Diffraction (2)

The divergence of a laser beam, or the way in which the beam's intensity profile changes as it propagates, is determined by diffraction. A laser beam with a planar, spatially coherent wavefront that approximates a Gaussian beam profile will have the lowest divergence for a given diameter. The smaller the output beam, the faster it will diverge. The divergence of a laser beam can be reduced by expanding it with a convex lens and then collimating it with a second convex lens whose focal point is coincident with the first lens. This results in a beam with a larger diameter and lower divergence. In some cases, the divergence of a laser beam can be reduced below the diffraction limit of a Gaussian beam or even reversed to convergence if the refractive index of the propagation media increases with the light intensity, leading to a self-focusing effect. When the wavefront of the emitted beam has perturbations, the transverse coherence length, or the distance over which the wavefront perturbation is less than a quarter wavelength, should be used to determine the divergence of the laser beam. If the transverse coherence length is higher in one direction than the other, the laser beam's divergence will be lower in that direction.

Diffraction ultimately limits the ability of an imaging system to resolve detail. When a plane wave is incident on a circular lens or mirror, it is diffracted, forming an Airy disk with a central spot in the focal plane. The radius of this central spot, as measured to the first null, is determined by the wavelength of the light and the f-number (the ratio of the focal length to the aperture diameter) of the imaging optics. In object space, the corresponding angular resolution is determined by the diameter of the entrance pupil of the imaging lens and the wavelength of the light. When two point sources are imaged, each will produce its own Airy pattern. As the point sources move closer together, the patterns will begin to overlap and eventually merge into a single pattern, at which point the two point sources cannot be resolved in the image. The Rayleigh criterion states that two point sources are considered resolved if the separation between their images is at least the radius of the Airy disk. This means that the larger the aperture of the lens compared to the wavelength, the finer the resolution of the imaging system. This is why astronomical telescopes require large objectives and microscope objectives require a high numerical aperture (a large aperture diameter relative to the working distance) to achieve the highest possible resolution.

The speckle pattern that appears when using a laser pointer is a result of the superposition of multiple waves with different phases. These waves are produced when a laser beam hits a rough surface and interfere with each other, causing the intensity of the resulting wave to vary randomly. Babinet's principle states that the diffraction pattern produced by an opaque object is identical to the pattern produced by a hole of the same size and shape, but with different intensities. This means that the interference conditions for a single obstruction are the same as those for a single slit. The knife-edge effect, or knife-edge diffraction, occurs when a portion of incident radiation is truncated by a sharp, well-defined obstacle like a mountain range or the wall of a building. It can be explained using the Huygens-Fresnel principle, which states that a well-defined obstruction to an electromagnetic wave acts as a secondary source, creating a new wavefront that propagates into the shadow area behind the obstacle. The knife-edge effect is related to the "half-plane problem," which was originally solved using a plane wave spectrum formulation by Arnold Sommerfeld. The "wedge problem," a generalization of the half-plane problem, can be solved as a boundary value problem in cylindrical coordinates and was later extended to the optical regime by Joseph B. Keller through his development of the geometrical theory of diffraction (GTD). The Keller coefficients were later generalized further through the uniform theory of diffraction (UTD) by Pathak and Kouyoumjian.

There are several key things to know about diffraction. It refers to the bending or interference of waves as they pass around an obstacle or through an opening. This can happen with any kind of wave, including light, sound, water, and even subatomic particles. The size of the diffraction effect depends on the size of the obstacle or opening in relation to the wavelength of the wave. When a wave from a coherent source, such as a laser, encounters an obstacle or opening that is similar in size to its wavelength, the diffraction effect is particularly pronounced. This can lead to the formation of patterns with maxima and minima, or the appearance of fringes. The diffraction effect is important in various applications, as it sets a fundamental limit on the resolution of imaging systems like cameras, telescopes, and microscopes. It is also responsible for phenomena like the rainbow pattern on CDs and DVDs, the bright rings around bright light sources, and the speckle pattern on rough surfaces when illuminated with laser light.

As is well-known in the realm of quantum theory, every particle exhibits wave-like properties. This becomes particularly evident when observing massive particles, which have the ability to interfere with themselves and therefore diffract. The diffraction of electrons and neutrons has long been hailed as one of the strongest arguments in support of quantum mechanics.

The wavelength of a particle is described by the de Broglie equation, in which the wavelength is equal to Planck's constant divided by the momentum of the particle (the product of mass and velocity for slow-moving particles). For most macroscopic objects, the resulting wavelength is so small that it is practically imperceptible. However, for smaller particles such as electrons, neutrons, atoms, and small molecules, the wavelength is more significant and allows for the observation of diffraction.

This phenomenon has proven particularly useful in the study of the atomic crystal structure of solids and large molecules like proteins, due to the suitability of matter waves with short wavelengths for this purpose. Even relatively larger molecules like buckyballs have been shown to exhibit diffraction.

When waves are diffracted by a three-dimensional periodic structure, such as the atoms in a crystal, the resulting diffraction is known as Bragg diffraction. It is similar to the scattering of waves by a diffraction grating. This phenomenon is a result of the interference between waves that are reflected from different crystal planes. The condition for constructive interference is described by Bragg's law, which states that the product of the wavelength and an integer known as the diffracted beam's order, must be equal to twice the distance between crystal planes, multiplied by the sine of the angle of the diffracted wave.

Bragg diffraction can be carried out using either electromagnetic radiation with a very short wavelength, such as X-rays, or matter waves like neutrons or electrons, which have wavelengths on the same order as or smaller than the spacing between atoms. The resulting diffraction pattern provides information about the separation of crystallographic planes, allowing one to deduce the crystal structure. In electron microscopes and x-topography devices, diffraction contrast is also used to examine individual defects and local strain fields in crystals.

The usual explanation of diffraction involves the interference of waves emanating from a single source, which take different paths to the same point on a screen. This description assumes that the phase difference between these waves is solely determined by the difference in their path lengths. However, this ignores the fact that the waves arriving at the screen at the same time were actually emitted by the source at different times. The initial phase of the emitted waves can change randomly over time, meaning that waves emitted by the source at times that are too far apart can no longer produce a consistent interference pattern due to the changing relationship between their phases.

The length over which the phase of a beam of light is correlated is known as the coherence length. In order for interference to occur, the difference in path lengths must be smaller than the coherence length. This is sometimes referred to as spectral coherence, as it is related to the presence of different frequency components in the wave. In the case of light emitted from an atomic transition, the coherence length is determined by the lifetime of the excited state from which the atom made its transition.

If waves are emitted from an extended source, this can lead to incoherence in the transverse direction. The length over which the phase is correlated in a cross section of a beam of light is known as the transverse coherence length. In the case of Young's double-slit experiment, if the transverse coherence length is smaller than the spacing between the two slits, the resulting pattern on a screen would resemble two single-slit diffraction patterns.

For particles such as electrons, neutrons, and atoms, the coherence length is related to the spatial extent of the wave function that describes the particle.

In recent years, a new method for imaging single biological particles has emerged, making use of bright X-rays generated by X-ray free electron lasers. These pulses, which last for only a few femtoseconds, have the potential to enable the imaging of individual biological macromolecules. Because the duration of these pulses is so short, radiation damage can be avoided, allowing for the creation of diffraction patterns from single biological macromolecules.