Diffraction of light refers to the phenomenon where light waves bend or spread out as they encounter the edge of a material or pass through a slit opening. Light behaves like a quantized wave in the electromagnetic field. When it travels, it also shows internal diffraction. Internal diffraction is the interaction of the light waves with the diffracting object or aperture, resulting in deviations from the straight path. This phenomenon arises from the fact that light continuously interacts with itself, causing internal interference among the different wave components. Consequently, light waves bend or spread. It results in a distinctive pattern of bright and dark regions, known as a diffraction pattern. This diffraction pattern can be observed on a screen situated behind the obstacle or aperture. The extent of bending depends upon the diameter of the slit. It is also influenced by several factors, including the size, shape, and wavelength of the waves. Fraunhofer and Fresnel diffraction are two distinct forms of diffraction that occur when light waves interact with an obstacle or aperture.

Figure: Diffraction pattern

Diffraction creates interference patterns with bright and dark fringes. The central maximum has the highest intensity due to constructive interference from all the parts of slit. Moving away from it, fringe intensity decreases as destructive interference from different parts of slit increases. Secondary maxima occur at specific angles with constructive interference from specific fractions of the slit. Intensity decreases further for successive bright fringes.

Fresnel Diffraction

Fresnel diffraction refers to the phenomenon of light waves encountering an obstacle or passing through an aperture at finite distances. It occurs when the distances between the diffracting object, the light source, and the observation plane are comparable to the wavelength of light. Fresnel diffraction was initially observed and described by Augustin-Jean Fresnel, a French physicist, in the early 1800s.

It is formed when a coherent beam of light passes through a small aperture or around an obstacle that is comparable in size to the wavelength of the light. As the light passes through the aperture or around the obstacle, the wavefronts interfere with each other, causing the light to spread out and deviate from its original path. This interference of wavefronts gives rise to a distinct pattern known as a Fresnel diffraction pattern. This pattern can be observed as regions of bright and dark fringes, representing constructive and destructive interference, respectively. In regions where the constructive interference is strong, the fringes appear brighter due to the amplified intensity resulting from the superposition of the waves. Conversely, regions with destructive interference exhibit darker fringes as the interfering waves cancel each other out, leading to reduced intensity.

In Fresnel diffraction, both incoming and outgoing wavefronts are either spherical or cylindrical. The wavefront curvature and amplitude variations across the wavefront are considered, leading to complex interference patterns and bending of the light waves. The complexity of the interference patterns arises from the intricate superposition of waves with different phases, amplitudes, and spatial distributions across the wavefront.

The resulting interference pattern produces a series of bright and dark fringes on a screen placed at a finite distance. The intensity of the fringes is directly related to the amplitude of the waves interfering at a particular point on the screen.

For the central maximum (the maxima) of the Fresnel diffraction pattern, all the wavelets from the entire aperture add up in phase, resulting in constructive interference. This leads to a bright fringe at the center of the pattern.

As we move away from the central maximum towards the bright fringes, the secondary sources (wavelets from different points on the aperture) have slightly different path lengths to the point of observation on the screen. These path length differences lead to phase differences between the wavelets. In some regions, the wavelets may interfere constructively, leading to bright fringes, while in other regions, they may interfere destructively, leading to dark fringes.

Figure: Fresnel diffraction

The diffraction pattern created by Fresnel diffraction can be observed by placing a screen at a finite distance away from the aperture or obstacle and allowing the diffracted light to fall on it. The diffraction pattern consists of circular bright and dark fringes, which become progressively wider (the separation or spacing between the individual fringes increases) as the distance from the aperture or obstacle increases.

Fresnel utilized the principle of superposition along with Huygens' principle to explain this phenomenon. Huygens' principle states that each point on a wavefront generates secondary waves, and it is the interference of these secondary waves that result in diffraction. The interference produces a resultant intensity of light at any point.

Figure: Huygen’s principle

In Fresnel diffraction, the diffraction pattern typically consists of circular bright and dark fringes, also known as Fresnel zones or rings. The zones are defined based on the path length difference between adjacent points on the wavefront, as measured from the aperture to the observation point. The path length difference determines the phase difference between the waves reaching those adjacent points, and this phase difference leads to constructive or destructive interference.

The first Fresnel zone is the region closest to the aperture, where the path length difference between adjacent points on the wavefront is zero or an integral multiple of the wavelength. In this zone, the waves arriving at adjacent points are in phase, resulting in constructive interference, and a bright fringe is observed.

As we move outward from the first zone, the path length difference increases, and at some distance, it becomes equal to half of the wavelength. At this distance, the waves arriving at adjacent points are out of phase by half a wavelength, leading to destructive interference and a dark fringe.

The second zone starts where the path length difference is half a wavelength and extends until the path length difference reaches one wavelength, where the second bright fringe is formed due to constructive interference. This pattern continues with alternating bright and dark fringes for each successive zone.

The resultant intensity of the diffracted wave depends on the degree of interference between the wavefronts in the Fresnel zones. If the aperture or object is small, the zones will be narrow, and the interference will be strong, leading to a more distinct diffraction pattern with sharp fringes. Conversely, if the aperture or object is large, the zones will be broad, and the interference will be weak, leading to a less distinct diffraction pattern with broad fringes.

The intensity of the diffraction pattern created by Fresnel diffraction can be calculated using the Fresnel-Kirchhoff diffraction integral:

Where U(P) is the complex amplitude of the diffracted light at a point P on the screen, U(Q) is the complex amplitude of the incident light at a point Q on the aperture or obstacle, λ is the wavelength of the incident light, k is the wave number (2π/λ), z is the distance between the aperture or obstacle and the screen, |P-Q| is the distance between the points P and Q, and θ is the angle between the normal to the aperture or obstacle and the direction of the diffracted light.

The grating equation of Fresnel diffraction relates the wavelength of the incident light to the distance between the slits and the position of the bright fringes on the screen. It can be expressed as:

Where m is the order of the bright fringe, λ is the wavelength of the incident light, z is the distance between the slits & the screen, θ is the angle between the incident beam & the normal to the grating and α is the angle between the diffracted beam & the normal to the grating

This equation shows that the position of the bright fringes depends on the wavelength of the incident light, the distance between the slits and the screen, and the angles of incidence and diffraction. By adjusting any of these parameters, the position of the bright fringes can be manipulated.

Applications of Fresnel diffraction

The applications of Fresnel diffraction are numerous and diverse, spanning multiple fields of science and engineering. Fresnel diffraction is used in optics to create various types of lenses and optical devices, and it is also used in acoustics and electromagnetic waves. In addition, Fresnel diffraction has applications in diffraction tomography, particle sizing, microscopy, and holography.

The understanding of Fresnel diffraction has played a significant role in the development of modern optics and has paved the way for the creation of innovative optical devices, such as Fresnel lenses, diffraction gratings, and holographic lenses.

• It is used in diffraction gratings in optical instruments to disperse and analyze light.
• In optical microscopy to study the diffraction pattern produced by small objects, such as cells or bacteria, Fresnel diffraction is used.
• This kind of diffraction is used to shape laser beams by passing them through a specially designed aperture or lens.
• Fresnel diffraction can also be used in radio communication to analyze the behavior of radio waves as they diffract around obstacles.
• It is also used in holography to create three-dimensional images.

Fraunhofer Diffraction

Fraunhofer diffraction is a specific type of diffraction that occurs when light waves pass through an aperture or encounters an obstacle at infinite distances. In this diffraction, the observation point is located in the far-field region, relatively far away from the aperture or object, so that the incoming and outgoing waves are effectively planar. Since the source is at infinity, the rays of light which pass through the slit are parallel rays of light. In order to make these parallel rays to focus on the screen, a converging lens is used. It is named after the German physicist Joseph von Fraunhofer, who first observed the phenomenon in the early 19th century.

Fraunhofer diffraction is formed when a beam of light is incident upon a slit or a series of slits that are much smaller in size than the wavelength of the light. The amount of diffraction is influenced by the size of the aperture or slit relative to the wavelength of the light.

As the light passes through the slit, it is diffracted, meaning it spreads out and bends around the edges of the slits. This diffracted light from different points on the slits then interferes with each other, creating an interference pattern on a screen placed some distance away. The diffraction pattern is a result of the interference between the diffracted waves that pass through the edge of the slit and the non-diffracted waves that pass directly through the slit.

The resulting diffraction pattern is characterized by a series of bright and dark fringes that are parallel to the slit. The position of the fringes depends on the wavelength of the incident light, the distance between the slit and the screen, and the width of the slit.

Figure: Fraunhofer diffraction

Fraunhofer explained the diffraction pattern using the principle of Fourier transform. He showed that the Fraunhofer diffraction pattern equation of a complex object could be obtained by taking the Fourier transform of the object's amplitude transmittance function. Amplitude transmittance function is a mathematical function that describes how a transparent aperture or object, affects the amplitude of light passing through it.

u,v are the spatial frequency variables corresponding to x and y directions, respectively.

The Fourier transform is a mathematical operation that splits a complex function into a set of simpler functions, known as frequency components, which can be visualized as a series of interference fringes. By applying the Fourier transform to the amplitude transmittance function, Fraunhofer was able to calculate the spatial intensity distribution of the diffracted light of the object.

The spatial intensity distribution (I(x, y)) of the diffracted light on the screen is obtained by taking the squared magnitude of the Fourier transform:

The intensity of the diffraction pattern created by Fraunhofer diffraction can be calculated using the following equation:

Where I₀ is the intensity of the incident beam, w is the width of the slit, λ is the wavelength of the incident light, and θ is the angle between the center of the diffraction pattern and the fringe under consideration.

Another important equation related to Fraunhofer diffraction is the grating equation, which relates the wavelength of the incident light to the distance between the slits and the position of the bright fringes:

Where d is the distance between the slits, θ is the angle between the direction of the incident light and the direction of the diffracted light, λ is the wavelength of the incident light, and m is the order of the bright fringe.

Applications of Fraunhofer diffraction

Fraunhofer diffraction is a fundamental phenomenon in physics that has many practical applications in various fields, including optics, physics, chemistry, and materials science. It is a powerful tool that allows scientists and engineers to analyze the properties of light and other waves, such as X-rays and sound waves. This information is used in a wide range of applications, from X-ray crystallography to laser technology, and from astronomy to microscopy.

• Fraunhofer diffraction is used in optical spectroscopy to analyze the spectral composition of light.
• The Fresnel diffraction patterns generated by X-rays passing through a crystal are used to determine the crystal's structure.
• It is used in astronomy to analyze the spectra of stars and galaxies.
• In laser technology, Fraunhofer diffraction is used to shape and control the output beam of lasers.
• To determine the size of small particles, such as those found in aerosols, Fresnel diffraction is used.

Differences Between Fraunhofer and Fresnel Diffraction

The main difference between Fraunhofer and Fresnel diffraction is the distance between the diffracting object and the screen on which the diffraction pattern is observed. In Fraunhofer diffraction, the diffracting object is placed far from the screen, which allows for a simpler mathematical analysis of the diffraction pattern. In contrast, Fresnel diffraction involves placing the diffracting object closer to the screen, which results in a more complex diffraction pattern that depends on the distance between the object and the screen. Another difference is the size of the diffracting object or aperture.