Resource: List of Biotechnology Companies to Watch


PDF version: List of Biotechnology Companies to Watch – by Logan Thrasher Collins

I created this list to serve as a resource to help people learn about and keep track of key biotechnology companies. Some of these are emerging startups, some are established giants, and some provide useful services. Though this list is far from comprehensive, I have tried to cover as many of the key players as possible. It is also important to realize that this landscape is constantly changing, so some of the information on this list will eventually transition into antiquity. This version was written over the course of 2021 and updated during the summer of 2022. I hope you enjoy delving into the exciting world of biotechnology! 

Nanobodies as therapeutics and as laboratory reagents.
AgeX Therapeutics
Treating aging using stem cell therapies, induced tissue regeneration, related methods.
Engineering microorganisms and enzymes to degrade environmental pollutants.
Funded by the Ferment Consortium of Ginkgo Bioworks.
Engineering salt-tolerant rice via CRISPR for ocean agriculture to feed the world.
Formerly known as Agrisea.
Early stage: raised a $1.4M seed round as of September 2022.
Developing computer aided design tools for synthetic biology, making host cell lines for viral vector and biologics manufacturing, constructing genetic parts database.
One of the co-founders is Christopher Voigt.
James Collins is on the scientific advisory board.
Beam Therapeutics
Developing base editor technologies towards therapeutic applications.
David Liu and Feng Zhang are among the co-founders.
Has developed a peptide called xB3 that facilitates transcytosis across the blood-brain barrier.
Working towards applications in glioblastomas, brain metastases, and neurodegenerative diseases.
Large pharmaceutical company focusing on developing treatments for neurological diseases.
Has made moves towards developing gene therapy pipelines for treating neurological diseases, though the company has experienced some setbacks in this space (i.e. failed clinical trials).
BioMarin Pharmaceutical
Enzyme replacement therapies for rare diseases.
During April 2021, announced a collaboration with the Allen Institute to develop AAV gene therapies for rare diseases of the brain.
Bionaut Labs
Microrobotics as a new paradigm for drug delivery.
Developing gene therapies to treat aging, offers tests for determining biological age.
Elizabeth Parrish (the company’s CEO) tested an experimental gene therapy on herself and reports positive results, though she did not intend for this information to go public.
George Church and Aubrey de Grey are on the scientific advisory board.
Anders Sandberg is the company’s ethics advisor.
Capsida Biotherapeutics
Developing targeted AAV gene therapies for a variety of brain diseases.
Has made blood-brain barrier crossing AAVs that are liver untargeted and brain targeted.
Founded by Viviana Gradinaru.
Engineering superior AAV gene therapy vectors through a proprietary method called Transcription-Dependent Directed Evolution (TRADETM).
Have developed greatly improved neurotrophic AAVs.
Entered into a partnership with Biogen during May of 2021 to develop AAV gene therapies that treat various brain and neuromuscular disorders.
Building a DNA-based platform for massive digital data storage and computation.
Centered on moonshot projects that are using advanced CRISPR methods to bring back the Wooly Mammoth, the Thylacine (Tazmanian Tiger), and other extinct animals.
Aims to reintroduce lost biodiversity and thus repair ecosystems.
Will develop biomedical technologies such as artificial wombs in conjunction with its de-extinction research, providing additional benefits to humanity and acting as a way to bring in funding.
Cofounded by George Church, Ben Lamm, and Andrew Busey.
Cortical Labs
Developing hybrid bioelectronic devices which incorporate cultured biological neurons to perform computational tasks. These devices are power efficient, scalable, robust to physical damage, and have the potential for fluid adaptation to many different computational problems.
Creative Biolabs
Custom services for antibody engineering, membrane protein production and characterization, bioconjugation, gene therapy development, viral vector engineering, cell therapy development, molecular dynamics simulations, drug development consulting, and more.
Developing molecular techniques, hardware platforms, and software tools to accelerate adoption of non-model microorganisms for biotechnology.
Cultivarium is a focused research organization (FRO), so it possesses a distinct funding approach and different goals compared to traditional startups. For more information, see this open access article describing FROs in Nature.
Dyno Therapeutics
Using deep learning to improve properties of AAV capsids as a platform technology for gene therapy.
George Church is one of the co-founders.
E11 Bio
Building moonshot technologies involving superior molecular barcoding, spatial -omics, and viral circuit tracing to help neuroscientists map the brain. Has a long-term goal of mapping brains at the one-hundred billion neuron scale.
E11 Bio is a focused research organization (FRO), so it possesses a distinct funding approach and different goals compared to traditional startups. For more information, see this open access article describing FROs in Nature.
Editas Medicine
CRISPR-based gene therapy.
George Church, David Liu, Jennifer Doudna, Feng Zhang, and J. Keith Joung are the co-founders.
Eikon Therapeutics
Superior drug discovery platform which leverages high-throughput automated super-resolution microscopy for tracking single protein movements in living cells.
Eric Betzig is one of the advisors.
Emerald Cloud Lab
Remote automated laboratory as a service for researchers.
Has a large array of automated equipment for synthetic biology and genetic engineering, physical and biophysical chemistry, structural biology, biochemistry, analytical chemistry, etc.
Provides a software interface for users to instruct the automated equipment.
DNA origami imaging probes, fluorescence microscopy reagents.
First commercial application of DNA origami.
Services in artificial DNA synthesis, synthetic biology, antibodies, cell therapies, enzyme engineering, etc.
Ginkgo Bioworks
Synthetic biology, biomanufacturing, microorganism design, enzyme engineering, etc.
Acquired Gen9 in 2017.
Developing an mRNA-based SARS-CoV-2 vaccine which might protect from all possible variants of the virus.
Pivoted from original plan of developing cancer vaccines using the same technology.
Co-founded by Hannu Rajaniemi, who is also a successful science fiction author.
George Church is an advisor.
Combining multi-omic single cell profiling technologies and machine learning to comprehensively map the immune system and thereby enable greatly improved immunotherapies as well as accelerate clinical trials and avoid costly failures.
Impossible Foods
Uses synthetic biology and biochemical engineering to develop plant-based substitutes for meat products.
Their signature product is the Impossible Burger. They also make a product which mimics sausages.
One notable strategy employed by Impossible Foods is production of leghemoglobin in yeast. This compound gives a meaty flavor when added to their food products. They also add other plant-based compounds to mimic the fats found in animal meat.
Intellia Therapeutics
Developing therapies which employ CRISPR gene editing technology.
Has conducted some successful clinical trials using CRISPR gene therapy to treat transthyretin amyloidosis (as of February 2022, this is not yet FDA approved though).
Also working on CRISPR therapeutics for engineering T cells towards targeting acute myeloid leukemia.
Partnered with Regeneron, Novartis, and others.
Jennifer Doudna was one of the co-founders.
Neurotechnology, noninvasive brain-computer interfaces, invasive neural prostheses.
Some noninvasive products anticipated to be released during 2021.
Founded by Bryan Johnson who personally invested $54 million.
Raised an additional $53 million from outside investors.
Early goal is to help treat brain disease, has ambitions to enable human enhancement.
Developing therapies which utilize circular RNAs (Laronde calls these “endless RNAs”) as expression vehicles for proteins. Such circular RNAs are much more stable and less immunogenic than linear RNAs.
Peptide nanoparticles for targeted CRISPR-Cas gene therapy delivery, immunotherapy, hematological gene therapy, aging treatments.
Founded by Andre Watson.
Allogenic cell therapy that uses host lymph nodes as bioereactors to grow ectopic replacement organs.
Has developed a method for generating ectopic livers via patient lymph nodes that is in early clinical trials as of September 2022.
Mammoth Biosciences
CRISPR-based diagnostics.
Jennifer Doudna is one of the co-founders.
System for barcoding protein therapeutics to enable high-throughput design and testing in complex environments, machine learning to optimize drug design.
George Church is one of the co-founders.
Biomedical technologies which utilize mRNA inside of lipid nanoparticles; application areas include drug discovery, drug development, and vaccines.
Major player in COVID-19 pandemic since it was one of the first companies which developed and distributed SARS-CoV-2 vaccines to the world.
Nautilus Biotechnology
Developing a high-throughput single-molecule proteomics platform which integrates many novel techniques to decipher protein networks and thereby help accelerate basic science, new therapeutics, and new diagnostics.
Developing a non-invasive brain-computer interface based on headphones that use electroencephalography to record brain signals, allowing people to control devices like phones with their minds.
As of September 2022, the company appears fairly far along in its product development process and is likely to release their headphones within a year or so.
High-bandwidth brain-machine interfaces, surgical robots which implant the interfaces in a manner resembling a sewing machine.
Early goal is to help treat brain disease, has ambitions to enable human enhancement.
Founded by Elon Musk and others, highly publicized by Elon Musk.
Has done testing on rats, pigs, monkeys, and other animals as of April 2021.
Portable medical imaging technologies which employ novel optoelectronics, lasers, and holographic systems.
Wearable imaging technologies which could be 1,000x cheaper than MRI and achieve similar or better results.
Has speculated that their technology might eventually allow telepathic communication.
3D tissue bioprinting for in vivo clinical applications, in vitro tissue models for disease modeling and toxicology.
Long-term goal is to print entire human organs for transplants.
Oxford Nanopore Technologies
Portable nanopore sequencing devices, high-throughput desktop nanopore sequencing devices, sample preparation kits.
The company states that they have the first and only nanopore DNA and RNA sequencing platform as of May 2021.
Genetically modified male insects which curb the reproduction of populations of their species in the wild, acting as a precise and environmentally friendly way of controlling dangerous pests that spread disease or destroy crops.
After years of battles with activists and regulatory bodies, the company will release 750 million genetically modified mosquitos in the Florida Keys (the first time this has been done in the U.S.) with the goal of reducing rates of illnesses such as yellow fever and dengue. 
Panacea Longevity
Enhancing longevity and health using a fasting-mimetic metabolite supplementation.
Early stage as of May 2021.
Prime Medicine
Developing CRISPR Prime editing technology as a novel therapeutic modality.
David Liu and Andrew Anzalone are co-founders.
Mass-produced insect larvae as an affordable way of manufacturing recombinant proteins.
Early stage as of May 2021.
Repair Biotechnologies
Developing a cholesterol degrading platform therapy which can reverse atherosclerosis.
The CEO, who is known as Reason, is outspoken about the need to combat aging.
Has preclinical proof-of-concept as of May 2021.
New manufacturing platforms to service partners for development and scaling of gene therapies, cell therapies, vaccines, protein therapies, and more.
Received $800 million in funding during 2020.
Sherlock Biosciences
CRISPR-based diagnostics.
Feng Zhang is one of the co-founders.
Proteomics platform called SomaScan for protein biomarker discovery which aids researchers in the development of new diagnostics.
SomaScan is an aptamer-based platform which can simultaneously measure 7,000 protein biomarkers.
Founded by Larry Gold, who is the inventor of SELEX.
Offers R&D services through remotely controlled automated laboratories.
Has extensive automated equipment for research in drug discovery, synthetic biology, imaging, cell and gene therapy, etc.
Endovascular brain-computer interfaces as a minimally invasive approach for neural prosthetics, neuromodulation, and neurodiagnostics.
Has developed the strentrode, an endovascular electrode array that can record or stimulate neurons from within blood vessels.
As of September 2022, a technology called (that employs stentrodes) is in early clinical trials and gives paralyzed patients the ability to control digital devices.
CRISPR genome engineering services, custom cell lines, custom screening libraries, CRISPR reagents and kits, aiding both academic researchers and clinical drug developers.
Syzygy Plasmonics
Developing a photocatalytic reactor system which leverages a nanoparticle-based plasmonic photocatalyst. The photocatalyst consists of a larger light-harvesting plasmonic nanoparticle decorated with smaller catalytic nanoparticles. Their first product will be a clean hydrogen fuel production system which does not rely on petroleum.
More of a chemical engineering company than a biotechnology company, but their technology may eventually have applications in biology.
Tilibit Nanosystems
Service which gives researchers predesigned and custom DNA origami nanostructures, including ones with chemical modifications.
Founded by Hendrik Dietz, who was CEO from 2012-2014. He is now a scientific advisor.
Twist Bioscience
Artificial DNA synthesis services. Synthetic biology towards insulin manufacturing in yeast, scalable spider silk manufacturing, combating malaria, and DNA data storage.
Emily Leproust is a co-founder.
Vault Pharma
Protein vault nanocompartments as a drug delivery platform to treat cancers and other diseases, protein vaults as a vaccine platform.
Co-founded by Leonard Rome.
Services in vector cloning, virus packaging, library construction, cell lines, etc.
Verve Therapeutics
Developing CRISPR base editing therapies to turn off key genes (e.g. PCSK9 and ANGPTL3) involved in atherosclerotic plaque formation and thus to combat cardiovascular disease.
The delivery mechanism involves lipid nanoparticles carrying gRNA and mRNA encoding a base editor protein.
Has potential to save tens of millions of lives due to the status of heart disease as one of the most common causes of death.
Early clinical trials began in July 2022.
Synthetic biology, metabolic engineering, biomanufacturing of materials and compounds as a substitute for chemical engineering practices.
4D Molecular Therapeutics
Using high-throughput screening and recombination methods to develop novel AAV serotypes that evade immune responses and that target and transduce specific organs.
Clinical trials for several new AAV vectors that treat pulmonary, cardiac, and eye diseases are ongoing as of September 2022
10x Genomics
Spatial transcriptomics, genomics, proteomics, immune cell profiling, etc.
Acquired ReadCoor and Cartana in 2020.
64x Bio
High-throughput screening and computational design of new mammalian cell lines for manufacturing gene and cell therapies.
George Church and Pamela Silver are among the co-founders.

Cover image is a photograph of a part of Emerald Cloud Lab modified from .

Notes on x-ray physics

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PDF version: Notes on x-ray physics – Logan Thrasher Collins

Thomson scattering and Compton scattering

  • Electrons are the main type of particle that can scatter x-rays. Elastic or Thomson scattering occurs when a non-relativistic electron is accelerated by the electrical component of an incoming electromagnetic field from an x-ray. The accelerated electron then reradiates light at the same frequency. Since the frequency of the input light and output light are the same, this is an elastic process.
  • The intensity of the re-emitted radiation at an observer’s location depends on the angle Χ between the incident light and the observer. Because of the sinusoidal wave character of light, the scattered intensity at the observer’s location is given by the proportionality equation below.


  • Light that encounters the electron is scattered if it is incident on the region defined by the electron’s classical radius. This region is called the Thomson scattering length r0. For a free electron, r0 = 2.82×10-5 Å.


  • Compton scattering occurs when an electron scatters a photon and the scattered photon has a lower energy than the incident photon (an inelastic process). For Compton scattering, a fraction of the incident photon’s energy is transferred to the electron.


  • The amount of energy lost via Compton scattering where the incident photon has energy E0 = hc/λ0 and the scattered photon has energy E1 = hc/λ1 is described by the following equation. Here, ψ represents the angle between the paths of the incident photon and the scattered photon.


Scattering from atoms

  • X-rays are scattered throughout the volumes of atomic electron clouds. For x-rays that scattered in the same direction as the incident x-rays, the strength of scattering is proportional to the atom’s Z-number. In the case of an ionic atom, this value is adjusted to equal the atom’s number of electrons. Note that this assumes free electron movement within the cloud.
  • By contrast, x-rays that are scattered at some angle 2θ relative to the incident x-rays exhibit lower scattering magnitudes. Each of the x-rays scattered at angle 2θ will possess different magnitudes and phases depending on where they were scattered from within the atomic cloud. As a result, the scattering amplitude for the x-rays at angle 2θ will be a vector sum of these waves with distinct magnitudes and phases.


  • A wavevector k is a vector with magnitude 2π/λ that points in the direction of a wave’s propagation. The difference between the wavevector of the incident wave k0 and the wavevector of the scattered wave k1 is equal to a scattering vector Q (that is, Q = k0k1). The magnitude of Q is given by the following equation.


  • The atomic scattering factor f describes the total scattering amplitude for an atom as a function of sin(θ)/λ. By assuming that the atom is spherically symmetric, f will depend only on the magnitude of Q and not on its orientation relative to the atom. Values for f can be found in the International Tables for Crystallography or computed using nine known coefficients a1,2,3,4, b1,2,3,4, and c (which can also be looked up) and the following expression. The coefficients vary depending on the atom and ionic state. The units of f are the scattering amplitude that would be produced by a single electron.


  • If the incident x-ray has an energy that is much less than that of an atom’s bound electrons, the response of the electrons will be damped due to their association with the atom. (This no longer assumes free electron movement within the cloud). As a result, f will be decreased by some value fa. The value fa increases when the incoming x-ray’s energy is close to the energy level of the electron and decreases when the incoming x-ray’s energy is far above the energy levels of the electrons.
  • When the incident x-ray’s energy is close to an electron’s energy level (called an absorption edge), the x-ray is partially absorbed. With this process of partial absorption, some of the radiation is still directly scattered and another part of the radiation is re-emitted after a delay. This re-emitted radiation interferes with the directly scattered radiation. To mathematically describe the effect of the re-emitted radiation’s phase shift and interference, f is adjusted by a second term fb (which is an imaginary value). Far from absorption edges, fb has a much weaker effect (it decays by E-2). The total atomic scattering factor is then given by the following complex-valued equation.


Refraction, reflection, and absorption

  • A material’s index of refraction can be expressed as a complex quantity nc = nRe + inIm. The real part represents the rate at which the wave propagates through the material and the imaginary part describes the degree of attenuation that the wave experiences as it passes through the material.
  • The reason that a material can possess a complex refractive index involves the complex plane wave equation. The wavenumber k = 2π/λ0 is the spatial frequency in wavelengths per unit distance and it is a constant within the complex plane wave equation (λ0 is the wave’s vacuum wavelength). The complex wavenumber kc = knc is the wavenumber multiplied by the complex refractive index. As such, the complex refractive index can be related to the complex wavenumber via kc = 2πnc0 where λ0 is the vacuum wavelength of the wave. After inserting 2π(nRe + inIm)/λ0 into the complex plane wave equation, a decaying exponential can be simplified out as a coefficient for the rest of the equation. The decaying exponential represents the attenuation of the wave in the material. Once this simplification is performed, the equation’s complex wavenumber is converted to a real-valued wavenumber.


  • For x-rays, a material’s complex refractive index for wavelength λ is related to the atomic scattering factors of atoms in the material using the following equation. Ni represents the number of atoms of type j per unit volume and fj(0) is the atomic scattering factor in the forward direction (angle of zero) for atoms of type j. Recall that r0 is the Thomson scattering length.


  • The refractive index is a function of the wavelength. For most optical situations, as the absorption maximum of a material is approached from lower frequencies, the refractive index increases. But when the radiation’s frequency is high enough that it passes the absorption maximum, the refractive index decreases to a value of less than one.
  • The refractive index is defined by n = c/v, where v is the wave’s phase velocity. Phase velocity is the rate at which a wave’s phase propagates (i.e. how rapidly one of the wave’s peaks moves through space). Rearranging the equation, v = c/n is obtained. When the refractive index is less than one, the phase velocity is greater than the speed of light. However, this does not violate relativity because the group velocity (not the phase velocity) carries the wave’s energy and information. For comparison, group velocity is the rate at which a change in amplitude of an oscillation propagates.
  • Anomalous dispersion occurs when the radiation’s frequency is high enough that the refractive index of a material is less than one. As a result, x-rays entering a material from vacuum are refracted away from the normal of the refracting surface. This is in contrast to the typical case where the radiation would be refracted toward the normal of the refracting surface. In addition, the refracted wave is phase shifted by π radians.
  • The complex refractive index is often expressed using the equation below. Here, δ is called the refractive index decrement and β is called the absorption index. Note that nRe = 1 – δ and nIm = β (as a comparison to the previously used notation). Recall that nIm = β describes the degree of a wave’s attenuation as it moves through a material.


  • The refractive index decrement can be approximately computed using the average density of electrons ρ, the Thomson scattering length r0, and the wavenumber k = 2π/λ0. Note that this approximation is better for x-rays that are far from an absorption edge.


  • With most materials, the resulting real part of the index of refraction is only slightly less than one when dealing with x-rays. For example, a typical electron density of one electron per cubic Angstrom yields a δ value of about 5×10-6.
  • Snell’s law applies to the index of refraction for x-rays and is given as follows.


  • Because the index of refraction for x-rays is slightly less than one, total external reflection can occur when x-rays are incident on a surface at angles less than the critical angle θcritical. This stands in contrast with the total internal reflection that commonly occurs with visible light.


  • The critical angle can be approximated with a high level of accuracy using the following equation (derived from the Taylor expansion of the cosine function). With typical values of δ on the order of 10-5, θcritical is often equal to just a few milliradians (or a few tenths of a degree). These small angles relative to the surface are called grazing angles.


  • Because grazing incident angles facilitate x-ray reflection, special curved mirrors can be used to focus x-rays. The curvature of these mirrors must be small enough that the steepest incident angle is less than θcritical. It should be noted that, even when undergoing total external reflection, x-rays do penetrate the reflecting material to a depth of a few nanometers via an evanescent wave.


  • The absorption index β is related to the value fb using the following equation where r0 is the Thomson scattering length. Recall that fb represents the effects of scattering from absorption and remission of x-rays with energies that are close to the absorption edges of a material.


  • Using the process explained earlier for computing the decaying exponential exp(-2πnImx/λ0) that represents the attenuation of a wave’s amplitude as it travels through a material, the decay of a wave’s intensity as it travels through a material can also be found. Recall that λ0 is the wavelength in a vacuum. Because intensity is proportional to the square of the amplitude, the equation below describes the exponential decay of a wave’s intensity in a material. (This decaying exponential function is multiplied by the equation of the wave). Here, μ is called the absorption coefficient and is defined as the reciprocal of the thickness of a material required to decrease a wave’s intensity by a factor of 1/e. The absorption coefficient is a rough indication of a material’s electron density and electron binding energy.


  • The correspondences between the atomic configurations associated with an x-ray absorption edge and the commonly used name for said absorption edge are given in the following table. The subscripts used with the configurations represent the total angular momenta.


X-ray fluorescence and Auger emission

  • Materials fluoresce after bombardment with x-rays or high-energy electrons. If electrons are used, the emitted light consists of Bremsstrahlung radiation (which comes from the deacceleration of the electrons) and fluorescence lines. The Bremsstrahlung radiation includes a broad spectrum of wavelengths and has low intensity while the fluorescence lines are sharp peaks and exhibit high intensity. If x-rays are used to bombard a material, there is no Bremsstrahlung radiation, but fluorescence lines occur.
  • Different materials exhibit different characteristic fluorescence lines. These x-ray fluorescence lines are caused by outer-shell electrons relaxing to fill the holes left after the ejection of photoelectrons. However, not all electronic transitions are allowed, only those which follow the selection rules for electric dipoles. These selection rules are given below. J is the total angular momentum and can be computed from the sum of the Azimuthal quantum number L (which determines the type of atomic orbital) and the spin quantum number S (which determines the direction of an electron’s spin).


  • The nomenclature for x-ray fluorescence lines is based on the shell to which an electron relaxes. If an excited electron relaxes to the 1s shell state, then the fluorescence line is part of the K series. For an excited electron that relaxes to the 2s or 2p state, the fluorescence line is part of the L series. The M series includes relaxations to 3s, 3p, and 3d. The N series includes relaxations to 5s, 5p, 5d, and 5f. As such, the Azimuthal quantum number determines if the fluorescence line falls into the K, L, M, or N series (there are some series beyond these as well which follow the same pattern). The transition within each series that exhibits the smallest energy difference is labeled with α (i.e. Kα), the transition with the next smallest energy difference is labeled with β, and so on. It should be noted that the fluorescence lines are further split by the effects of electron spin and angular momentum and so are labeled with suffixes of 1, 2, etc.
  • Auger emission is the process where a photoelectron is ejected, an outer shell electron relaxes to fill the hole, and the released energy causes ejection another electron instead of emitting a photon. The energies of emitted Auger electrons are independent of the energies of the incident photons.
  • The excess energy released by the relaxation of the outer shell electron is equal to |Ecore – Eouter|. In order for the last electron ejection to occur, the electron must have a binding energy that is less than the excess released energy from the relaxation. The kinetic energy of the ejected Auger electron is |Ecore – Eouter – Ebinding|. Note that Ebinding is the binding energy of the Auger electron in the ionized atom (which is different from the binding energy in the neutral form of the atom).
  • Auger emission and x-ray fluorescence are competitive with each other. Fluorescence is stronger for heavier atoms (higher Z-number) since they exhibit larger energy differences between adjacent shells as well as binding electrons more tightly. For the same reasons, Auger emission is stronger from atoms with lower Z-numbers.



Reference: Willmott, P. (2011). An Introduction to Synchrotron Radiation: Techniques and Applications. Wiley.

Cover image courtesy of: Asia Times






Notes on wave optics

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PDF version: Notes on wave optics – Logan Thrasher Collins

The wave equation

  • Because light exhibits wave-particle duality, wave-based descriptions of light are often appropriate in optical physics, allowing the establishment of an electromagnetic theory of light.
  • As electric fields can be generated by time-varying magnetic fields and magnetic fields can be generated time-varying electric fields, electromagnetic waves are perpendicular oscillating waves of electric and magnetic fields that propagate through space. For lossless media, the E and B field waves are in phase.
  • By manipulating Maxwell’s equations of electromagnetism, two relatively concise vector expressions that describe the propagation of electric and magnetic fields in free space are found. Recall that the constants ε0 and μ0 are the permittivity and permeability of free space respectively.


  • Since an electromagnetic wave consists of perpendicular electric and magnetic waves that are in phase, light can be described using the wave equation (which is equivalent to the expressions above). Note that the speed of light c = (ε0μ0)-1/2. Electromagnetic waves represent solutions to the wave equation.


  • Either the electric or the magnetic field can used to represent the electromagnetic wave since they propagate with the same phase and direction. With the exception of the wave equation above, the electric field E will instead be used to represent both waves. Note that either the electric or magnetic field can be employed to compute amplitudes.

Solutions to the wave equation

  • Plane waves represent an important class of solutions to the wave equation. The parameter k is the wavevector (which points in the direction of the wave’s propagation) with a magnitude equal to the wavenumber 2π/λ. In a 1-dimensional system, the dot product k•r is replaced by kx. The parameter ω is the angular frequency 2πf and φ is a phase shift.


  • To simplify calculations, Euler’s formula can be used to convert the equation above into complex exponential form. Only the real part describes the wave as the real part corresponds to the cosine term.


  • Spherical waves are another useful solution to the wave equation (though they are an approximation and truly spherical waves cannot exist). Because of their geometry, the electric field of a spherical wave is only dependent on distance from the origin. As such, the equation for a spherical wave can be written as seen below with origin r0.


  • Gaussian beams are a solution to the wave equation that can be used to model light from lasers or light propagating through lenses. If a Gaussian beam propagates in the z direction, then from the perspective of the xy plane, it shows a Gaussian Fig. 1intensity distribution. For a Gaussian beam, the amplitude decays over the direction of propagation according to some function A(z), R(z) represents the radius of curvature of the wavefront, and w(z) is the radius of the wave on the xy plane at distance z from the emitter. Often these functions can be approximated as constants.


Intensity and energy of electromagnetic waves

  • The Poynting vector S is oriented in the direction of a wave’s propagation (assuming that the wave’s energy flows in the direction of its propagation).


  • The magnitude of the Poynting vector represents the power per unit area (W/m2) or intensity crossing a surface with a normal parallel to S. Note that this is an approximation since, according to a quantum mechanical description of electromagnetic waves, the energy should be quantized.


  • Power per unit area (intensity) of plane waves, spherical waves, and Gaussian beams can also be calculated using the equations below. The formula for the Gaussian beam’s power represents the power at a plane perpendicular to the direction of light propagation z.


  • For electromagnetic waves, instantaneous energy per unit area is difficult to measure, so the average energy per unit area over a period of time Δt is often worked with instead. Since waves are continuous functions, taking their time-average requires an integral.


  • After using the above integral on the function eiωt and then taking the real and imaginary parts of the result, the time-averages of the functions cos(ωt) and sin(ωt) are found.


Superposition of waves

  • Let two waves E1 and E2 of the same frequency traveling in the same direction undergo superposition. E1 and E2 may or may not possess the same amplitude or phase. The substitution α = –(kx+φ) will be carried out.


  • If the phases of the waves are different, some special equations are necessary to find the amplitude E0 and the phase α of the resulting wave.


  • For the superposition of any number of waves, the equations above can be extended.


Polarization of light

  • The waves comprising linearly polarized light are all oriented at the same angle which is defined by the direction of the electric field of the light waves. For linearly polarized plane waves with electric fields oriented along the x or y axes that propagate in the z direction, the following equations describe their electric fields.


  • The superposition of two linearly polarized plane waves that are orthogonal to each other (and out of phase) is the vector sum of each electric field.


  • The superposition of two linearly polarized plane waves that are orthogonal to each other (and in phase) is computed via the following equation and has a tilt angle θ determined by the ratio of amplitudes of the original waves. This process can also be performed in reverse with a superposed polarized wave undergoing decomposition into two orthogonal waves.

eq18Fig. 2

  • When two constituent waves possess equal amplitudes and a phase shift of nπ/2, the superposed wave is circularly polarized (as it can be expressed using a sine and a cosine term). Equations for the constituent waves and the superposed wave are given below.


  • When circularly polarized light propagates, it takes a helical path and so rotates. As such, a full rotation occurs after one wavelength. If a circularly polarized wave rotates clockwise, it is called right-circularly polarized and has a positive sine term. If a circularly polarized wave rotates counterclockwise, it is called left-circularly polarized and has a negative sine term.


  • If a right-circularly polarized light wave and a left-circularly polarized light wave of equal amplitude are superposed, then they create a linearly polarized light wave with twice the amplitude of the individual waves.


  • Linearly polarized and circularly polarized light are special cases of elliptically polarized light. For elliptically polarized light, the amplitudes of the superposed waves may differ and the relative phase shift does not need to be nπ/2. As such, the electric field traces an elliptical helix as it propagates along the z direction.


  • For elliptically polarized light with a positive phase shift φ, it is called right-elliptically polarized if E0x > E0y and left-elliptically polarized if E0x < E0y.
  • Most light is unpolarized (or more appropriately, a mixture of randomly polarized waves). To obtain polarized light, polarizing filters are often used.



Boudoux, C. (2017). Fundamentals of Biomedical Optics. Blurb, Incorporated.

Degiorgio, V., & Cristiani, I. (2015). Photonics: A Short Course. Springer International Publishing.

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