Measurement of the speed of light according to Rœmer and Huygens and more.

Introduction 

Measurement is essential in physics because it allows us to quantify and comprehend the characteristics and behaviour of physical systems. The measurement of the speed of light is one of the most significant measurements in physics. The speed of light is a basic constant in physics, and it is important in many fields such as optics, electromagnetic, and contemporary physics. The measurement of the speed of light has a lengthy history, extending back to the efforts of Danish astronomer Ole Rømer and Dutch scientist Christian Huygens in the 17th century. 

Rømer and Huygens were among the first to use a range of methods and procedures to attempt to measure the speed of light. Rømer estimated the speed of light by observing eclipses of Jupiter's moons, but Huygens employed a technique known as time-of-flight measurement, in which he measured the movement of light between two mirrors.  

While not particularly precise by current standards, these results established the groundwork for subsequent measurements of the speed of light and proved the capability of measuring the speed of light with respectable precision. Many scientists, like James Bradley and Albert Michelson, have refined and improved on the measurement of the speed of light over the ages, eventually resulting to the contemporary, accurate number of 299,792,458 metres per second. 

Electromagnetic wave field distributions and the Poynting vector

The field distributions and Poynting vectors of electromagnetic waves describe the behaviour and attributes of these waves as they travel over space in electromagnetic theory. The electric field is perpendicular to the propagation path and oscillates in the plane perpendicular to the propagation direction of the wave. The magnetic field is also perpendicular to the direction of wave propagation, and it oscillates in a plane perpendicular to both the electric field and the propagation direction. 

The Poynting vector is a vector that describes the flow of energy through a specific point in space. It is named after British physicist John Henry Poynting. It is defined as the product of the electric and magnetic fields at that place, and it is always perpendicular to both fields. The size of the Poynting vector is proportional to the strength of the wave and points in the direction of wave propagation.

The electric and magnetic field distributions, as well as the Poynting vector, together offer a comprehensive description of the behaviour and characteristics of electromagnetic waves as they propagate across space. They are useful in understanding phenomena like as electromagnetic wave reflection, refraction, and diffraction, and they play an important role in the design and analysis of electromagnetic devices such as antennas, waveguides, and optical fibres. 

In terms of the electric and magnetic field distributions, it is important to note that these fields are transverse in nature, which means they oscillate in directions that are perpendicular to the direction of wave propagation. This is a key characteristic of electromagnetic waves, as opposed to longitudinal waves such as sound waves. The electric and magnetic fields are also in phase with each other, meaning that they oscillate at the same frequency and with a phase difference of 90 degrees. This relationship is known as the phase relationship between the E and H fields, and it is a fundamental property of electromagnetic waves. The Poynting vector, which is the cross product of the electric and magnetic fields, is a measure of the energy density of an electromagnetic wave. The Poynting vector's magnitude is proportional to the wave strength and its direction is the direction of wave propagation. The scalar product of the electric and magnetic fields gives the time averaged energy density from the Poynting vector.

Electromagnetic waves, which are commonly referred to as "photons," may display both wave-like and particle-like behaviour. The wave-particle duality of electromagnetic waves is a key notion in quantum physics that defines particle behaviour at the subatomic level. The Photoelectric effect, Compton scattering, and the double-slit experiment have all been used to demonstrate this duality. Furthermore, electromagnetic waves can display polarisations, which relate to the orientation of the electric field oscillations. Polarizations are classified into three types: linear, circular, and elliptical. They may be understood by examining the orientation of the E-field vector in reference to the propagation direction of the wave.

Laguerre Gaussian modes

Laguerre-Gaussian (LG) modes are a form of paraxial optical wave with cylindrical symmetry. They are a type of solution to the paraxial wave equation, which explains the behaviour of optical waves near the optical axis. These modes are frequently used to describe laser beams and other forms of optical beams with cylindrical symmetry, which are frequently employed in applications such as optical trapping, micromanipulation, and quantum optics. 

Two parameters describe LG modes: the radial index p and the azimuthal index l. The radial index p describes the mode's radial distribution and is proportional to the number of radial nodes in the mode. The azimuthal index l characterises the mode's azimuthal distribution and is proportional to the number of azimuthal nodes in the mode.

The Laguerre and Hermite polynomials may be used to express the LG modes numerically. These polynomials are used to explain the mode's radial and azimuthal fluctuations. The LG modes' resultant electric field distribution is given by: 

Where r, angular, and longitudinal coordinates are used, w(z) is the beam waist, R(z) is the radius of curvature, and (z) is the Gouy phase shift.

Many relevant aspects of LG modes include the capacity to transport orbital angular momentum (OAM) and a doughnut-shaped intensity distribution. OAM is an additional degree of freedom that may be employed for optical information multiplexing and manipulation. The intensity distribution's doughnut shape makes it suitable for optical trapping and manipulation of tiny particles. 

What are Orbital angular momentum and Laguerre Gaussian modes?

M = r×E×H.

One of the most important aspects of Laguerre-Gaussian (LG) modes is their ability to transport orbital angular momentum (OAM). OAM is a feature of a light beam that describes the degree of rotation of the phase front of the beam around its propagation axis. A beam's OAM is quantized and measured in angular momentum units per photon. The azimuthal index l, which defines the number of azimuthal nodes in the mode, determines the OAM of an LG mode. A LG mode's OAM per photon is determined by l, where is the reduced Planck constant. This implies that LG modes with a positive azimuthal index l rotate the phase front counter-clockwise and have a positive OAM, whereas LG modes with a negative azimuthal index l rotate the phase front clockwise and have a negative OAM.

LG modes may contain OAM, allowing them to be employed in a variety of applications such as optical communication, quantum information, and microscopy. In optical communication, for example, various LG modes can be used to multiplex distinct information channels. OAM may be employed in quantum information to expand the dimensionality of a quantum system, resulting in an exponential rise in the number of states accessible for encoding information. OAM beams can be utilised to increase image resolution in microscopy.

One key feature of OAM is that it is a discrete variable, which means that it can only take on particular values rather than any arbitrary value. This is in contrast to other angular momentum variables that are continuous, such as spin. Because of its discreteness, OAM provides a considerable degree of freedom for encoding and altering information. Another key feature of OAM is that it is a "hidden variable," meaning that the OAM of a beam is not evident in the intensity distribution of the beam. The OAM of a beam, on the other hand, is encoded in its phase and can only be exposed by interference experiments or other phase-sensitive tests. Because of its concealed nature, OAM provides a desirable degree of freedom for secure communication and other applications where the information being encoded must remain secret.

It is crucial to note that Laguerre-Gaussian modes may be created and controlled in a variety of methods, including spiral phase plates, q-plates, and computer generated holograms. These methods can be used to produce LG modes with a given OAM value or to convert between LG modes. Another key feature of LG modes is the presence of a helical wavefront, which is a phase front that revolves around the propagation axis. This helical wavefront is proportional to the beam's OAM. This characteristic enables LG modes to be employed for optical trapping and manipulation of tiny particles such as atoms and molecules, as well as micrometer-scale object manipulation. 

Furthermore, LG modes may be utilised to generate structured light beams with singularities, in which the intensity of the beam is zero but the phase is unknown. These singularities have a variety of uses, including optical tweezers and optical vortices. 

Microwave Gaussian beam

The transverse mode structure of Gaussian beams, which represents the spatial distribution of the electric and magnetic fields in the beam, is another essential characteristic. Depending on the number and distribution of the electric and magnetic field components, Gaussian beams can be classed as TEM (transverse electromagnetic) modes or higher-order modes. The transverse mode structure of Gaussian beams, which represents the spatial distribution of the electric and magnetic fields in the beam, is another essential characteristic. Depending on the number and distribution of the electric and magnetic field components, Gaussian beams can be classed as TEM (transverse electromagnetic) modes or higher-order modes. 

The intensity, electric field, and magnetic field distribution all follow a Gaussian profile, which is an essential feature of Gaussian beams. Gaussian beams are therefore an useful approximation for many optical systems since they provide for a straightforward mathematical explanation of the beam's behaviour.

Another essential feature of Gaussian beams is that they are solutions to the paraxial wave equation, which is a simplified form of the complete wave equation that is valid for beams that are collimated or have modest divergence angles. This suggests that Gaussian beams are a fair approximation for beams with modest divergence angles and large propagation lengths. In the case of microwave Gaussian beams, they are microwave-frequency solutions to the paraxial wave equation. They also have a Gaussian intensity profile and little divergence, which makes them similar to optical gaussian beams.

Microwave Gaussian beams can be formed by a number of sources, including horn antennas, parabolic reflectors, and dielectric lenses. These sources are intended to provide a Gaussian intensity profile beam. Different equipment, including as lenses, phase plates, and polarizers, can be used to modulate these beams. Furthermore, Gaussian beams may be used to analyse the behaviour of other types of beams, such as Bessel beams and Airy beams, by manipulating the Gaussian beam solutions mathematically. 

The equation below describes the electric field of a Gaussian beam:

E(r,z) = E0(r) * exp(ikz - (r^2)/(w(z)^2)) 

where r is the transverse coordinate, z is the propagation distance, k is the wave number, and w(z) is the beam waist, and E0(r) is the transverse electric field distribution (the location where the beam has the smallest radius). As the distance from the beam waist grows, the strength of the electric field falls exponentially.

The square of the electric field determines the Gaussian beam's intensity:

I(r,z) = (c/2) * n * (E(r,z))^2

where n denotes the medium's refractive index and c the speed of light, respectively, through which the beam is travelling.

The two primary parameters that determine the behaviour of the Gaussian beam are the beam waist, or w(z), and the beam radius, or w(z)*sqrt(1 + (z/z R)2, where z R is the Rayleigh length. The distance over which the beam radius increases by a factor of two is indicated by the Rayleigh length and the beam waist, respectively. The beam waist indicates where the beam's minimum radius is located. 

A Gaussian beam's Gouy phase, or the difference in phase between the beam's centre and edges, may also be calculated. This phase difference is significant for the trapping and manipulation of particles in various Gaussian beam applications, such as optical tweezers.

Finally, the angular spectrum of Gaussian beams is well described. This implies that there is a frequency or wavelength that corresponds to the angular deviation from the beam axis of each point in the beam. This characteristic enables the employment of Gaussian beams, among other things, in holography and Fourier optics. 

Phython code 

Python can be used to build and analyse Gaussian beams. Here's an example of how to use the NumPy and Matplotlib tools to construct a basic simulation of a Gaussian beam:

import numpy as np

import matplotlib.pyplot as plt

# Define beam waist and wavelength

w0 = 1.0

wavelength = 1.0

# Define grid for transverse coordinates

x = np.linspace(-5*w0, 5*w0, 1000)

y = np.linspace(-5*w0, 5*w0, 1000)

X, Y = np.meshgrid(x, y)

# Calculate electric field distribution

E = np.exp(-2*(X**2 + Y**2)/w0**2)

# Plot electric field distribution

plt.figure()

plt.imshow(E, extent=[x[0], x[-1], y[0], y[-1]])

plt.title('Electric field distribution')

plt.xlabel('X (wavelengths)')

plt.ylabel('Y (wavelengths)')

plt.colorbar()

plt.show()

Explanation:- 

The provided Python code creates a 2D grid of transverse coordinates and calculates the electric field distribution of a Gaussian beam at the beam waist w0 and wavelength, then it plots the result. In addition, there are other libraries such as scikit-image, scikit-beam, and pypolar that can be used to analyze and manipulate Gaussian beams, these libraries can be used for calculating beam radius, Rayleigh length, Gouy phase and other characteristics of the beam. Furthermore, there are libraries that are specifically developed for simulation of electromagnetic waves, like the Meep library, which allows simulating electromagnetic waves in the time and frequency domain, written in C++ and has a python wrapper, it is a powerful tool for simulating Gaussian beams and its propagation in different media. Mentioning that while it's possible to simulate Gaussian beams using Python, commercial software like ZEMAX and CODE V which are designed for optical design and analysis, can also be used and have built-in simulation of Gaussian beams and other optical elements with a user-friendly interface.

Gouy phase

A phase shift that happens in the transverse plane of a Gaussian beam as it propagates along the optical axis is known as the Gouy phase. The difference in the phase velocities of the light in the centre of the beam and at the beam's borders causes the Gouy phase. 

The Gouy phase is given by the following equation:

Φ_Gouy = arctan(z/z_R)

where z is the distance along the optical axis from the waist of the beam and z_R is the Rayleigh range. It can also be written in term of the waist radius w(z) as :

Φ_Gouy = arctan(z/z_R) = arctan(z/(π*w(z)^2/λ))

The Gouy phase has a substantial influence on beam behaviour, affecting the position of the beam waist, phase dispersion, and beam intensity. The Gouy phase, for example, can impact particle entrapment and manipulation in optical tweezers. 

Experiment 

Q. Devise an experiment using an interferometric setup to prove the existence of the Gouy phase by measuring the position of the interference fringes. Compare the results to the expected behavior of the Gouy phase. Determine if there is an analogue of the Gouy phase in ray optics by analyzing the position of the beam waist and its divergence in the geometric optics limit.

An interferometric setup can be used to experimentally verify the presence of the Gouy phase. The following stages might be included in one conceivable experiment:

1. Generate a Gaussian beam using a laser and a focusing lens.
2. Use a beam splitter to split the beam into two paths.
3. In one path, introduce a phase shift by passing the beam through a plate of transparent material with a known refractive index.
4. In the other path, leave the beam undisturbed.
5. Use a second beam splitter to recombine the two beams and form an interference pattern.
6. Observe the interference pattern on a screen or detector, and measure the position of the interference fringes.
7. Repeat the experiment for different values of the phase shift, and observe how the position of the interference fringes changes.
8. Compare the observed changes in the position of the fringes to the expected behavior of the Gouy phase.

The position of the interference fringes in this experiment is dependent on the phase shift produced in one of the routes, which corresponds to the Gouy phase. In terms of ray optics, the Gouy phase is a phenomenon that arises due to the wave nature of light; ray optics has no direct analogue to the Gouy phase. In the geometric optics limit, when the beam is heavily collimated, Gouy phase can be seen by measuring the position of the beam waist and its divergence; the findings would be identical to the wave optics case.