Orbital Angular Momentum Multiplexing over Visible Light ...

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Theses

11-2017

Orbital Angular Momentum Multiplexing over Visible Light Orbital Angular Momentum Multiplexing over Visible Light

Communication Systems Communication Systems

Hardik Rameshchandra Tripathi [email protected]

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Orbital Angular Momentum Multiplexing over

Visible Light Communication Systems Hardik Rameshchandra Tripathi

MS Telecommunications Engineering Technology Program Electrical, Computer, and Telecommunications Engineering Technology

College of Applied Science and Technology Rochester Institute of Technology

Rochester, New York

MS Thesis Supervisor: Drew Maywar, PhD

MS Thesis Defense: 30 November 2017

Approved by

Drew N. Maywar, Associate Professor

Electrical, Computer, and Telecommunications Engineering Technology

William Johnson, Professor


Mark J. Indelicato, Associate Professor


Abstract

This thesis propose and explores the possibility of using Orbital Angular

Momentum multiplexing in Visible Light Communication system. Orbital

Angular Momentum is mainly applied for LASER and optical ber trans-

missions, while Visible Light Communication is a technology using the light

as a carrier for wireless communication. In this research, study of the s-

tate of art and experiments showing some results on multiplexing based on

Orbital Angular Momentum over Visible Light Communication system were

done. After completion of the initial stage; research work and simulations

were performed on spatial multiplexing over Li-Fi channel modeling. Simula-

tion scenarios which allowed to evaluate the Signal-to-Noise Ratio, Received

Power Distribution, Intensity and Illuminance were dened and developed.

Contents

1 Visible Light Communication 9

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 State of Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.1 VLC Transmitters . . . . . . . . . . . . . . . . . . . . 12

1.2.2 VLC Receivers . . . . . . . . . . . . . . . . . . . . . . 14

1.2.3 VLC Channel Characteristics . . . . . . . . . . . . . . 15

1.3 Modulation Techniques for VLC System . . . . . . . . . . . . 17

1.4 Advantages and Disadvantages of VLC System . . . . . . . . . 19

1.5 Potential Applications for VLC System . . . . . . . . . . . . . 20

1.6 VLC Channel Modeling with MATLAB . . . . . . . . . . . . . 22

1.6.1 Illuminance Distribution . . . . . . . . . . . . . . . . . 22

1.6.2 Received Power Distribution . . . . . . . . . . . . . . . 24

1.6.3 SNR Distribution . . . . . . . . . . . . . . . . . . . . . 29

1.7 Environment for VLC System MATLAB Simulation . . . . . . 31

1.8 Derivation of Transmitted Optical Power . . . . . . . . . . . . 35

1.8.1 Quantum Eciency and LED Power . . . . . . . . . . 35

1

1.9 MATLAB Simulation of VLC System . . . . . . . . . . . . . . 41

1.9.1 Illuminance Distribution . . . . . . . . . . . . . . . . . 41

1.9.2 Calculation of Angle of Incidence . . . . . . . . . . . . 44

1.9.3 Total Horizontal Illuminance . . . . . . . . . . . . . . . 45

1.9.4 Transmitted Optical Power for VLC System . . . . . . 48

1.9.5 Received Power Distribution for VLC System . . . . . 49

1.9.6 SNR Distribution of VLC System . . . . . . . . . . . . 55

2 Orbital Angular Momentum Multiplexing 57

2.1 Electromagnetic Fields and Angular Momentum . . . . . . . . 57

2.2 Gaussian Beam . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.2.1 Paraxial Equation . . . . . . . . . . . . . . . . . . . . . 61

2.2.2 MATLAB Simulation of Gaussian Beam . . . . . . . . 70

2.3 Laguerre Gaussian Beam . . . . . . . . . . . . . . . . . . . . . 72

2.3.1 Expression of Laguerre Gaussian Beam . . . . . . . . . 72

2.3.2 MATLAB Simulation of LG / OAM Beam . . . . . . . 78

3 Dierent Techniques for OAM Beam Generation 86

3.1 Holographic Plate . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.1.1 Mathematical Modeling of Holographic Plate . . . . . . 89

3.2 Spiral Phase Plate . . . . . . . . . . . . . . . . . . . . . . . . 91

4 Interconnection of OAM & VLC System 95

4.1 Generation of OAM Beam . . . . . . . . . . . . . . . . . . . . 95

2

4.2 Luminous Intensity & Horizontal Illuminance . . . . . . . . . 99

4.3 Received Power Distribution . . . . . . . . . . . . . . . . . . . 102

4.3.1 Transmitted Optical Power by OAM Beam . . . . . . . 102

4.3.2 Received Power Distribution for LOS Path . . . . . . . 103

4.3.3 Received Power Distribution for Diused Path . . . . . 104

4.3.4 SNR & Noise Distribution . . . . . . . . . . . . . . . . 105

4.4 Comparison of Various Parameters . . . . . . . . . . . . . . . 106

4.4.1 Impact of L & Beam Width . . . . . . . . . . . . . . . 106

4.4.2 Impact of Distance, H . . . . . . . . . . . . . . . . . . 110

4.5 Reasons for Developing OAM over VLC System . . . . . . . . 111

5 Conclusion 113

3

List of Figures

1.1 Scheme of establishment of communication in VLC System . . 10

1.2 VLC Circuit Scheme . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 VLC Channel Congurations . . . . . . . . . . . . . . . . . . 16

1.4 VLC Applications : Optical Wi-Fi and VLP (Visible Light

Positioning System) for accurate positioning . . . . . . . . . . 21

1.5 Model Room / Position of Transmitters on the ceiling . . . . . 23

1.6 Indoor Non-Directed LOS VLC Link (Directed Path) . . . . . 26

1.7 Propagation Model of Diused Channel . . . . . . . . . . . . . 27

1.8 (a) 1 Transmitter, (b) 4 Transmitters . . . . . . . . . . . . . . 33

1.9 Scenario 1 for simulation : 1 Transmitter . . . . . . . . . . . . 34

1.10 Scenario 2 for simulation : 4 Transmitters . . . . . . . . . . . 34

1.11 Lambertian Emission Order (m) of LED . . . . . . . . . . . . 42

1.12 Normalization of Luminous Intensity for dierent m values . . 43

1.13 Calculation of Angle of Incidence for VLC System . . . . . . . 44

1.14 Normalization of Horizontal Illuminance for dierent m values 46

1.15 Luminous Intensity of VLC System in [lx] . . . . . . . . . . . 47

4

1.16 Total Horizontal Illuminance of VLC System in [lx] . . . . . . 47

1.17 Received Power for Directed/LOS Path of VLC System in [mW] 50

1.18 Received Power for Directed/LOS Path of VLC System in [dBm] 51

1.19 Total Received Power for VLC System in [mW] . . . . . . . . 54

1.20 Total Received Power for VLC System in [dBm] . . . . . . . . 54

1.21 SNR Distribution for VLC System in [dB] . . . . . . . . . . . 56

2.1 Angular Momentum Subdivision . . . . . . . . . . . . . . . . . 58

2.2 Linear Polarization (Left) and Circular Polarization (Right) . 59

2.3 Helical Phase Front . . . . . . . . . . . . . . . . . . . . . . . . 60

2.4 (a) Phase front, (b) Cross Intensity Distribution, (c) Intensity

prole of Gaussian Beam . . . . . . . . . . . . . . . . . . . . . 69

2.5 Intensity Prole of Gaussian Beam in 3-D Plot . . . . . . . . . 71

2.6 Transversal View of Gaussian Beam Intensity Prole . . . . . 71

2.7 (a) Phase Front, (b) Cross Intensity Distribution, (c) Intensity

Prole of Beam with OAM for l = 1. . . . . . . . . . . . . . . 76

2.8 Cross Section of LG Beam Intensities for dierent values of l&p. 77

2.9 Intensity Prole of OAM Beam in 3-D Plot, l = 1 . . . . . . . 81

2.10 Transversal View of OAM Beam Intensity Prole, l = 1 . . . . 81





5





3.1 Intensity pattern of interference between helical with a planar

beam for several dierent value of the charges singularities . . 87

3.2 Beams generated by a Hologram Plate with L = 2 . . . . . . . 88

3.3 Dierent grating shape: (a): Sinusoidal, (b): Blazed, (c): Bi-

nary, (d): Triangle . . . . . . . . . . . . . . . . . . . . . . . . 89

3.4 Holographic Plate for l = 1 . . . . . . . . . . . . . . . . . . . . 90

3.5 Schematic of SPP illuminated by TEM00 beam & output OAM 92

3.6 Creation of Laguerre Gaussian Beam using SPP . . . . . . . . 93

4.1 Generation of OAM Beam for, L = 1 . . . . . . . . . . . . . . 97

4.2 Luminous Intensity of OAM over VLC System [lx] for, L = 1 . 100

4.3 Cross-Sectional View of Luminous Intensity for, L = 1 . . . . 100

4.4 Total Horizontal Illuminance of OAM over VLC System for,

L = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.5 Transversal View of Total Horizontal Illuminance for, L = 1 . 102

4.6 Received Power (LOS/Directed) [mW] for, L = 1 . . . . . . . 103

4.7 Received Power (LOS/Directed) [dBm] for, L = 1 . . . . . . . 103

4.8 Total Received Power [mW] for, L = 1 . . . . . . . . . . . . . 104

4.9 Total Received Power [dBm] for, L = 1 . . . . . . . . . . . . . 104

6

4.10 SNR Distribution of OAM over VLC System [dB] for, L = 1 . 105

4.11 Noise Distribution [dB] for, L = 1 . . . . . . . . . . . . . . . . 105

4.12 Noise Distribution of VLC System [dB] for, L = 1 . . . . . . . 106

4.13 Table: Simulation Results, Prx v/s L & w0 . . . . . . . . . . . 107

4.14 Simulation Result Comparison, Prx v/s L & w0 . . . . . . . . 108

4.15 Transversal View of Beam Width, Prx v/s L & w0 . . . . . . . 108

4.16 Impact of Beam Width on Intensity, Iφ v/s w0 . . . . . . . . . 109

4.17 Impact of Distance [m], Prx v/s H & w0 . . . . . . . . . . . . 110

7

List of Tables

1.1 Parameters for VLC System MATLAB Simulation . . . . . . . 32

4.1 Parameters for OAM over VLC System MATLAB Simulation 98

8

Chapter 1

Visible Light Communication

1.1 Introduction

The Visible Light Communication (VLC) is new technology that uses light

for sending information. Light is used rstly for illumination needs but can

now be used as medium for communication and allows achieving high data

rate. The visible spectrum used by the light is license free and not much used

today while the radio spectrum is saturated which adds an undue advantage

to this new technology for further development.

Therefore, the VLC technology is one advanced optical wireless commu-

nication and is very promising as compared to the other wireless communi-

cation technologies which use radio waves as a medium for communication

(WI-FI, Bluetooth, Wi-MAX, etc.).

Because of the continuous development of this technology, a lot of compa-

9

nies, research institutes and also universities are interested in VLC systems

and the current research is concentrated in the area of increasing the data

rate and for long range communications.

For example, OMEGA (Home Gigabit Access) home area-network, Euro-

pean project's main objective was to provide [Gb/s] connectivity to users in

home for access network via communication technologies including wireless

connections (RF and VLC) and power line communication. They exper-

imented an indoor visible light wireless communication including a MAC

layer protocol adapted to optical wireless communication systems and a data

rate of 73 [Mb/s] was achieved [1].

There are many opportunities and application domains for VLC (intelli-

gent trac system, indoor networking, underwater communication, etc.) and

these applications domain are experiencing tremendous growth.

Figure 1.1: Scheme of establishment of communication in VLC System

10

1.2 State of Art

In VLC, the communication is established by modulating the LED light

intensity in such a way that is undetectable by the human eyes as the human

eye doesn't see any change up to 60 [Hz] -100 [Hz] for central vision and

peripheral vision respectively [2].

LED is used to transmit information and at the receiver side a photo

sensitive detector demodulates the light signal into electronic form. The

transmission chain of VLC signal consists of a LED source and an appropriate

modulation technique for optical channel. It is possible to use laser like light

source because it can be modulated at high frequency. In this case, white

light can produced by combining yellow, blue, green and red lasers [3].

The transmission chain of a Visible Light Communication system is sum-

marized below :

Figure 1.2: VLC Circuit Scheme

At the input data of the circuit, we have one type of data (text, image,

audio, and video) provided at the input of the driver circuit. For example

a MATLAB program can be used for processing signal at the transmission

11

and reception side.

The driver circuit includes an amplier, signal is then sent to the LED

and light is emitted. Typically a photodiode is used to detect the emitted

light and convert it in electrical signal and forward it to the receiver circuit.

The receiver circuit is composed by an amplier (Trans- impedance amplier)

and a lter.

In some experimentation, it is possible to put a lens before the photodiode

to focal the maximum of signal on the photodiode.

1.2.1 VLC Transmitters

To transmit data in free space in VLC technology, LED (Light Emitting

Diode) can be used in two dierent ways :

LEDLight : The LED is a semiconductor light source. It is a basic PN

- junction diode. LED can emit light when current ows. During electrons

and holes recombination, we have emission of photons formed by the energy

released.

• Phosphorescent: To emit white light, a blue LED chip is coated with

a yellow phosphor. This kind of LED is used for inexpensive installations

because the driver design is simple but the response time of yellow phosphor

is slow. However, it is possible to enhance the bandwidth to 20MHz by

eliminating phosphor material using blue lter.

• Multi-color (RGB): Mainly constituted of red color, with blue and

green colors, three color channels individually allow bandwidth of 15MHz. A

12

wavelength-division multiplexing (WDM) can be realized when using three

drivers in parallel.

The RGB LEDs are better because they allow increasing the data rate.

Experiments show data rate of up to 1.5 [Gb/s] with single channel and 3.4

[Gb/s] by implementing WDM transmission.

LEDs are used rstly for illumination but also for communication. Be-

cause of their capacity of fast switching, LEDs have dual functionality of

brightness and communication, and allows many applications.

The two basic properties of LEDs are :

• Luminous intensity and,

• Transmitted Optical Power.

The luminous intensity indicates the energy ow per angle and is function

of illuminance at an illuminated area. Luminous intensity is used to represent

the illumination of LED and is given by [4] :

I =d Φ

d Ω(1.1)

where, Φ is the light ow and, Ω is the spatial angle.

The light ow i.e. Φ can further be written as follows:

Φ = Km

∫ 780

380

V (λ)Φe(λ)dλ (1.2)

where, Φe is the energy ow and, V (λ) is the standard luminosity. Km

is the maximum visibility i.e. 683 [lm/W ] at λ = 555 [nm], as you can

13

see the wavelength falls in the visible spectrum range.

The transmitted optical power indicates the total energy radiated from a

LED and it represents the integral of the energy ow Φe in all directions. It

is given as [4] :

Pt =

∫ Λmax

Λmin

∫ 2π

0

Φe dθ dλ (1.3)

where, Λmax and Λmin are determined by the sensitivity curve of the

Photodiode.

MobileDevices : In VLC technology, some mobile devices can be used

like transmitters because of the presence of LED in these devices. We have

TV screens, advertising panels, and projectors.

1.2.2 VLC Receivers

The VLC receivers can be classied according to two categories :

Photodiode : The photodiode is a PN or PIN junction semiconductor

component able to detect light and convert it in electrical signal. When in

the photodiode photons are absorbed, current is generated.

Photons are absorbed if Eph > Eg, that corresponds to the absorbed

energy by photon to move from valence band to conduction band. An electron

/ hole pair is created, this mechanism is photoelectric eect and photo current

is produced when holes move toward the anode and electron towards the

cathode.

14

The received power is given by [4] :

Prx = (PLOS + Pdiff) ? Ts (ψ) ? g (ψ) (1.4)

where, Ts(ψ) is the transmission coecient of the optical ber and g(ψ)

is the concentrator gain.

Then, the output current will be written as :

i = R ? Prx (1.5)

where, R is the responsivity of the photodiode [A/W]. To obtain high

data rate in visible light communication, the photodiode should be used.

Devices :

• Camera Sensor : A two dimensional array of photodiode, each photo-

diode represent one Pixel.

• Smartphones or Tablets : They have two cameras; a front camera is

used for indoor positioning (facing the ceiling light, the transmitted VLC

signals from it can be decoded). The rear camera is useful for decoding

signals from LED screen TV, advertising panels, and mobile devices.

1.2.3 VLC Channel Characteristics

The VLC links are classied in three links types :

• Directed LOS (Line of Sight)

• Non-Directed LOS

15

• Directed NLOS (Non-Line of Sight)

Utilizing reected paths of the light from indoor surfaces of wall, ceiling

and furniture can be easily used and have strong robustness to blocking, but

can suer from multi path eect and non-directed NLOS depending on the

existence of an unobstructed path between the transmitter and receiver, can

maintain small path loss, but are susceptible to blockage as there are multiple

reections.

In VLC, the non-directed is an important point because general lighting

work in LOS environments and light is not directed. One important param-

eter in non-directed optical wireless communication is the directivity of light

source and the photodetector, more the Field of Viewer (FOV) is large more

the system robustness. To have a very robust system, we can use a diusion

lens or an array of LEDs for transmitters and an array of photodetectors for

the reception because it's allowing wider FOV.

Figure 1.3: VLC Channel Congurations

16

1.3 Modulation Techniques for VLC System

In VLC system, there are many types of modulations to use. We will discuss

them based on their uses in various journal papers.

• OOK (On-O Keying) : It is used for inexpensive and simple systems.

This modulation scheme operation is a presence of a carrier represented by a

binary one while absence is represented by a binary zero. For information of

positioning, the modulation is simple to generate and to detect. This work

is comprehensively discussed in this journal paper which VLC system based

on OOK modulation technique [5].

• CSK (Color Shift Keying) : It uses red, green and blue LEDs of a

white luminance and varies there intensities in an order dependent on the

transmitted bit sequence [6]. Color icker can be present which is discussed

by the work done in journal paper based on CSK modulation scheme for

VLC system [7].

• Metameric Modulation : LEDs are mathematically equivalent, data

sequences are encoded to combinations of LEDs and these LEDs have a

dierence in their SPDs (Spectral Power Distribution). The color icker

eects can be eliminated by metameric modulation.

• Pulse Position based Modulation : To implement this modulation

scheme, LEDs are turning on and o at time intervals. They are sensitive to

the slow rise-time of LED, and are not aected by their nonlinearity [8].

PPM (Pulse Position Modulation) : Symbol duration is divided into

17

q time slots, during one time slot, one pulse is transmitted. Symbols are

identied by pulse position. This modulation scheme is inappropriate for

high speed due to the spectral eciency which decreases when the size of

the constellation increases. However PPM is useful to reduce the interfering

eects of the background light.

MPPM (Multiple PPM): modulation scheme proposed to increase the

spectral eciency of PPM. In each symbol time, a lot of pulses are sent. The

minimum hamming distance is 2.

EPPM (Expurgated PPM) : Modied form PPM modulation scheme

useful to enhance its performances in optical systems. To maximize the

hamming distance, symbols are expurgated. In EPPM, the error probability

decreases to attenuate icker because many pulses are transmitted on one

period. However, there is low spectral eciency.

MultilevelEPPM : To increase the size of constellation and rate.

This modulation scheme is more adapted for indoor VLC.

• OFDM (Orthogonal Frequency Division Multiplexing) : It is used to in-

crease the data rate in bandwidth-constrained channels, it is a multi-carriers

modulation technique. At the rst time, OFDM was developed for RF com-

munication systems, and now, modied forms (DC biased OFDM, asymmet-

rically clipped optical OFDM) have been proposed for indoor VLC systems.

However, OFDM suers from the nonlinearity of diodes, the PAPR is high

and this can cause the distortion of output signal.

• Spatial Modulation : It is a multipoint system (MIMO). Used to in-

18

crease the data rate transmission, spectral eciency, and total received pow-

er. The system is sensitive to multipath interferences. To maximize the data

rate up to 1Gbits, OFDM and MIMO can be combined together [8].

An experiment on VLC MIMO using VCSEL (Vertical Cavity Surface

Emitting Laser) and OFDM 16-QAM modulation showed that it is possible

to have data rate of 10 [Gb/s] with link of 15 [m] in free space [9].

In another experiment on VLC with QBD-OFDM (Quasi-Balanced De-

tection) using RGB LED, a data rate of 2.1 [Gb/s] with 2.5 [m] of distance

was achieved [10].

1.4 Advantages and Disadvantages of VLC Sys-

tem

The adoption of LED light sources has made opportunities for Visible Light

Communication using LED lights and photodetectors to establish communi-

cation links. The primary usefulness of LED is lighting, so using LED for

data transmission shall not have an impact on feature.

The following are the benets of the VLC system :

• Very large available bandwidth (< 320 THz)

• No license for visible spectrum

• Long life of the LEDs

• Low cost

• Low power consumption

19

• Harmless for human body

• Can be used in places where RF could be prohibited : aircrafts, hospitals

• High spectral density

• VLC signals are in closed environment

• Capacity to modulate LED intensity quickly in a way that is unde-

tectable by human eye

• Security (need a eld of view between transmitter and receiver)

Although the technology presents many advantages, there are also certain

disadvantages as well :

• When distance of the link increase, data rate decrease

• Ambient light noise

1.5 Potential Applications for VLC System

The indoor Visible Light Communication has been the subject of several

researches. The channel characteristics and modeling was studied in stated

papers [11] [12] [13] [14] [15] [16] [17] [18].

Applications of indoor positioning using lighting LEDs are a lot of, and

in some of them, it is necessary to have an accurate localization.

Hence based on the comprehensive study of the stated journal papers,

some of the highlighted applications are as follows:

20

• VLC for Indoor Applications

Accurate Positioning

Trading

Medical Domain

Indoor Internet Navigation

• VLC for Transportation Applications

Collision Warning and Avoidance

Lane Changing Assistance / Warning

Cooperative Adaptive Cruise Control

• VLC System for Under Water Applications

• Text Transmission, Real-time Audio and Video Transmission System

based on VLC technology

Figure 1.4: VLC Applications : Optical Wi-Fi and VLP (Visible Light Posi-tioning System) for accurate positioning

21

1.6 VLC Channel Modeling with MATLAB

The Visible Light Communication channel is a subject of numerous research-

es. In many journal papers, a VLC channel modeling is proposed. In [4] and

[19] programs for indoor visible light communication environment based on

MATLAB and Simulink are proposed. To improve data rate of indoor com-

munication links, an equalization system can be used, papers [11] and [13]

have discussed about it in depth.

In a journal paper, the simulations show that a SNR (Signal-to-Noise

Ration) of up to 81 [dB] can be achieved and data rate can be improved

from 16 [Mb/s] to 32 [Mb/s] NRZ-OOK at BER (Bit Error Rate) of 10−6

[13].

The main objective is to model a room in where, a communication link

is established between LEDs (used like a communication device) and user

terminal is placed on the desk. We will see the distribution of illuminance

at the desk surface where the user's machine is installed, the received power

and the signal to noise ratio.

1.6.1 Illuminance Distribution

LEDs are used rstly for illumination but also for communication. Because

of their capacity of fast switching, LEDs have dual functionality of brightness

and communication and allow many applications.

The illuminance expresses the brightness of an illuminated surface. To

22

Figure 1.5: Model Room / Position of Transmitters on the ceiling

express the distribution of illuminance in room, it is assumed a LED chip

has a Lambertian radiation pattern, so the radiant intensity depends on the

Angle of Irradiance (Angle of Emission).

The luminous intensity in angle Φ is given by [4] :

I (Φ) = I (0) cosm (φ) (1.6)

where, I (0) is the Center Luminous Intensity of a LED. The maximum

of intensity is obtained at 0.

Ehor is the horizontal illuminance at a point (x, y) and is given by the

following equation [4] :

Ehor =I (0) cosm (φ)

d2 cos (Ψ)(1.7)

where,

• I (0) is the Center Luminous Intensity of a LED

• φ is the Angle of Irradiance

23

• Ψ is the Angle of Incidence

• d is the Distance between LED and Detector's Surface

• m is the Order of Lambertian Emission, where

m =ln 2

ln (cos Φ1/2)(1.8)

where, Φ1/2 is the Semi-Angle at Half Power of a LED

1.6.2 Received Power Distribution

The signal emitted by LEDs is sent to the receiver by the optical channel in

free space.

The VLC systems use intensity modulation, then, intensity of the back-

ground light is an important point to take into account for the desired channel

model.

We have two conguration types:

For a channel with low background light, the received signal can be mod-

eled like a Poisson process:

λr (t) = λs (t) + λn (t) (1.9)

where λs and λn are proportional to the instantaneous optical power of

received signal and the power of background light, respectively.

However for channel with high background light it means that, λn is high,

in environment where the ambient light is important (articial and natural

24

sources); the photodiode shot noise must be modeled as an additive white

Gaussian noise.

The Channel DC Gain is given by the equation [4] :

For, 0 ≤ Ψ ≤ Ψc :

H (0) =(m + 1)

2 π d2A cosm (φ) Ts (Ψ) g (Ψ) cos (Ψ) (1.10)

For, Ψ ≥ Ψc :

H (0) = 0 (1.11)

where,

• A is the Physical Area of the Detector in Photodiode.

• Ts (Ψ) is the Gain of an Optical Filter.

• g (Ψ) is the Gain of an Optical Concentrator.

• Ψc is the Width of FOV (Field of Viewer) at the Receiver.

Now, the equation for g (Ψ) is given as :

For, 0 ≤ Ψ ≤ Ψc :

g (Ψ) =n2

sin2 (Ψc)(1.12)

For, Ψ ≥ Ψc :

g (Ψ) = 0 (1.13)

25

The photodiode captures the light sent from the LEDs in form of optical

power. In case of directed path, it is given as [4] :

Pr = H (0) ? Pt (1.14)

where,

• Pr is the Received Power

• Pt is the Transmitted Power

Figure 1.6: Indoor Non-Directed LOS VLC Link (Directed Path)

In case of reections of light by other surfaces (walls, ceiling, oor, win-

dows), it is called a diuse channel. The received power has a new expression

given by the channel DC gain on directed path Hd (0) and reected path

Href (0).

The Channel DC Gain on the rst reection is given by the equation as

26

follows [4] :

For, 0 ≤ Ψ ≤ Ψc :

dHref (0) =(m+ 1)A

2 π2 D21 D

22

ρdAwall cosm(φ) cos(α) cos(β)Ts(Ψ) g(Ψ)

(1.15)

For, Ψ ≥ Ψc :

dHref (0) = 0 (1.16)

where,

• ρ is the Reectance Factor.

• dAwall is the a Reective Area of Small Region.

Figure 1.7: Propagation Model of Diused Channel

It is possible to represent the optical power in diuse channel using other

method. Another solution would be, an integrating sphere model for the

27

optical wireless where diuse signal is used [13].

Here we use this method to do simulation of diuse channel.

The rst diuse reection of a wide-beam optical source, in room, emits

intensity I1 over the whole room surface Aroom and is given by the following

equation :

I1 = ρ1

PtotalLED

Aroom

(1.17)

where,

• ρ1 is the Reectivity of the Surface.

• PtotalLED is the Total Power of all the LEDs.

The average reectivity, < ρ > is noted by [13] :

< ρ >≥1

Aroom

∑i

Ai ρi (1.18)

where,

• ρi is the Individual Reectivity of walls, windows and other objects.

• Ai is the Individual Area of walls, windows and other objects.

So, the sum of geometrical series gives the total intensity noted by [13] :

I = I1

∞∑j=1

< ρ >j−1 =I1

1− < ρ >(1.19)

where, j is the Number of Reections.

Hence, the Received Diused Power is given by the equation as follows

28

[13] :

Pdiff = Arx ? I (1.20)

where, Arx is the Area of Photodiode.

So at the receiver end, the Total Received Power is given by the equation

as follows [13] :

Prx = (PLOS + Pdiff) ? Ts(Ψ) ? g(Ψ) (1.21)

1.6.3 SNR Distribution

To express the quality of communication link, a Signal-to-Noise Ratio is used

which expresses the power of the signal with respect to the power of noise.

The SNR calculated for VLC system is given by the following equation

[13] :

SNR =(R Prx)

2

σ2total

(1.22)

where,

• σ2total is given by the equation as follows :

σ2total = σ2

shot + σ2amplifier (1.23)

Now, the equation of σ2shot is given as follows :

29

σ2shot = 2 q R (Prx + Pn)Bn (1.24)

and the equation of σ2amplifier is given as follows :

σ2amplifier = i2amplifier Ba (1.25)

Bn is given by the following equation :

Bn = I2 Rb (1.26)

where,

• σ2total is the Total Noise Variance.

• σ2shot is the Shot Noise Variance.

• σ2amplifier is the Amplier Noise Variance.

• Bn is the Noise Bandwidth.

• Pn is the Noise Power.

• Rb is the Data Rate.

• I2 is the Noise Bandwidth Factor.

• Ba is the Amplier Bandwidth.

30

1.7 Environment for VLC System MATLAB

Simulation

In this section of the report, we will discuss about the parameters / values

taken for MATLAB simulation of two types of technologies namely: Visi-

ble Light Communication System (VLC) and Orbital Angular Momentum

Multiplexing over Visible Light Communication System (OAM over VLC

System).

We will also represent the dimensions of the room and various aspects

in out simulation environment based on the research done in the previous

sections of OAM and VLC.

First, we will display the table of various parameters considered for a

VLC System. It is shown in the following table:

31

Table 1.1: Parameters for VLC System MATLAB Simulation

Parameters Dimensions

Room Size 5 * 5 * H [m]

Height of Desk 0.5 [m]

Transmitted Optical Power Pt

Semi-Angle Half Power 30

Center Luminous Intensity 0.73 [cd]

Number of LEDs in a group 3600 (60 * 60)

Number of LED Lights 4

LED Interval 0.01 [m]

Size of LED 0.59 * 0.59 [cm]

FOV at Receiver 70

Detector Physical Area of Photodiode 1 [cm2]

Optical Filter Gain 1

Refractive Index of Lens at Photodiode 1.5

Photodiode Responsivity 0.8

Floor Reectivity 0.15

Ceiling Reectivity 0.8

Wall Reectivity 0.7

Amplier Noise Density 5 [pA/√Hz]

Ambient Light Photocurrent 5840 [µA]

Noise Bandwidth Factor 0.562

32

In a normal VLC system, the Center Luminous Intensity is considered to be

0.73 [cd]. The transmitted optical power, Pt of the VLC System is considered

and will be derived in the next section. Rest of the parameters / values of

various quantities will remain same for both the systems, i.e. VLC System

and OAM over VLC System.

Now, we will discuss about the position of the transmitters on the ceiling

and receiver in our simulation test.

Figure 1.8: (a) 1 Transmitter, (b) 4 Transmitters

Hence, the 1 and 4 Transmitter scenario can be better explained in a three

dimensional gure as follows :

33

Figure 1.9: Scenario 1 for simulation : 1 Transmitter

Figure 1.10: Scenario 2 for simulation : 4 Transmitters

34

1.8 Derivation of Transmitted Optical Power

For optical to free space communication systems requiring bit rates less than

approximately 100-200 [Mb/s] together with Multimode ber (MMF) coupled

optical power in the tens of microwatts, LEDs are usually the best light source

choice [20].

LEDs require less complex circuit drivers than Laser diodes as no thermal

or optical stabilization circuits are needed and they can be fabricated less

expensively with higher yields.

1.8.1 Quantum Eciency and LED Power

Excess carrier density of semiconductor material decays exponentially with

time according to the relation as follows [20] :

n = n0 e− tτ (1.27)

where,

• n0 is the Initial injected excess electron density.

• τ is Time Constant which signies Carrier Lifetime. It's value can range

from [ms] to fractions of a [ns] depending on the material composition and

device defects of semiconductor in LEDs.

Now, we will discuss important aspects related to quantum eciency and

LED power :

35

• RadiativeRecombination : A photon of energy, h υ approximately

equal to bandgap energy of semiconductor material is emitted.

• Non−RadiativeRecombination : These eects include optical ab-

sorption in active region of semiconductor material (Self Absorption).

When there is constant current ow into LED, equilibrium condition is estab-

lished, i.e. excess density of electron, (e−) denoted by n and holes denoted

by p is equal.

Since injected carriers are created and recombined in pairs such that charge

neutrality is maintained within device [20].

Total rate at which carriers are generated is sum of externally supplied and

thermally generated rates. Now,

• Externally Supplied Rate = J / q d

where,

J is the Current Density [A/cm2]

q is the Electron Charge

d is the Thickness of Recombination Region

• Thermal Generation Rate = n / τ

Hence, Rate equation for carrier recombination in a LED can be written as

[20] :

d n

d t=

J

q d−n

τ(1.28)

36

The equilibrium condition is found by setting above equation [1.28] equal to

a value of zero;

=⇒J

q d−n

τ= 0 (1.29)

Hence, Excess Density of electrons, n can be written as :

n =J τ

q d(1.30)

This relation gives steady state electron density in active region of semicon-

ductor material when constant current is owing through it.

Internal Quantum Eciency

The internal quantum eciency in active region is fraction of electron -hole

pairs that recombine radiatively. It is the ratio of radiative recombination

rate to the total recombination rate [20].

ηint =Rr

Rr + Rnr

(1.31)

where,

• Rr is the Radiative Recombination Rate.

• Rnr is the Non-Radiative Recombination Rate.

For exponential decay of excess carriers, the radiative recombination lifetime

is given as :

37

τr =n

Rr

(1.32)

and, the non-radiative recombination lifetime is given as :

τnr =n

Rnr

(1.33)

Hence, the Internal Quantum Eciency, ηint can be written as follows [20]:

=⇒ ηint =1

1 + τrτnr

=τ

τr(1.34)

where, the Bulk Recombination Lifetime, τ is given as follows :

1

τ=

1

τr+

1

τnr(1.35)

If the current injected into the LED is I, the total number of recombination

per seconds is given by following equation [20] :

Rr + Rnr =I

q(1.36)

Now, by substituting the equation [1.36] into the above equation [1.31], will

yield to the following :

Rr = ηintI

q(1.37)

where, Rr is the Total number of Photons generated per second.

38

Each photon has an energy which is equivalent to hυ, then the optical power

generated internally to LED is given by the following equations [20] :

Pint = ηintI

qh υ (1.38)

or can be rewritten as follows :

Pint = ηinth c

λ

I

q(1.39)

where,

• h is the Planck's Constant.

• c is the Speed of Light.

• λ is the Wavelength in Visible Spectrum Range.

• q is the Electron Charge.

• I is the Current Injected into LED.

External Quantum Eciency

Ratio of photons emitted from LED to the number of internally generated

photons is known as the External Quantum Eciency, ηext which is needed

to nd the emitted power. For this, we need to take into account the reection

eects at surface of LED [20].

Light falling within cone is dened by φc, known as Critical Angle will cross

the interface and will result in following equation [20] :

39

=⇒ φc = sin− 1 (n1

n2

) (1.40)

where,

• n1 is the Refractive Index of Semiconductor material.

• n2 is the Refractive Index of Outside material / Air.

Now, the External Quantum Eciency, ηext is dened by the equation as

follows [20] :

ηext =1

4 π

∫ φc

0

T (φ) (2π sin φ) d π (1.41)

where,

• T (π) is the Fresnel Transmission Coecient / Fresnel Transmissivity. It

depends on incidence angle, φ. To simplify, we can use the expression for

normal incidence which is given as follows [20] :

T (0) =4 n1 n2

(n1 + n2)2(1.42)

Now, assume that outside medium is air, and let, n1 = n :

=⇒ T (0) =4 n

(n + 1)2(1.43)

Therefore, the External Quantum Eciency, ηext can be rewritten as follows

[20] :

40

ηext ≈1

n (n + 1)2(1.44)

Hence, the derived Transmitted Optical Power emitted from LED will be

given by the following equation [20] :

Pt = ηext Pint ≈Pint

n (n + 1)2(1.45)

1.9 MATLAB Simulation of VLC System

1.9.1 Illuminance Distribution

In VLC system, Luminous Intensity at Angle of Irradiance, φ is given by the

following equation [4] :

I(φ) = I(0) cosm(φ) (1.46)

where, the Center Luminous Intensity, I(0) is considered to be 0.73[cd] and,

Order of the Lambertian Emission, m is given as follows :

m =− ln 2

ln cos(φ1/2)(1.47)

where, φ1/2 is the Semi-Angle at Half Illuminance.

41

Hence, we have plotted the graph for Order of the Lambertian Emission, m

:

Figure 1.11: Lambertian Emission Order (m) of LED

The Order of Lambertian Emission is higher as the Semi-Angle at Half Illu-

minance decreases. The decrease in φ1/2 means that the LED light is more

intensied or concentrated over a small regional area.

Hence, larger the value ofm, the intensity of the LED light beam is more at

a concentrated area as the distribution is less.

In the next plot, we will study the behavior of Luminous Intensity from

equation [1.46] on Normalization of the LED Beam.

Center Luminous Intensity, I(0) = 0.73[cd],

Normalization of Luminous Intensity will be given by following equation :

42

Normalization :I(φ)

I(0)= cosm(φ) (1.48)

Figure 1.12: Normalization of Luminous Intensity for dierent m values

Hence, by this plot we can understand that m with higher values are more

concentrated towards the center i.e. high intensity over a small central con-

centrated illumination region. And, them with lower values are more spread

out from the center i.e. low intensity over a large central spread out illumi-

nation region.

43

1.9.2 Calculation of Angle of Incidence

The Angle of Irradiance, φ is calculated by the following formula :

φ = cos−1(H

D) (1.49)

where,

• H = Height of the Room - h (Height of Receiver Plane)

• Distance between Transmitter and Receiver, D is calculated as follows :

d =√

(X1−Xp1)2 + (Y 1− Y p1)2 + H2 (1.50)

where,

• Position of the Transmitter : [Xp1 Yp1 Zp1]

• Position of the Receiver : [X1 Y1 Z1]

Condition : Suppose the transmitter is pointed straight down from the

ceiling, and as receiver is moved about the room, it is pointed straight up at

the ceiling. Then, condition α = φ is maintained.

Figure 1.13: Calculation of Angle of Incidence for VLC System

44

Now, Using Alternate Interior Angles Property :

Angle of Irradiance, φ = Angle of Incidence, α. Hence calculated Angle of

Incidence will be written as follows :

α = cos−1(H

D) (1.51)

where,

• D is the Distance between Transmitter and Receiver

• H = Height of the Room (S[m]) - Height of Receiver Plane (h[m])

1.9.3 Total Horizontal Illuminance

Ehor is the Horizontal Illuminance at a point (x, y) and is given by :

Ehor =I(0) ? cosm(φ)

D2 ? cos(α)(1.52)

And, hence the Total Horizontal Illuminance is given as follows :

Ehort1 = Ehor ? nLED ? nLED (1.53)

where, Number of LEDs in each group : nLED = 60.

45

First, we plot the Normalized Horizontal Illuminance Pattern which is given

as follows :

Normalized Ehor = cosm(φ) (1.54)

Figure 1.14: Normalization of Horizontal Illuminance for dierent m values

As you can observe, the illumination for the lower values of m is more as

compared to the higher values of m.

Even though the intensity of LED beam is more for higher m values, the

spread across detector surface is less as it is just concentrated in a small

region at the center.

While for lowerm values, the LED beam concentration at the central region

would be less but the spread would more evenly distributed throughout the

detector surface resulting in more illumination.

46

Therefore, using the scenario for 4 Transmitters, we will plot the Luminous

Intensity Iφ and Total Horizontal Illuminance Ehort1 of VLC System.

Figure 1.15: Luminous Intensity of VLC System in [lx]

Figure 1.16: Total Horizontal Illuminance of VLC System in [lx]

47

1.9.4 Transmitted Optical Power for VLC System

The equation for Transmitted Optical Power, Pt is given as follows from the

previous derivation :

Pt =Pint

n (n+ 1)2(1.55)

where, n is the Refractive Index of Semiconductor Material and,

Pint = ηinth c

λ

I

q(1.56)

where,

• Planck's Constant, h = 6.62607004 e−34 [m2kg/s]

• Speed of Light, c = 3 e8 [m/s]

• Electron Charge, q = 1.60217662 e−19 [C]

• λ is Wavelength with Visible Spectrum Range : 370− 780 [nm]

• Current injected into LED, I [mA]

Now, the equation for Internal Quantum Eciency, ηint is given as follows :

=⇒ ηint =1

1 + τrτnr

=τ

τr(1.57)

where, Bulk Recombination Lifetime, τ [ns] is given as follows :

1

τ=

1

τr+

1

τnr(1.58)

48

• τr : Radiative Recombination Lifetime [ns]

• τnr : Non-Radiative Recombination Lifetime [ns]

The Radiative Recombination Rate, Rr can also be written as :

Rr = ηintI

q(1.59)

where, Rr : Total number of Photons generated per second

1.9.5 Received Power Distribution for VLC System

For Directed / LOS Path :

The Channel DC Gain is given by the equation :

For, 0 ≤ Ψ ≤ Ψc :

H (0) =(m + 1)

2 π d2A cosm (φ) Ts (Ψ) g (Ψ) cos (Ψ) (1.60)

For, Ψ ≥ Ψc :

H (0) = 0 (1.61)

Now, the equation for Optical Concentrator g (Ψ) is given as :

For, 0 ≤ Ψ ≤ Ψc :

g (Ψ) =n2

sin2 (Ψc)(1.62)

For, Ψ ≥ Ψc :

g (Ψ) = 0 (1.63)

49

Hence, the Received Power for Directed / LOS Path is given by :

Pr = Ptotal ? H(0) (1.64)

where, Ptotal = Pt ? nLED ? nLED

• Physical area of the detector in Photodiode : A = 1 [cm2]

• Gain of an Optical Filter : Ts(ψ) = 1

• Gain of Optical Concentrator : g(ψ) = n2

sin2(ψc)

• Width of FOV at the Receiver : ψc = 70

• Refractive Index of a lens at Photodiode : n = 1.5

Therefore, the graph for Received Power LOS Path in [mW] and [dBm] is

plotted as follows for the scenario of 4 Transmitters :

Figure 1.17: Received Power for Directed/LOS Path of VLC System in [mW]

50

Figure 1.18: Received Power for Directed/LOS Path of VLC System in [dBm]

For Diused / Reected Path :

The room dimensions are considered as follows :

Area of the oor and ceiling of the test room is calculated as follows :

Areafloor = Areaceiling = lx ? ly (1.65)

Area of the wall of the test room is calculated as follows :

Areawall = lx ? lz (1.66)

Therefore, the Total Area of the Room is calculated as follows :

Aroom = 2 ? (Areafloor) + 4 (Areawall) (1.67)

51

where, the room dimensions are :

• Length = lx [m]

• Width = ly [m]

• Height = lz [m]

The equation for Average Reectivity, ρ is calculated as follows :

ρ =1

Aroom

?[(Areafloor?0.15)+(Areaceiling?0.8)+(Areawall?0.7)]

(1.68)

where,

• Floor Reectivity = 0.15

• Ceiling Reectivity = 0.8

• Wall Reectivity = 0.7

First diused reection of a wide-beam optical source emits an intensity,

Iprime over whole room surface, Aroom given as follows :

Iprime = ρ ?Ptotal

Aroom

(1.69)

Hence, the Total Intensity, It is given as follows :

It =Iprime

(1− ρ)(1.70)

Therefore, the Received Power for Diused / Reected Path is calculated by

52

following equation :

Pdiff = A ? It ? Ts(ψ) ? g(ψ) (1.71)

The Total Received Power for VLC System in [mW] is calculated as follows :

Prx = Pr + Pdiff (1.72)

And the Total Received Power for VLC System in [dBm] is calculated as

follows :

Prx[dBm] = 10 ? log10(Prx) (1.73)

Therefore, the graph for Total Received Power of VLC System in [mW] and

[dBm] is plotted as follows for the scenario of 4 Transmitters :

53

Figure 1.19: Total Received Power for VLC System in [mW]

Figure 1.20: Total Received Power for VLC System in [dBm]

54

1.9.6 SNR Distribution of VLC System

Denition of Parameters is as follows :

• Charge of Electron, q = 1.6 ? e−19 [C]

• Photodiode Responsivity, ρpd = 0.4

• Ambient Light Photocurrent, Iamb = 5840 [µA]

• Noise Bandwidth Factor, I2 = 0.562

• Amplier Bandwidth, Ba = 50 [MHz]

• Data Rate, Rb = 1 [Mb/s]

• Amplier Noise Density, q = 5 ? e−12 [pA/√Hz]

The Noise Power of Ambient Light is calculated as follows :

Np =Iamb

ρpd(1.74)

The Noise Bandwidth is calculated as :

Bn = Rb ? I2 (1.75)

The Shot Noise Variance is calculated as follows :

σ2shot = 2 ? q ? ρpd ? (Prx +Np) ? Bn (1.76)

55

The Amplier Noise Variance is calculated as follows :

σ2amplifier = I2

amplifier ? Ba (1.77)

Hence, the Signal to Noise Ratio (SNR) for VLC System is calculated as

follows :

SNR =(ρpd ? Prx)

2

σ2total

(1.78)

and, SNR in [dB] is calculated as follows :

SNR[dB] = 10 ? log10(SNR) (1.79)

Therefore, the graph for SNR Distribution of VLC System in [dB] is plotted

as follows for the scenario of 4 Transmitters :

Figure 1.21: SNR Distribution for VLC System in [dB]

56

Chapter 2

Orbital Angular Momentum

Multiplexing

2.1 Electromagnetic Fields and Angular Mo-

mentum

An electromagnetic wave is characterized by it's amplitude, wave vector,

frequency, but also by its angular momentum. The angular momentum of

an electromagnetic wave is divided into two parts [21] :

• Spin Angular Momentum (SAM) associated with Polarization.

• Orbital Angular Momentum (OAM), associated with the Spatial Distribu-

tion of the electric eld around its one axis of propagation.

The following gure will help us to understand this division better :

57

Figure 2.1: Angular Momentum Subdivision

We can also dene Angular momentum in terms of photon :

Many researchers discovered that a light beam carries a linear momentum

equal to hk0 and if circularly polarized a Spin angular momentum (SAM)

equivalent of ±δh per photon [22].

In 1992, Al and Allen recognized that beams with an azimuthal dependence,

e(−ilψ) carry OAM (Orbital Angular Momentum) that can be many times

greater than SAM and these beams are realizable [22].

This OAM is completely distinct from the familiar SAM, most usually asso-

ciated with the photon spin that is manifest as in circular polarization. The

spin angular momentum (SAM) of light is connected to the polarization of

the electric eld.

The given gure below shows light with linear polarization (left) carries no

58

SAM, whereas right or left circularly polarized light (right) carries a SAM of

±h per photon.

Figure 2.2: Linear Polarization (Left) and Circular Polarization (Right)

Thus, relationship between linear momentum and angular momentum is giv-

en by [22] :

L = r ? p (2.1)

where, r is the particle's position from the origin which is distinguished as

follows [22] :

p = m v (2.2)

here, • m is the Mass,

• v is the Velocity,

• p is the Linear Momentum, and

• ? is the Cross Product.

Any angular momentum component in the z direction, by denition, requires

a component of linear momentum in the x, y plane, i.e., a light beam with

59

transverse momentum components [22].

The linear momentum of a transverse plane wave is then in the propagation

direction, z, and there cannot be any component of angular momentum in

the same direction.

Hence it follows that, at the most fundamental level, an angular momentum

in the z direction requires a component of the electric and/or magnetic eld

also in the z direction.

The origin of OAM is easier to understand; light beam carrying OAM possess

a phase of e−ilψ, where φ is the angular coordinate and l can be any integer

value, positive or negative.

As shown in gure below, such beams have helical phase fronts with the num-

ber of intertwined helices and the handedness depending on the magnitude

and the sign of l.

Figure 2.3: Helical Phase Front

60

2.2 Gaussian Beam

2.2.1 Paraxial Equation

To obtain the z components in ~E and ~B elds we can start with a vector

potential in the form,

A = u(x, y, z) e(−ikz), (2.3)

where, u(x, y, z) , is a function describing the Field Amplitude Distribu-

tion. This vector potential is valid for linearly polarized waves. Now, we

can consider monochromatic waves with frequency. The solutions are also

solution to the Helmholtz equation, so we resolve the Helmholtz equations

to obtain u as follows :

(∇2 + k2) u(x, y, z) e(−ikz) = 0, (2.4)

where, k = 2Π/Λ. Assume variation in x and y directions are larger than

the variation in z direction, the variations are small with z, so we can neglect

the d2/dz2 part;

|2kdu

dz| >> |

du2

dz2|, (2.5)

This is called Paraxial Wave Approximation, since it basically says that the

beam does not diverge much from the beam axis. So, the equation becomes,

61

∇2tu − 2ik

d

dzu = 0, (2.6)

where, ∇2t represents the transverse part of the Laplacian and this equation

is called the Paraxial Helmholtz equation.

A well known solution of Paraxial Helmholtz equation is a spherical wave is

given by,

u(r) =Ae(−ikr)

r, (2.7)

where, A is a constant and r is distance from the source. Now, we rewrite

this in terms of Cartesian coordinates,

u(x, y, z) =A e(−ikz

√1+x2+y2

z2)

z√

1 + x2+y2

z2

, (2.8)

Hence, the approximation would be,

u(x, y, z) ≈A e(−ikz) e(

−ik(x2+y2)2z

)

z(2.9)

We use this approximation and try to form our Gaussian beam with cylin-

drical ansatz,

u(ρ, ψ, z) = A e[−if(z)] e(−xkρ2)

2g(z) (2.10)

where, ρ2 = x2 +y2 and f(z) & g(z) are analytic function of z. Amplitude

62

function u must be a solution to Paraxial Helmholtz equation.

∇2t [e−if(z) e

(−xkρ2)2g(z) ] − 2ik

d

dz[e−if(z) e

(−xkρ2)2g(z) ] = 0 (2.11)

Now, further solving rst part of the above equation :

∇2t [e−if(z) e

(−xkρ2)2g(z) ] = e[−if(z)]

1

ρ

d

dρ[ρ

d

dρe

(−xkρ2)2g(z) ] (2.12)

∇2t [e−if(z) e

(−ikρ2)2g(z) ] = e[−if(z)] e

(−ikρ2)2g(z) [

−2ik

g(z)−k2ρ2

g2z] (2.13)

and then the ddz

term, i.e. second part of the equation is solved,

−2ikd

dz[e−if(z)e

(−xkρ2)2g(z) ] = −2ike−if(z)e

(−xkρ2)2g(z) [−i

d

dzf(z)+

ikρ2

2g2(z)

d

dzg(z)]

(2.14)

Now, we combine the rst and second parts of the solved equation i.e. [2.13]

and [2.14] then, substitute it into the previous one i.e. equation [2.11]. There-

fore, the equation will be as follows :

e−if(z)e(−ikρ2)2g(z) [(

−2ik

g(z)−k2ρ2

g2(z))−2ik(−i

d

dzf(z)+

ikρ2

2g2(z)

d

dzg(z))] = 0

(2.15)

Further solving the above equation, we get

63

2ik

g(z)+ 2k

d

dzf(z) +

K2ρ2

g2(z)(1

d

dzg(z)) = 0 (2.16)

First two terms in L.H. S of the equation are independent of ρ, which means

for any ρ we can separate equation into new ones,

2ik

g(z)+ 2 k

d

dzf(z) = 0 (2.17)

and,

K2ρ2

g2(z)(1

d

dzg(z)) = 0 (2.18)

Now, solving the two terms of L.H.S of the equation :

2ik

g(z)+ 2 k

d

dzf(z) = 0⇒

d

dzf(z) =

−ig(z)

(2.19)

and,

K2ρ2

g2(z)(1

d

dzg(z)) = 0⇒

d

dzg(z) = 1 (2.20)

Both the terms of L.H.S of the equation are integrated and we obtain,

f(z) = −i ln(z + g0) (2.21)

and,

64

g(z) = z + g0 (2.22)

where, g0 is a constant. Further we substitute these solutions in the Gaussian

Beam approximation equation [2.10],

u(ρ, ψ, z) = A1

z + g0

e( −ikρ22(2+g0)

)(2.23)

Before solving further, we would discuss certain important aspects of Gaus-

sian Beam Paraxial equation which are as follows :

• Compare characteristics of Gaussian beam with those of a spherical wave.

• Relation: f(z)→ Complex Phase Shift, and g(z)→ Intensity Variation.

• Width, w(z) is a measure of how electric eld amplitude falls o when

moving out from the beam axis. This fall o is Gaussian andw is the distance

from the axis where the amplitude is 1eof the maximum value. Hence, Width

of a Beam⇒ 1etimes of the Electric Field.

• Intensity, I is proportional to E2. Hence, I ∝ E2.

• Width of the Beam contains, I(1 − 1E2 ), or 86% of the total intensity.

• At z = 0, Beam Width is minimum and, w0 will be Beam Waist.

Now, we assume and solve further,

1

g(z)=

1

Rz

−i c

h(w(z))(2.24)

where, c is a Constant. At z = 0, wave front is plane, so radius of curvature

is∞. Hence,

65

1

R(0)= 0, (2.25)

Therefore, at z = 0, 1g(z)

is purely imaginary. Hence,

1

g(z)=

1

g(0), (2.26)

So, at z = 0, we have the following equation :

u(ρ, ψ, z) = A e( ikρ2

zic

h(w(z))) (2.27)

Insert ρ = w(z) and, Amplitude will 1eof the maximum value. Hence,

taking R.H.S of the above equation,

=⇒ e[ ikρ2

zic

h(w(z))] = e(−1), (2.28)

and,

=⇒ h(w(z)) = w2(z), (2.29)

and,

c =2

k=

λ

π(2.30)

Hence,

66

g(z) = z +i πw2

0

λ(2.31)

and,

1

g(z)=

1

R(z)−

i λ

π w2(z)(2.32)

Therefore, now we can calculate and obtain the expression for w and R,

=⇒ w(z) = w0

√1 + (

λ z

π w20

)2 (2.33)

and,

=⇒ R(z) = z [1 + (π w2

0

λ z)2] (2.34)

Now, we replace z =π w2

0

λ, in the above equation which consists of w(z).

We get,

w(z) =√

2 w0, (2.35)

It means at this z, the cross section of the beam is twice as large as the cross

section at z = 0. This is known as the Rayleigh Range z, or zR. Rayleigh

Range gives information of the spread of the beam. A short zR means a

beam diverges rapidly.

Now, insert zR in the previous equations of w(z) and R(z) and we obtain

the following results,

67

w(z) = w0

√1 + (

z

zR)2, (2.36)

and,

R(z) = z [1 + (zR

z)2], (2.37)

Now, we replace w and R in the previous equation [2.21] of f(z) and we

can rewrite it as follows,

f(z) = −i ln(z + g0), (2.38)

⇒ f(z) = −i ln(z + izR), (2.39)

⇒ f(z) = −i ln(√z2 + z2

R e[i arctan(

zRz

)]), (2.40)

⇒ f(z) = −i ln(√z2 + z2

R) + arctan(zR

z), (2.41)

Now, further solving the above equation, we obtain following results,

⇒ e[−if(z)] =e[−i arctan(

zRz

)]√z2 + z2

R, (2.42)

⇒ e[−if(z)] =w0

zRw(z)e[−i arctan(

zRz

)], (2.43)

Hence, the previous equation of u(ρ, ψ, z) which is as follows:

u(ρ, ψ, z) = A1

z + g0

e( i k ρ2

2 (2 + g0)), (2.44)

68

The above equation [2.44] of u(ρ, ψ, z) can also be written as follows :

uG(ρ, ψ, z) = Aw0

zR w(z)e[−i arctan(

zRz

)] e

[ −i k ρ2

2z (1 +z2Rz2)

]e[−ρ

2

wz] (2.45)

Hence, we have derived the Paraxial equation for the complex amplitude of

the Gaussian beam. In the next section, we will move forward to derive and

obtain the ansatz expression for MATLAB simulation of the Gaussian beam

intensity.

where,

• w0 is Beam Waist at z = 0.

• Rayleigh Range, zR = π w0

λ

• Beam Width at z : w(z) = w0

√1 + z2

z2R

• Radius of Curvature : R(z) = z (1 +z2Rz2

)

The following gure represents the intensity prole of Gaussian Beam using

the nal expression in the equation [2.45],

Figure 2.4: (a) Phase front, (b) Cross Intensity Distribution, (c) Intensityprole of Gaussian Beam

69

2.2.2 MATLAB Simulation of Gaussian Beam

In order to understand better the form of Gaussian beam and the associat-

ed physical phenomena, we will derive the mathematical expression of the

Gaussian beam in MATLAB and model as well as plot the intensity prole

of Gaussian beam.

To plot the intensity prole of Gaussian beam, a square modulus of the

complex amplitude, u(ρ, ψ, z) is taken. Hence, the intensity of Gaussian

Beam is as follows :

Intensity Profile, I = |Complex Amplitude|2, (2.46)

=⇒ I = |u(ρ, ψ, z)|2, (2.47)

=⇒ I = (Aw0

zR w(z))2 e

[− 2 ρ2

w2(z)], (2.48)

where, frequency, f = 60 [GHz], corresponding to wavelength, λ = 5 [mm].

A is a constant and for MATLAB simulation, we x the value w0 = 50 λ.

w0 represent the BeamWaist i.e. the size of focal spot of the Gaussian Beam.

The following MATLAB simulation graphs will show the intensity prole of

a Gaussian beam with dierent representations, i.e. a normal 3-D plot of

Gaussian beam intensity prole and a transversal view of the same plot to

understand the concentration of the beam intensity at center.

The red colored regions will show the maximum intensity, which is mostly

concentrated at the central region and the blue colored regions will show

70

minimum intensity.

Figure 2.5: Intensity Prole of Gaussian Beam in 3-D Plot

Figure 2.6: Transversal View of Gaussian Beam Intensity Prole

71

2.3 Laguerre Gaussian Beam

2.3.1 Expression of Laguerre Gaussian Beam

Laguerre Gaussian (LG) beam is one of the forms of electromagnetic beam

which carries Orbital Angular Momentum Multiplexing (OAM).

According to Simpson & Al, in order to endow our electromagnetic beam

with orbital angular momentum, we need to search for a solution of the

Helmholtz equation with an azimuthal phase e(− L ψ) dependence [23].

We will use the Gaussian beam expression, uG to determine the expression

for Laguerre Gaussian beam, uLG. Now, the LG beam expression carrying

OAM is derived as follows :

u(ρ, ψ, z) = c h (ρ

w) e(−iLψ) e[−iφ(z)] uG(ρ, ψ, z), (2.49)

u (ρ, ψ, z) = F (ρ, ψ, z) uG (ρ, ψ, z), (2.50)

Substitute this equation into the Helmholtz equation, we obtain

F ∇2t uG + 2[∇tF ×∇tuG] + uG∇2

tF − 2ikuG∂F

∂z−2ikF

∂uG

∂z= 0,

(2.51)

where,

72

∇2t uG − 2 i k

∂uG

∂z= 0, (2.52)

Therefore, we obtain the following equation

⇒ 2 [∇t F × ∇t uG] + uG ∇2t F − 2 i k uG

∂F

∂z= 0, (2.53)

Now,

2[ikρ

2(1 +z2Rz2

)+

2ρ

w2(z)]∂h

∂ρ+(∂h2

∂ρ2+

1

ρ

∂h

∂ρ−l2

ρ2)−2ik(ih

∂φ

∂z+∂h

∂z) = 0,

(2.54)

Now, we replace the Laplacian ∇t, by this expression in spherical coordi-

nates, uG by the expression of Gaussian Beam [2.45] and also F by the

equivalent in the equation [2.50] and the one found in [24],

Therefore,

h (ρ

w) = [

√2 (ρ

w)]l × LlP (2

ρ2

w2) (2.55)

where,

LlP is Generalized Laguerre Polynomial which is dened as regular solutions

to dierential equation. It is dened by following solution,

73

xd2

dx2LlP + (l + 1− x)

d

dxLlP + p LlP = 0, (2.56)

where, 1 and p are integers greater than −1. And, the Gouy Phase is given

as,

φ(z) = (2p + l) arctan (zR

z) (2.57)

Now, we can rewrite the nal expression of Laguerre Gaussian Beam in the

following solution. Hence the equation is,

uLGpl (ρ, ψ, z) = Cplw0

zRw(z)[√

2( ρw(z)

)]|l| × L|l|P (2 ρ2

w2(z))e(−ilψ) ×

e[−i(2p+ |l|+ 1) arctan(zRz

)]e

[ −ikρ2

2z(1+z2Rz2

]

e[ −ρ2

W2(z)]

where,

Cpl = A

√p!

(p+ |l|)!; (2.58)

And, A is a Constant.

Also, there is a special case that, when the value of p and l is equal to zero,

our LG Beam equation becomes Gaussian Beam equation, i.e.

uLG00 = uG, (2.59)

Here,

• z : Distance from the Transmitting Antenna

74

• ψ : Azimuthal Phase

• l : Number of twists in Helical Wavefront

• k : Wavenumber

• p : Number of Radial Nodes in mode

• w(z) : Beam Width

• w0 = w(0), i.e. Beam Width at z = 0

• zR : Rayleigh Range

• R(z) : Radius of Curvature

• Cpl : Normalization Constant

According to [25], beams of millimeter-wave radiation carrying OAM can be

described in terms of LG (Laguerre Gaussian) modes that contain an e(−ilφ)

term describing an on-axis phase singularity. The phase term e(−ilφ) creates

the helical wave front.

The LG modes form a complete innite-dimensional basis for the solutions

of the paraxial wave equation; thus any eld distribution can be represented

as a vector state in that basis.

The topological charge l represents the number of twists in the helical wave-

front; when l 6= 0 , the wave beam of LG modes has l intertwined helical

phase front. The sign of the topological charge is important since it will

result in a right hand spiral for negative l and left hand spiral for positive.

As shown in the following gure given below, their intensity cross sections

consist of a radially symmetric shape with zero value at the center of the

gure.

75

Figure 2.7: (a) Phase Front, (b) Cross Intensity Distribution, (c) IntensityProle of Beam with OAM for l = 1.

Beams with OAM are characterized by an azimuthal phase dependence

e(−ilφ) where l is an integer dierent to zero corresponding to the topo-

logical charge and φ is the azimuthal angle. The Poynting vector, which

indicates the direction of energy ows, has an azimuthal component produc-

ing an orbital angular momentum parallel to the beam axis as shown in the

above gure [2.7]. This family of beam possesses a phase singularity or an

optical vortex i.e. one zero at the center.

Now, in the following gure, we will discuss and depict the Laguerre Gaussian

beam expression for orbital angular momentum beam cross sections with

dierent values of l and p.

76

Figure 2.8: Cross Section of LG Beam Intensities for dierent values of l&p.

In the top row, p = 0, in the middle row, p = 1 and in the bottom row,

p = 2. in each row l changes from zero to three when going from left to

right. To understand better, we can observe that the intensity prole varies

dependeing on the values of l and p. the value of l = 0 gives us the Gaussian

Beam intensity prole but when l 6= 0, the intensity of the beam gives us

rings instead of circular spots. The increase of p brings more rings i.e. the

number of intensity peaks are p+ 1.

77

2.3.2 MATLAB Simulation of LG / OAM Beam

In this section, we will discuss the MATLAB simulation of Laguerre Gaussian

Beam using its complex amplitude.

Depending on the values of p and l the intensity prole of OAM beam will

vary. As discussed earlier, by setting l = 0 we will obtain a Gaussian Beam

intensity prole, but by changing the l values to a higher or lower range, the

beam will exhibit intensity rings of instead of a circular spot as observed in

Gaussian Beam. If p is given values other than zero, then it will give multiple

rings (the number of intensity peaks are p+ 1).

In order to know the behavior of intensity peak and the phase singularity,

we need to plot the beam intensity of the complex amplitude of Laguerre

Gaussian Beam which will exhibit OAM.

Now,

I = |ULG(ρ, ψ, z)|2 (2.60)

The Laguerre Polynomial is dened by:

Lαn(x) =x−αex

n!

dn

dxn(e−xxn+α) (2.61)

Then,

Lα0 (x) = 1, (2.62)

78

Lα1 (x) = −x+ α+ 1, (2.63)

Lα2 (x) =x2

2(α+ 2)x −

(α+ 2)(α+ 3)

2, (2.64)

and,

Lα3 (x) =−x3

6+

(α+ 3)x2

2−

(α+ 2)(α+ 3)x

2+

(α+ 1)(α+ 2)(α+ 3)

6(2.65)

Now, to be consistent with the notation in the formula of Laguerre Gaussian

equation, uLG, p = n and α = l. Using this we can compute the intensity

with dierent values of l and p.

Ip,l = |uLG(ρ, ψ, z)|2 (2.66)

Ip,l = |LGp,l|2 (2.67)

Therefore, for MATLAB simulation of OAM Beam and to calculate the com-

plex amplitude for intensity prole, the Generalized Laguerre Gaussian ex-

pression is considered,

uLGpl (ρ, ψ, z) = Cplw0

zRw(z)[√

2( ρw(z)

)]|l| × L|l|P (2 ρ2

w2(z))e(−ilψ) ×

e[−i(2p+ |l|+ 1) arctan(zRz

)]e

[ −ikρ2

2z(1+z2Rz2

]

e[ −ρ2

W2(z)]

In MATLAB simulation test, we neglect the phase terms of the generalized

LG Beam equation, which are as follows:

79

e[−i(2p + |l|+ 1) arctan(zR

z)] e[−ikρ

2

2R(z)], (2.68)

Also, we negate one part of the amplitude from the generalized LG Beam

equation in order to simulate, the neglected part is as follows:

Cplw0

zRw(z)[

√2

w(z)]|l| L|l|p [

2ρ2

w2(z)], (2.69)

Therefore, the approximate LG expression carrying orbital angular momen-

tum will be,

ul = ρl × e[ −ρ2

w2(z)]e[−ilψ] (2.70)

Hence, the above equation will be used to simulate LG Beam carrying or-

bital angular momentum in MATLAB software tool. We will be studying at

least ve simulation test results in order to draw certain conclusions for the

dierent values of l in a LG Beam which is responsible for orbital angular

momentum.

80

Figure 2.9: Intensity Prole of OAM Beam in 3-D Plot, l = 1

Figure 2.10: Transversal View of OAM Beam Intensity Prole, l = 1

81



82



83



84



85

Chapter 3

Dierent Techniques for OAM

Beam Generation

3.1 Holographic Plate

In order to plot a hologram that generates a helical phase front starting from

a planar reference we must consider the interference pattern of these two

beams in a given plane, for example in the (x,y) plan, let us consider a plane

reference beam [26] :

Er = E0 ei (kxx+ kzz) (3.1)

Where, |E0|2 is the intensity of the OAM beam and k is the wave vector.

At z = 0, I, is the interference pattern with a helical beam :

86

EI = E0 eilψ (3.2)

The above equation is given by :

I = |Er + El|2 = 2 E20 (1 + cos(kxx − lψ)) (3.3)

This equation can allow plotting the intensity pattern of interference as shown

in the below gure :

Figure 3.1: Intensity pattern of interference between helical with a planarbeam for several dierent value of the charges singularities

Photographic plate recordings of these patterns are now, as holograms, capa-

ble of reconstructing the original of a helical beam when they are illuminated

by a beam with a plane wavefront.

After traveling along the hologram of the input wave, the eld is given by:

Et = A0 T (x, y) =|A0|2

2(1 + cos(kxx − lψ)) (3.4)

87

where, |A0|2, is the intensity of illuminating beam and T (x, y) is the spa-

tial transmittance of the hologram. We can rewrite Et with the following

expression after calculation :

Et =A0

2[1 +

1

2e(kxx−lψ) +

1

2e(−kxx+lψ)] (3.5)

This expression is composed of three terms: The rst corresponds of the zero

order beams propagating along the axes, the two others terms, the rst order

diracted beam, are conjugate and propagate on either to others of the axe.

Figure 3.2: Beams generated by a Hologram Plate with L = 2

In the following gure, we are mentioning the dierent types of grating to

generate OAM Beams with dierent l values.

88

Figure 3.3: Dierent grating shape: (a): Sinusoidal, (b): Blazed, (c): Binary,(d): Triangle

3.1.1 Mathematical Modeling of Holographic Plate

The phase-only interference of an optical vortex e[i(kz−lψ+φ0)] with that

of an oblique plane reference wave e[i(kxx+kzz)] results in an interference

pattern that gives the grating structure needed to produce an optical or

radio vortex beam.

A holographic interference pattern is the product of the phase-only interfer-

ence of an indirect plane reference wave which generates a radio vortex beam

using the following formula :

I(x, y) =1

2[1 + cos(l arctan(

y

x) − 2π Kx + φ0)] (3.6)

Here, l = 0,±1,±2,±3, ...n, n, is the topological charge of the OV(Optical

Vortex) or RV(Radio Vortex) to be produced.

K = Kx2π

, is the number of grating lines per unit length and, φ0, is an

89

arbitrary phase factor. The previous relationships are commonly used in the

production of gratings.

For instance, for a binary grating we take the value of 1 when the interference

term is positive and the value zero elsewhere [27].

T (x, y) = 0, cos(φ) ≤ 0, (3.7)

and,

T (x, y) = 1, cos(φ) > 0, (3.8)

where,

φ = l arctan(y

x) − 2πkx + φ0 (3.9)

Hence, the gure for generated holographic plate will be as follows :

Figure 3.4: Holographic Plate for l = 1

90

A hologram is an interference pattern which is characterized by l dislocations.

Here, we have just one dislocation corresponding to l = 1.

The fork-grating dislocation in a Holographic Plate (HP) is the main feature

of hologram masks, which creates an OAM state from the plane input wave.

The results of passing a plane input wave in the center of the dislocation

is a diraction limited beam which creates OAM states in the far eld re-

gion. When this plane input wave is a Gaussian beam, this latter becomes a

Laguerre Gaussian beam which carries OAM information.

3.2 Spiral Phase Plate

The spiral phase plate (SPP) is a transparent dielectric plate with one plane

and one spiral surface. The thickness of the SPP increases proportional to

the azimuthal angle,φ, around a point at the center of plate [26].

The spiral surface forms a period of helix. Such a thin transparent plate

typically has strips or radial sectors that can be obtained by coating or

etching a substrate.

The following gure shows a schematic of a SPP illuminated by a TEM00

beam and its outgoing wave.

91

Figure 3.5: Schematic of SPP illuminated by TEM00 beam & output OAM

When a light beam of wavelength λ passes through the SPP, the helical

surface can be expected to give a helical structure to the beam wavefront.

In fact, the SPP introduces in the outgoing beam a phase shift, δ, which

depends on the azimuthal angle, φ by :

δ =(n1 − n2) h

λφ, (3.10)

where, n2 & n2 are the refractive indices of the SPP and surrounding medi-

um, respectively, and h is the physical step height at ψ = 0.

For generating a beam with a value of OAM, for example lh, the total phase

delay around the SPP must be an integer multiple of 2π, i.e. 2πl.

Thus, to produce this beam, the physical height of the step in the SPP is

given by :

92

h =l λ

(n1 − n2), (3.11)

where, ∆n = n1 − n2, Hence, the nal equation will be :

=⇒ l =∆ n h

λ(3.12)

The Laguerre Gaussian mode beam seen in previous section can be created

using spiral phase plates.

In [28], authors show that the phase fronts of the created LG beams are

helical in a sense that the phase front varies linearly with azimuthal angle,

as illustrated in the following gure :

Figure 3.6: Creation of Laguerre Gaussian Beam using SPP

The above gure illustrates the creation of Laguerre-Gaussian laser beams

93

from planar laser beams by using transparent spiral phase plates introducing

a linear phase delay with azimuthal angle.

OAM state l implies a 2πl phase delay over one revolution. Three dierent

phase plates are illustrated in gray, for OAM state 0, 1 and 2.

94

Chapter 4

Interconnection of OAM & VLC

System

4.1 Generation of OAM Beam

In this section, we will discuss about the main equations and it's parameters

which are chosen to derive and simulate a Laguerre Gaussian beam which

in turn carries orbital angular momentum multiplexing. Therefore, we will

mainly discuss those aspects where exactly we are substituting and modeling

the OAM multiplexing scheme in a VLC system.

The nal derived equation for MATLAB simulation of OAM beam will be

given as follows i.e. Complex Amplitude of LG beam carrying OAM :

uLG = ρ|L| e[ −ρ2

w2(z)]e[−iLψ] (4.1)

95

Now, the Beam Width of OAM beam at z is stated as follows :

w(z) = w0

√1 + (

λz

πw20

)2 (4.2)

The Rayleigh distance, z is stated as follows :

z =π w2

0

λ(4.3)

Distance from the source in terms of Cartesian Coordinates is stated as fol-

lows :

ρ2 = x2 + y2 (4.4)

The Azimuthal Phase, ψ on which the helical twist of OAM beam depends

is stated as follows :

ψ = tan−1(y

x) (4.5)

Now, the parameter values set for simulations are as follows :

• Frequency, f = 60 [GHz]

• Transmission Speed of Light, c = 3 e8 [m/s]

• Beam Width, w0 = 50λ (At z = 0), also known as Beam Waist

• Number of twists in the helical wavefront : L = N (N is an Integer)

The Wavelength, λ is specied by the equation :

λ =c

f(4.6)

96

Wavenumber, k is given by the equation as follows :

k =2π

λ(4.7)

Hence, the nal equation for the Intensity of LG Beam carrying OAM will

be as follows :

ILG = |uLG|2 (4.8)

Therefore, intensity plot for a LG Beam carrying will be generated as follows

:

Figure 4.1: Generation of OAM Beam for, L = 1

97

Table 4.1: Parameters for OAM over VLC System MATLAB Simulation

Parameters Dimensions

Room Size 5 * 5 * H [m]

Height of Desk 0.5 [m]

Transmitted Optical Power Pt

Semi-Angle Half Power 30

Center Luminous Intensity ILG

Number of LEDs in a group 3600 (60 * 60)

Number of LED Lights 4

LED Interval 0.01 [m]

Size of LED 0.59 * 0.59 [cm]

FOV at Receiver 70

Detector Physical Area of Photodiode 1 [cm2]

Optical Filter Gain 1

Refractive Index of Lens at Photodiode 1.5

Photodiode Responsivity 0.8

Floor Reectivity 0.15

Ceiling Reectivity 0.8

Wall Reectivity 0.7

Amplier Noise Density 5 [pA/√Hz]

Ambient Light Photocurrent 5840 [µA]

Noise Bandwidth Factor 0.562

98

4.2 Luminous Intensity & Horizontal Illumi-

nance

In a VLC system, Luminous Intensity at Angle of Irradiance, φ is given by :

I(φ) = I(0) cosm(φ) (4.9)

where,

• Order of the Lambertian Emission, m is given a follows :

m =− ln 2

ln cos(θ)(4.10)

where, θ is the Semi-Angle at Half Illuminance and, θ = 30

The Center Luminous Intensity, I(0) of VLC is replaced by LG beam inten-

sity i.e., ILG as it carries Orbital Angular Momentum Multiplexing.

Hence, the Luminous Intensity equation for OAM over VLC System will be

stated as follows :

I(φ) = ILG cosm(φ) (4.11)

Luminous Intensity and The Cross-Sectional view of Luminous Intensity for

OAM over VLC System is plotted as follows :

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Figure 4.2: Luminous Intensity of OAM over VLC System [lx] for, L = 1

Figure 4.3: Cross-Sectional View of Luminous Intensity for, L = 1

100

Therefore, the Horizontal Illuminance, Ehor at a point (x, y) is given as

follows :

Ehor =ILG cosm(φ)

d2 cos(α)(4.12)

and, the Total Horizontal Illuminance, Ehort1 is given as follows :

Ehort1 = Ehor ? nLED ? nLED (4.13)

Figure 4.4: Total Horizontal Illuminance of OAM over VLC System for,L = 1

The transversal view of Ehort1 will be plotted as follows :

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Figure 4.5: Transversal View of Total Horizontal Illuminance for, L = 1

4.3 Received Power Distribution

4.3.1 Transmitted Optical Power by OAM Beam

All the power equations for OAM over VLC system will be same as the case

of VLC system. The only equation which is dierent is the Transmitted

Optical Power equation, i.e. Pt.

The total power transmitted, Pt by LG Beam carrying OAM will be given

as follows :

Pt =1

2? π ? ILG ? w2

0 (4.14)

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4.3.2 Received Power Distribution for LOS Path

Figure 4.6: Received Power (LOS/Directed) [mW] for, L = 1

Figure 4.7: Received Power (LOS/Directed) [dBm] for, L = 1

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4.3.3 Received Power Distribution for Diused Path

Figure 4.8: Total Received Power [mW] for, L = 1

Figure 4.9: Total Received Power [dBm] for, L = 1

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4.3.4 SNR & Noise Distribution

Figure 4.10: SNR Distribution of OAM over VLC System [dB] for, L = 1

Figure 4.11: Noise Distribution [dB] for, L = 1

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Noise Distribution for VLC System

Figure 4.12: Noise Distribution of VLC System [dB] for, L = 1

4.4 Comparison of Various Parameters

4.4.1 Impact of L & Beam Width

The number of helical twists in the wavefront of OAM Beam, L and the

Beam width, w0 share an inverse relationship.

When the positive integer value of L is increased the maximum total received

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power, Prx reduces. This can be be depicted by the equation as follows :

Prx ∝1

L(4.15)

This can be further discussed by following plots and measurements of simu-

lation results :

Figure 4.13: Table: Simulation Results, Prx v/s L & w0

Now, the Beam Width, w0 is directly proportional to received power, Prx

and is depicted by the equation as follows :

Prx ∝ w0 (4.16)

We can check the degree of the impact by plotting a graph of various simu-

107

lation result measurements for w0 = 50λ, 100λ, 150λ, 200λ :

Figure 4.14: Simulation Result Comparison, Prx v/s L & w0

Figure 4.15: Transversal View of Beam Width, Prx v/s L & w0

108

The impact of Beam Width, w0 & number of helical twists in wavefront of

OAM Beam, L will be same on the Intensity, I(φ) as it was on Received

Power, Prx.

I(φ) ∝ w0 ∝1

L(4.17)

It can be depicted by the following plot of Intensity of various OAM Beam

with Beam Width, w0 = 50λ, 100λ, 150λ, 200λ :

Figure 4.16: Impact of Beam Width on Intensity, Iφ v/s w0

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4.4.2 Impact of Distance, H

As the Distance, H is increased the Maximum Total Received Power, Prx

will decrease. This can be depicted by the following equation :

Prx ∝1

H(4.18)

But, if the Beam Width, w0 is increased the impact of increasing distance,

H will reduce on the Received Power, Prx. We can understand it further by

observing simulation results as follows for distances, H = 1, 2, 3 [m] :

Figure 4.17: Impact of Distance [m], Prx v/s H & w0

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4.5 Reasons for Developing OAM over VLC

System

The main focus of this research is driven by the present challenges and de-

mands faced by the telecommunications industry. They are mainly excessive-

ly saturating radio frequency (RF) spectrum which is leading to exorbitant

costs of purchasing licensed spectrum and increasing usage of heavy web

browsing applications which is adding up to the demand for very high data

rates and link capacity for various applications. Also, it is expected that

by 2020 the number of telecommunications service subscribers are bound to

multiply by several folds [29][30].

The two technologies which are being considered for this research have several

advantages which can help to overcome these challenges and serve the growing

demands.

If we consider the technology of Orbital Angular Momentum Multiplexing

(OAM), one of the main advantages is that it will help to increase the data

rate enormously as the number of bits the OAM of a single photon can

represent is in principle, unlimited.

OAM also results in mathematical encryption of data which prevents da-

ta from being compromised by atmospheric scattering, hence providing a

secured free-space communication.

The advantages of Visible Light Communication (VLC) would be that it has

111

a very large available bandwidth which is greater than 320 [THz], low power

consumption, long life of Light Emitting Diodes (LEDs) which are used as

transmitters, and no license required for visible light spectrum.

However, there are several obstacles in establishing these technologies as VLC

system uses visible light it will encounter problems related to ambient light

noise, high dependency on link distance, and very low received power levels

in OAM beams.

112

Chapter 5

Conclusion

Received Power is very less for OAM over VLC system as compared to a

normal VLC system. On analysis and comparison of both technologies, VLC

& OAM over VLC, it is observed that the Max. Total Received Power, Prx

is very less in OAM over VLC System as compared to VLC System.

OAM over VLC Technology can be used for applications having a very short

range of distance. As the H ( Height Distance between Tansmitter & Re-

ceiver) is increased, the Max. Total Received Power, Prx is decreased, irre-

spective of OAM State (L Value) and Beam Width, w0.

OAM over VLC Tecchnology can increase the speed / bit-rate of the ap-

plication. Since this technology uses OAM technique, the number of bits

the OAM of a single photon can represent is hence, in principle, unlimited.

Therefore, more the number of OAM states (L Values) are carried by EM

beam, more number of bits are transmitted.

113

OAM over VLC Technology has a very large available bandwidth. Since, this

system also implements VLC technology, which has a very large / unlimited

unlicensed bandwidth, greater than 320 [THz].

Therefore, OAM over VLC System will also have a very large bandwidth

available for the applications.

Therefore, in order to enhance the performance of OAM over VLC System,

we need to investigate more about the areas mentioned as follows :

• Design improved lters, ampliers, etc. in order to receive more power over

greater distance.

• Attenuation at the Holographic / Spiral Phase Plate has not been consid-

ered in the simulation environment.

• Practical conditions need to be researched and developed for implementa-

tion of the OAM over VLC System.

114

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