increase omni-directional reflection band widths.

(Received December 18, 2009; accepted January 19, 2010) Keywords: Photonic crystal, Omnidirectional reflection, Multilayer system

1. Introduction Since the publication of the seminal works of

Yablonovitch and John, study of photonic crystals has drawn attention of many investigators [1-3]. Subsequently, a class of photonic crystals exhibiting photonic band gaps has become a field of intense research. Photonic band gap (PBG) materials which are nothing but photonic crystals exhibiting photonic band gaps may be designed in one, two and three dimensions. But one-dimensional PBG materials (i.e. multilayered periodic structure with different refractive index materials) are attractive because such structures can be fabricated more easily at any wavelength scale and their analytical study and numerical calculations are simpler. Omni directional dielectric reflector is a mirror having cent percent reflectivity at any angle of incidence for both TE and TM polarized electromagnetic waves. Recently, such mirrors are realized and have been manufactured; and the conditions for their existence have been formulated. Thus, systems with periodic structure have become significant structures in photonics [4-8].

In 1998, Fink et al. [9] reported for the first time that one-dimensional dielectric lattice displays total omni-directional reflection for incident light under certain conditions. They constructed a stack of nine alternate polystyrene/tellurium layers having a thickness of a few micrometres. Further works by various researchers found many interesting results. Gallas et al. [10] reported the annealing effect in the Si/SiO2 PBG based omni-directional reflectors. Chen et al. [11] fabricated six bi-layers of SiO2 and TiO2 quarter wave films using the sol gel method and found an omni-directional photonic band gap of about 70nm in near infrared region. Chigrin et al. [12,13] fabricated a lattice consisting of 19 layers of Na3AlF6/ZnSe and found that omni-directional photonic band gap exists in the spectral range 604.3 to 638.4nm.

Much later in 2003, Lee and Yao [14] studied theoretically and experimentally a wide range of structures for the realistic fabrication of omni-directional photonic crystals having photonic band gaps (PBGs) in one dimension. C. J. Wu [15] has theoretically studied microwave transmission and reflection in a periodic superconductor/dielectric film multilayer structure in mixed state.

Ojha et al. [16] theoretically studied omni-directional high reflectors for infrared wavelengths, large omni-directional reflection using combination of periodic and Fibonacci structures respectively. They found that the range of omni-directional reflection can be increased by overlapping these photonic crystals. J. Zi et al. [17] showed that it is possible to enlarge the range of low transmission in one-dimensional photonic crystals by introducing a defect in the photonic quantum well structures and found that defect modes have very high quality factor.

In the present communication, it is shown that by introducing a defect in photonic band gap materials, very large ranges of omni directional reflection with cent percent reflectivity can be realised. The defect in the normal photonic band gap structure can be created by removing a part of or the entire layer of a single material slab.

2. Theoretical analysis To study the propagation of electromagnetic waves in

one dimensional photonic crystal, let us consider a structure of alternate layers of two materials with different refractive indices n1 and n2 respectively having high refractive index contrast in which a and b are the thicknesses of the two layers respectively and also the unit cell thickness, d=a+b [Fig. 1(a)]. Suppose an

20 V. Kumar, Kh. S. Singh, S. K. Singh, S. P. Ojha

electromagnetic wave incident obliquely on the interface of a one dimensional photonic crystal with an incident angle 0. We assume that the wave vector has components only in the x and z directions, then according to the Bloch wave theory E(x+d) = exp(ikd).E(x), the dispersion relation for this periodic dielectric layers is given by

)sin()sin(121)cos()cos(cos1)( 2121

1 bkakbkakd

k

(1) where ki=[( ni/c)2- 2]1/2=( ni/c).cos( i), i=sin-

1[(n0/ni).sin( 0)], i=1,2 and 2

1

kk for TE mode and

212

221

nknk for TM mode of polarization

Fig. 1. Schematic diagram of periodic structure (a) normal structure (b) defect structure.

Dispersion relation for defect structure: The removal

of a part or all of a single dielectric slab may now be considered within the above analytical model. Identifying the defect region as region II, we note that, from region I proceeding to left, we have a semi infinite lattice with a solution that increases exponentially to the right. Thus, equation (1) continues to hold, with eikd replaced by eKd, where K is real and positive and the sign factor = 1 is determined by that of the right hand side of equation (1). Hence the dispersion relation for a defect PBG structure can be written as [19]

)sin()sin(21)cos()cos(cosh

.1)( 21

1

2

2

121

1 bkakkk

kkbkak

dK

(2)

The reflection and transmission can be related easily

between the plane wave amplifications.

r

Mt 10

(3)

and 2221

1211

MMMM

M with 211111 NN UUmM ,

12121 NUmM , 211212 NN UUmM , 212222 NN UUmM

and ]).(sin[

]).().1sin[(dK

dKNU N and transmission and

reflection coefficients are given by

22

211211

.M

MMMt (4)

22

21

MMr

(5) The associated reflectance (R) is obtained by taking the absolute square of r

2rR (6)

In the next section, we study the reflection properties

of one dimensional photonic crystal by using equation (6).

3. Result and discussion To evaluate the reflectivity of the defect photonic

crystal, we used the transfer matrix method introduced by P. Yeh [18]. For the numerical computation, we have considered [na/(n1n2)5/(n1na)n/(n1n2)5/na] multilayer system, where n1 and n2 are the materials with low and high refractive index materials respectively and na is the refractive index for air. We have used SiO2 (n1=1.5) as low refractive index material and Si (n2=3.7) as the high refractive material. The thickness of the two layers taken a=0.59d and b=0.41d respectively.

From Snell�s law11 sinsin nn ia, we can see that

the refracted angle 1 is restricted to a certain range, where na and n1 are the refractive indices of air and the dielectric layer adjacent to air, respectively, and i is the incident angle. If the maximal refracted angle is smaller than the internal Brewster angle )/arctan( 121 nnB , the incident wave from the outside can not couple to the Brewster window, leading to the total reflection for all incident angles.

From Fig. 2 (a) it is clear that, there is a region of unit reflectance with omni-directional reflection for TE polarization from 689nm to 772nm and for TM polarization from 689nm to 738nm. Fig. 2 (b) shows the combined photonic band structure for both polarizations. From Fig. 2 (b), it is clear that there is a common region of unit reflectance with omni-directional reflection both for the TE and the TM modes of polarization from 689nm to 738nm in the scale of wavelength. The total bandwidth of this region is 49nm.

Design of an omni directional reflector using one dimensional photonic crystal with a single defect 21

(a)

(b)

Fig. 2. (a) Reflectance spectra of 1-D PC (n=0) for TE and TM polarizations; (b) Photonic band structure of 1-

D PC for n=0. Fig. 3. (a) shows the reflectivity spectra for TE and

TM mode of polarizations for n=1 i.e. one defect introduced in the middle of conventional photonic structure containing 10 pairs of lattice period. From Fig. 3 (a) it is clear that, there is a region of unit reflectance with omni-directional reflection for TE polarization from 685nm to 918nm and for TM polarization from 585nm to 952nm. Fig. 3 (b) shows the combined photonic band gap structure for both polarizations. From Fig. 3 (b) it is clear that, there is a common region of unit reflectance with omni-directional reflection both for the TE and the TM modes of polarization from 585nm to 918nm. The total bandwidth of this region is 333nm.

(a)

(b)

Fig. 3. (a). Reflectance spectra of 1-D PC (n=1) for TE and TM polarizations; (b) Photonic band structure of 1-

D PC for n=1. So, we can say that we can increase the

omnidirectional band width by introducing a defect in the conventional PC. It is clear that by introducing a defect in a one-dimensional photonic crystal, the region of omni-directional refection with reflectance equal to unity can be enlarged to 6.8 times of that of an photonic crystal without a defect.

4. Conclusions It is possible to enlarge the range of omni directional

reflection in one dimensional photonic crystal by introducing the defect. By introducing some defects in the photonic crystal, defects modes with very large quality factor may appear. Such a structure may be used in the design of optical resonators and mirrors in which reflectivity is independent of the angle of incidence.

22 V. Kumar, Kh. S. Singh, S. K. Singh, S. P. Ojha

Acknowledgements One of the authors, Vipin Kumar wishes to express

sincere gratitude to Dr K. B. Thapa for their valuable suggestions and critical comments.

References

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[9] Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, Science 282, 1679 (1998). [10] B. Gallas S. Fission, E. Charron, A. Brunet-Bruneau, R. Vuye, J. Revory, Appl. Opt. 40, 5056 (2001). [11] K. M. Chen, A. W. Sparks, H. C. Luan, D. R. Lim, K. Wada, L. C. Kimerling, Appl. Phys. Lett. 73, 3330 (1998). [12] D. N. Chigrin, A. V. Levrinenko, D. A. Yarotsky, S. V. Gaponeko, Appl. Phys. A: Mater. Sci. Process. 68, 25 (1999). [13] D. N. Chigrin, A. V. Levrinenko, D. A. Yarotsky, S. V. Gaponeko, J. Lightwave Tech. 17, 2018 (1999). [14] H. Y. Lee, T. Yao, J. Appl. Phys. 93, 819 (2003). [15] C. J. Wu, J. Electromagn. Waves Appl. 19, 1991 (2006). [16] S. K. Singh, K. B. Thapa, S. P. Ojha, Optoelectron. Adv. Mater.-Rapid Comm. 1(2), 49 (2007). [17] J. Zi, J.Wan, C. Zhang, Appl. Phys. Lett. 73, 2084 (1998). [18] P. Yeh, Optical waves in layered media, John Wiley and Sons, New York, 1988. [19] D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, J. Opt. Soc. Am. 10(2), 314 (1993). _______________________ *Corresponding author: vrpcommon@yahoo.com