SimulationofSingleFrequency MillimeterWaveGenerationby...

Simulation of Single FrequencyMillimeter Wave Generation byFrequency Multiplication using

OptiSystem

Arnav Mukhopadhyaygudduarnav.com

January, 2019

Contents

1 Theoritical Fundamentals 51.1 Bessel Function and its Properties . . . . . . . . . . . . . . . . 51.2 Jacobi-Anger Expansion . . . . . . . . . . . . . . . . . . . . . 61.3 Mach-Zehnder Modulator (MZM) . . . . . . . . . . . . . . . . 6

1.3.1 Biasing of MZM . . . . . . . . . . . . . . . . . . . . . 91.4 Jones Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Jones vector representation of LASER . . . . . . . . . . . . . 121.6 PC (Rotator) . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.7 Linear Polarizer (Fixed) . . . . . . . . . . . . . . . . . . . . . 131.8 Pol (Rotatable Linear Polarizer) . . . . . . . . . . . . . . . . . 131.9 Polarization Beam Splitter (PBS) . . . . . . . . . . . . . . . . 141.10 Polarization Beam Combiner (PBC) . . . . . . . . . . . . . . 151.11 Polarization Modulator (PolM) . . . . . . . . . . . . . . . . . 161.12 Photodiode (PD) . . . . . . . . . . . . . . . . . . . . . . . . . 171.13 Sideband Suppression Ratio (SSR) . . . . . . . . . . . . . . . 171.14 (Required) Analysis and Results . . . . . . . . . . . . . . . . . 18

2 Microwave FMF-1 202.1 Remote Delivery of RF using Optical Fiber and Quadrature

biased MZM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1

http://gudduarnav.com/

OptiSystem Single Frequency Millimeter Wave Generation

3 Microwave FMF-6 213.1 [2017] Frequency Sextupling using Dual-Polarization Modu-

lator (Filterless Design) . . . . . . . . . . . . . . . . . . . . . 213.1.1 Optical Spectrum . . . . . . . . . . . . . . . . . . . . . 213.1.2 Optical Sideband Suppression Ratio (OSSR) . . . . . . 273.1.3 Output of Photodiode (Electrical RF Spectrum) . . . . 273.1.4 RF Spurious Suppression Ratio (RFSSR) . . . . . . . 283.1.5 Simulation in OptiSystem . . . . . . . . . . . . . . . . 29

4 Frequency Octupling 364.1 [2010] RF Frequency Octupling using Cascaded MZM, Optical

Phase Shifter and FBG Notch Filter . . . . . . . . . . . . . . 36

References 37

Arnav Mukhopadhyay (gudduarnav.com)



AcronymsCCW Counter-Clockwise Rotation

CW Clockwise Rotation

DC Direct Current (Bias Current)

DPPolM Dual-Parallel PolM

FMF Frequency Multiplication Factor

H-POL Horizontal Polarizer

IL Insertion Loss

LD (Continuous Wave) Laser Diode / Laser Source

MATP MAximum Transfer Point

MITP MInimum Transfer Point

MZM Mach-Zehnder Modulator

OptiSystem OptiWave OptiSystem Optical Simulation Software

OSSR Optical Sidebands Suppression Ratio

p-polarized Parallel (Horizontal) Polarized

PBC Polarization Beam Combiner

PBS Polarization Beam Splitter

PC Polarization Controller

PC(Rot) Polarization Controller (Rotator)

PM Phase Modulator

Pol Linear Polarizer (Rotatable)

PolM Polarization Modulator

QTP Quadrature Tranfer Point

RF Radio Frequency




RFSSR RF Spurious Suppression Ratio

s-polarized Perpendicular (German: Senkrecht) Polarized

SSR Sideband Suppression Ratio

TE Transverse Electric

TM Transverse Magnetic

V-POL Vertical Polarizer




1 Theoritical Fundamentals

1.1 Bessel Function and its PropertiesBessel Function of first kind Jn(z) is one of the solution obtained for solv-ing the differential equation that describes various physical processes. Theimportant equations whose solution can be interpreted using Bessel Func-tions are: Laplace, d’ Alembert (Wave), Poisson, Helmholtz, Heat (Diffu-sion) Equations. Some relevant properties of Bessel function, that will beused throughout the article are listed hereunder [1]:

J0(0) = 1 (1)J0(z → 0) → 1 (2)J1(z → 0) → z

2(3)

Jn(z → 0) → 1

n!(z

2)n (4)

J−n(z) = (−1)nJn(z) (5)Jn(−z) = ejnπJn(z) (6)

Jn(z) =∞∑k=0

(−1)k

k! (n+ k)!(z

2)(n+2k) (7)

Jn(x+ y) =∞∑

m=−∞

Jm(x)Jn−m(y) (8)

Another important property of Bessel Function is the Location of Zeroesor Root of Bessel Function. In Mathematica, root of equation Jn(z) = 0(find z) is obtained using the function BesselJZero[order, which zero] orBesselJZero[order, which zero, root must be above value]. The followingTable-1.1 the zeros of Bessel Function [1]:

Table 1: Zeros of Bessel Function [1]Zero J0(z) J1(z) J2(z) J3(z) J4(z) J5(z)

1 3.8317 1.8412 3.0542 4.2012 5.3175 6.41562 7.0156 5.3314 6.7061 8.0152 9.2824 10.51993 10.1735 8.5363 9.9695 11.3459 12.6819 13.98724 13.3237 11.7060 13.1704 14.5858 15.9641 17.31285 16.4706 14.8636 16.3475 17.7887 19.1960 20.5755




1.2 Jacobi-Anger ExpansionJacobi-Anger Expansion or Jacobi-Anger Identity is the expansion ofExponential of Trigonometric Functions, which finds applications inthe mathematical treatment of Angle Modulation (like Phase Modulation,Frequency Modulation) systems. The general identity is written as [1]:

(9)ejz cos θ =

∞∑n=−∞

jnJn(z)ejnθ

= J0(z) + 2∞∑n=1

jnJn(z) cos(nθ)

Similarly, the Jacobi-Anger expansion for −z cos θ, following equation-9,takes the form (using Bessel function property in equation-6):

(10)e−jz cos θ =

∞∑n=−∞

jnJn(−z)ejnθ

=∞∑

n=−∞

jnJn(z)ejn(θ+π)

Some other simplified form of Jacobi-Anger expansion are:

ejz sin θ =∞∑

n=−∞

Jn(z)ejnθ (11)

cos[z cos(θ)] = J0(z) + 2∞∑n=1

(−1)nJ2n(z) cos(2nθ) (12)

sin[z cos(θ)] = −2∞∑n=1

(−1)nJ(2n−1)(z) cos[(2n− 1)θ] (13)

cos[z sin(θ)] = J0(z) + 2∞∑n=1

J2n(z) cos(2nθ) (14)

sin[z sin(θ)] = 2∞∑n=1

J(2n−1)(z) sin[(2n− 1)θ] (15)

1.3 Mach-Zehnder Modulator (MZM)Mach-Zehnder Modulator (MZM), illustrated in Figure-1 is an Optical Mod-ulator, which takes advantage of the Interference of Light to achieve Optical




Figure 1: Illustration of a Mach-Zehnder Modulator (MZM)

Modulation [2]. The light from the source is applied to the Optical Input ofMZM, which is then split along two path using an Optical Y-Splitter. Thenthe light along each path is optically modulated using a Phase Modulator(PM1 and PM2), which works according to the principle of Electro-OpticModulation. An Electrical Signal consisting of a Direct Current (Bias Cur-rent) (DC) and Radio Frequency (RF) signal is applied to each arm of MZM,through Electrical Waveguides encompassing the Phase Modulator (PM) (oneach arm). The whole waveguide along with the PM (constituting the Inter-ferometric Structure) is constructed (fabricated) using Electro-Optic Mate-rial (like Lithium Niobate [3], SiP over SOI [4], GaAs [5], Graphene [6], etc.),thus causing the light beam to undergo a Phase Shift proportional to the ap-plied Electric Field. The resultant light, after undergoing Phase Shift in eachbranch are applied to a Y-Coupler, where they undergo Interference, causingthe Light Phase Modulation to be converted to Light Intensity Modulation.

Consider a Light from an Optical Source be represented by an opticalfield of form El be applied to the input of an MZM. This input light is splitalong two path using a Y-Splitter (1x2 Optical Splitter), whose output field




from each arm is written as:

(16)

EPM1,input = EPM2,input

=1√

Number of Output× Input Optical F ield

=1√2× El

=El√2

The optical field in equation-16, are Phase Modulated using PM1 andPM2, which are driven by Equal (Magnitude) and Opposite (Sign) Electric(RF + DC). Consider the Electric Field to be represented by V (t), then thecorresponding Phase Modulation Index for PM1 and PM2 are represented asβ2

= 12× π

VπV (t) and −β

2= −1

2× π

VπV (t), respectively; where β = π

VπV (t).

The Phase Modulated Optical Field at the output of PM1 and PM2 aremathematically obtained by:

(17)EPM1,output = EPM1,input × ej

β2

=El√2ej

β2

(18)EPM2,output = EPM2,input × e−j β

2

=El√2e−j β

2

The corresponding output of PM1 and PM2 are combined together usinga Y-Coupler (2x1 Optical Coupler), whose output field can be written as:

(19)Eout =

1√Number of Inputs

∑(Electric F ield from Each Input)

=1√2(EPM1,output + EPM2,output)

Substituting the values from equation-17 and 18 in equation-19, we ob-tain:

(20)

Eout =1√2(El√2ej

β2 +

El√2e−j β

2 )

=El

2(ej

β2 + e−j β

2 )

=El

2× 2 cos(

β

2)

= El cos(β

2)




The Phase Modulation Index β contains two components. The firstcomponent of modulation index is DC Modulation Index written as βDC

and the second component is RF Modulation Index βm cos(ωmt), ωm beingthe RF frequency applied (in radians). Therefore, the total electric field isβ = βDC + βm cos(ωmt). Therefore, equation-20 can be rewritten as:

(21)Eout = El cos[βDC

2+

βm

2cos(ωmt)]

1.3.1 Biasing of MZMGenerally, MZM can be biased in three different regions viz.:

1. MAximum Transfer Point (MATP)

2. MInimum Transfer Point (MITP)

3. Quadrature Tranfer Point (QTP)

When the MZM is biased at MATP, the output intensity should be max-imum. Hence both the arms of MZM must be In-Phase, so as to offer Con-structive Interference between the Light arriving from the individual arms.This can only happen when the MZM is biased at Zero voltage or at theintegral multiple of 2 Vπ voltage. Consider that the MZM be biased at 2NVπ

voltage, then the corresponding βDC

2= 2NVπ×π

2Vπ= Nπ. Now substituting

this value in equation-21, we get (and using Jacobi-Anger Expansion fromequation-12):

(22)

Eout = El cos[Nπ +βm

2cos(ωmt)]

= (−1)NEl cos[βm

2cos(ωmt)]

= (−1)NEl[J0(βm

2) + 2

∞∑n=1

(−1)nJ2n(βm

2) cos 2n(ωmt)]

Therefore, an alternative definition of MATP is the bias point of MZM,which results in output optical spectrum to retain Optical Carrier and theEven Harmonics of Input RF as the Optical Sidebands. A special cause isthe Null Bias, when VDC = 0 V , when the above equation-22 can be written




as:

Eout = El[J0(βm

2) + 2

∞∑n=1

(−1)nJ2n(βm

2) cos 2n(ωmt)]

= El[∞∑n=1

(−1)nJ2n(βm

2)e−j2nωmt + J0(

βm

2) +

∞∑n=1

(−1)nJ2n(βm

2)ej2nωmt]

(23)

When the MZM is biased at MITP, the output optical carrier must bediminished. In order to do this, the light phase at the individual arms of theMZM must be complementary in phase, so that the Constructive Interferencetakes place at the output. Therefore, from equation-21, setting RF term,βm = 0, we get the MITP point as: Eout = El cos[

βDC

2] = 0; or, βDC

2=

(2N +1)π2; or, βDC = (2N +1)π. This relation shows that, when the MZM is

biased at an Integral multiple of Vπ, then the bias point is known as MITP.Now substituting the value of the bias point in equation-21, we get (alsoexpanding using Jacobi-Anger Expansion from equation-13):

(24)

Eout = El cos[(2N + 1)π

2+

βm

2cos(ωmt)]

= El cos[Nπ +π

2+

ωm

2cos(ωmt)]

= (−1)NEl cos[π

2+ cos(ωmt)]

= (−1)NEl{− sin[cos(ωmt)]}= (−1)(N+1)El sin[cos(ωmt)]

= (−1)(N+2)El[2∞∑n=1

(−1)nJ(2n−1)(βm

2) cos(2n− 1)(ωmt)]

= (−1)NEl[2∞∑n=1

(−1)nJ(2n−1)(βm

2) cos(2n− 1)(ωmt)]

Therefore, MITP bias point can be defined as the MZM Bias point wherethe Optical Carrier is absent, and the Odd order harmonics of RF appear asthe Optical Sidebands. As a special case, if the MZM is biased at first MITPpoint given by Bias Voltage of Vπ, then the form for output optical field is




given by:

Eout = El[2∞∑n=1

(−1)nJ(2n−1)(βm

2) cos(2n− 1)(ωmt)]

= El[∞∑n=1

(−1)nJ(2n−1)(βm

2)e−j(2n−1)ωmt

+∞∑n=1

(−1)nJ(2n−1)(βm

2)ej(2n−1)ωmt]

(25)

QTP is another bias point which lies half-way between MATP and MITP.The bias point is such that, βDC

2= π

4or, βDC = π

2, which on substituting

in equation-21, (and expanding using Jacobi-Anger expansion in equation-12and 13):

Eout = El cos[π

4+

βm

2cos(ωmt)]

=El√2[cos{ωm

2cos(ωmt)} − sin{ωm

2cos(ωmt)}]

=El√2[J0(

βm

2) + 2

∞∑n=1

(−1)nJ(2n−1)(βm

2) cos(2n− 1)(ωmt)

+ 2∞∑n=1

(−1)nJ2n(βm

2) cos 2n(ωmt)]

=El√2[∞∑n=1

(−1)nJ2n(βm

2)e−j2nωmt +

∞∑n=1

(−1)nJ(2n−1)(βm

2)e−j(2n−1)(ωmt)

+ J0(βm

2) +

∞∑n=1

(−1)nJ(2n−1)(βm

2)ej(2n−1)(ωmt)

+∞∑n=1

(−1)nJ2n(βm

2)ej2nωmt]

(26)

Therefore, QTP is the Bias Point of MZM, whereby the Optical Carrier aswell as all the RF harmonics appear as the Optical Sideband in the OpticalSpectrum.




1.4 Jones CalculusA completely polarized light can be treated using Jones Calculus, wherebythe Electric Field of the Light can be represented as a Column Vector calledJones Vector. Jones Vector is written as

(ExEy

). Further, the translation

of Polarization State can be altered by multiplying the Jones Vector with a2x2 matrix called Jones Matrix. The Jones Matrix is written as ( A B

C D ).However, the user should be careful when expressing Light wave using JonesCalculus as Jones Matrix and Vectors are only valid when the Optical Waveis completely polarized. But whenever the Light wave consists of unpolarizedcomponent, it can only be treated using Mueller Calculus.

1.5 Jones vector representation of LASERA LASER is generally considered as a Horizontally polarized Optical Wave-form in literature. It is therefore, useful to represent Laser Optical Field usingJones Vector as:

Elaser = E0ejω0t ( 1

0 ) (27)

The above equation-27 is generally used to represent a Laser beam, whichis Parallel (Horizontal) Polarized (p-polarized). The optical intensity, Pl isrelated to the optical field by the relation, Pl = |E0|2

2=

E0×E∗0

2. But

OptiWave OptiSystem Optical Simulation Software (OptiSystem) representsthe Laser in following format shown in equation-28 below:

Elaser = E0ejω0t ( −1

0 ) (28)

In order to represent 28 in terms of equation-27 (which is used throughout thedocument), we will connect a Polarization Controller (Rotator) (PC(Rot))with rotation angle of 180◦. This PC(Rot) will rotate the direction of po-larization of the OptiSystem Laser component, in equation-28 to that ofequation-27.

1.6 PC (Rotator)In Optics, the Plane of Polarization can be rotated in either of two waysfollowing the movement of clock-hand, with direction of propagation formingthe center of clock. Therefore, if the light propagates in a such way that itseems like coming out of the clock, then the hand movement of the clockwill dictate Clockwise Rotation (CW) rotation of the plane of polarization.Keeping the direction of propagation of Light unaltered, the movement of




Plane of Polarization in the opposite direction will form Counter-ClockwiseRotation (CCW) rotation.

It is a generic convention (in this document) to represent the rotationangle in CW direction. Therefore, if a optical polarization rotator is usedto rotate the plane of polarization by θ CW angle, then the correspondingJones matrix will be represented as:

JPC(θ CW ) =(

cos(θ) sin(θ)− sin(θ) cos(θ)

)(29)

An opposite rotation of plane of polarization is in CCW, which can bedenoted by Jones matrix:

JPC(θ CCW ) =(

cos(θ) − sin(θ)sin(θ) cos(θ)

)(30)

A Polarization Controller (PC) is an Optical Device that can alter Stateof Polarization of an optical beam. The PC is designed using a cascade ofHalf and Quarter waveplates to obtain a desired Polarization state. PC canalso act as a Polarization Rotator and the Jones vectors shown in equation-29and 30 can be used to obtain the output state of polarization for a given inputstate of polarization. Further, PC exhibit very low Insertion Loss (IL), whichis of the order of 0.004 dB within the wavelength of 1250 – 1600 nm.

1.7 Linear Polarizer (Fixed)A Linear Polarizer can take any state of polarization as input and produce alinearly polarized output, which is either p-polarized or Perpendicular (Ger-man: Senkrecht) Polarized (s-polarized). Hence, the Jones matrix that canrepresent the fixed Linear Polarizer will possess non-zero elements along thediagonal and all-zeros on the non-diagonal section. The purpose of usingLinear Polarizer is to extract the light-component which is the projection ofthe input light state on the required plane of polarization.

A Linear Polarizer which outputs only p-polarized light will be repre-sented by Jones Matrix ( 1 0

0 0 ), and is called Horizontal Polarizer (H-POL).However, the Linear Polarizer which outputs only s-polarized light is repre-sented by Jones Matrix ( 0 0

0 1 ), and is called Vertical Polarizer (V-POL).

1.8 Pol (Rotatable Linear Polarizer)The Linear Polarizer (Rotatable) (Pol) is an optical device which is actuallya Linear Polarizer which can be rotated by a user specific angle θ. The plane




Figure 2: Illustration of a Pol (Rotatable Linear Polarizer)

of polarization of input light is first rotated clockwise by an angle θ CW.Then a H-POL linear polarizer will only retain the p-polarized component ofthe rotated light. The resultant light is then re-rotated by an angle θ CCWto produce the output light.

Consider the optical field at input be given by a Jones Vector Ein. Thenthe output of the ROT θ CW can be written as Jones Matrix of form(

cos(θ) sin(θ)− sin(θ) cos(θ)

). The output is passed through a H-POL whose Jones Ma-

trix is of form ( 1 00 0 ). Then the output light is rotated by θ CCW, using the

matrix(

cos(θ) − sin(θ)sin(θ) cos(θ)

). The resultant Jones Matrix of the complete system

shown in figure-2 can be obtained by multiplying the matrices of the subsys-tems from output to input (moving from right to left), thus output opticalfield Eout can be written as:

(31)Eout =

(cos(θ) − sin(θ)sin(θ) cos(θ)

)( 1 00 0 )

(cos(θ) sin(θ)

− sin(θ) cos(θ)

)Ein

=(

cos2(θ) sin(θ) cos(θ)

sin(θ) cos(θ) sin2(θ)

)Ein

In OptiSystem, the component with this transfer function is Linear Po-larization. Inline Polarizer are designed completely with optical fiber andoffers a typical IL of 0.4 dB.

1.9 Polarization Beam Splitter (PBS)A Polarization Beam Splitter (PBS) is an optical device with 1-input and2-outputs, with schematic as shown in Figure-3. A Polarized Light is appliedat the input of the PBS. The light is split along two paths. In the firstpath, the light will be passed through a Rotatable Linear Polarizer, Polwhich is rotated at an angle θ CW. The Jones Matrix along this path (calledtransmission path) is written as:

(cos2(θ) sin(θ) cos(θ)


). Along the second

path the light is passed through a Rotatable Linear Polarizer, which is rotatedby an angle (θ + 90◦) CW, to output a perpendicular polarized light. The




Figure 3: Schematic of a Polarization Beam Splitter (PBS)

Jones Matrix along the second path (called reflected path) is written as:(sin2(θ) − sin(θ) cos(θ)

− sin(θ) cos(θ) cos2(θ)

). Therefore, if the Jones Vector representation of

the input light is Ein. Then the Jones Vector for p-polarized light is:

Ep−pol =(



)Ein (32)

The corresponding Jones Vector along s-polarized path is:

Es−pol =(

sin2(θ) − sin(θ) cos(θ)


)Ein (33)

Note that, equation-33 is obtained by substituting (θ + 90◦) CW for θCW in equation-32.

OptiSystem provides a component PBS which obeys the above equations32 and 33. The commercial PBS offers typical IL of 0.4 dB and the maximumIL will be atmost 0.7 dB.

1.10 Polarization Beam Combiner (PBC)A Polarization Beam Combiner (PBC) operates in converse to PBS, withcommercial PBC offerring an IL of 0.4 dB (typical) and 0.7 dB (maximum).The device has 2-inputs and 1-output for optical signal. The p-polarized ands-polarized signals are passed through Linear Polarizer which are oriented atrepective angles θ CW and (θ + 90◦) CW. The resultant light is combinedand output light is obtained. OptiSystem provides a component named PBC,which obeys equation-34.

Consider two light beam at p-polarized and s-polarized be representedby Jones Vectors Ep,in and Es,in, respectively. The output Jones Vector is




Figure 4: Schematic of a Polarization Beam Combiner (PBC)

written using the following equation:

Eout =(



)Ep,in +

(sin2(θ) − sin(θ) cos(θ)


)Es,in (34)

1.11 Polarization Modulator (PolM)

Figure 5: Schematic of Polarization Modulator (PolM)

A Polarization Modulator (PolM) is an electro-optic Phase Modulatorthat can independently phase modulate the Transverse Electric (TE) andTransverse Magnetic (TM) mode of the incident light, but with oppositephase modulation index, as illustrated in figure-5 [7, 8]. The advantageof PolM is that it is free from bias drift problem that is sufferred by MZM[7] and low IL of the order of 3.5 dB. The Jones Matrix that represents the




PolM can be written as:(

ejβ(t) 00 e−jβ(t)

)[8]. If we consider a polarized light

given by Jones Vector Ein, then the polarized light at the output of PolM isgiven by [8]:

Eout = Ein

(ejβ(t) 0

0 e−jβ(t)

)(35)

In equation-35, β(t) = πVπ

×Vrf (t),is the modulation index of the PolM.Vπ is the half-wave voltage of the phase modulator, which will produce a180◦ phase shift along TE (or TM) axis. And Vrf (t) is the instantaneous RFvoltage applied to the PolM input.

OptiSystem does not contain a standard model for PolM. Hence, to sim-ulate a PolM, the figure-5 is used throughout the article.

1.12 Photodiode (PD)A Photodiode/Photodetector is an optoelectronic device that aborbs incidentoptical energy and produces photocurrent at the output [9]. Photodiode isgenerally used for photodetection of optical signal, in optical communica-tion systems. Since, the output photocurrent is proportional to the incidentoptical power, we can write:

(36)Iph = R Pop

= R |E|2= R E × E∗

Here, Iph is the resultant photocurrent at the output of PhotoDiode, R isResponsivity of Photodiode, Pop is Incident Optical Power, E is correspond-ing optical field (which can be real or complex number).

In OptiSystem, PiN Photodiode is used for simulation, which does notpossess external gain, but can be represented by mathematical expressionin equation-36. The Responsivity must be set to R = 0.82 A/W duringsimulation.

1.13 Sideband Suppression Ratio (SSR)When using a nonlinear frequency generation technique, it is generally seenthat apart from the required frequency there will be several sidebands, whichoccurs due to nonlinearity itself. Hence, in order to characterize such anon-linear system, a new factor called Sideband Suppression Ratio (SSR) isdefined as the ratio of the power in the wanted frequency to the ratio of the




power in the unwanted frequency sideband. SSR is generally expressed indB, and is obtained by the relation:

(37)SSR = 10 log10(Power in Wanted Frequency (W )

Power in Unwanted Frequency Sideband (W ))

Whenever the spectrum consists of multiple unwanted sidebands, thentake the unwanted sideband which has maximum power. Further, note that,in an ideal frequency generator, all the unwanted sidebands are absent, hencethe denominator in equation-37 in zero, which results in SSR value to beinfinity. Since, for a practical system the sidebands are present with finitepower value, thus, SSR value is finite. However, it is always desirable todesign a frequency generator with very high SSR value, to suppressthe sideband and attain a nearly clean spectrum.

When Photonics is used to generate RF, the optical spectrum is generatedfirst. Hence, the optical spectrum must be characterized first. Therefore, thea quantity called Optical Sidebands Suppression Ratio (OSSR) is used forcharacterizing the optical spectrum, which is defined in a manner similar toequation-37 as:

OSSR

= 10 log10(Optical Power in Wanted Frequency (W )

Optical Power in Unwanted Frequency Sideband (W ))

(38)

Finally, the optical spectral components are heterodyned with each otherusing a Photodiode to produce the desired RF spectrum. Hence, a new quan-tity called RF Spurious Suppression Ratio (RFSSR), which is RF spectralequivalent to SSR, is defined as:

(39)RFSSR

= 10 log10(Power in Wanted Radio Frequency (W )

Power in Unwanted Radio Frequency Sideband (W ))

It should be noted that, the Microwave Frequency Generator mustpossess a very high value of High value of OSSR and RFSSR.

1.14 (Required) Analysis and ResultsFollowing are the list of analysis that must be included when presentingan article on the Design of Frequency Generation Systems using MicrowavePhotonics:




• Theory and Mathematical expression for Electric and Optical field [10]

• Derivation of any particular condition required to generate the requiredfrequency, like Phase and Modulation Index [10]

• OSSR and RFSSR from Mathematical Expression [10]

• Required Spectrum and brief explanation about the plots (only if re-quired) [10]

• OSSR and RFSSR values must be marked (value written on plot alongwith the respective label: OSSR or RFSSR) properly in the spectralplots, and it must be matched with the corresponding theoretical value[10]

• Effect of Nonideality of components on the value of OSSR and RFSSR[10]. The Effect of Microwave Drive Voltage or Microwave Modula-tion Index (keeping Phasors constant) on OSSR, RFSSR [10]; Effect ofPhase Offset (a 5− 15◦ and (−15)− (−5)◦ in steps of 1◦ ) on OSSR orRFSSR keeping Modulation Index Fixed [10]; Keeping Modulation In-dices and Phasors constant, the Effect of non-ideality of Optical and/orRF components (like PBS [10]) may also be studied. In all the cases,it is not important to provide mathematical justification (or numeri-cal expression) of the non-ideality studies performed (and is strictly,optional). However, a small text briefly explaining the effect of non-idealities (individually or as a collective) should be included in thetext.




2 Microwave FMF-1

2.1 Remote Delivery of RF using Optical Fiberand Quadrature biased MZM

Figure 6: Schematic of a Remote RF delivery mechanism using a QuadratureBiased MZM and Optical Fiber. LD: Laser Diode, PD: Photo Diode, MZM:Mach-Zehnder Modulator, QTP: Quadrature Transfer Point Biasing of MZM.




3 Microwave FMF-6

3.1 [2017] Frequency Sextupling using Dual-Polarization Modulator (Filterless Design)

Figure 7: Schematic of a Frequency Sextupling system using DPPolM. LD:Laser Diode, PD: Photo Diode, PC: Polarization Controller (Rotator), PolM:Polarization Modulator, PBS: Polarization Beam Splitter, PBC: PolarizationBeam Combiner, OSC: RF Oscillator (Signal Generator), TEPS: TunableElectrical Phase Shifter.

High RF can be obtained from low frequency RF using Frequency Multi-plication Factor (FMF), achieved using a photonics scheme proposed in [10].In [10], Dual-Parallel PolM (DPPolM) is used to achieve an FMF of 6, with-out using optical or RF filters. In the following section, the scheme will bediscussed, OSSR, RFSSR values are obtained and compared against the sim-ulation and theory, along with the results obtained in [10]. The advantage ofthis design is the absence of Filters, which will allow the design to be tunedto a desired RF without replacing components.

3.1.1 Optical SpectrumIn the scheme shown in Figure-7, the (OptiSystem) (Continuous Wave) LaserDiode / Laser Source (LD) is the source of light, represented by the JonesVector of form (equation-27):

El = ( −10 ) E0 ejω0t (40)




The light output of LD laser source is rotated by 180◦ CW using aPC1 (PC as Polarization Rotator). The relation connecting input field inequation-40 to output field is given by (using equation-29):

(41)EPC1 =

(cos(180◦) sin(180◦)

− sin(180◦) cos(180◦)

)El

=( −1 0

0 −1

)( −1

0 )E0ejω0t

= ( 10 )E0e

jω0t

The output from PC-1 (equation-41) is applied to PBS, whose outputfields are given by equation-32 and 33. Here, PBS is rotated by −45◦ CW inp-polarized direction, thus the output optical field in p-polarized direction isgiven by (and using equation-41):

(42)

EPBS1,p =(

sin2(−45◦) sin(−45◦) cos(−45◦)

sin(−45◦) cos(−45◦) cos2(−45◦)

)EPC1

=1

2

(1 −1

−1 1

)( 10 )E0e

jω0t

=E0

2

(ejω0t

−ejω0t

)In the s-polarized direction of PBS, the output is followed from using

equation-41 as:

(43)

EPBS1,s =(

cos2(−45◦) − sin(−45◦) cos(−45◦)

− sin(−45◦) cos(−45◦) sin2(−45◦)

)EPC1

=1

2( 1 11 1 ) (

10 )E0e

jω0t

=E0

2

(ejω0t

ejω0t

)The output of PBS along p-polarized path is passed through PolM1. Also

a RF input of frequency ωm and RF voltage m1 = πVπ

× Vrf,1 is applied toRF input of PolM1. The output optical field from PolM1 can be written(using equation-35, 42) as:

(44)

EPolM1 =(

ejβ(t) 00 e−jβ(t)

)EPBS,p

=(

ejm1 cos(ωmt) 00 e−jm1 cos(ωmt)

) E0

2

(ejω0t

−ejω0t

)=

E0

2

(ej[ω0t+m1 cos(ωmt)]

−ej[ω0t−m1 cos(ωmt)]

)The s-polarized output of PBS (equation-43) is applied to PolM2. The

RF input of PolM1 is first phase shifted by an angle φ, but its frequency




ωm remains the same. Although the RF voltage may differ, and let it bem2 = π

Vπ× Vrf,2. Then similar to equation-44, we can write the expression

for the output optical field for PolM2 as (using equation-43):

(45)

EPolM2 =(

ejm2 cos(ωmt+φ) 00 e−jm2 cos(ωmt+φ)

)EPBS,s

=(

ejm2 cos(ωmt+φ) 00 e−jm2 cos(ωmt+φ))

) E0

2

(ejω0t

ejω0t

)=

E0

2

(ej[ω0t+m2 cos(ωmt+φ)]

ej[ω0t−m2 cos(ωmt+φ)]

)The output of PolM1 (equation-44) is then passed through PC2, which

rotates the state of polarization by +90◦ CW. The output optical field fromPC2 is given by (using equation-29):

(46)

EPC2 =(

cos(90◦) sin(90◦)− sin(90◦) cos(90◦)

)EPolM1

= ( 0 1−1 0 )

E0

2

(ej[ω0t+m1 cos(ωmt)]

−ej[ω0t−m1 cos(ωmt)]

)=

E0

2

(−ej[ω0t−m1 cos(ωmt)]

−ej[ω0t+m1 cos(ωmt)]

)The output of PolM2 (equation-45) is passed through PC3, which is used

to rotate the state of polarization by +90◦ CW. The output optical field fromPC3 is given by (using equation-29):

(47)

EPC3 =(

cos(90◦) sin(90◦)− sin(90◦) cos(90◦)

)EPolM2

= ( 0 1−1 0 )

E0

2

(ej[ω0t+m2 cos(ωmt+φ)]

ej[ω0t−m2 cos(ωmt+φ)]

)=

E0

2

(ej[ω0t−m2 cos(ωmt+φ)]

−ej[ω0t+m2 cos(ωmt+φ)]

)The output of PC2 (equation-46) is applied as input to p-polarized path

of PBC, and the output of PC3 (equation-47) is applied to the input to s-polarized of PBC. The PBC is rotated to an angle −45◦ CW. The output ofPBC can be written as (using equation-34) as:

(48)

EPBC =(

cos2(−45◦)) sin(−45◦) cos(−45◦)

sin(−45◦) cos(−45◦) sin2(−45◦)

)EPC2

+(

sin2(−45◦) − sin(−45◦) cos(−45◦)

− sin(−45◦) cos(−45◦) cos2(−45◦)

)EPC3

=1

2

(1 −1

−1 1

)EPC2 +

1

2( 1 11 1 )EPC3

=1

2

(EPC2,p−EPC2,s+EPC3,p+EPC3,s

−EPC2,p+EPC2,s+EPC3,p+EPC3,s

)Arnav Mukhopadhyay (gudduarnav.com)



The output of PBC (given by equation-48), is passed through a Pol whichis set at 0◦ CW. Using equation-31, we can write:

(49)Epol =

(cos2(0◦) sin(0◦) cos(0◦)

sin(0◦) cos(0◦) cos2(0◦)

)EPBC

= ( 1 00 0 )EPBC

= EPBC,p

Now from equation-48, expression for EPBC,p is obtained by retaining thefirst row only. Therefore, equation-49 can be written using equation-48 as:

EPol =1

2[EPC2,p − EPC2,s + EPC3, p+ EPC3,s]

=1

2E0 ×

1

2[−ej(ω0t−m1 cos(ωmt)) + ej(ω0t+m1 cos(ωmt)) + ej(ω0t−m2 cos(ωmt+φ))

− ej(ω0t+m2 cos(ωmt+φ))]

=1

4E0e

jω0t[−e−jm1 cos(ωmt) + ejm1 cos(ωmt) + e−jm2 cos(ωmt+φ)

− ejm2 cos(ωmt+φ)](50)

Using Jacobi-Anger expansion (equations-9 and 10), equation-50 can bewritten as:

(51)

EPol =1

4E0e

jω0t[−n=∞∑n=−∞

jnJn(m1)ejn(ωmt+π) +

∞∑n=−∞

jnJn(m1)ejnωmt

+∞∑

n=−∞

jnJn(m2)ejn(ωmt+φ+π) −

∞∑n=−∞

jnJn(m2)ejn(ωmt+φ)]

=1

4E0e

jω0t[{∞∑

n=−∞

jnJn(m1)ejnωmt −

∞∑n=−∞

jnJn(m1)ejn(ωmt+π)}

− {∞∑

n=−∞

jnJn(m2)ejn(ωmt+φ) −

∞∑n=−∞

jnJn(m2)ejn(ωmt+φ+π)}]




Setting m1 = m2 = m, in equation-51, we have:

Epol =1

4E0e

jω0t[{∞∑

n=−∞

jnJn(m)ejnωmt −∞∑

n=−∞

jnJn(m)ejn(ωmt+π)}

− {∞∑

n=−∞

jnJn(m)ejn(ωmt+φ) −∞∑

n=−∞

jnJn(m)ejn(ωmt+φ+π)}]

=1

4E0e

jω0t

∞∑n=−∞

jnJn(m)[{ejnωmt − ejn(ωmt+π)}

− {ejn(ωmt+φ) + ejn(ωmt+φ+π)}]

=1

4E0e

jω0t

∞∑n=−∞

jnJn(m)ejnωmt[{1− ejnπ} − {ejnφ + ejn(φ+π)}]

=1

4E0e

jω0t

∞∑n=−∞

jnJn(m)ejnωmtejnπ2 [{e−j nπ

2 − ejnπ2 }

− {ej(nφ−nπ2) + ejn(φ+

nπ2)}]

=1

4E0e

jωmt

∞∑n=−∞

jnJn(m)ejn(ωmt+π2)[e−j nπ

2 (1− ejnφ) + ejnπ2 (1− ejnφ)]

=1

4E0e

jωmt

∞∑n=−∞

jnJn(m)ejn(ωmt+π2)[e−j nπ

2 − ejnπ2 ][1− ejnφ]

=1

4E0e

jωmt

∞∑n=−∞

jnJn(m)ejn(ωmt+π2)[−2j sin(

nπ

2)][1− ejnφ]

= −j

2E0e

jωmt

∞∑n=−∞

jnJn(m)ejn(ωmt+π2)(1− ejnφ) sin(

nπ

2)

(52)

In figure-52, n = 0 that is, 0th sideband is absent. Again for n = Even,then sin(nπ

2) = 0, and therefore, in equation-52. Therefore, only Odd

Sidebands will be present in the optical spectrum. The higher order oddsidebands will be decreasing in power. In order to eliminate the first ordersideband, find the value of m for which J1(m) = 0, and from table-1.1, wefind that m = 3.8 (approx.). Thus the dominant sidebands with appreciablepower are n = ±3, ±5, ±7.

In order to eliminate n = ±5 sideband, solve equation-52, to find a value




of φ, for which Epol = 0. From equation-52, we find:

−j

2E0e

jωmtj5J5(m = 3.83)ej5(ωmt+π2)(1− ej5φ) sin(

5π

2) = 0 (53)

(1− ej5φ) = 0 (54)(1− cos 5φ+ j sin 5φ) = 0 (55)

Therefore we must find an angle φ such that the condition, cos 5φ = 1and simultaneously sin 5φ = 0 . The conditions are obeyed by the solution,φ = (some integer) × 2π

5. Thus some possible values for electrical phase

shifter φ are 2π5

and 4π5

, which will eliminate ±5 optical sideband from theoptical spectrum.

The optical spectrum contains ±3, ±5 optical sideband. Thus, the termwith n = −3, obtained from equation-52 is of form:

(56)Epol,−3 = j−3J−3(m)ej(−3)(ωmt+π2)(1− e−j3φ) sin(−3π

2)

= J3(m)e−j3ωmt(1− e−j3φ)

Thus, the term with n = +3, obtained from equation-52 is of form:

(57)Epol,+3 = j3J3(m)ej(3)(ωmt+π2)(1− ej3φ) sin(

3π

2)

= J3(m)ej3ωmt(1− ej3φ)

Thus, the term with n = −7, obtained from equation-52 is of form:

(58)Epol,−7 = j−7J−7(m)ej(−7)(ωmt+π2)(1− e−j7φ) sin(

−7π

2)

= J7(m)e−j7ωmt(1− e−j7φ)

Thus, the term with n = +7, obtained from equation-52 is of form:

(59)Epol,+7 = j7J7(m)ej(7)(ωmt+π2)(1− ej7φ) sin(

7π

2)

= J7(m)ej7ωmt(1− ej7φ)

Hence, combining equations-56, 57, 58, 59, the expression for opticalspectrum becomes:

(60)Epol = −j

2E0e

jω0t[J3(m)e−j3ωmt(1− e−j3φ) + J3(m)ej3ωmt(1− ej3φ)

+ J7(m)e−j7ωmt(1− e−j7φ) + J7(m)ej7ωmt(1− ej7φ)]




3.1.2 Optical Sideband Suppression Ratio (OSSR)OSSR is calculated from its definition in equation-38 and using the mathe-matical expression in equation-60, noting the fact that ±3 and ±7 sidebandsin the optical spectrum, are the wanted and unwanted optical sidebands,respectively. We can write:

(61)

OSSR = 10 log10 [P±3

P±7

]

= 10 log10 [E(pol,+3) × E∗

(pol,+3)

E(pol,+7) × E∗(pol,+7)

]

= 10 log10 [J23 (m)(1− ej3φ)(1− e−j3φ)

J27 (m)(1− ej7φ)(1− e−j7φ)

]

= 10 log10 [2J2

3 (m)(1− cos 3φ)

2J27 (m)(1− cos 7φ)

]

= 10 log10 [J23 (m)(1− cos 3φ)

2J27 (m)(1− cos 7φ)

]

• With m = 3.83 and φ = 2π5

: Substituting these values in equation-61,the value of OSSR = 31.1064 dB.

• With m = 3.83 and φ = 4π5

: Substituting these values in equation-61,the value of OSSR = 31.1064 dB.

3.1.3 Output of Photodiode (Electrical RF Spec-trum)

The output Photocurrent from Photo Diode is written using equation-36 as:

(62)I(t) = R|E|2

Herein equation-62, Responsivity R is specified in Ampere/Watts.Further, let us write the simplified form for optical field incident on Photo

Diode, from equation-60 as:

(63)EPol = −j

2E0e

jω0t[J3(m){e−j3ωmt(1− e−j3φ) + ej3ωmt(1− ej3φ)}

+ J7(m){e−j7ωmt(1− e−j7φ) + ej7ωmt(1− ej7φ)}]




The value in equation-63 is substituted in equation-62, and simplifiedusing Mathematica, to obtain photocurrent as:

I = R1

4|E0|2[{J2

3 (m) + J27 (m)− J2

3 (m) cos(3φ)− J27 cos(7φ)}

+ {J3(m)J7(m) cos(4ωmt− 3φ) + J3(m)J7(m) cos(4ωmt)+ J3(m)J7(m) cos(4ωmt+ 4φ) + J3(m)J7(m) cos(4ωmt+ 7φ)}

+{12J23 (m) cos(6ωmt)−J2

3 (m) cos(6ωmt+3φ)+1

2J23 (m) cos(6ωmt+6φ)}

+ {−J3(m)J7(m) cos(10ωmt)− J3(m)J7(m) cos(3ωmt+ 3φ)− J3(m)J7(m) cos(10ωmt+ 7φ) + J3(m)J7(m) cos(10ωmt+ 10φ)}

+ {12J27 (m) cos(14ωmt)− J2

7 (m) cos(14ωmt+ 7φ)

+1

2J27 (m) cos(14ωmt+ 7φ)}]

(64)

From equation-64, it can be seen that the RF spectrum is composed ofDC, 4rd„ 6th, 10th, 14th order frequency multiples of modulating RF frequen-cies. Among these, the 6th order is wanted and remaining are sideband ofconsiderably lower power.

3.1.4 RF Spurious Suppression Ratio (RFSSR)RFSSR is obtained from using the definition in 39 and power of wanted6th order frequency and the spurious frequency 10th order frequency, fromequation-64. Therefore, the expression finally becomes,

(65)

RFSSR = 10 log10[P(el,6th)

P(el,10th)

]

= 10 log10[12J23 (m)

J3(m)J7(m)]2

= 10 log10[J3(m)

2J7(m)]2

Finally substituting the value of modulation index m = 3.83 in equation-65,we find:

(66)RFSSR = 10 log10[

J3(3.83)

2J7(3.83)]2

= 25.0858 dB= 25.09dB




3.1.5 Simulation in OptiSystemThe Model in 7 is simulated in OptiSystem, and the parameters used forvarious components are specified below in tabular format.

Table 2: CW LASER SpecificationsParameter Name ValuesCenter Frequency 193.1 THzPower 0 dBmLinewidth 0 Hz (Ideal)

Table 3: PC1 SpecificationsParameter Name ValuesIL 0.004 dBRotation Angle 0◦ CW

Table 4: PBS SpecificationsParameter Name ValuesIL 0.4 dBRotation Angle −45◦ CW

Table 5: PolM1 and PolM2 SpecificationsParameter Name ValuesIL 3.5 dBVπ 3.5 V

With the above parameters, the corresponding optical and electrical spec-trum are shown in the following figures.

From optical spectrum shown in figure-15, the value of OSSR = 31 dB, isagreeable with the value of OSSR = 31.1064 dB, obtained from equation-61,by substituting the values: m = 3.83 and TEPS angle φ = 2π

5.




Table 6: PC2 and PC3 SpecificationsParameter Name ValuesIL 0.004 dBRotation Angle 90◦ CW

Table 7: PBC SpecificationsParameter Name ValuesIL 0.4 dBRotation Angle −45◦ CW

Table 8: Pol SpecificationsParameter Name ValuesIL 0.4 dBRotation Angle 0◦ CW

Table 9: PiN Photodetector SpecificationsParameter Name ValuesResponsivity (R) 0.82 A/WDark Current 10 nA

Table 10: RF Source SpecificationsParameter Name ValuesRF Frequency 10 GHzAmplitude (A) 4.267 VModulation Index( π

V(π,PolM)× A) 3.83

TEPS (Phase Shift by Angle) 72◦ (2π5rad)

From the RF spectrum shown in figure-16, the value of RFSSR = 25dB which closely agrees with the value of RFSSR = 25.09 dB obtained inequation-66.

This method can therefore, be used to generate high frequency usingFMF of 6 to translate a 10 GHz RF to produce 60 GHz RF output, withappreciable values of OSSR = 31 dB and RFSSR = 25 dB. Without usage




Figure 8: Laser Output: (a) Optical Spectrum, (b) Polarization State

Figure 9: Output of PC1: (a) Optical Spectrum, (b) Polarization State

of any external Optical or Electrical Amplifiers, the high frequency RF (of60 GHz) will have an (Electrical) power of -54 dBm (in Simulation).




Figure 10: Polarization State at the Output of PBS, along: (a) p-polarizedpath, (b) s-polarized path

Figure 11: RF of 10 GHz applied at PolM1 and PolM2 (through TEPS) RFinput




Figure 12: Output of Polarization Modulator. For PolM1: (a) Optical Spec-trum, (b) Polarization State. For PolM2: (a) Optical Spectrum, (b) Polar-ization State.




Figure 13: Output of PC2: (a) Optical Spectrum, (b) Polarization State.Output of PC3: (a) Optical Spectrum, (b) Polarization State.

Figure 14: Output of PBS: (a) Optical Spectrum, (b) Polarization State.




Figure 15: Output of Pol (incident on Photodetector PD): (a) Optical Spec-trum, (b) Polarization State.

Figure 16: RF spectrum at the output of Photodetector.




4 Frequency Octupling

4.1 [2010] RF Frequency Octupling using Cas-caded MZM, Optical Phase Shifter andFBG Notch Filter

here




References[1] Eric W. Weisstein. CRC Concise Encyclopedia of Mathematics. CRC,

2nd edition, 2003.

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[3] Le Nguyen Binh. Optical Modulation: Advanced Techniques and Appli-cations in Transmission Systems and Networks. CRC Press, 1st edition,2017.

[4] Miaofeng Li, Lei Weng, Xiang Li, Xi Xiao, and Shaoyua Yu. Siliconintensity mach–zehnder modulator for single lane 100 gb/s applications.Chinese Laser Press, 6(2):109–116, 2018.

[5] R. G. Walker, N. Cameron, and S. Clements. Electro-optic modulatorsfor space using gallium arsenide. International Conference on SpaceOptics (ICSO), 2016.

[6] Mohsen Sabbaghi, Hyun-Woo Lee, and Tobias Stauber. Electro-opticsof current-carrying graphene. arXiv, 2018.

[7] Chi H. Lee. Microwave Photonics. CRC Press, 2nd edition, 2013.

[8] V. J. Urick, Keith J. Williams, and Jason D. McKinney. Fundamentalsof Microwave Photonics. John Wiley and Sons, 2015.

[9] Pallab Bhattacharya. Semiconductor Optoelectronics Devices. PHILearning Private Limited, 2012.

[10] Zihang Zhu, Shanghong Zhao, Xuan Li, Kun Qu, and Tao Lin. Pho-tonic generation of frequency-sextupled microwave signal based on dual-polarization modulation without an optical filter. Optics and LaserTechnology, 22(1):1–6, 2017.



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