Optics Primer Series Stable

OptiX Optics, from first principles to all-optical logic gates Foundations: the mathematics of light. The complex-amplitude language and the maths each optics idea needs.. Write a monochromatic field as a complex amplitude and read its phase.. Map each optics tool (phasors, vector calculus, Fourier, linear algebra) to its prerequisite.. State the engineering sign convention used throughout the course.. Why complex amplitudes, Phasors and the exponential, The maths-to-optics map, Conventions Module 0, Foundations: the mathematics of light OptiX (codename LUMEN), Module 0. Source language: English (authoritative). A short pointer module: it sets the language and conventions the rest of the course assumes, and maps each tool to where you already learned it. Learning outcomes Write a monochromatic field as a complex amplitude and read its phase.. Map each optics tool to its mathematical prerequisite.. State the sign convention used throughout OptiX. 1. Why complex amplitudes Concept. Optics is the arithmetic of waves, and waves carry two numbers at every point: how big they are and where they are in their cycle, the amplitude and the phase. Tracking both with sines and cosines is clumsy; a single complex number carries both at once. This is not a trick, it is the natural bookkeeping of anything that oscillates. Mathematics. A monochromatic field is written say: E of r and t equals the real part of E tilde of r times e to the minus i omega t; and E tilde equals its magnitude times e to the i phi. The modulus $|\tilde{E}|$ is the amplitude; the argument $\phi$ is the phase. Multiplying by $e^{i\delta}$ advances the phase by $\delta$ without touching the amplitude, which is exactly what a delay, a path, or a phase shifter does. Hands-on check. In the phasor view, confirm that adding two unit phasors at angle $\delta$ apart gives a resultant of length $2\cos(\delta/2)$. 2. Phasors and the exponential A phasor is the complex amplitude drawn as an arrow in the plane: length is amplitude, angle is phase. Adding waves of the same frequency is adding arrows tip to tail. The whole of interference (Module 2) is this addition followed by taking the modulus squared, because the detector reports $|\tilde{E}|^2$. 3. The maths-to-optics map If any row is shaky, revisit it before the named module; the maths is layered, not gating. 4. Conventions OptiX uses the time factor $e^{-i\omega t}$ (the engineering convention is $e^{+j\omega t}$; both appear in the literature and differ only by the sign of $i$). A forward-travelling plane wave is therefore $e^{i(kz-\omega t)}$. Intensity is always the time-averaged squared field, $I\propto|\tilde{E}|^2$. Hold these fixed and every later sign will land correctly. Appendix: References Hecht, E. (2017). *Optics* (5th ed.). Boston: Pearson. https://www.pearson.com/en-us/subject-catalog/p/optics/P200000006793/9780137526420 Saleh, B. E. A., and Teich, M. C. (2019). *Fundamentals of Photonics* (3rd ed.). Hoboken: Wiley. https://www.wiley.com/en-us/Fundamentals+of+Photonics%2C+3rd+Edition-p-9781119506874 Related CCI capabilities Computer Architecture (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/computer-architecture/). Optics Primer Series (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/optics/). Maths Refresher Series, Finance (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/maths-finance/). System Dynamics (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/system-dynamics/). CCI Lab: Run it, build with it, read the thinking, reuse the data. (https://www.cambridgecyberinternational.com/en/insights/lab/) Electromagnetic waves and the nature of light. Maxwell's equations give a transverse wave at speed c/n carrying energy.. Derive the wave equation from Maxwell's equations in a source-free medium.. Relate wavelength, frequency, refractive index and the speed of light.. Identify intensity as the time-averaged squared field.. Maxwell to the wave equation, Plane waves, Energy, intensity, the Poynting vector, Photons in one paragraph Module 1, Electromagnetic waves and the nature of light OptiX (codename LUMEN), Module 1. Source language: English (authoritative). This lesson follows the course spine: the concept in plain language, then the precise mathematics, then a hands-on check the learner runs in the engine. Learning outcomes On finishing this module you can: Derive the wave equation from Maxwell's equations in a source-free medium.. Relate wavelength, frequency, refractive index, and the speed of light, and predict what changes when light crosses into a denser medium.. Identify intensity as the time-averaged squared field, and name what a detector actually measures. --- 1. Maxwell to the wave equation Concept. Light is not a substance travelling through space; it is a self-sustaining disturbance of the electric and magnetic fields. A changing electric field creates a magnetic field, and a changing magnetic field creates an electric field. Once started, the two fields regenerate each other and the disturbance propagates on its own. That mutual regeneration is the whole reason light moves, and it is why its speed is fixed by two constants of the vacuum rather than by any source (Hecht 2017). Mathematics. In a source-free, linear, isotropic medium with permittivity $\varepsilon$ and permeability $\mu$, Maxwell's equations are say: The divergence of E is zero, and the divergence of B is zero. The curl of E equals minus the partial derivative of B with respect to time, and the curl of B equals mu epsilon times the partial derivative of E with respect to time. Taking the curl of the third equation and substituting the fourth removes the magnetic field and leaves a wave equation for the electric field, say: The Laplacian of E equals mu epsilon times the second partial derivative of E with respect to time. with an identical equation for $\mathbf{B}$. Comparing this with the standard wave equation $\nabla^2 u = (1/v^2)\,\partial^2 u/\partial t^2$ identifies the propagation speed as $v = 1/\sqrt{\mu\varepsilon}$. In vacuum $v=c=1/\sqrt{\mu_0\varepsilon_0}\approx 2.998\times 10^{8}\ \text{m/s}$, and in a medium $v=c/n$ with refractive index $n=\sqrt{\mu\varepsilon/\mu_0\varepsilon_0}$ (Born and Wolf 1999). Hands-on check. In the Finite-Difference Time-Domain (FDTD) ripple-tank lab, launch a disturbance and confirm it propagates outward at a fixed speed set only by the medium, not by how hard you drove the source. --- 2. Plane waves Concept. The simplest solution of the wave equation is a plane wave: a field whose value is the same everywhere on any plane perpendicular to the direction of travel, oscillating in step as it advances. It is transverse, meaning the fields point across the direction of travel, not along it. Mathematics. A monochromatic plane wave is written compactly with a complex amplitude, say: E of r and t equals the real part of E nought times e to the i, times the quantity k dot r minus omega t. where $\mathbf{k}$ is the wavevector, $|\mathbf{k}|=k=2\pi/\lambda=\omega/v$, and $\omega=2\pi f$. The phase $\mathbf{k}\cdot\mathbf{r}-\omega t$ is the bookkeeping that the rest of the course turns into interference; this module only asks that you read it. Two relations matter most. The dispersion relation $k=n\omega/c$ ties the spatial and temporal oscillations together, and transversality $\mathbf{k}\cdot\mathbf{E}_0=0$ follows from $\nabla\cdot\mathbf{E}=0$. The consequence for crossing into a denser medium is decisive. The frequency $f$ is set by the source and cannot change at a passive interface, so it is conserved. Since $v=c/n$ drops and $v=f\lambda$, the wavelength must shrink by the same factor: $\lambda_{\text{medium}}=\lambda_0/n$. Frequency is the invariant; wavelength and speed are not. Hands-on check. Drive a plane wave, read its wavelength, then raise the medium index and confirm the wavelength compresses by exactly $n$ while the frequency stays put. --- 3. Energy, intensity, and the Poynting vector Concept. A light wave carries energy, and the rate at which it delivers energy to a surface is what a detector, an eye, or a solar cell responds to. No ordinary detector follows the field oscillating at hundreds of terahertz; it reports a time average. Mathematics. The instantaneous energy flow per unit area is the Poynting vector $\mathbf{S}=\mathbf{E}\times\mathbf{H}$. For a plane wave the time-averaged magnitude, which we call the intensity, is say: The intensity I equals the time average of the magnitude of the Poynting vector, which is one half c epsilon nought n times the magnitude of E nought squared, and is proportional to the magnitude of E nought squared. The single most important fact carried into the rest of OptiX is the proportionality $I\propto|E_0|^2$: the detector reports the modulus squared of the field amplitude. Every interference result later in the course is this statement applied to a sum of amplitudes. Hands-on check. Double the field amplitude in a lab and confirm the reported intensity quadruples, not doubles. --- 4. Photons, in one paragraph The same light is also a stream of photons, each carrying energy $E=hf$ and momentum $p=hf/c$, where $h$ is Planck's constant (Saleh and Teich 2019). The wave picture and the photon picture are two descriptions of one reality; which one is convenient depends on the question. OptiX uses the wave picture for everything up to the all-optical gate capstone, because interference, diffraction, guiding, and switching are all wave phenomena, and notes the photon picture only where energy quantisation or detection statistics genuinely matter. For this course, treat light as a wave whose intensity is a photon arrival rate. --- 5. Worked example A laser of vacuum wavelength $\lambda_0=1550\ \text{nm}$ enters glass of refractive index $n=1.5$. Frequency in vacuum: $f=c/\lambda_0=(2.998\times 10^8)/(1550\times 10^{-9})\approx 1.93\times 10^{14}\ \text{Hz}$. This frequency is unchanged inside the glass. Speed in glass: $v=c/n=(2.998\times 10^8)/1.5\approx 2.00\times 10^8\ \text{m/s}$. Wavelength in glass: $\lambda=\lambda_0/n=1550/1.5\approx 1033\ \text{nm}$. The wave slows and its wavelength shrinks by the factor $n$, while its colour, set by frequency, is unchanged. This is exactly why a prism does not change the identity of a colour, only its direction. --- 6. Readiness check before Module 2 You are ready to move on when you can, without help: derive $v=1/\sqrt{\mu\varepsilon}$ from Maxwell's equations; state which of wavelength, frequency, and speed are conserved across an interface and why; and explain why a detector reports $|E_0|^2$ rather than $E_0$. The last point is the bridge into Module 2, where two amplitudes are added before the squaring and interference appears. --- Appendix: References Born, M., and Wolf, E. (1999). *Principles of Optics* (7th ed.). Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9781139644181 Hecht, E. (2017). *Optics* (5th ed.). Boston: Pearson. https://www.pearson.com/en-us/subject-catalog/p/optics/P200000006793/9780137526420 MIT OpenCourseWare (2016). *8.03SC Physics III: Vibrations and Waves, Fall 2016.* https://ocw.mit.edu/courses/8-03sc-physics-iii-vibrations-and-waves-fall-2016/ Saleh, B. E. A., and Teich, M. C. (2019). *Fundamentals of Photonics* (3rd ed.). Hoboken: Wiley. https://www.wiley.com/en-us/Fundamentals+of+Photonics%2C+3rd+Edition-p-9781119506874 Related CCI capabilities Computer Architecture (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/computer-architecture/). Optics Primer Series (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/optics/). Maths Refresher Series, Finance (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/maths-finance/). System Dynamics (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/system-dynamics/). CCI Lab: Run it, build with it, read the thinking, reuse the data. (https://www.cambridgecyberinternational.com/en/insights/lab/) The wave description: phasors and superposition. Linear fields add; the detector squares the sum, giving interference.. Add two waves as phasors and compute the resultant intensity.. State the two-beam interference law and its limits.. Explain why the cross term is the whole of interference.. Superposition, Phasor addition, The interference law, Constructive and destructive limits Module 2, The wave description: phasors and superposition OptiX (codename LUMEN), Module 2. Source language: English (authoritative). This is the hinge of the course: Module 1 ended with the detector reporting the squared field; here two fields are added before the squaring, and interference appears. Learning outcomes Add two waves as phasors and predict the resultant intensity.. State the two-beam interference law and its limits.. Explain why the cross term is the whole of interference. 1. Superposition Concept. Maxwell's equations are linear, so when two light fields occupy the same place their amplitudes add. Crucially, fields add, not intensities. The detector then squares the summed field, and that squaring of a sum is where interference is born (Hecht 2017). This additivity holds regardless of the beams' relative amplitude, polarization state, or angle of incidence, and it is the reason two overlapping laser pointers never simply add their powers on a screen. Mathematics. For two scalar fields of the same frequency, $\tilde{E}=\tilde{E}_1+\tilde{E}_2$. The detected intensity is $I=|\tilde{E}_1+\tilde{E}_2|^2$, not $|\tilde{E}_1|^2+|\tilde{E}_2|^2$. 2. Phasor addition Concept. Represent each field as a phasor (Module 0). Adding the two waves is adding the two arrows; the resultant length squared is the intensity. Mathematics. With $\tilde{E}_1=\sqrt{I_1}$ and $\tilde{E}_2=\sqrt{I_2}\,e^{i\delta}$, expanding the modulus squared gives say: The intensity I equals I one plus I two, plus two times the square root of I one I two, times the cosine of delta. Hands-on check. In the interference lab, set two beams and sweep $\delta$ from $0$ to $2\pi$; the readout intensity must trace a raised cosine. 3. The interference law The first two terms, $I_1+I_2$, are what you would get if the beams did not interfere (incoherent addition). Everything optical lives in the third term, the cross term $2\sqrt{I_1 I_2}\cos\delta$. It is positive (brightening) when the phasors align and negative (darkening) when they oppose. No cross term, no interference. Physically, the cross term redistributes energy across the observation plane rather than creating or destroying it, so a bright fringe on one side of the pattern is always paid for by a dark fringe elsewhere, an accounting that follows directly from energy conservation applied to the summed field. 4. Constructive and destructive limits For equal beams ($I_1=I_2=I_0$): at $\delta=0$ the output is $4I_0$ (constructive, twice the field gives four times the intensity); at $\delta=\pi$ it is $0$ (destructive, the energy is redirected, not destroyed). The cross term requires the two beams to keep a stable phase relationship, which is the topic of coherence in Module 5. In practice this stability window is set by the source linewidth and by path-length matching between the two arms, both of which reappear as quantitative constraints once coherence is treated properly. Readiness check You are ready for Module 3 (and the wave-optics arc) when you can derive the interference law from $|\tilde{E}_1+\tilde{E}_2|^2$ and explain, in one sentence, why two equal beams can give zero. Appendix: References Hecht, E. (2017). *Optics* (5th ed.). Boston: Pearson. https://www.pearson.com/en-us/subject-catalog/p/optics/P200000006793/9780137526420 Related CCI capabilities Computer Architecture (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/computer-architecture/). Optics Primer Series (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/optics/). Maths Refresher Series, Finance (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/maths-finance/). System Dynamics (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/system-dynamics/). CCI Lab: Run it, build with it, read the thinking, reuse the data. (https://www.cambridgecyberinternational.com/en/insights/lab/) Geometrical optics. Rays, Snell's law, ABCD matrices, imaging, and total internal reflection.. Apply Fermat's principle and Snell's law at an interface.. Propagate a ray with ABCD matrices and use the lens equation.. Explain total internal reflection as the basis of waveguiding.. Fermat and Snell, Ray-transfer (ABCD) matrices, Imaging and the lens equation, Total internal reflection Module 3, Geometrical optics OptiX (codename LUMEN), Module 3. Source language: English (authoritative). When features are large compared with the wavelength, light travels in rays. This module gives the ray tools and, crucially, the total internal reflection that confines light in a waveguide (the bridge to Module 8). Learning outcomes Apply Fermat's principle and Snell's law at an interface.. Propagate a ray with ABCD matrices and use the lens equation.. Explain total internal reflection as the basis of waveguiding. 1. Fermat and Snell Concept. A ray takes the path of stationary travel time between two points. At an interface this bends the ray toward the normal when entering a slower (denser) medium. Mathematics. Fermat's principle yields the law of reflection and Snell's law, say: n one sine theta one equals n two sine theta two. Hands-on check. Send a ray from air into glass at 30 degrees and confirm the refracted angle satisfies Snell's law (about 19.5 degrees for n = 1.5). 2. Ray-transfer (ABCD) matrices Concept. A paraxial ray near the axis is two numbers: its height $y$ and its slope $\theta$. Each optical element multiplies that vector by a 2x2 matrix, so a whole system is one matrix product. This is the first appearance of the matrix bookkeeping that returns for Jones calculus (Module 4) and couplers (Module 8). Mathematics. Free space of length $d$ and a thin lens of focal length $f$ are say: The first matrix has top row one and d, and bottom row zero and one. The second matrix has top row one and zero, and bottom row minus one over f and one. 3. Imaging and the lens equation The imaging condition is the Gaussian lens equation, say: One over s o plus one over s i equals one over f. Hands-on check. For $f=10$ cm and an object at $s_o=30$ cm, solve for the image distance ($s_i=15$ cm) and confirm with a ray trace. 4. Total internal reflection Concept. Going from a dense to a rare medium, beyond a critical angle the ray cannot leave and is totally reflected. This is exactly how a high-index core traps light, the principle underneath every optical fibre and integrated waveguide. Mathematics. The critical angle is say: The critical angle theta c equals the arcsine of n two over n one, where n one is greater than n two. For glass to air ($n_1=1.5$), $\theta_c\approx 41.8$ degrees. Readiness check You are ready for the wave-optics device modules when you can apply Snell's law, propagate a ray through a lens with ABCD matrices, and state why total internal reflection confines a guided mode. Appendix: References Hecht, E. (2017). *Optics* (5th ed.). Boston: Pearson. https://www.pearson.com/en-us/subject-catalog/p/optics/P200000006793/9780137526420 Related CCI capabilities Computer Architecture (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/computer-architecture/). Optics Primer Series (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/optics/). Maths Refresher Series, Finance (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/maths-finance/). System Dynamics (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/system-dynamics/). CCI Lab: Run it, build with it, read the thinking, reuse the data. (https://www.cambridgecyberinternational.com/en/insights/lab/) Polarization and Jones calculus. The field is a vector; 2x2 Jones matrices control its state.. Represent a polarization state as a Jones vector.. Apply polariser and wave-plate Jones matrices and recover Malus's law.. Explain birefringence as polarization-dependent index.. Polarization states, Jones vectors and matrices, Polarisers, Malus's law, Wave plates and birefringence Module 4, Polarization and Jones calculus OptiX (codename LUMEN), Module 4. Source language: English (authoritative). The electric field is a vector; controlling where it points is half of controlling light. The algebra is a 2x2 matrix formalism you already own from linear algebra. Learning outcomes Represent a polarization state as a Jones vector.. Apply polariser and wave-plate Jones matrices and recover Malus's law.. Explain birefringence as a polarization-dependent index. 1. Polarization states Concept. As a wave advances, the tip of its electric-field vector traces a line, a circle, or an ellipse in the transverse plane. These are linear, circular, and elliptical polarization. The state is set by the relative amplitude and phase of the two transverse components. When the two components are in phase the tip traces a straight line (linear polarization); when they are in quadrature with equal amplitude it traces a circle (circular polarization); every other amplitude and phase combination traces an ellipse, with the line and circle as its two limiting cases. 2. Jones vectors and matrices Concept and mathematics. A polarization state is a two-component complex vector, and each optical element is a 2x2 Jones matrix acting on it; cascading elements is matrix multiplication (Jones 1941). say: The Jones vector J is the column vector with E x on top and E y below. Horizontal linear light is $(1,0)^\top$; a quarter-wave plate with its axis at 45 degrees turns that into circular light, $(1,i)^\top/\sqrt2$. Hands-on check. Apply the quarter-wave-plate matrix to linear light at 45 degrees and confirm the output is circular. 3. Polarisers and Malus's law Concept. A linear polariser passes only the component along its axis. The transmitted intensity follows Malus's law, say: The intensity I equals I nought times the cosine squared of theta. where $\theta$ is the angle between the input polarization and the analyser axis. At 45 degrees, half passes; at 60 degrees, a quarter passes; at 90 degrees, nothing. This cosine-squared falloff is the basis of variable optical attenuators built from two rotatable polarisers, and it is why crossed polarisers placed either side of a birefringent sample form the classic photoelastic stress-viewing setup. 4. Wave plates and birefringence Concept. A birefringent material has a refractive index that depends on polarization, so the two components accumulate different phase. A slab cut to give a quarter-cycle relative delay is a quarter-wave plate; a half-cycle is a half-wave plate, which rotates linear polarization. Birefringence is the controllable phase-between-components that makes wave plates and many modulators work. Because the fast and slow axes see different indices, rotating the wave plate about the beam axis changes the orientation of the output polarization ellipse without changing its shape, which is the mechanical equivalent of what an electro-optic modulator does electrically. Readiness check You are ready to continue when you can write a Jones vector for a given state, apply a polariser and a wave plate by matrix multiplication, and recover Malus's law. Appendix: References Jones, R. C. (1941). A new calculus for the treatment of optical systems. I. *Journal of the Optical Society of America (JOSA)*, 31(7), 488-493. https://doi.org/10.1364/JOSA.31.000488 Hecht, E. (2017). *Optics* (5th ed.). Boston: Pearson. https://www.pearson.com/en-us/subject-catalog/p/optics/P200000006793/9780137526420 Related CCI capabilities Computer Architecture (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/computer-architecture/). Optics Primer Series (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/optics/). Maths Refresher Series, Finance (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/maths-finance/). System Dynamics (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/system-dynamics/). CCI Lab: Run it, build with it, read the thinking, reuse the data. (https://www.cambridgecyberinternational.com/en/insights/lab/) Interference and coherence. Coherence sets when two paths interfere; interferometers exploit it.. Define fringe visibility and relate it to coherence.. Measure the output of a two-arm interferometer versus path difference.. Estimate a temporal coherence length.. Coherence, Visibility, Michelson and Mach-Zehnder, Coherence length Module 5, Interference and coherence OptiX (codename LUMEN), Module 5. Source language: English (authoritative). Module 2 gave the bare interference law; here we add the missing ingredient (coherence) and turn the law into the interferometers that recur for the rest of the course. Learning outcomes Define fringe visibility and relate it to coherence.. Predict the output of a two-arm interferometer versus path difference.. Estimate a temporal coherence length. 1. Coherence Concept. Interference needs two copies of a wave that keep a stable phase relationship. Coherence measures how well a wave stays in step with a delayed copy of itself. Real sources are only partly coherent, which is why interference washes out once the two paths differ by too much. Two common measures capture this: temporal coherence describes how long a wave stays correlated with a time-delayed copy of itself, while spatial coherence describes how well two points across the wavefront stay correlated at the same instant; an interferometer with matched arm lengths mainly tests the first. 2. Visibility Concept and mathematics. The contrast of the fringes is the visibility, say: The visibility V equals I max minus I min, divided by I max plus I min. which equals the magnitude of the complex degree of coherence. $V=1$ for equal, fully coherent beams; $V=0$ for incoherent light. Hands-on check. With $I_{\max}=9$ and $I_{\min}=1$, confirm $V=0.8$. 3. Michelson and Mach-Zehnder Concept. A two-arm interferometer splits a wave, gives each copy a path, and recombines them. The output follows the Module 2 cosine with phase $\delta=2\pi\Delta/\lambda$ set by the path difference $\Delta$. The Michelson folds the arms with mirrors; the Mach-Zehnder (MZI) runs two separate arms between two splitters, and its bar and cross outputs are $\cos^2(\delta/2)$ and $\sin^2(\delta/2)$ (the device skeleton of Modules 9 and 11). The Michelson is preferred when a simple, adjustable reference arm is needed, for example in spectroscopy and metrology, while the Mach-Zehnder's separated arms make it the natural layout for an integrated modulator or switch, since each arm can be addressed independently. Hands-on check. In the interference lab, sweep one arm and watch the two outputs breathe in antiphase. 4. Coherence length Concept and mathematics. A wave interferes with its delayed copy only while the path difference stays within the coherence length, say: The coherence length ell c is approximately equal to lambda squared divided by delta lambda. set by the spectral width $\Delta\lambda$. A laser (narrow $\Delta\lambda$) has a long coherence length; a white-light source has a very short one. A He-Ne laser with a linewidth near 1 MHz has a coherence length of hundreds of metres, while a white-light source with a linewidth spanning the visible band has a coherence length of only a few micrometres, which is why white-light interferometry is used to profile surfaces with micrometre-scale height sensitivity. Readiness check You are ready for diffraction and the device modules when you can compute visibility, predict an interferometer output versus path difference, and estimate a coherence length. Appendix: References Hecht, E. (2017). *Optics* (5th ed.). Boston: Pearson. https://www.pearson.com/en-us/subject-catalog/p/optics/P200000006793/9780137526420 Born, M., and Wolf, E. (1999). *Principles of Optics* (7th ed.). Cambridge University Press. https://doi.org/10.1017/CBO9781139644181 Related CCI capabilities Computer Architecture (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/computer-architecture/). Optics Primer Series (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/optics/). Maths Refresher Series, Finance (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/maths-finance/). System Dynamics (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/system-dynamics/). CCI Lab: Run it, build with it, read the thinking, reuse the data. (https://www.cambridgecyberinternational.com/en/insights/lab/) Diffraction. Apertures spread light; the far field is the aperture's transform.. Apply the Huygens-Fresnel principle qualitatively.. Predict single-slit zeros and grating maxima.. State the Rayleigh resolution limit.. Huygens-Fresnel, Single slit (sinc), Gratings, Resolution Module 6, Diffraction OptiX (codename LUMEN), Module 6. Source language: English (authoritative). Waves bend around obstacles and spread through apertures. Diffraction sets the resolution limit of every optical system and explains the grating, the workhorse of spectral control. Learning outcomes Apply the Huygens-Fresnel principle qualitatively.. Predict single-slit zeros and grating maxima.. State the Rayleigh resolution limit. 1. Huygens-Fresnel Concept. Treat every point of a wavefront as a source of secondary wavelets; the new wavefront is their interference. Narrow the aperture and the wavelets spread more, so a smaller opening gives a broader pattern. Diffraction is just interference from a continuum of sources. This picture also explains why light appears to travel in straight lines through a large aperture: away from the edges, wavelets from neighbouring points interfere constructively only along the forward direction, and it is only near an edge or through a small opening that the spreading becomes visible. 2. Single slit (sinc) Concept and mathematics. In the far field (Fraunhofer), the pattern is the Fourier transform of the aperture. A slit of width $a$ gives say: The intensity I of theta equals I nought times the sinc squared of pi a sine theta over lambda. with first zeros at $\sin\theta=\lambda/a$. Hands-on check. For $a=8\ \mu$m and $\lambda=0.5\ \mu$m, the first zero is near $3.6$ degrees. 3. Gratings Concept and mathematics. Many equally spaced slits of period $d$ sharpen the bright orders to the grating equation, say: d sine theta equals m lambda, where m is zero, plus or minus one, plus or minus two, and so on. so different wavelengths leave at different angles. This is how a grating disperses light into a spectrum. The angular separation between adjacent orders grows with the order number and with the ratio of wavelength to groove spacing, so a grating with a finer period gives coarser (more widely separated) orders and, for a fixed number of illuminated grooves, higher resolving power. Hands-on check. For $d=2\ \mu$m, $\lambda=0.6\ \mu$m, first order is near $17.5$ degrees. 4. Resolution Concept and mathematics. Two points are just resolvable when one's diffraction peak falls on the other's first zero (Rayleigh). For a circular aperture of diameter $D$, say: The minimum resolvable angle theta min equals one point two two times lambda over D. A larger aperture resolves finer detail; this is why telescopes and microscope objectives are large. The same Rayleigh criterion sets the smallest feature a microscope can distinguish, the angular resolution of a camera lens, and the diffraction-limited spot size a focusing lens can deliver, all from the identical ratio of wavelength to aperture. Readiness check You are ready for Fourier optics when you can locate a single-slit zero, a grating order, and state how resolution scales with aperture and wavelength. Appendix: References Goodman, J. W. (2017). *Introduction to Fourier Optics* (4th ed.). New York: W. H. Freeman. https://www.worldcat.org/title/introduction-to-fourier-optics/oclc/958780856 Hecht, E. (2017). *Optics* (5th ed.). Boston: Pearson. https://www.pearson.com/en-us/subject-catalog/p/optics/P200000006793/9780137526420 Related CCI capabilities Computer Architecture (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/computer-architecture/). Optics Primer Series (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/optics/). Maths Refresher Series, Finance (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/maths-finance/). System Dynamics (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/system-dynamics/). CCI Lab: Run it, build with it, read the thinking, reuse the data. (https://www.cambridgecyberinternational.com/en/insights/lab/) Fourier optics. A lens Fourier-transforms the field; imaging is linear filtering.. Explain why a lens performs a spatial Fourier transform.. Describe low-pass and high-pass spatial filtering.. Model how the optical transfer function connects to image sharpness.. The lens as a transformer, Spatial frequency, Spatial filtering (4f), Transfer functions Module 7, Fourier optics OptiX (codename LUMEN), Module 7. Source language: English (authoritative). One fact unifies imaging, diffraction, and filtering: a lens performs a Fourier transform of the field. This gives the transfer-function language the device modules reuse. Learning outcomes Explain why a lens performs a spatial Fourier transform.. Describe low-pass and high-pass spatial filtering.. Connect the optical transfer function to image sharpness. 1. The lens as a transformer Concept and mathematics. The field at the back focal plane of a lens is the spatial Fourier transform of the field at the front focal plane. Position in the focal plane maps to spatial frequency, say: The spatial frequency f x equals x divided by lambda F. with $F$ the focal length. Diffraction (Module 6) was the far-field transform; a lens brings that transform to a reachable plane. This is why a single lens, not a lens system, is enough to display a diffraction pattern on a screen: the same physics that produces the far-field pattern in free space happens at the focal plane in one focal length instead of over an arbitrarily long propagation distance. 2. Spatial frequency Concept. A spatial frequency is a rate of variation across the image: low frequencies are broad, slowly varying brightness; high frequencies are fine detail and sharp edges. Any image is a sum of spatial-frequency components, just as a sound is a sum of tones. A photograph of a brick wall, for instance, is dominated by a strong spatial frequency component at the spacing of the bricks, riding on a broad low-frequency background set by the overall lighting gradient across the scene. 3. Spatial filtering (4f) Concept. In a 4f system (two lenses one focal length apart), the middle plane is the Fourier plane. Place a mask there and you filter spatial frequencies directly: a small central hole passes only low frequencies and blurs the image (low-pass); a central block removes low frequencies and leaves edges (high-pass). Hands-on check. Conceptually, block the high frequencies and predict a blurred image; block the low frequencies and predict edge enhancement. 4. Transfer functions Concept. An imaging system is a linear, shift-invariant filter described by an optical transfer function: how strongly it passes each spatial frequency. Cutting off high frequencies limits sharpness; this is the same transfer-function thinking used for the interferometers and resonators of Module 9, now in two spatial dimensions. A diffraction-limited system has a transfer function that falls smoothly to zero at a cutoff frequency set by the aperture and wavelength, while aberrations or defocus depress the transfer function at intermediate frequencies first, which is why a slightly out-of-focus image looks soft before it looks obviously blurred. Readiness check You are ready for guided-wave optics when you can explain why a lens transforms, what low-pass and high-pass filtering do, and how the transfer function limits sharpness. Appendix: References Goodman, J. W. (2017). *Introduction to Fourier Optics* (4th ed.). New York: W. H. Freeman. https://www.worldcat.org/title/introduction-to-fourier-optics/oclc/958780856 Related CCI capabilities Computer Architecture (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/computer-architecture/). Optics Primer Series (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/optics/). Maths Refresher Series, Finance (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/maths-finance/). System Dynamics (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/system-dynamics/). CCI Lab: Run it, build with it, read the thinking, reuse the data. (https://www.cambridgecyberinternational.com/en/insights/lab/) Guided-wave optics. Waveguides carry modes; evanescent overlap couples neighbours.. Describe single-mode guiding and cutoff qualitatively.. Apply coupled-mode theory: P2 = P0 sin^2(kappa z).. Identify the 50:50 coupler and its cross-port phase.. Modes and cutoff, Coupled-mode theory, Directional and MMI couplers, Scattering matrix Module 8, Guided-wave optics OptiX (codename LUMEN), Module 8. Source language: English (authoritative). Confine light in a high-index core and it travels as discrete modes; bring two guides close and power sloshes between them. This is the directional coupler, the beam splitter of integrated optics and the input and output stage of every interferometer. Learning outcomes Describe single-mode guiding and cutoff qualitatively.. Use coupled-mode theory, P2 = P0 sin^2(kappa z).. Identify the 50:50 coupler and its cross-port phase. 1. Modes and cutoff Concept. Total internal reflection (Module 3) traps light in a core, but only certain transverse field patterns, the modes, propagate without changing shape. A guide that is narrow enough supports a single mode; wider guides support several. Single-mode operation holds below a cutoff set by the core size, index contrast, and wavelength (the V-number). Below cutoff only the fundamental mode propagates without loss, which is the regime single-mode fibre and most integrated waveguide circuits are designed to operate in, since a single, well-defined field pattern is what makes phase and interference-based devices predictable. 2. Coupled-mode theory Concept and mathematics. Place two single-mode guides close enough that their evanescent tails overlap, and light launched in one leaks into the other. The cross-guide power is say: The power P two of z equals P nought times the sine squared of kappa z. where the coupling coefficient $\kappa$ grows as the guides get closer. Full transfer occurs at $\kappa z=\pi/2$. Hands-on check. In the coupler lab, the Bit Pattern Mask (BPM) solver reproduces this $\sin^2$ transfer; verify full transfer at $\kappa L=\pi/2$ and a balanced split at $\pi/4$. 3. Directional and MMI couplers Concept. A coupler of length $L$ with $\kappa L=\pi/4$ splits power 50:50, the fundamental building block. A multimode-interference (MMI) coupler does the same job by self-imaging in a wide section, and a Y-junction is the simplest splitter. Directional couplers are compact and give a controllable, wavelength-dependent split ratio, MMI couplers are more tolerant of fabrication error because the self-imaging condition depends mainly on the width of the multimode section, and Y-junctions are simplest to build but always split power evenly and cannot be tuned. Hands-on check. For $\kappa=0.2/\text{mm}$, a 50:50 split needs $L=\pi/(4\kappa)\approx 3.93$ mm. 4. Scattering matrix Concept and mathematics. A lossless 2x2 coupler is described by a unitary scattering matrix that imposes a 90-degree phase (a factor $i$) on the cross port relative to the bar port. That cross-port phase is not a detail: it decides the bar and cross transmissions of the Mach-Zehnder in Module 9. Readiness check You are ready for the device blocks when you can use the $\sin^2(\kappa z)$ transfer, find the 50:50 length, and state the cross-port phase of a lossless coupler. Appendix: References Saleh, B. E. A., and Teich, M. C. (2019). *Fundamentals of Photonics* (3rd ed.). Hoboken: Wiley. https://www.wiley.com/en-us/Fundamentals+of+Photonics%2C+3rd+Edition-p-9781119506874 Yariv, A., and Yeh, P. (2006). *Photonics* (6th ed.). Oxford University Press. https://www.worldcat.org/title/photonics-optical-electronics-in-modern-communications/oclc/58648003 Related CCI capabilities Computer Architecture (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/computer-architecture/). Optics Primer Series (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/optics/). Maths Refresher Series, Finance (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/maths-finance/). System Dynamics (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/system-dynamics/). CCI Lab: Run it, build with it, read the thinking, reuse the data. (https://www.cambridgecyberinternational.com/en/insights/lab/) The device building blocks. Interferometers, resonators, gratings: each a compact transfer function.. Model a Mach-Zehnder output as cos^2 / sin^2 in the phase difference.. Describe a micro-ring resonance and free spectral range.. State a Bragg reflection condition.. Interferometers (MZI, Michelson, Sagnac), Micro-ring and Fabry-Perot resonators, Gratings (Bragg, AWG), Composing transfer functions Module 9, The device building blocks OptiX (codename LUMEN), Module 9. Source language: English (authoritative). The implementation diagrams you must read are assembled from a finite set of recurring blocks, each with a compact transfer function. The Mach-Zehnder is one worked example here, not the destination; the point is fluency across the whole set. Learning outcomes Read a Mach-Zehnder as cos^2 / sin^2 in the phase difference.. Describe a micro-ring resonance and its free spectral range.. State a Bragg reflection condition. 1. Interferometers (Mach-Zehnder Interferometer (MZI), Michelson, Sagnac) Concept and mathematics. The Mach-Zehnder interferometer is two 50:50 couplers joined by two arms. Cascading the coupler scattering matrices with an arm phase difference $\Delta\phi$ gives say: The bar transmission equals the cosine squared of delta phi over two, and the cross transmission equals the sine squared of delta phi over two. so power switches output as $\Delta\phi$ moves from $0$ to $\pi$. The Michelson is the folded, reflective relative; the Sagnac loop interferes counter-propagating beams. Because a Sagnac loop sends both directions through the same physical path, it is intrinsically insensitive to slow drifts in the arm length or temperature, which is why it is the workhorse configuration for fibre-optic gyroscopes rather than for switching. Hands-on check. In the gates lab, sweep $\Delta\phi$ and confirm the two outputs trade as $\cos^2$ and $\sin^2$; at $\Delta\phi=\pi/2$ each is $0.5$. 2. Micro-ring and Fabry-Perot resonators Concept and mathematics. A ring coupled to a bus transmits sharply near resonance; a small index shift moves the resonance and switches transmission. Its resonance spacing is the free spectral range (FSR), say: The free spectral range, delta lambda, equals lambda squared divided by n g times L. with $L$ the round-trip length and $n_g$ the group index. The Fabry-Perot etalon is the straight-cavity analogue with its Airy transmission. 3. Gratings (Bragg, AWG) Concept and mathematics. A Bragg grating reflects strongly at say: The Bragg wavelength lambda B equals two times n effective times capital lambda. with period $\Lambda$. An arrayed-waveguide grating (AWG) routes wavelengths by engineered path differences, the basis of optical multiplexers. 4. Composing transfer functions Concept. An implementation diagram is the composition of these one-line transfer functions: read each block, write its transfer function, and multiply. That is the reading skill the capstone (Module 11) exercises on unfamiliar diagrams. Two blocks in series multiply their transfer functions, while two blocks in parallel add their fields before the final detector squares the sum, so the order of multiplication and addition in a diagram is fixed entirely by whether the paths recombine coherently or are read out independently. Readiness check You are ready for nonlinearity and the capstone when you can derive the MZI bar and cross transmissions, state a ring's FSR, and write a Bragg condition. Appendix: References Saleh, B. E. A., and Teich, M. C. (2019). *Fundamentals of Photonics* (3rd ed.). Hoboken: Wiley. https://www.wiley.com/en-us/Fundamentals+of+Photonics%2C+3rd+Edition-p-9781119506874 Related CCI capabilities Computer Architecture (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/computer-architecture/). Optics Primer Series (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/optics/). Maths Refresher Series, Finance (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/maths-finance/). System Dynamics (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/system-dynamics/). CCI Lab: Run it, build with it, read the thinking, reuse the data. (https://www.cambridgecyberinternational.com/en/insights/lab/) Optical nonlinearity. Light controlling light: Kerr, cross-phase, gain saturation, wave mixing.. Explain the intensity-dependent index (Kerr) and cross-phase modulation.. Distinguish XGM, XPM, FWM and saturable absorption.. Compute a Kerr nonlinear phase shift.. Why nonlinearity is needed, Kerr and cross-phase modulation, SOA: gain and index, Wave mixing and saturation Module 10, Optical nonlinearity OptiX (codename LUMEN), Module 10. Source language: English (authoritative). A passive device routes light by a phase you set with a voltage or heater. An all-optical gate routes light by light: one beam changes the medium another beam travels through. That needs an optical nonlinearity, the hard prerequisite for the capstone. Learning outcomes Explain the intensity-dependent index (Kerr) and cross-phase modulation.. Distinguish Cross-Gain Modulation (XGM), XPM, FWM, and saturable absorption.. Estimate a Kerr nonlinear phase shift. 1. Why nonlinearity is needed Concept. Linear optics obeys superposition: beams pass through each other unchanged. To make one beam control another, you need a medium whose response depends on intensity. Without nonlinearity there is no gain, no thresholding, and no general Boolean logic with intensity-encoded, cascadable signals. A linear medium can still attenuate, delay, or redirect light by a fixed amount, but it cannot make the presence of one signal change how a second, independent signal is treated, and that control function is exactly what a logic gate requires. 2. Kerr and cross-phase modulation Concept and mathematics. The optical Kerr effect makes the index depend on intensity, say: The refractive index n equals n nought plus n two times I. so a control beam of intensity $I_c$ adds a nonlinear (NL) phase to a co-propagating signal, say: The nonlinear phase shift delta phi equals two pi over lambda, times n two, times I c, times L. This is cross-phase modulation (XPM): the control writes a phase the signal carries, which an interferometer then turns into an intensity change. Because the Kerr effect responds on a femtosecond timescale in most solid-state media, XPM-based switching can in principle run far faster than electronic switching, though the achievable device speed in practice is usually limited by how quickly the control pulse itself can be generated and detected. 3. SOA: gain and index Concept. In a semiconductor optical amplifier (SOA), a strong beam depletes carriers, lowering the gain seen by a second beam (cross-gain modulation, XGM), and shifting the index at the same time (a large, fast cross-phase effect). The SOA is the workhorse nonlinear element of integrated all-optical gates. 4. Wave mixing and saturation Concept. Four-wave mixing (FWM) lets beams beat to generate new frequencies, an ultrafast near-instantaneous process. A saturable absorber becomes transparent only above an intensity threshold, giving thresholding logic. These mechanisms, plus Kerr/XPM and XGM, are the toolkit the capstone draws on. Saturable absorption and gain saturation are close cousins: one clips a signal above a threshold by absorbing less as intensity rises, the other clips it by amplifying less, and both give the sharp, threshold-like transfer curve that a clean logic gate needs. Readiness check You are ready for the capstone when you can explain how a control beam writes a phase on a signal, name the main nonlinear mechanisms, and estimate a Kerr phase shift. Appendix: References Saleh, B. E. A., and Teich, M. C. (2019). *Fundamentals of Photonics* (3rd ed.). Hoboken: Wiley. https://www.wiley.com/en-us/Fundamentals+of+Photonics%2C+3rd+Edition-p-9781119506874 Related CCI capabilities Computer Architecture (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/computer-architecture/). Optics Primer Series (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/optics/). Maths Refresher Series, Finance (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/maths-finance/). System Dynamics (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/system-dynamics/). CCI Lab: Run it, build with it, read the thinking, reuse the data. (https://www.cambridgecyberinternational.com/en/insights/lab/) All-optical passive components and logic gates (capstone). Read any optical gate diagram by its mechanism and platform.. Name each block of an implementation diagram and its transfer function.. Map a gate to its mechanism (interference, resonance, gain, mixing) and platform, then justify the choice.. Explain why inverting gates need an always-on reference, and why passive AND needs a nonlinearity.. The mechanism-by-platform matrix, Reading an implementation diagram, Where the photons come from, Latency and the time domain Module 11, All-optical passive components and logic gates (capstone) OptiX (codename LUMEN), Module 11. Source language: English (authoritative). Everything converges. The goal is not one device but the skill of reading any optical gate diagram by its mechanism and platform. Learning outcomes Name each block of an implementation diagram and its transfer function.. Map a gate to its mechanism (interference, resonance, gain, mixing) and platform.. Explain why inverting gates need an always-on reference, and why passive AND needs a nonlinearity. 1. The mechanism-by-platform matrix Concept. A small set of mechanisms (interference, resonance, gain and absorption saturation, wave mixing, polarization rotation, nonlinear phase) is realised across many platforms (couplers, several interferometer types, micro-ring and Fabry-Perot resonators, photonic-crystal cavities, loop mirrors, gratings, semiconductor optical amplifiers). The same logic function appears many ways, so you classify a device by its mechanism and its platform rather than memorising devices. 2. Reading an implementation diagram Concept. Decompose the diagram into known blocks (Module 9): a coupler is a scattering matrix, an arm or ring is a phase or resonance element, a semiconductor optical amplifier (SOA) is an intensity-controlled gain-and-index element. Compose their transfer functions and you have the device, named or not. In the gates lab, the demonstrator does this for AND, OR, eXclusive OR (XOR), NOT across six platforms. 3. Where the photons come from Concept. This is the rule that catches most mistakes. A passive device makes no light of its own, so any gate that must emit when all data inputs are dark (NOT, NOT-AND (logic gate) (NAND), NOT-OR (logic gate) (NOR), eXclusive NOR (logic gate) (XNOR)) needs an always-on reference or carrier beam; the data interferes with or gates that beam. Likewise, a purely linear device cannot threshold, so a clean intensity AND needs a nonlinearity. XOR is the one gate that passive interference does naturally, because equal inputs cancel. 4. Latency and the time domain Concept and mathematics. The input-to-output latency of a gate is the group delay of its structure, say: The delay tau equals n g times L, divided by c. set by the platform geometry and wavelength, not by the logic function, so a platform's gates share one latency. A resonator adds slow-light delay near resonance, set by its radius and coupling (finesse). Integrated photonic gates live in the picosecond range; the gates lab plots this across colours and platforms. Readiness gate (course exit) You have finished OptiX when, given an unfamiliar optical gate diagram on any platform, you can name each component, identify its mechanism, compose its transfer function, reproduce its truth table, say where its output photons come from, and estimate its latency. At that point a Mach-Zehnder Interferometer (MZI) is a single recognisable word in a much larger language. Appendix: References Saleh, B. E. A., and Teich, M. C. (2019). *Fundamentals of Photonics* (3rd ed.). Hoboken: Wiley. https://www.wiley.com/en-us/Fundamentals+of+Photonics%2C+3rd+Edition-p-9781119506874 Nanomaterials (2026). *Advances in Semiconductor Optical Amplifier Technologies for All-Optical Logic Gate Implementations: A Comprehensive Review.* https://www.mdpi.com/2079-4991/16/3/202 Related CCI capabilities Computer Architecture (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/computer-architecture/). Optics Primer Series (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/optics/). Maths Refresher Series, Finance (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/maths-finance/). System Dynamics (Course) (https://www.cambridgecyberinternational.com/en/insights/academy/system-dynamics/). CCI Lab: Run it, build with it, read the thinking, reuse the data. (https://www.cambridgecyberinternational.com/en/insights/lab/)