Magneto-optics

An introduction

Paul Fumagalli, Joachim Schoenes

• Verlag: De Gruyter
• Jahr: 2021
• ISBN:
  9783110635454
• Auflage: 2019
• Thema: Physical Sciences
• Thema: Optics & Light

Magneto-optics describes in general any interaction between electromagnetic radiation and a material which is magnetized. The book gives a concise but comprehensive introduction to theory, calculus, and typical experimental set-up used in magneto-optics. It includes a variety of practice problems with detailed solutions. The focus lies on the spectral range between near-infrared and near ultraviolet light because it is easily accessible in experiment using standard quartz optics.

1 Introduction

1.1 Definition of magneto-optics

Magneto-optics (MO) is a wide field in physics and describes, in general, any interaction between electromagnetic radiation and a material which is magnetized. The application of a magnetic field to induce a magnetization is in principle not necessary in case of the existence of spontaneous magnetization. The local magnetic field produced by the spontaneous magnetization will lead to similar MO effects as an external magnetic field. Nevertheless, a magnetic field is usually applied even in the case of spontaneous magnetization to avoid indeterminacies due to the formation of magnetic domains. If the area that is illuminated by the incoming light covers several magnetic domains, the MO effect measured is the average of the signal produced by the individual domains. By applying a magnetic field, the magnetic domains will be aligned. On the other hand, one can take advantage of this behavior when studying MO hysteresis curves.

MO effects are not limited to the energy range of visible light but exist in principle in the whole spectral range of electromagnetic waves: MO phenomena are known in the microwave as well as in the X-ray range and the DC limit relates to the Hall effect. In the microwave range, predominantly free carriers give a contribution. In X-ray range, transitions from inner shells to the conduction band are important. This is known as X-ray magnetic circular dichroism (XMCD) and, for example, has been successfully used for the proof of an induced magnetic moment in Co/Pt multilayers.

1.2 History of magneto-optics

To understand the importance and impact of Faraday’s discovery of the first MO effect, one has to know that in the beginning of the nineteenth century the true nature of light was still unknown. Émile Verdet has given an elaborate survey on the history of wave optics in his lectures on optics published in 1869 (1). Light was believed to spread by means of an ether in a very mechanistic way, like sound waves in fluids. But a corpuscular nature was assumed possible, as well. Yet another property of light was not really understood: polarization. This property appears when light is reflected from transparent surfaces at a specific angle (which is known today as Brewster angle), or, when light is transmitted through birefringent materials such as calcareous spar. Here, the light splits into an ordinary and an extraordinary ray, traveling in different directions, which are both linearly polarized with planes of polarization orthogonal to each other. It is the merit of Augustin Fresnel to have contributed in his short life to a very deep understanding of the behavior of linear polarization without the knowledge of the true nature of light. Being an engineer by training, he occupied himself more and more with optics. In 1819, he published with his mentor Arago four rules about how linearly polarized light rays will interfere with each other (2). He was able to calculate for the first time the intensity of reflected light by combining his knowledge about how to add up polarized waves with the Huygens principle (3, 4). In a second note to his note on the calculation of the hues that thin layers of birefringent crystals produce (5), Fresnel developed his well-known equation describing the intensity of a light ray (linearly polarized and perpendicular to the plane of incidence) that is reflected from a transparent material (6): “Je vais faire voir maintenant comment on peut calculer l’intensité de la lumière réfléchie sous une incidence quelconque pour le faisceau polarisé suivant le plan de réflexion.” The amazing fact is that Fresnel was using a purely mechanical picture of excited ether molecules in a fluid producing transversal light waves (discounting longitudinal waves completely, which made him feel uncomfortable). Moreover, he did not use boundary conditions at all (6): “Soit m la masse d’un élément différentiel du premier milieu, qui, en glissant sur lui-même, met en mouvement l’élément différentiel contigu m’ du milieu réfléchissant, que je suppose de même élasticité. Dans le premier instant, m’ était en repos, et m avait une vitesse v; un instant après, les deux élémens ont la même vitesse, et c’est alors que s’arrête le déplacement du premier par rapport au second; mais, en raison du déplacement effectué, le premier doit recevoir après, en sens contraire, toute la partie de la vitesse initiale qu’il a perdue.” In his picture, the ether molecule of the first material accelerates the molecule of the second material at the interface until they reach a common velocity. Then, the second molecule transfers the velocity difference back to the first molecule. The whole process is mathematically equivalent to an elastic collision of two masses. He assumes the intensity of the light to be proportional to the square of the velocity of the ether molecules. Therefore, the intensity ratio between reflected and incoming ray is equal to the ratio of the square of the velocities of the ether molecules after the interaction. To get the intensity of the reflected ray, Fresnel replaced the masses of the ether molecules by a product of volume and density. The volumes he deducted from the width and wavelength of the incoming and the refracted ray and the density he assumed proportional to the inverse square of the propagation velocity of the light (like sound waves in fluids). Putting all this together, he finally derived what is called today the first Fresnel equation. Being an engineer, he carefully tested the equation by comparing with very accurate measurements and found excellent agreement. The same year, Fresnel also published the formula for light polarized in the plane of incidence, this time without presenting a derivation (7). With our knowledge of the correct physics, it seems like a miracle that Fresnel was able to derive the correct formulas with his improper assumptions. Nevertheless, his formulas together with his work on the interference of polarized light prepared the ground for a better understanding of the interaction of polarized light with material and of the role the refractive index plays.

When in 1845 Michael Faraday observed a rotation of the plane of polarization of light passing through lead borosilicate glass, to which he had applied a magnetic field along the light propagation direction (8), his discovery triggered a deeper understanding of the nature of light and its relation to electromagnetism. Faraday notes in his famous paper (8): “I believe that, in the experiments I describe in the paper, light has been magnetically affected, i.e. that that which is magnetic in the forces of matter has been affected, and in turn has affected that which is truly magnetic in the force of light.” Faraday examined a variety of transparent diamagnetic materials, some fluids such as water and alcohol and about 150 different aqueous solutions such as soluble acids, alkalis, and salts. In all of them, he found the new effect. In this way, he could show that a change of state of matter under the influence of magnetic forces is a general property not limited to paramagnetic or magnetically ordering materials. In the same study, Faraday revealed the following qualitative relations for the rotation of the plane of polarization, since then known as Faraday rotation θ_F:

1. θ_F ∝ l, where l is the distance of the light traveled in the material;
2. θ_F ∝ B, where B is the magnetic field applied to the material;
3. the rotation is strongest when the force lines of the magnetic field are parallel to the light ray, no effect happens when they are perpendicular;
4. in a material (e.g., a sugar solution), showing optical activity, or, as Faraday calls it, a natural rotation, which means the plane of polarization rotates without magnetic field, the Faraday effect just adds to it;
5. a positive rotation θ_F > 0 is defined as a rotation of the plane of polarization in the same direction as the positive current must flow in a coil to produce the magnetic field;
6. an empty coil does not generate a rotation of the plane of polarization.

Faraday thought that all forces have a common origin or are directly related and mutually dependent. With his discovery, he could for the first time interconnect magnetism and light and proof that they are not independent (8): “Thus is established, I think for the first time, a true, direct relation and dependence between light and the magnetic and electric forces; and thus, a great addition made to the facts and considerations which tend to prove that all natural forces are tied together, and have one common origin.” Because Faraday found for all the materials he investigated the same sign of the rotation, he assumed that all materials behave like that. However, as we will see later, there is no direct connection between these quantities. We know today that dia-, para-, and ferromagnetism differ in the temperature dependence and dispersion of the Faraday effect. From Faraday’s last relation it follows that the magnetic force is not acting directly on the light but only through the material. From the third relation follows the possibility of constructing a light trap, the so-called Rayleigh light trap (9), as sketched in Fig. 1.1.

Fig. 1.1: Rayleigh light trap. A piece of glass, with a length such that it rotates the polarization in a magnetic field by 45° (called Faraday rotator), is placed between two polarizers set at 0° and 45°, respectively. Light traveling to the right is transmitted while it is blocked when travelling to the left due to the 45° rotation independent of travelling direction. The system therefore acts as an optical isolator which transmits radiation in one direction only.

Faraday was very good at finding the fundamental dependencies of the effect in a rather qualitative way. The person who proofed his findings quantitatively was Émile Verdet. He developed an exact measurement of the magnetic-field strength and carefully verified the proportionality between magnetic field and rotation (10). Thereafter, Verdet established the angular dependence between magnetic field and light-propagation direction. In another investigation, he discovered materials with a negative Faraday rotation: iron salts (11). Most importantly, he mentioned for the first time the phenomenological model that the Faraday rotation could originate from the difference of the reciprocal propagation velocities of right and left circularly polarized light (12). This is identical to the difference of the corresponding refractive indices. He wrote that Fresnel had already used a similar argument to explain the rotation of linearly polarized light in optically active materials (13). In a succeeding publication (14), Verdet restates a theory proposed by Airy (15) put forward shortly after the discov...