All posts by mbenkerumass

Methods of Calculation for Signal Envelope

The envelope of a signal is an important concept. When a signal is modulated, meaning that information is combined with or embedded in a carrier signal, the envelope follows the shape of the signal on it’s upper and lower most edges.

There are a number of methods for calculating an envelope. When given an in-phase and quadrature signal, the envelope is defined as:

E = sqrt(I^2  + Q^2).

This envelope, if plotted will contain the exact upper or lower edge of the signal. An exact envelope may be sought, depending on the level of detail required for the application.

Here, this data was collected as a return from a fiber laser source. We seek to identify this section of the data to determine if the return signal fits the description out of a number of choices. The exact envelope using the above formula is less useful for the application.

The MATLAB formula is used to calculate the envelope:

[upI, lowI] = envelope(I,x,’peak’);

And this is plotted below with the I and Q signals:


Here are two envelopes depicted without the signal shown. By selecting the range of interpolation, this envelope can be smoother. Typically it is less desirable for an envelope to contain so many carrier signals, as is the following where x=1000, the range of interpolation.


Further methods involving the use of filters may also be of consideration. Below, the I and Q signals are taken through a bandpass filter (to ensure that the data is from the desired frequency range) and finally a lowpass filter is applied to the envelope to remove higher frequency oscillation.


Semiconductor Growth Technology: Molecular Beam Epitaxy and MOCVD

The development of advanced semiconductor technologies presents one important challenge: fabrication. Two methods of fabrication that are being used to in bandgap engineering are Molecular Beam Epitaxy (MBE) and Metal organic chemical vapour deposition (MOCVD).

Molecular Beam Epitaxy uses high-intensity vacuums to fabricate compound semiconductor materials and compounds. Atoms or molecules containing the desired atoms are directed to a heated substrate. Molecular Beam Epitaxy is highly sensitive. The vacuums used make use of diffusion pumps or cryo-pumps; diffusion pumps for gas source MBE and cryo-pumps for solid source MBE. Effusion cells are found in MBE and allow the flow of molecules through small holes without collusion. The RHEED source in MBE stands for Reflection Hish Energy Electron Diffraction and allows for information regarding the epitaxial growth structure such as surface smoothness and growth rate to be registered by reflecting high energy electrons. The growth chamber, heated to 200 degrees Celsius, while the substrate temperatures are kept in the range of 400-700 degrees Celsius.

MBE is not suitable for large scale production due to the slow growth rate and higher cost of production. However, it is highly accurate, making it highly desired for research and highly complex structures.



MOCVD is a more popular method for growing layers to a semiconductor wafer. MOCVD is primarily chemical, where elements are deposited as complex chemical compounds containing the desired chemical elements and the remains are evaporated. The MOCVD does not use a high-intensity vacuum. This process (MOCVD) can be used for a large number of optoelectronic devices with specific properties, including quantum wells. High quality semiconductor layers in the micrometer level are developed using this process. MOCVD produces a number of toxic elements including AsH3 and PH3.

MOCVD is recommended for simpler devices and for mass production.



Heterostructures & Carrier Recombination

Heterojunction is the term for a region where two different materials interact. A Heterostructure is a combination of two or more materials. Here, we will explore several interesting cases.


The AlGaAs-InGaAs interaction is interesting due to the difference in energy bandgap levels. It was found that AlGaAs has a higher bandgap level, while InGaAs has a lower bandgap. By layering these two materials together with a stark difference in bandgap levels, the two materials make for an interesting demonstration of a heterostructure.

The layering of a smaller bandgap material between a wider bandgap material has an effect of trapping both electrons and holes. As shown on the right side of the below picture, the center region, made of AlGaAs exibits high concentrations of both electrons and holes. This leads to a higher rate of carrier recombination, which can generate photons.


Here, the lasing profile of the material under bias:






A commonly used group of materials is InGaAsP, InGaAs and InP. Unlike the above arrangements, these materials may be lattice-matched. Lattice-matching may be explored in depth later on.Simulations suggest low or non-existent recombination rates. Although this is a heterostructure, one can see that there are no jagged or sudden drastic movements in the conduction and valence band layers with respect to each other to create a discontinuity that may result in a high recombination rate.



Materials & Photogeneration Rate at 1550 nm

We now seek to understand how different materials respond and interact with light. Photogeneration is the rate at which electrons are created through the absorption of light.

A program is built in ATLAS TCAD to simulate a beam incident on a block of material. A PN junction is used, similar to previous iterations. An example of the code for the Photogeration Simulator will be provided at the end of this article.

The subject of photogeneration certainly can see a more thorough examination that is provided here. Consider this as an introduction and initial exploration.

GaAs-InP-GaAs PN Junction


Here we see that a cross section of this unintentionally doped InP region, sandwiched between a GaAs PN junction exhibits a level of photogeneration, while the GaAs regions do not.

Adding more layers of other materials, as well as introducing a bias of the structure, we notice that the InP region still exhibits the highest (only) level of photogeneration of the materials tested in this condition. Interestingly, this structure emits light under the conditions tested.


Also consider that a photogeneration effect may not be sought. If, for instance, a device is supposed to act as a waveguide, there will be no benefit to having a photogeneration effect, let alone losses in the beam that result from it.


InGaAsP-InP-InGaAs Heterostructure

A common set of materials for use in Photodetectors is InGaAsP, InP and InGaAs. This particular structure features a simple, n-doped InGaAsP, unintentionally doped InP and p-doped InGaAs. The absorption rate of InP was already demonstrated above. InGaAs proves also to exhibit absorption at 1500 nm.



go atlas

Title Photogeneration Simulator

#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 = GaAs NY = 20 acceptor = 1e17

region num=3 bottom thick = 0.5 material = InP NY = 10

region num=2 bottom thick = 0.5 material = GaAs NY = 20 donor = 1e17

#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 fermi

models region=2 srh optr print conmob fldmob srh optr fermi

models material=GaAs fldmob srh optr fermi print \

laser gainmod=1 las_maxch=200. \

las_xmin=-0.5 las_xmax=0.5 las_ymin=0.4 las_ymax=0.6 \

photon_energy=1.43 las_nx=37 las_ny=33 \

lmodes las_einit=1.415 las_efinal=1.47 cavity_length=200

beam     num=1 x.origin=0 y.origin=4 angle=270 wavelength=1550 min.window=-1 max.window=1

output band.param ramptime TRANS.ANALY photogen opt.intens e.mobility h.mobility band.param photogen opt.intens recomb u.srh u.aug u.rad flowlines

method newton autonr trap  maxtrap=6 climit=1e-6



solve    init

SOLVE B1=1.0

output band.param ramptime TRANS.ANALY photogen opt.intens e.mobility h.mobility band.param photogen opt.intens recomb u.srh u.aug u.rad flowlines

outf=diode_mb1.str master

tonyplot diode_mb1.str

method newton autonr trap  maxtrap=6 climit=1e-6

LOG outf=electrooptic1.log

solve vanode = 0.5

solve vanode = 1.0

solve vanode = 1.5

solve vanode = 2.0

solve vanode = 2.5

save outfile=diode_mb2.str

tonyplot diode_mb2.str

tonyplot electrooptic1.log


HF Antenna Matched Network for a Radio Broadcasting Station

The goal of this demonstration is to explain the importance of a matched network and the role of transmission lines (coax) for an HF Antenna matched network. This network is designed for the 20-meter band in the HF domain of the radio frequency region of the electromagnetic spectrum.

Consider you have an HF antenna load, which is positioned on a tower. The tower height is a consideration as a feed coax line will be connected to the antenna from the bottom (roughly) of the tower. Secondly, another coax line will be connected from the base of the tower to the radio station.

The reflection coefficient is the measure for an impedance matched network. A matched network will mean that loss will be minimal. SimSmith is a free tool that is useful for smith chart matching. In SimSmith, the load (left), transmission lines (as mentioned in the previous paragraph) and the radio are plotted on the smith chart.


The length chosen for T1 was chosen at 18.23 feet, which gives a clear shot for an impedance match towards the center using a stub transmission line.


We now add a shorted stub between both coax lines and adjust the length of the excess line until the impedance is matched at the radio station.


As shown above, the the excess length on the stub is about 6′. Plotting the SWR shows that the system is matched well for the whole band, meaning that this station is set up well for an HF radio broadcasting station for extra class amateur radio broadcasters.


Conduction & Valence Band Energies under Biasing (PN & PIN Junctions)

Previously, we discussed the effect of doping concentrations on the energy band gap. The conclusion of this process was that the doping concentration alone does not alter the band gap. The band gap is the difference between the conduction band and valence bands. Under biasing, the conduction and valence bands are in fact affected by doping concentration.

One method to explain how the doping level will influence the conduction band and valence band under bias is by demonstrating the difference between the energy bands of a PN Junction versus that of a PIN Junction. Simulations of both are presented below. The intermediate section found between the p-doped and n-doped regions of the PIN junction diode offer a more gradual transition between the two levels. A PN junction offers a sharper transition at the conduction and valence band levels simulatenously. A heterostructure, which is made of more than one material (which will have different band gaps) may produce even greater discontinuities. Depending on the application, a discontinuity may be sought (think, Quantum well), while in other situations, it may be necessary to smooth the transition between band levels for a desired result.

The conduction and valence bands are of great importance for determining the carrier concentrations and carrier mobilities in a semiconductor structure. These will be discussed soon.

PN Junction under biasing (conduction and valence band energies):


Code Used (PN Junction):

#TOP TO BOTTOM – Structure Specification
region num=1 bottom thick = 0.5 material = GaAs NY = 20 acceptor = 1e18
region num=2 bottom thick = 0.5 material = GaAs NY = 20 donor = 1e18


PIN Junction Biased:


PIN Junction Unbiased:


Code Used (PIN Junction):

#TOP TO BOTTOM – Structure Specification
region num=1 bottom thick = 0.5 material = GaAs NY = 20 acceptor = 1e18
region num=3 bottom thick = 0.2 material = GaAs NY = 10
region num=2 bottom thick = 0.5 material = GaAs NY = 20 donor = 1e18

Here, the carrier concentrations are plotted:


Energy Bandgaps

Previously, a PN Junction Simulator in ATLAS program was posted. Now, we will use and modify this program to explore more theory in respect to semiconductor materials, high speed electronics and optoelectronics.

The bandgap, as mentioned previously is the difference between the conduction band energy and valence band energy. The materials GaAs, InP, AlGaAs, InGaAs and InGaAsP are simulated and the bandgap values for each are estimated (just don’t use these values for anything important).

  • GaAs: ~ 1.2 eV
  • InP: ~ 1.35 eV
  • AlGaAs: ~ 1.8 eV
  • InGaAs: ~0.75 eV
  • InGaAsP: 1.1 eV


Here the conduction band and valence band are shown.


The structure used in the PN Junction Simulator is found below:

#TOP TO BOTTOM – Structure Specification
region num=1 bottom thick = 0.5 material = GaAs NY = 20 acceptor = 1e17
region num=3 bottom thick = 0.001 material = InP NY = 10
region num=4 bottom thick = 0.001 material = GaAs NY = 10
region num=5 bottom thick = 0.001 material = AlGaAs NY = 10 x.composition=0.3 grad.3=0.002
region num=6 bottom thick = 0.001 material = GaAs NY = 10
region num=7 bottom thick = 0.001 material = InGaAs NY = 10 x.comp=0.468
region num=8 bottom thick = 0.001 material = GaAs NY = 10
region num=9 bottom thick = 0.001 material = InGaAsP NY = 10 x.comp=0.145 y.comp = 0.317
region num=2 bottom thick = 0.5 material = GaAs NY = 20 donor = 1e17

Is the bandgap affected by doping the concentration level?

A quick simulation (below) will tell us that the answer is no. What might influence the bandgap however? And what could the concentration level change?


This (above) is a simulation of GaAs with layers at different doping concentration levels. The top is a contour of the bandgap, which is constant, as expected. The top right is a cross section of this GaAs structure (technically still a pn junction diode); the bandgap is still constant. The bottom two images are the donor and acceptor concentrations.

The bandgap energy E_g is the amount of energy needed for a valence electron to move to the conduction band. The short answer to the question of how the bandgap may be altered is that the bandgap energy is mostly fixed for a single material. In praxis however, Bandgap Engineering employs thin epitaxial layers, quantum dots and blends of materials to form a different bandgap. Bandgap smoothing is employed, as are concentrations of specific elements in ternary and quarternary compounds. However, the bandgap cannot be altered by changing the doping level of the material.

PN Junction Simulator in ATLAS

This post will outline a program for ATLAS that can simulate a pn junction. The mesh definition and structure between the anode and cathode will be defined by the user. The simulator plots both an unbiased and biased pn junction.

go atlas


#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 = GaAs NY = 20 acceptor = 1e17
region num=2 bottom thick = 0.5 material = GaAs NY = 20 donor = 1e17

#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 init outf=diode_mb1.str master
tonyplot diode_mb1.str

method newton autonr trap maxtrap=6 climit=1e-6
solve vanode = 2.5 name=anode
save outfile=diode_mb2.str
tonyplot diode_mb2.str

This program may also be useful for understanding how different materials interact between a PN junction. This simulation below is for a simple GaAs pn junction.

The first image shows four contour plots for the pn junction with an applied 2.5 volts. With an applied voltage of 2.5, the recombination rate is high at the PN junction, while there is low recombination throughout the unbiased pn junction. The hole and electron currents are plotted on the bottom left and right respectively.


Here is the pn junction with no biasing.


The beam profile can also be obtained:


ATLAS TCAD: Simulation of Frequency Response from Light Impulse

Recently a project was posted for a high speed photodetector. Part of that project was to develop a program that takes the frequency response of a light impulse. My thought is to create a program that can perform these tasks, including an impulse response for any structure.

Generic Light Frequency Response Simulator Program in ATLAS TCAD

The first part of the program should include all the particulars of the structure that is being simulated:

go atlas

[define mesh]

[define structure]

[define electrodes]

[define materials]

Then, the beam is defined. x.origin and y.origin describes from where the beam is originating on the 2D x-y plane. The angle shown of 270 degrees means that the beam will be facing upwards. One may think of this angle as starting on the right hand sixe of the x-y coordinate plane and moves clockwise. The wavelength is the optical wavelength of the beam and the window defines how wide the beam will be.

beam num=1 x.origin=0 y.origin=5 angle=270 wavelength=1550 min.window=-15 max.window=15

The program now should run an initial solution and set the conditions (such as if a voltage is applied to a contact) for the frequency response.


solve init
outf = lightpulse_frequencyresponse.str
LOG lightpulse_frequencyresponse.log

[simulation conditions such as applied voltage]

LOG off

Now the optical pulse is is simulated as follows:

LOG outf=transient.log

tonyplot transient.log

outf=lightpulse_frequencyresponse.str master onefile
log off

The optical pulse “transient.log” is simulated using Tonyplot at the end of the program. It is a good idea to separate transient plots from frequency plots to ensure that these parameters may be chosen in Tonyplot. Tonyplot does not give the option to use a parameter if it is not the object that is being solved before saving the .log file.

log outf=frequencyplot.log
FOURIER INFILE=transient.log OUTFILE=frequencyplot.log T.START=0 T.STOP=20E-9 INTERPOLATE
tonyplot frequencyplot.log
log off

output band.param ramptime TRANS.ANALY photogen opt.intens e.mobility h.mobility band.param photogen opt.intens recomb u.srh u.aug u.rad flowlines

save outf=lightpulse_frequencyresponse.str
tonyplot lightpulse_frequencyresponse.str


Now you can focus on the structure and mesh for a light impulse frequency response. Note that adjustments may be warranted on the light impulse and beam.

And so, here is a structure simulation that could be done easily using the process above.



High Speed UTC Photodetector Simulation with Frequency Response in TCAD

The following is a TCAD simulation of a high speed UTC photodetector. An I-V curve is simulated for the photodetector, forward and reverse. A light beam is simulated to enter the photodetector. The photo-current response to a light impulse is simulated, followed by a frequency response in TCAD.



I-V Curve


Beam Simulation Entering Photodetector:



Light Impulse:


Frequency Response in ATLAS:


The full project (pdf) is here: ece530_final_mbenker


Sinusoidal and Exponential Sequences, Periodicity of Sequences

Continuing our discussion on discrete-time sequences, we now come to define exponential and sinusoidal sequences. The general formula for a discrete-time exponential sequence is as follows:

x[n] = Aα^n.

This exponential behaves differently according to the value of α. If the sequence starts at n=0, the formula is as follows:

x[n] = Aα^n * u[n].


If α is a complex number, the exponential function exhibits newer characteristics. The envelope of the exponential is |α|. If |α| < 1, the system is decaying. If |α|> 1, the system is growing.


When α is complex, the sequence may be analyzed as follows, using the definition of Euler’s formula to express a complex relationship as a magnitude and phase difference.

Captu56 ma

Where ω0 is the frequency and φ is the phase, for n number of samples, a complex exponential sequence of form Ae^jw0n may be considered as a sinusoidal sequence for a set of frequencies in an interval of 2π.

A sinusoidal sequence is defined as follows:

x[n] = Acos(ω0*n + φ), for all n, and A, φ are real constants.

Periodicity for discrete-time signals means that the sequence will repeat itself for a certain delay, N.

x[n] = x[n+N] : system is periodic.

t = (-5:1:15)’;

impulse = t==0;
unitstep = t>=0;
Alpha1 = -0.5;
Alpha2 = 0.5;
Alpha3 = 2.5;
Alpha4 = -2.5;
cAlpha1 = -0.5 – 0.5i;
cAlpha2 = 0.5 + 0.5i;
cAlpha3 = 2.5 -2.5i;
cAlpha4 = -2.5 + 2.5i;
A = 1;

Exp1 = A.*unitstep.*Alpha1.^t;
Exp2 = A.*unitstep.*Alpha2.^t;
Exp3 = A.*unitstep.*Alpha3.^t;
Exp4 = A.*unitstep.*Alpha4.^t;

cExp1 = A.*unitstep.*cAlpha1.^t;
cExp2 = A.*unitstep.*cAlpha2.^t;
cExp3 = A.*unitstep.*cAlpha3.^t;
cExp4 = A.*unitstep.*cAlpha4.^t;

stem(t, impulse)

stem(t, unitstep)
title(‘Unit Step’)
stem(t, cExp1)
title(‘Exponential: alpha = -0.5 – 0.5i’)

stem(t, cExp2)
title(‘Exponential: alpha = 0.5 + 0.5i’)

stem(t, cExp3)
title(‘Exponential: alpha = 2.5 -2.5i’)

stem(t, cExp4)
title(‘Exponential: alpha = -2.5 + 2.5i’)
stem(t, Exp1)
title(‘Exponential: alpha = -0.5’)

stem(t, Exp2)
title(‘Exponential: alpha = 0.5’)

stem(t, Exp3)
title(‘Exponential: alpha = 2.5’)

stem(t, Exp4)
title(‘Exponential: alpha = -2.5’)



Discrete-Time Impulse and Unit Step Functions

Discrete-Time Signals are understood as a set or sequence of numbers. These sequences possess magnitudes or values at a given index.

One mark of Discrete-Time Signals is that the index value is an integer. Thus, the sequence will have a magnitude or value for a whole number index such as -5, -4, 0, 6, 10000, etc.

A discrete-time signal represented as a sequence of numbers takes the following form:

x[n] = {x[n]},          -∞ < n < ∞,

where n is any real integer (the index).

An analog representation describes values of a signal at time nT, where T is the sampling period. The sampling frequency is the inverse of the sampling period.

x[n] = X_a(nT),      -∞ < n < ∞.


Common Sequences

Both a very simple and important sequence is the unit sample sequence, “discrete time impulse” or simply “impulse,” equal to 1 only at zero and equal to zero otherwise.


The discrete time impulse is used to describe an entire system using a delayed impulse. An entire sequence may also be shifted or delayed using the following relation:

y[n] = x[n – n0],

where n0 is an integer (which is the increment of indices by which the system is delayed. The impulse function delayed to any index and multiplied by the value of the system at that index can describe any discrete-time system. The general formula for this relationship is,


The unit step sequence is related to the unit impulse. The unit step sequence is a set of numbers that is equal to zero for all numbers less than zero and equal to one for numbers equal and greater than zero.


The unit step sequence is therefore equal to a sequence of delta impulses with a zero and greater delay.

u[n] = δ[n] + δ[n-1] + δ[n-2] + . . .


The unit impulse can also be represented by unit step functions:

δ[n] = u[n] – u[n-1].

Below I’ve plotted both the impulse and unit step function in matlab.


t = (-10:1:10)';

impulse = t==0;
unitstep = t>=0;

stem(t, impulse)
stem(t, unitstep)
title('Unit Step')



Image Resolution

Consider that we are interested in building an optical sensor. This sensor contains a number of pixels, which is dependent on the size of the sensor. The sensor has two dimensions, horizontal and vertical. Knowing the size of the pixels, we will be able to find the total number of pixels on this sensor.

The horizontal field of view, HFOV is the total angle of view normal from the sensor. The effective focal length, EFL of the sensor is then:

Effective Focal Length: EFL = V / (tan(HFOV/2)),

where V is the vertical sensor size in (in meters, not in number of pixels) and HFOV is the horizontal field of view. Horizontal field of view as an angled is halved to account that HFOV extends to both sizes of the normal of the sensor.

The system resolution using the Kell Factor: R = 1000 * KellFactor * (1 / (PixelSize)),

where the Pixel size is typically given and the Kell factor, less than 1 will approximate a best real case result and accounts for aberrations and other potential issues.

Angular resolution: AR = R * EFL / 1000,

where R is the resolution using the Kell factor and EFL is the effective focal length. It is possible to compute the angular resolution using either pixels per millimeter or cycles per millimeter, however one would need to be consistent with units.

Minimum field of view: Δl = 1.22 * f * λ / D,

which was used previously for the calculation of the spatial resolution of a microscope. The minimum field of view is exactly a different wording for the minimum spatial resolution, or minimum size resolvable.

Below is a MATLAB program that computed these parameters, while sweeping the diameter of the lens aperture. The wavelength admittedly may not be appropriate for a microscope, but let’s say that you are looking for something in the infrared spectrum. Maybe you are trying to view some tiny laser beams that will be used in the telecom industry at 1550 nanometer.

Pixel size: 3 um. HFOV: 4 degrees. Sensor size: 8.9mm x 11.84mm.


Spatial Resolution of a Microscope

Angular resolution describes the smallest angle between two objects that are able to be resolved.

θ = 1.22 * λ / D,

where λ is the wavelength of the light and D is the diameter of the lens aperture.

Spatial resolution on the other hand describes the smallest object that a lens can resolve. While angular resolution was employed for the telescope, the following formula for spatial resolution is applied to microscopes.

Spatial resolution: Δl = θf = 1.22 * f * λ / D,

where θ is the angular resolution, f is the focal length (assumed to be distance to object from lens as well), λ is the wavelength and D is the diameter of the lens aperture.



The Numerical Aperture (NA) is a measure of the the ability to of the lens to gather light and resolve fine detail. In the case of fiber optics, the numerical aperture applies to the maximum acceptance angle of light entering a fiber. The angle by the lens at its focus is θ = 2α. α is shown in the first diagram.

Numerical Aperture for a lens: NA = n * sin(α),

where n is the index of refraction of the medium between the lens and the object. Further,

sin(α) = D / (2d).

The resolving power of a microscope is related.

Resolving power: x = 1.22 * d * λ / D,

where d is the distance from the lens aperture to the region of focus.


Using the definition of NA,

Resolving power: x = 1.22 * d * λ / D = 1.22 * λ / (2sin(α)) = 0.61 * λ / NA.


Telescope Resolution & Distance Between Stars using the Rayleigh Limit

Previously, the Rayleigh Criterion and the concept of maximum resolution was explained. As mentioned, Rayleigh found this formula performing an experiment with telescopes and stars, exploring the concept of resolution. This formula may be used to determine the distance between two stars.

θ = 1.22 * λ / D.

Consider a telescope of lens diameter of 2.4 meters for a star of visible white light at approximately 550 nanometer wavelength. The distance between the two stars in lightyears may be calculated as follows. The stars are approximately 2.6 million lightyears away from the lens.

θ = 1.22 * (550*10^(-9)m)/(2.4m)

θ =2.80*10^(-7) rad

Distance between two objects (s) at a distance away (r), separated by angle (θ): s = rθ

s = rθ = (2.0*10^(6) ly)*(2.80*10^(-7)) = 0.56 ly.

This means that the maximum resolution for the lens size, star distance from the lens and wavelength would be that two stars would need to be separated at least 0.56 lightyears for the two stars to be distinguishable.