Chapter 27: Wave Optics

axis of a polarizing filter

the direction along which the filter passes the electric field of an EM wave

birefringent

crystals that split an unpolarized beam of light into two beams

Brewster’s angle

𝜃b=tan−1⁢(𝑛2𝑛1), where 𝑛2 is the index of refraction of the medium from which the light is reflected and 𝑛1 is the index of refraction of the medium in which the reflected light travels

Brewster’s law

tan⁡𝜃b=𝑛2𝑛1, where 𝑛1 is the medium in which the incident and reflected light travel and 𝑛2 is the index of refraction of the medium that forms the interface that reflects the light

coherent

waves are in phase or have a definite phase relationship

confocal microscopes

microscopes that use the extended focal region to obtain three-dimensional images rather than two-dimensional images

constructive interference for a diffraction grating

occurs when the condition 𝑑⁢sin⁡𝜃=𝑚𝜆⁢(for𝑚=0,1,–1,2,–2,…) is satisfied, where 𝑑 is the distance between slits in the grating, 𝜆 is the wavelength of light, and 𝑚 is the order of the maximum

constructive interference for a double slit

the path length difference must be an integral multiple of the wavelength

contrast

the difference in intensity between objects and the background on which they are observed

destructive interference for a double slit

the path length difference must be a half-integral multiple of the wavelength

destructive interference for a single slit

occurs when 𝐷⁢sin⁡𝜃=mλ,(for𝑚=1,–1,2,–2,3,…), where 𝐷 is the slit width, 𝜆 is the light’s wavelength, 𝜃 is the angle relative to the original direction of the light, and 𝑚 is the order of the minimum

diffraction

the bending of a wave around the edges of an opening or an obstacle

diffraction grating

a large number of evenly spaced parallel slits

direction of polarization

the direction parallel to the electric field for EM waves

horizontally polarized

the oscillations are in a horizontal plane

Huygens’s principle

every point on a wavefront is a source of wavelets that spread out in the forward direction at the same speed as the wave itself. The new wavefront is a line tangent to all of the wavelets

incoherent

waves have random phase relationships

interference microscopes

microscopes that enhance contrast between objects and background by superimposing a reference beam of light upon the light emerging from the sample

optically active

substances that rotate the plane of polarization of light passing through them

order

the integer 𝑚 used in the equations for constructive and destructive interference for a double slit

phase-contrast microscope

microscope utilizing wave interference and differences in phases to enhance contrast

polarization

the attribute that wave oscillations have a definite direction relative to the direction of propagation of the wave

polarization microscope

microscope that enhances contrast by utilizing a wave characteristic of light, useful for objects that are optically active

polarized

waves having the electric and magnetic field oscillations in a definite direction

Rayleigh criterion

two images are just resolvable when the center of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other

reflected light that is completely polarized

light reflected at the angle of reflection 𝜃b, known as Brewster’s angle

thin film interference

interference between light reflected from different surfaces of a thin film

ultraviolet (UV) microscopes

microscopes constructed with special lenses that transmit UV rays and utilize photographic or electronic techniques to record images

unpolarized

waves that are randomly polarized

vertically polarized

the oscillations are in a vertical plane

wavelength in a medium

𝜆n=𝜆/𝑛, where 𝜆 is the wavelength in vacuum, and 𝑛 is the index of refraction of the medium

Like all EM waves, the following relationship is valid in vacuum:

𝑐=𝜈𝜆, where 𝑐 is the speed of light, 𝜈 is the frequency, and 𝜆 is the wavelength.

The wavelength 𝜆n of light in a medium with index of refraction

is given by the formula 𝜆n=𝜆/𝑛, indicating how the speed of light is altered in different media.

constructive interference (Young’s Double Slit Experiment)

occurs when waves combine in phase, resulting in increased amplitude and brightness at certain points.

destructive interference (Young’s Double Slit Experiment)

occurs when waves combine out of phase, resulting in a lower intensity or cancellation of light at certain angles.

constructive interference for a diffraction grating

occurs when light waves from multiple slits align in phase, producing bright fringes and enhanced light intensity at specific angles.

destructive interference for a single slit

occurs when light waves from a single slit combine out of phase, leading to a reduction in intensity and dark regions in the diffraction pattern.

The intensity 𝐼 of polarized light after passing through a polarizing filter

is described by Malus's Law, which states that 𝐼 = 𝐼₀ cos²(θ), where 𝐼₀ is the initial intensity and θ is the angle between the light's polarization direction and the filter's axis.