Ring Resonators for Wavelength Division Multiplexing

The ring resonator is a rather simple passive photonic component, however the uses of it are quite broad.

The basic concept of the ring resonator is that for a certain resonance frequency, those frequencies entering port 1 on the diagram below will be trapped in the ring of the ring resonator and exit out of port 3. Frequencies that are not of the resonance frequency will pass through to port 2.

ringres

Ring resonators can be used for Wavelength Division Multiplexing (WDM). WDM allows for the transmission of information allocated to different wavelengths simultaneously without interference. There are other methods for WDM, such as an Asymmetric Mach Zehnder Modulator.

Here I present one scheme that will utilize four ring resonators to perform wavelength division multiplexing. The fifth output will transmit the remaining wavelengths after removing the chosen wavelengths dependent on the resonating frequency (and actually, the radius) of the ring resonators.

wdm

 

 

 

Quantum Well: InP-InGaAsP-InP

Quantum wells are widely used in optoelectronic and photonic components and for a variety of purposes. Two materials that are often used together are InP and InGaAsP. Two different models will be presented here with simulations of these structures. The first is an InP pn-junction with a 10 nm InGaAsP (unintentionally doped) layer between. The second is an InP pn-junction with 10 nm InGaAsP quantum wells positioned in both the positive and negative doped regions.

Quantum Well between pn-junction

quantum well

The conduction band and valence band energies are depicted below for the biased case:

quantum well2

The conduction current vector lines:

qwell1

ATLAS program:

go atlas
Title Quantum Wells
# Define the mesh
mesh auto
x.m l = -2 Spac=0.1
x.m l = -1 Spac=0.05
x.m l = 1 Spac=0.05
x.m l = 2 Spac =0.1
#TOP TO BOTTOM – Structure Specification
region num=1 bottom thick = 0.5 material = InP NY = 10 acceptor = 1e18
region num=3 bottom thick = 0.01 material = InGaAsP NY = 10 x.comp=0.1393  y.comp = 0.3048
region num=2 bottom thick = 0.5 material = InP NY = 10 donor = 1e18
# Electrode specification
elec       num=1  name=anode  x.min=-1.0 x.max=1.0 top
elec       num=2  name=cathode   x.min=-1.0 x.max=1.0 bottom

#Gate Metal Work Function
contact num=2 work=4.77
models region=1 print conmob fldmob srh optr
models region=2 srh optr
material region=2

#SOLVE AND PLOT
solve    init outf=diode_mb1.str master
output con.band val.band e.mobility h.mobility band.param photogen opt.intens recomb u.srh u.aug u.rad flowlines
tonyplot diode_mb1.str
method newton autonr trap  maxtrap=6 climit=1e-6
solve vanode = 2 name=anode
save outfile=diode_mb2.str
tonyplot diode_mb2.str
quit
Quantum Well layers inside both p and n doped regions of the pn-junction
Structure:
qwell3
Simulation results:
qwell2
#TOP TO BOTTOM – Structure Specification
region num=1 bottom thick = 0.25 material = InP NY = 10 acceptor = 1e18
region num=3 bottom thick = 0.01 material = InGaAsP NY = 10 x.comp=0.1393  y.comp = 0.3048
region num=4 bottom thick = 0.25 material = InP NY = 10 acceptor = 1e18
region num=2 bottom thick = 0.25 material = InP NY = 10 donor = 1e18
region num=6 bottom thick = 0.01 material = InGaAsP NY = 10 x.comp=0.1393  y.comp = 0.3048
region num=2 bottom thick = 0.25 material = InP NY = 10 donor = 1e18

Capacitance and Parallel Plate Capacitors

Capacitance relates two fundamental electric concepts: charge and electric potential. The formula that relates the two is Capacitance = charge / electric_potential.

The term equipotential surface refers to how a charge, if moved along a particular path or surface, the work done on the field is equal to zero. If there are many charges along the surface of a conductor (along an equipotential surface), then the potential energy of the charged conductor will be equal to 1/2 multiplied by the electric potential φ and the integral of all charges along this surface.

Ue = ½ φ ∫ dq.

Given a scenario in which both charge and electric potential are related, we may introduce capacitance. The following formula proves important for calculating the energy of a charged conductor:

Ue = ½ φ q = ½ φ2 C = q2 / (2C).

A parallel plate capacitor is a system of metal plates separated by a a dielectric. One plate of the capacitor will be positively charged, while the other is negatively charged. The potential difference and charge on the capacitor places causes a storage of energy between the two plates in an electric field.

caps

Electric Potential and Electric Potential Energy

Electric potential can be summarized as the work done by an electric force to move a charge from one point to another. The units are in Volts. Electric potential is not dependent on the shape of the path that the work is applied. Being a conservative system, the amount of energy required to move a charge in a full circle, to return it back to where it started will be equal to zero.

The work of an electrostatic field takes the formula

W12 = keqQ(1/r1 – 1/r2),

which is found by integrating the the charge q times the electric field. The work of an electrostatic field also contains both the electric potential and electric potential energy. Electric potential energy, U is equal to the electric potential φ multiplied by the charge q. Electric potential energy is a difference of potentials, while electric potential uses the exact level of electric potential in the given case.

0001

To calculate electric potential energy, it is convenient to assume that the potential energy is zero at a distance of infinity (and surely it should be). In this case, we can write the electric potential energy as equal to the work needed to move a charge from point 1 to infinity.

0010

We’ll consider a quick application related to both the dipole moment and the electric potential. The dipole potential takes the formula in the figure below. Dipole potential decreases faster with distance r than it would for a point charge.

0011

Dipole Moment

Consider we have both a positive and negative charge, separated by a distance. When applying supperposition of the electric force and electric field generated by the two charges on a target point, it is said that the positive and negative charges create an effect called a dipole moment. Let’s consider a few example of how an electric field will be generated for a point charge in the presence of both a positive and negative charge. Molecules also often have a dipole moment.

Here, the target point is at distance b at the center between the negative and positive charges. Where both charges are of the same magnitude, both the vertical attraction and repulsion components are cancelled, leaving the electric field to be generated in a direction parallel to the axis of the two charges.

Capture

Now, we’ll consider a target point along the axis of the two charges. Remember that a positive charge will produce an electric force and electric field that radiates from itself outward, while the force and field is directed inwards towards a negative charge. We can expect then, that the electric field will be different on either side. We can expect that the side of the positive charge will repel and the negative side will attract. This works, because the distance inverse proportionality is squared, making it so that the effect from the other charge will be less. This is a dipole.

Given how a dipole functions, it would be nice to have a different set of formulas and a more refined approach to solving electric field problems with dipoles. The dipole moment p is found using the formula, p=qI with units Couolumb*meter. I is the vector which points from the negative charge to the positive charge. The dipole moment is drawn as one point at the center of the dipole with vector I through it.dipole

In order to treat the two charges as a center of a dipole, there should be a minimum distance between the dipole and the target point. The distance between the dipole and the target should be much larger than the length l of the magnitude of vector I.

dipole2

Finally, the formula for these electric fields using a dipole moment are

E1 = 2kep/b13

E2 = 2kep/b23

Electric Force & Electric Field

While the electric force describes the exertion of one charge or body to another, we also have to remember that the two objects do not need to be touching physically for this force to be applied. For this reason, we describe the force that is being exerted through empty space (i.e. where the two objects aren’t touching) as an electric field. Any charge or body or thing that exerts an electrical force, generated most importantly by the distance between the objects and the amount of charge present, will generate an electric field.

The electric field generated as a result of two charges is directly proportional to the electric force exerted on a charge, or Coulomb force and inversely proportional to the charge of the particle. In other words, if the Coulomb force is greater, then the electric field will be stronger, but it will also be smaller if the charge it is applied to is smaller. Coulomb force as mentioned previously is inversely proportional to the distance between the charges. The electric field, E then uses the formula E = F/q and the units are Volts per meter.

By combining both Coulomb’s Law and our definition for the electric field, the electric field can be written as

E1 = ke * q1/r2 er

where er again is the unit vector direction from charge q1.

Capturl

When drawing electric field lines, there are three rules pay attention to:

  1. The direction is tangent to the field line (in the direction of flow).
  2. The density of the lines is proportional to the magnitude of the electric field.
  3. Vector lines emerge from positive charges and sink towards negative charges.

CapturX

Adding electric fields to produce a resultant electric field is simple, thanks to the property of superposition which applies to electric fields. Below is an example of how a resultant electric field will be calculated geometrically. The direction of each individual field from the charges is determined by the polarity of the charge.

CapturMPNG

Coulomb Force

Electric charge is important in determining how a body or particle will behave and interact electromagnetically. It is also key for understanding how electric fields, electric potentials and electromagnetic waves come into existence. It starts with the atom and it’s number of protons and electrons.

Charges are positive or negative. In a neutral atom, the number of protons in a nucleus is equal to the number of electrons. When an atom loses or gains an electron from this state, it becomes a negatively or positively charged ion. When bodies or particles exhibit a net charge, either positive or negative, an electric force arises. Charges can be caused by friction or irradiation. Electrostatic force functions similar to the gravitational force – in fact the formulas look very similar! The difference between the two is most importantly that electrostatic force can be attraction or repulsion, but gravitational force is always attraction. However for small bodies, the electrostatic force is primary and the gravitational force is negligible.

Charles Coloumb conducted experiments around 1785 to understand how electric charges interact. He devised two main relations that would become Coulomb’s Law:

The magnitude of the force between two stationary point charges is

  1. proportional to the product of the magnitude of the charges and
  2. inversely proportional to the square of the distance between the two charges.

The following expression describes how one charge will exert a force on another:

coulomb

The unit vector in the direction of charge 1 to charge 2 is written as e12 and the position of the two numbers indicates the direction of the force, moving from the first numbered position to the second. Reversing the direction of the force will result in a reversed polarity, F12 = -F21.

The coefficient ke will depend on the unit system and is related to the permittivity:

coulomb2

The permittivity of vacuum, ε0 = 8.85*10^(-12)  C^2N*m^2.

Coulomb forces obey superposition, meaning that a series of charges may be added linearly without effecting their independent effects on it’s ‘target’ charge. Coulomb’s Law extends to bodies and non-point charges to describe an applied electrostatic force on an object; the same first equation may be used in this scenario.

Noise Figure

Electrical noise is unwanted alterations to a signal of random amplitude, frequency and phase. Since RADAR is typically done at microwaves frequencies, the noise contribution of most RADAR receivers is highest at the first stages. This is mostly thermal noise (Johnson noise). Each component of a receiver has its own Noise Figure (dB) which is typically kept low through the use of a LNA (Low Noise amplifier). It is important to know that all conductors generate thermal noise when above absolute zero (0K).

Noise Power

Noise Power is the product of Boltzman’s constant, temperature in Kelvin and receiver bandwidth (k*t0*B). This is typically also expressed in dBm. This value is -174 dBm at room temperature  for a 1 Hz bandwidth. If a different receiver bandwidth is present, you can simply add the decibel equivalent of the bandwidth to this value. For example, at a 1MHz bandwidth, the bandwidth ratio is 60 dB (10*log(10^6) = 60). This value can be added to the standard 1Hz bandwidth to arrive at -114 dBm. For a real receiver, this number is scaled by the Noise Figure.

 

The Noise Figure is defined as 10*log(Na/Ni) where Na is the noise output of an actual receiver and Ni is the noise output of an ideal receiver. Alternatively these can be converted to dB and subtracted. It can also be defined as the rate at which SNR degrades. For systems on earth, Noise Figure is quite useful as temperature tends to stay around 290K (room temperature). However, for satellite communication, the antenna temperature tends to be colder than 290K and therefore effective noise temperature would be used instead.

Noise Factor is the linear equivalent of Noise Figure. For cascaded systems, the noise factor gradually decreases and decreases as shown. This explains why in a receiver chain, the initial components have a much higher effect on the Noise Figure.

noisefactor

Noise Figure is a very important Figure of Merit for detection systems where the input signal strength is unknown. For example, it is necessary to decrease the Noise Figure in the electromagnetic components of a submarine in order to detect communication and RADAR signals.

Dispersion in Optical Fibers

Dispersion is defined as the spreading of a pulse as it propagates through a medium. It essentially causes different components of the light to propagate at different speeds, leading to distortion. The most commonly discussed dispersion in optical fibers is modal dispersion, which is the result of different modes propagating within a MMF (multimode fiber). The fiber optic cable supports many modes because the core is of a larger diameter than SMF (single mode fibers). Single mode fibers tend to be used more commonly now due to decreased attenuation and dispersion over long distances, although MMF fibers can be cheaper over short distances.

Let’s analyze modal dispersion. When the core is sufficiently large (generally the core of a SMF is around 8.5 microns or so), light enters are different angles creating different modes. Because these modes experience total internal reflection at different angles, their speeds differ and over long distances, this can have a huge effect. In many cases, the signal which was sent is completely unrecognizable. This type of dispersion limits the bandwidth of the signal. Often GRIN (graded index) fibers are employed to reduce this type of dispersion by gradually varying the refractive index of the fiber within the core so that it decreases as you move further out. As we have learned, the refractive index directly influences the propagation velocity of light. The refractive index is defined as the ratio of the speed of light to the speed of the medium. In other words, it is inversely proportional to the speed of the medium (in this case silica glass).

modal

In order to mitigate the effects of intermodal distortion in multimode fibers, pulses are lengthened to overlap components of different modes, or even better to switch to Single mode fibers when it is available.

The next type of dispersion is chromatic dispersion. All lasers suffer from this effect because no laser is comprised of a single frequency. Therefore, different wavelengths will propagate at different speeds. Sometimes chirped Bragg gratings are employed to compensate for this effect. Doped fiber lasers and solid state lasers tend to have much thinner linewidths than semiconductor PIN lasers and therefore tend to have less chromatic dispersion, although the semiconductor lasers has several advantages such as lesser cost and smaller size.

Another dispersion type is PMD (Polarization mode dispersion) which is caused by different polarizations travelling at different speeds within a fiber. Generally, these travel at the same speed however spreading of pulses can be caused by imperfections in the material.

For SMF fibers, it is important to cover waveguide dispersion. It is important to note that since the cladding of the fiber is doped differently than the core, the core has a higher refractive index than the cladding (doping with fluorine lowers refractive index and doping with germanium increases it). As we know, a lower refractive index indicates faster speed of propagation. Although most of the light stays within the core, some is absorbed by the cladding. Over long distances this can lead to greater dispersion as the light travels faster in the core leading to different propagation velocities.

RF Over Fiber Links

The basic principle of an RF over Fiber link is to convey a radio frequency electrical signal optically through modulation and demodulation techniques. This has many advantages including reduced attenuation over long distances, increased bandwidth capability, and immunity to electromagnetic interference. In fact, Rf over fiber links are essentially limitless in terms of distance of propagation, whereas coaxial cable transmission lines tend to be limited to 300 ft due to higher attenuation over distance.

The simple RFoF link comprises of an optical source, optical modulator, fiber optic cable and a receiver.

rfof

The RF signal modulates the optical signal at its frequency (f_opt) with sidebands at the sum and difference of the RF frequency and optical signal frequency. These beat against the carrier in the photodetector to reproduce and electrical RF signal. The above picture shows amplitude modulation and direct detection method. Also, impedance matching circuitry is generally included to match the ports of the modulator to the demodulator as well as amplifiers.

Before designing an RFoF link, it must be essential to bypass a transmission line in the first place. Will the system benefit from having a lower size and weight or immunity to electromagnetic interference? Is a wide bandwidth required? If not, this sort of link may not be necessary. It also must be determined the maximum SWaP of all the hardware at the two ends of the link. Another important consideration is the temperature that the link will be exposed to (or even pressure, humidity or vibration levels) that the link will be exposed to. The bandwidth of the RF and distance of propagation must be considered, finally.

The Following Figures of Merit can be used to quantify the RFoF link:

Gain

In dB, this is defined as the Signal out (in dBm) – Signal in (dBm) or 10log(g) where g is the small signal gain (gain for which the amplitude is small enough that there is no amplitude compression)

Noise Figure

For RADAR and detection systems where the input signal strength is unknown, Noise Figure is more important than SNR. NF is the rate at which SNR degrades from input to output and is given as N_out – kTB – Gain (all in dB scale).

Dynamic Range

It is known that the Noise Floor defines the lower end of dynamic range. The higher end is limited by spurious frequencies or amplitude compression. The difference between the highest acceptable and lowest acceptable input power is the dynamic range.

For example, if defined in terms of full compression, the dynamic range would be (in dB scale) : S_in.max – MDS. where MDS is the minimum detectable signal strength power.

Scattering Parameters

Scattering parameters are frequency dependent parameters that define the loss or gain at various ports. For two port systems, this forms a 2×2 matrix. In most Fiber Optic links, the backwards isolation S_12 is equal to zero due to the functionality of the detectors and modulators (they cannot perform each other’s functions). Generally the return losses at port 2 and 1 are what are specified to meet the system requirements.

 

 

Erbium Doped Fiber Amplifiers (EDFA)

EDFA

The above figure demonstrates the attenuation of optical fibers relative to wavelength. It can be seen that Rayleigh Scattering is more prevalent at higher frequencies. Rayleigh scattering occurs when minute changes in density or refractive index of optical fibers is present due to manufacturing processes. This tends to scatter either in the direction of propagation within the core or not. If it is not, this leads to increased attenuation. This accounts for 96% of attenuation in optical fibers. It can also be noted that lattice absorption varies wildly with the wavelength of light. From the graph, it is apparent that 1550 nm wavelength this value (and also Rayleigh Scattering) is quite low. It is for this reason that 1550 nm is a common wavelength of propagation with silica glass optical fibers. Although this wavelength allows for greater options in design, shorter wavelengths (such as 850 nm) are also used when distance of propagation is short. However, 1550 is the common wavelength due to the development of dispersion shifted fibers as well as something called the EDFA (Erbium doped fiber amplifier).

EDFAs operate around the 1550 nm region (1530 to 1610 nm) and work based on the principle of stimulated emission, in which a photon is emitted within a optical device when another photon causes electrons and holes to recombine. The stimulated emission creates a photon of the same size and in the same direction (coherent light). The EDFA acts as an amplifier, boosting the intensity of light with a heavily doped core (erbium doped). As discussed earlier, the lowest power loss for silica fibers tends to occur at 1550 nm, which is the wavelength that this stimulated emission occurs. The excitation, however, occurs at 980 or 1480 nm, which is shown to have high loss.

The advantages of the EDFA is high gain and availability to operate in the C and L bands of light, It is also not polarization dependent and has low distortion at high frequencies. The major disadvantage is the requirement of optical pumping.

EDFA

RSoft Tutorials 9. Using Real Materials and Multilayer Structures

Rsoft comes with a number of libraries for real materials. To access these materials, we can add them at any time from the Materials button on the side. However, to build a Multilayer structure that can utilize many materials, select “Multilayer” under 3D Structure Type.

rsoft17.2

Now, select “Materials…” to add desired materials. Move through the RSoft Libraries to chose a material and use the button in the top right (not the X button, silly) to use the material in the project. Now select OK to be brought back to the Startup Window, where we must now design a layered structure using these materials. Note that while building the layers, you can add more materials.

rsoft17.1

Selecting “Edit Layers…” on the Startup window brings you to the following window. Here, you can define your layers by selecting “New Layer”. Enter the Height and Material of the layer and select “Accept Layer” and repeat the process until the structure is finished. Select OK when done and select OK on the Startup window if all other settings are complete. This is my structure. Note that my structure size adds up to 1. Remember what the size of your layers are.

rsoft17.3

Now, design the shape of the structure. I’ve made a rectangular waveguide. What is also important to consider is where the beam should enter the structure. By default, the beam is focused across the entire structure. In the case where a particular layer is meant to be a waveguide, this should be reduced in size. By remembering the sizes of the layers however it will not be difficult to aim the beam at a particular section of the waveguide. For my structure, I will aim my beam at the 0.2 GaInAsP layer. The positioning, width, height, angle and more of the launch beam can be edited in the “Launch Parameters” window, accessible through “Launch Fields” on the right side.

rsoft17.4

Finally, run a simulation with your structure!

rsoft17.5

rsoft17.7

 

 

 

 

 

 

 

Rsoft Tutorials 8. Air Gaps

There are cases where you may want to simulate a region of air in between two components. A simple way of approaching this task is by creating a region with the same refractive index as air. The segment between the two waveguides (colored in gray) will serve as the “air” region. Right-click on the segment to define properties and under “Index Difference”, chose the value to be 1 minus the background index.

rsoft14.1

Properties for the segment:

rsoft14.2

Symbol Table Editor:

rsoft14.3

Notice that in the “air” region, the pathway monitor detects the efficiency to be zero, though the beam reconvenes in the waveguide, if the gap is short and the waveguide continues at the same angle, but with losses.

rsoft14.0

 

Rsoft Tutorials 7. Index Grating

Index grating is a common method to alter the frequency characteristics of light. In Rsoft, a graded index component is found under the “Index Taper” tab when right-clicking on a component. By selecting the tab “Tapers…”, one can create a new index taper.

rsoft12.1

Here, the taper is called “User 1” and defined by an equation step(M*z), with z being the z-coordinate location.

rsoft12.2

Selecting “Test” on the User Taper Editor will plot the index function of the tapered component:

rsoft12.6

The index contour is plotted below:

rsoft12.5

Here, the field pattern:

rsoft12.4

Light contour plot:

rsoft12.3

 

Rsoft Tutorials 6. Multiple Launch Fields, Merging Parts

Launch Fields define where light will enter a photonic device in Rsoft CAD. An example that uses multiple launch fields is the beam combiner.

rsoft11.1

rsoft11.2

 

On the sidebar, select “Edit Launch Fields”. To add a new lauch, select New and chose the pathway. A waveguide will be selected by default. Moving the launch to a new location however will place it elsewhere. Input a parameter other than “default” to change the location, and other beam parameters.

rsoft11.5

Choosing “View Launch” will plot the field amplitude of the launches. For the plot below, the third launch was removed.

rsoft11.4

Merging Waveguides

Right-clicking on the structure will give the option to chose the “Combine Mode.” Be sure that Merge is selected to allow waveguides to combine.

rsoft11.3

 

Electrical Engineering Students at University of Massachusetts Dartmouth