Notes on wave optics


No Comments

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.

eq1

  • 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.

eq2

  • 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.

eq3

  • 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.

eq4

  • 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.

eq5

  • 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.

eq6

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).

eq7

  • 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.

eq8

  • 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.

eq9

  • 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.

eq10

  • 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.

eq11

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.

eq12

  • 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.

eq13

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

eq14

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.

eq15

  • 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.

eq16

  • 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.

eq19

  • 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.

eq20

  • 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.

eq21

  • 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.

eq22

  • 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.

 

References

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

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

Hecht, E. (2017). Optics. Pearson Education, Incorporated.

 

Notes on fiber biomechanics


No Comments

PDF version: Notes on fiber biomechanics – Logan Thrasher Collins

Elastic fiber models

  • For an elastic fiber in which a linear relationship between force and change in length is assumed, the force is given by F = k(L – L0).
  • To normalize for other elastic fibers with different starting lengths, this equation is divided by L0 to give F = k(L/L0 – 1). It is common practice to represent L/L0 as a parameter λ (called the stretch ratio).
  • As such, F is found using the formula below. Note that the quantity λ – 1 is referred to as the strain.

Eq1

  • While linear models are often useful, many real fibers exhibit finite extensibility (a nonlinear phenomenon) after exceeding a certain critical strain value λc. That is, the force necessary to extend the fiber farther after exceeding λc increases rapidly. Finite extensibility can be modeled using the following equation which divides k by a term dependent on λ and λc.

Eq2

  • To model a muscle, let L0 represent the muscle’s length in its inactive state and Lcontracted represent the muscle’s length in its contracted state. Unlike the spring, the contracted state is used as the reference length. The contraction stretch is described by the ratio λcontracted = Lcontracted/L0 while the stretch ratio remains as λ = L/L0.

Eq3

  • If this muscle is contracted without carrying a load such that F = 0, then λ = λcontracted. If the muscle acquires a load and so must maintain a constant length equal to its original length L0 (to “hold the load steady”), then the force in the muscle is F = k(1/λcontracted – 1).
  • To generalize this model for 3-dimensional space, the locations of the fiber’s endpoints A and B are used. The fiber’s length and orientation are given below.

Eq4

  • The following force vectors can act on point B and on point A. The stretch ratio is still λ = L/L0.

Eq5

Viscous fiber models

  • Purely viscous behavior (as with liquids) can be described 1-dimensionally using the equation below where cη is a damping coefficient.

Eq6

  • The normalized rate of deformation is equivalent to the above formula without the damping coefficient. In addition, the rate of deformation can be written in terms of the stretch ratio λ = L/L0.

Eq7

  • If one endpoint of a filament of fluid is moved with constant velocity, its position is given by xB = L0 + vt. This means that the rate of deformation is v/xB.

Eq8

  • Solving the above equation gives the following result. For a constant rate of elongation, the point xB must be displaced exponentially over time.

Eq9

  • Given endpoint displacements uA and uB, the total displacement is ΔL = uB – uA. Using this quantity, the stretch λ and the strain ε can be written using the equations below.

Eq10

  • For the small strains (i.e. |ε| is much less than 1) that result from small stretches, some approximations can be made which reduce the force equation to the following form.

Eq11

Viscoelastic fiber models

  • Many biological materials exhibit viscoelastic behavior rather than elastic behavior. In viscoelastic systems, the force on a fiber with a constant length decreases over time and applying constant force causes the length to increase.
  • The strain response of a fiber to force is given as ε(t). The creep function J(t) describes the fiber’s tendency to permanently deform as strain is applied. Many possible creep functions can be devised depending on the system. The creep function is related to the strain by a factor of force increase F0.

Eq12

  • Strain responses follow the principle of superposition. That is, if a force is applied at τ1 and then another force is applied at τ2, a total strain response can be expressed as a sum of the two individual strain responses.

Eq14

  • For an arbitrary history of applied forces, an integral formulation of strain response is used. In this equation, the change in force at each infinitesimal time interval is multiplied by the creep function.

Eq15

  • Similarly, the force resulting from an imposed strain history can be expressed as an integral where G is a function that describes the relaxation of the fiber with time (analogous to the creep function, but the opposite concept).

 

Reference: Oomens, C., Brekelmans, M., & Baaijens, F. (2009). Biomechanics: Concepts and Computation. Cambridge University Press.

 

Notes on nanoparticle self-assembly


No Comments

PDF version: Notes on nanoparticle self-assembly – Logan Thrasher Collins

Preparation of nanoparticle superlattices

  • Nanoparticle superlattices can be prepared using solvent evaporation, solvent destabilization, or gravitational sedimentation methods.
  • Solvent evaporation involves evaporating a nanoparticle-containing solvent to induce ordered aggregation of the particles. Note that many inorganic nanoparticles are insoluble in polar solvents and soluble in nonpolar solvents, though the presence of polar surface ligands can alter this behavior. Fig. 1
  • Solvent evaporation techniques include (i) placing a small droplet of nanoparticle-containing solvent on a solid substrate and allowing for evaporation to occur, (ii) evaporating a nanoparticle-containing solvent from a tilted vial so as to control the orientation of the meniscus, (iii) placing a small droplet of polar solvent on a solid substrate and then adding a nonpolar nanoparticle-containing solvent over the top to facilitate aggregation in the thin layer of nonpolar solvent, and (iv) filling a tray with a polar solvent and adding nonpolar nanoparticle-containing solvent over the top to facilitate aggregation in the thin layer of nonpolar solvent.
  • Solvent destabilization promotes gradual clustering of nanoparticle in solution via slowly changing the solvent conditions. Solvent destabilization techniques include (i) allowing for a polar and a nonpolar solvent to gradually intermingle and so facilitate a steady increase in the favorability of nanoparticle-nanoparticle interactions and (ii) heating a premixed solvent mixture that includes both polar and nonpolar components and so facilitating a controlled enrichment of the solvent with a higher boiling point. This increases the favorability of nanoparticle-nanoparticle interactions in a controlled fashion. Because many nonpolar liquids possess lower boiling points, the nanoparticle lattices prepared in this way may require nanoparticles equipped with polar surface ligands.
  • Gravitational sedimentation is less common than the other techniques since many nanoparticles are small enough to remain dissolved in spite of gravitational forces. But very large nanoparticles (100-1,000 nm) often do sediment, facilitating close-packing and the assembly of superlattices.

Characterization of nanoparticle superlattices

  • Transmission electron microscopy (TEM) is used to visualize nanoparticle superlattices directly. As TEM requires very thin slices, it makes 2-dimensional images of nanoparticle superlattices.
  • TEM operates best when there is a high contrast between the atomic number of the nanoparticles and the atomic number of the background support structure. For instance, PbS is easily imaged on a carbon support.
  • To circumvent issues that arise with atomic number contrast, ultrathin (i.e. graphene) supports or supports that possess numerous holes can be used. Ultrathin supports absorb less electrons while “holey” supports allow some nanoparticles to be positioned over the holes during imaging, preventing background absorption.
  • TEM often requires a vacuum chamber and so necessitates dry samples, meaning  that superlattice structure can be visualized after removal of the solvent, but snapshots of the self-assembly process cannot be taken. However, recent investigations into designing liquid-cell TEM may circumvent this problem.
  • Scanning electron microscopy (SEM) generates 3-dimensional images via a scanning electron beam and so is useful for imaging nanoparticles and Fig. 2nanoparticle superlattices that exhibit some kinds of notable 3-dimensional geometric characteristics.
  • Atomic force microscopy, a technique in which a nanoscale probe is moved across a sample to reconstruct its shape via a “sense of touch,” has also been used for nanoparticle superlattice characterization.
  • Images of repetitive superlattices are amenable to processing with two-dimensional fast Fourier transforms (FFTs) that can reveal insights about the lattice’s characteristics. Performing this form of FFT upon an image of a repetitive crystal structure creates a plot of spatial frequencies. This plot is said to display reciprocal space (or Fourier space).
  • Distinct points on the reciprocal space plot correspond to certain properties of the lattice that are sometimes not apparent from the image prior to the FFT. In this way, very similar lattices can be clearly distinguished.

Kinetics of nanoparticle superlattice formation

  • Homogenous nucleation occurs in solution and requires overcoming a nucleation barrier while heterogenous nucleation occurs as nanoparticles are added to a preexisting seed crystal. Homogenous nucleation typically leads to disordered solids and is typically much slower than heterogenous nucleation. Fig. 3
  • In heterogenous nucleation, crystal growth occurs at differing rates depending on how many new contacts are formed (assuming attractive interparticle interactions). If more new contacts occur upon the addition of a nanoparticle, the process will exhibit grater energetic favorability and occur at a faster rate.
  • This means that adding nanoparticles to kinks and vacancies happens more rapidly than the adsorption of nanoparticles to steps, terraces, and “adatoms” (see figure at right). As such, large scale structures that minimize surface energy tend to form.

Thermodynamics of nanoparticle superlattices

  • As mentioned, if superlattice assembly occurs rapidly, disordered aggregates can form. Allowing the process to occur more slowly facilitates sampling of many states as assembly proceeds. As such, the most thermodynamically stable structures can form when gradual assembly is performed.
  • Van der Waals interactions between nanoparticles are often approximated using the following pair potential equation. U is the potential energy for the interparticle interaction, C represents a proportionality constant for the interparticle interaction, ρ1 and ρ2 are the number of atoms per unit volume in two interacting bodies, and r is the distance between the bodies.

Eq.1

  • For nanoparticles with volumes V1 and V2, the total van der Waals energy of attraction is obtained by the following integral that performs a pairwise summation of all the atomic van der Waals interactions.

Eq.2

  • If two nanoparticles are spherical with radii R1 and R2, the integral can be solved analytically to give their interparticle potential energy.

Eq.3

  • When the distance d between two nanoparticles is much less than the radius of either nanoparticle, the above equation can be approximated using the following formula.

Eq.4

  • For many nanoparticles without chemical ligands, these van der Waals interactions would cause rapid aggregation in solution. However, the presence of certain surface ligands gives repulsion that maintains colloidal solutions of nanoparticles.
  • In order to convey repulsive effects between nanoparticles, the surface ligands require a proper solvent. Such solvents exhibit negative free energy upon ligand-solvent mixing. That is, interactions between the surface ligand and the solvent are energetically favorable.
  • Surface ligand repulsion includes an osmotic component and an elastic component. As the solvent molecules are sterically blocked by surface ligands, when the surface ligands of two nanoparticles start to interact, the volume that the solvent cannot enter increases. This situation is osmotically unfavorable, so osmotic repulsion occurs. When surface ligands are compressed because two nanoparticles are close to each other, elastic repulsion occurs.
  • When a solution of nanoparticles with surface ligands that are attracted to each other (i.e. hydrophobic chains) is dried, the ligands begin to freeze together rather than experiencing repulsion.
  • Equilibrium superlattice structures minimize the free energy G in terms of enthalpy U and entropy S according to Gibb’s equation ΔG = ΔU – TΔS.
  • The contributions of cores and ligands can be decomposed into the terms ΔUcores, ΔUligands, ΔScores, and ΔSligands. The energy terms can be further broken down into the components of van der Waals interactions (London dispersion forces, dipole-induced dipole interactions, and dipole-dipole interactions). The entropy terms can be further broken down into configurational, rotational, and translational components.

 

Reference: Boles, M. A., Engel, M., & Talapin, D. V. (2016). Self-Assembly of Colloidal Nanocrystals: From Intricate Structures to Functional Materials. Chemical Reviews, 116(18), 11220–11289. https://doi.org/10.1021/acs.chemrev.6b00196