# Refractive index

Wavefronts emanating from a point

The refractive index , also the refractive index or optical density , formerly also called the refractive index , is an optical material property . It is the ratio of the wavelength of the light in the vacuum to the wavelength in the material, and thus also the phase velocity of the light in the vacuum to that in the material. The refractive index is a quantity in the dimension number , and it is generally dependent on the frequency of the light, which is called dispersion .

Light is refracted and reflected at the interface between two media with different refractive indices . The medium with the higher refractive index is called the optically denser one .

Note that “optical density” is sometimes used to denote a measure of extinction .

## Physical basics

Influence of the complex refractive index of a material on the reflection behavior of a light beam when it hits the air / material interface${\ displaystyle n + \ mathrm {i} k}$
Course of the wavelength-dependent complex index of refraction in the visual field for a semiconductor having band transitions in this area

The term "refractive index" comes from the term refraction and its appearance in Snellius' law of refraction . The refractive index is a quantity in the dimension number . It indicates the ratio of the speed of light in a vacuum to the speed of light propagation in the medium: ${\ displaystyle n}$ ${\ displaystyle c_ {0}}$ ${\ displaystyle c _ {\ mathrm {M}}}$

${\ displaystyle n = {\ frac {c_ {0}} {c _ {\ mathrm {M}}}}}$

### Complex index of refraction

Describes the temporal and spatial propagation of an electromagnetic wave of the circular frequency with the help of the wave equation ${\ displaystyle \ omega}$

${\ displaystyle E (t, x) = E \; \ mathrm {e} ^ {- \ mathrm {i} {\ big (} \ omega t - {\ frac {\ omega} {c}} x {\ varvec symbol {n}} {\ big)}}}$,

It is found that both the classical refractive index and the attenuation of the wave can be combined in a complex-valued refractive index and both the temporal and spatial progression of the wave and its absorption can be described using an equation . The real-valued part , which is usually greater than 1, shortens the wavelength in the medium, and the complex-valued part damps the wave . ${\ displaystyle {\ boldsymbol {n}} = n _ {\ mathrm {r}} + \ mathrm {i} \, n _ {\ mathrm {i}}}$${\ displaystyle n _ {\ mathrm {r}}}$${\ displaystyle E (x) = \ mathrm {e} ^ {- \ mathrm {i} {\ big (} - {\ frac {\ omega} {c}} x \, n _ {\ mathrm {r}} { \ big)}} = \ mathrm {e} ^ {\, \ mathrm {i} {\ frac {\ omega} {c}} x \, n _ {\ mathrm {r}}}}$${\ displaystyle n _ {\ mathrm {i}}}$${\ displaystyle E (x) = \ mathrm {e} ^ {- \ mathrm {i} {\ big (} - {\ frac {\ omega} {c}} x \, \ mathrm {i} n _ {\ mathrm {i}} {\ big)}} = \ mathrm {e} ^ {- {\ frac {\ omega} {c}} x \, n _ {\ mathrm {i}}}}$

Different, equivalent representations for the complex-valued refractive index are common here:

• as the sum of the real part and the imaginary part of a complex number multiplied by the imaginary unit  : ${\ displaystyle \ mathrm {i}}$
${\ displaystyle {\ boldsymbol {n}} = n _ {\ mathrm {r}} + \ mathrm {i} \, n _ {\ mathrm {i}}}$ or
${\ displaystyle {\ boldsymbol {n}} = n '+ \ mathrm {i} \, n' '}$ or
${\ displaystyle {\ boldsymbol {n}} = n + \ mathrm {i} \, K}$
• as the difference between the real part and the  multiplied by the imaginary part of a complex number: ${\ displaystyle \ mathrm {i}}$
${\ displaystyle {\ boldsymbol {n}} = n _ {\ mathrm {r}} - \ mathrm {i} \, k}$ or
${\ displaystyle {\ boldsymbol {n}} = n '- \ mathrm {i} \, n' '}$
• as the product of the real refractive index and a complex number: ${\ displaystyle n}$
${\ displaystyle {\ boldsymbol {n}} = n \ left (1- \ mathrm {i} \, \ kappa \ right)}$

The minus sign in front of the imaginary part in some representations is chosen so that the imaginary part ( , or or ) has a positive sign in the case of absorbent material . This imaginary part is called the extinction coefficient or absorption index. In contrast to this, authors who use the representation as a product refer to the size , i.e. the imaginary part divided by , as the absorption index . ${\ displaystyle n _ {\ mathrm {i}}}$${\ displaystyle n ''}$${\ displaystyle K}$${\ displaystyle k}$${\ displaystyle \ kappa}$${\ displaystyle n}$

Both the real part and the imaginary part of the refractive index, if they are not equal to 1, depend on the frequency and thus on the wavelength. This effect, known as dispersion , is unavoidable and enables white light to be broken down into its spectral colors on a prism . The frequency dependence of the refractive index in matter can be described quite well using the model of the Lorentz oscillator .

Since the reaction of an optical medium to an electromagnetic wave must be causal, the complex-valued refractive index is a meromorphic function, real and imaginary parts are linked via the Kramers-Kronig relationships .

#### Anisotropic index of refraction

In anisotropic media the refractive index is not a scalar, but a second order tensor. The wave vector and the direction of propagation then no longer match.

#### Birefringence

If the refractive index depends on the polarization (and therefore inevitably also on the direction), one speaks of birefringence .

#### Link with permittivity and permeability

The complex refractive index is linked to the permittivity number (dielectric function) and the permeability number : ${\ displaystyle \ varepsilon _ {\ mathrm {r}}}$ ${\ displaystyle \ mu _ {\ mathrm {r}}}$

${\ displaystyle {\ boldsymbol {n}} = {\ sqrt {\ varepsilon _ {\ mathrm {r}} \ cdot \ mu _ {\ mathrm {r}}}}}$

All sizes are generally complex and frequency-dependent. The permittivity and the permeability number are approximations which, depending on the system, are better or worse for describing the polarization and magnetization effects.

The wavelength dependency of the refractive index of a material can be determined theoretically using the electrical susceptibility . This quantity records the contributions of the various mechanisms in the material to its properties and leads to the complex permittivity . In the case of non-magnetic material , and the complex refractive index can be specified directly from the real ( ) and imaginary ( ) part of the permittivity number: ${\ displaystyle \ mu _ {r} \ approx 1}$${\ displaystyle \ varepsilon _ {1}}$${\ displaystyle \ varepsilon _ {2}}$

${\ displaystyle {\ boldsymbol {n}} \ approx {\ sqrt {\ varepsilon _ {r}}} = {\ sqrt {{\ varepsilon} _ {1} + \ mathrm {i} {\ varepsilon} _ {2 }}}}$

By comparison with the complex refractive index in both of the above Representations 1 and 2 (sum or difference) you can calculate the sizes and : ${\ displaystyle n}$${\ displaystyle k}$

${\ displaystyle n ^ {2} = {\ frac {1} {2}} \ left ({\ sqrt {{\ varepsilon _ {1}} ^ {2} + {\ varepsilon _ {2}} ^ {2 }}} + \ varepsilon _ {1} \ right)}$
${\ displaystyle k ^ {2} = {\ frac {1} {2}} \ left ({\ sqrt {{\ varepsilon _ {1}} ^ {2} + {\ varepsilon _ {2}} ^ {2 }}} - \ varepsilon _ {1} \ right)}$

### Group refractive index

The ratio of the vacuum speed of light to the group speed of light in the medium is the group refractive index . This material property is dependent on the wavelength of the light via the group speed : ${\ displaystyle c_ {0}}$ ${\ displaystyle c _ {\ mathrm {g}}}$ ${\ displaystyle n _ {\ mathrm {g}}}$${\ displaystyle \ lambda}$

${\ displaystyle n _ {\ mathrm {g}} (\ lambda) = {\ frac {c_ {0}} {c _ {\ mathrm {g}} (\ lambda)}}}$

In a vacuum, the group velocity has the same value as the phase velocity, and this value is also independent of the wavelength of the light. In the medium this is not necessarily the case; there are differences especially at wavelengths for which the material shows great dispersion.

## Other definitions

Refraction in a medium 2 with a higher refractive index; the lower ray shows the behavior in a metamaterial with a sign that changes compared to medium 1

The definition of the refractive index was made above via the speed at which light propagates in the material. This approach is obvious, but not applicable in all cases. For example, metamaterials can have a negative refractive index according to the geometric beam path (see below). However, a negative value for the speed of light is not meaningfully defined.

Alternative definitions of the refractive index that do not have this problem are:

• Via Fermat's principle , according to which light travels the path between two points for which it needs an extreme value of time.
• About the Huygens principle , which says that every point of a wave front can be viewed as the starting point of a spherical wave and the interference of all these waves results in the wave front propagating further.
• About the ray optics . According to the aforementioned Snellius law of refraction, n corresponds to the sine ratio of the angle of incidence and the refracted angle .

All of these definitions provide the same value for common optical materials.

## Refractive index of air and other substances

Refractive index of selected substances at the wavelength 589 nm (yellow-orange) of the sodium D line .
material Refractive index n
vacuum exactly 1
Helium (standard conditions) 1,000 034 911
Air (standard conditions) 1,000 292
Sulfur hexafluoride (standard conditions) 1,000 729
Airgel 1.007 ... 1.24
ice 1.31
Water (liqu.) 20 ° C 1.3330
human Eye lens 1.35 ... 1.42
Ethanol (liqu.) 1.3614
Magnesium fluoride 1.38
Fluorspar ( calcium fluoride ) 1.43
human epidermis 1.45
Carbon tetrachloride (liqu.) 1.4630
Quartz glass 1.46
Glycerine (liqu.) 1.473 99
Cellulose acetate (CA) 1.48
PMMA (plexiglass) 1.49
Crown glass 1.46 ... 1.65
Benzene (liqu.) 1.5011
Window glass 1.52
Microscopic cover slips 1.523
COC (a plastic) 1.533
PMMI (a plastic) 1.534
quartz 1.54
Halite (rock salt) 1.54
Polystyrene (PS) 1.58
Polycarbonate (PC) 1.585
Epoxy resin 1.55 ... 1.63
Flint glass 1.56 ... 1.93
Carbon disulfide (liqu.) 1.6319
Plastic lens for glasses to 1.76
Diiodomethane (liqu.) 1.7425
Ruby ( aluminum oxide ) 1.76
Mineral glass for glasses (polarizing) up to 1.9 (1.5)
Glass 1.45 ... 2.14
Zircon 1.92
sulfur 2.00
Zinc sulfide 2.37
diamond 2.42
Titanium dioxide (anatase) 2.52
Silicon carbide 2.65 ... 2.69
Titanium dioxide (rutile) 3.10

### Orders of magnitude

Refractive index of water between 3 nm and 300 m

By definition, the vacuum has a refractive index of exactly 1. On the one hand, this represents a reference value; on the other hand, it results from the propagation speed of light in the vacuum, which corresponds exactly to the vacuum speed of light.

In “normal” substances there are mobile electrical charge carriers (and mobile magnetic dipoles). By compensating for the electric (and magnetic) field, these cause the electromagnetic field to spread more slowly. This is described by the refractive index . However, this compensation behavior is frequency-dependent, since the charge carriers (and magnetic dipoles) can only follow the electric field up to a certain frequency. With a certain refractive index, substances start at very low frequencies (e.g. water at ) and reduce this value towards high frequencies. Every reduction takes place in the vicinity of an electron resonance (or magnetic dipole resonance) of the substance and leads to an initially increased refractive index, which then decreases and then levels off again at a lower level. ${\ displaystyle \ mathbf {n}}$${\ displaystyle n \ approx 9}$

In the visible range, the refractive indices of transparent or weakly (to medium) absorbing materials are generally greater than 1. In the case of electrically conductive and therefore strongly absorbing materials such as metals, different physical conditions prevail. Visible light can only penetrate a few nanometers into such materials. The above-mentioned relationship with permittivity and permeability often results in a real part of the refractive index between 0 and 1, but this cannot be interpreted in the same way as with transparent materials (reference to the speed of light), because the complex refractive index in this Case is dominated by the imaginary part.

In addition, there are wavelength ranges for every substance (e.g. above the visible range) in which the real part of the refractive index is less than 1 (but remains positive). So for very small wavelengths ( X-rays , gamma rays ) the refractive index is always smaller than 1 and approaches 1 from below as the wavelength decreases. Therefore, the representation has established itself in the X-ray area, for example , with typical values ​​for between 10 −9 and 10 −5 (strongly dependent on the wavelength, depending on the atomic number and density of the material). ${\ displaystyle n = 1- \ delta}$${\ displaystyle \ delta}$

### air

The refractive index for visible light from air is 1,00028 at sea ​​level (dry air in a normal atmosphere ). It depends on the density and thus on the temperature of the air, as well as on the special composition of the air - especially the humidity . Since the air density  decreases exponentially upwards - according to the gas laws in a gravitational field , see barometric altitude formula - the refractive index at an altitude of 8 km is only 1,00011. Because of this astronomical refraction , stars seem to be higher than they would be without an atmosphere. In the technical field, the refractive index of materials is sometimes referred to that of air to simplify matters.

### Wavelength dependence

Figure 1: Refractive index of selected types of glass as a function of wavelength. The visible range from 380 nm to 780 nm is marked in red.

Since, as described in the introduction, the refractive index of every material depends on the wavelength of the incident light (which also applies to electromagnetic radiation outside the visible range), it would also be necessary to specify this as a function of the wavelength (in a table or as a function). However, since this is not necessary for many simple applications, the refractive index is usually given for the wavelength of the sodium D line (0.589 µm). Figure 1 shows curves of the wavelength-dependent refractive index of some types of glass as an example. They show the typical course of a normal dispersion.

The strength of the dispersion in the visible spectral range can be described as a first approximation by the Abbe number ; a more precise estimate can be obtained by using the Sellmeier equation .

## Refractive index of the plasma

Every linearly polarized wave can be interpreted as a superposition of two circular waves with opposite directions of rotation. If the direction of propagation runs parallel to the magnetic field lines, the following formulas result for the refractive indices n :

${\ displaystyle n _ {\ text {left}} = {\ sqrt {1 - {\ frac {f_ {P} ^ {2}} {f (f + f_ {B})}}}}}$
${\ displaystyle n _ {\ text {right}} = {\ sqrt {1 - {\ frac {f_ {P} ^ {2}} {f (f-f_ {B})}}}}}$

Here is the frequency of the wave, the plasma frequency of the free electrons in the plasma and the gyration frequency of these electrons. The difference between the two formulas disappears if the wave vector encloses a right angle with the direction of the magnetic field, because then is. ${\ displaystyle f}$${\ displaystyle f_ {P}}$${\ displaystyle f_ {B}}$${\ displaystyle f_ {B} = 0}$

If is positive, it can be used to determine the phase velocity of the wave ${\ displaystyle n}$

${\ displaystyle v _ {\ text {phase}} = {\ frac {c} {n}}}$

and thus in turn the wavelength

${\ displaystyle \ lambda = {\ frac {c} {nf}}}$

to calculate. Because the right and left rotating circular waves differ in their wavelengths, one of them is rotated a small angle further than the other after a certain distance. The resulting vector (and thus the plane of polarization) as the sum of the two components is therefore rotated when passing through the plasma, which is known as the Faraday rotation . After a long distance the total rotation can be very large and changes constantly because of the movement of the ionosphere. A transmission with vertical polarization can also reach the receiver with horizontal polarization at irregular time intervals. If the receiving antenna does not react to this, the signal strength changes very drastically, which is known as fading .

When radio communications with satellites differ and because of the much higher frequencies only slightly, according to lower the Faraday rotation. ${\ displaystyle n _ {\ text {left}}}$${\ displaystyle n _ {\ text {right}}}$

### Polarization-dependent absorption

The unbound free electrons of the ionosphere can move helically around the magnetic field lines and thereby withdraw energy from a parallel electromagnetic wave if the frequency and direction of rotation match. This cyclotron resonance can only be observed with the right circularly polarized extraordinary wave , because the denominator in the above formula becomes zero. The left circularly polarized ordinary wave cannot lose any energy in the plasma in this way. ${\ displaystyle f = f_ {B}}$

The field lines of the earth's magnetic field are oriented in such a way that they point from the ionosphere to the earth in the northern hemisphere, one “looks” towards them, so to speak, which is why left and right have to be swapped. Therefore, here an upwardly radiated linkszirkuläre shaft absorbed in HAARP the ionosphere is so heated.

If, on the other hand, a wave in the lower shortwave range is radiated vertically upwards in the right direction of rotation (in the northern hemisphere) , it does not lose any energy through cyclotron resonance in the ionosphere and is reflected by the ionosphere at a height of several hundred kilometers if the plasma frequency is not exceeded. If a linearly polarized wave is radiated upwards, half of the transmission energy heats the ionosphere and only the rest arrives back down here in a left-circular polarized manner, because the direction of rotation changes when it is reflected.

In radio communications with satellites, the frequencies are well above the plasma frequency of the ionosphere in order to avoid similarly serious phenomena.

## Measurement in the optical range

For the experimental determination of the refractive index of a medium with (for example non-magnetic) one can, for example, measure the Brewster angle at the transition from air into this medium. In this case applies ${\ displaystyle \ mu _ {r2} \ ({\ text {med}}) = \ mu _ {r1} \ ({\ text {air}})}$${\ displaystyle n _ {\ text {med}}}$

${\ displaystyle \ tan (\ alpha _ {\ text {Brewster}}) = {\ frac {n _ {\ text {med}}} {n _ {\ text {air}}}} \ approx n _ {\ text {med }}}$.

A refractometer is used for the measurement .

An estimate of the refractive index is possible with the so-called immersion method by immersing an object in transparent liquids with different densities. When the refractive index of the object and the liquid are identical, the contours of the object disappear. This method can easily be used to identify, for example, rubies or sapphires with a refractive index of around 1.76 by immersing them in a suitable heavy liquid such as diiodomethane (refractive index = 1.74).

## application

The refractive index is one of the central parameters for optical lenses . The art of optics calculation for the design of optical instruments ( lenses , measuring instruments, exposure systems for photolithography ) is based on the combination of different refractive lens surfaces with suitable types of glass.

In chemistry and pharmacy , the refractive index is often used at a certain temperature to characterize liquid substances. The temperature and the wavelength at which the refractive index was determined are added to the symbol for the refractive index, for 20 ° C and the sodium D line z. B. . ${\ displaystyle n_ {D} ^ {20}}$

The determination of the refractive index allows a simple determination of the content of a certain substance in a solvent:

## Connection with the atomic structure

### With crystalline materials

The refractive index of a crystalline material depends directly on its atomic structure, since the degree of crystallinity and the crystal lattice of a solid have an effect on its band structure . In the visible spectrum to the shows for example in the shift of the band gap .

An anisotropic crystal structure can also produce effects such as birefringence , where the material has different refractive indices for differently polarized light. In this case, the indicatrix a triaxial ellipsoid ( index ellipsoid ), and there are the principal refractive indices , and (also as  n 1 , n 2 and  n 3 denotes), the indexing is always performed so that: . ${\ displaystyle n _ {\ alpha}}$${\ displaystyle n _ {\ beta}}$${\ displaystyle n _ {\ gamma}}$${\ displaystyle n _ {\ alpha}

In the whirling crystal systems ( trigonal , tetragonal and hexagonal ) the main axis of the tensor, which is also referred to as the optical axis , coincides with the crystallographic c-axis. With these optically uniaxial materials

• corresponds to the refractive index of ordinary ray (engl. ordinary ray ) and is usually with  n o , n or , n ? or designated.${\ displaystyle n _ {\ alpha} = n _ {\ beta}}$${\ displaystyle n _ {\ perp}}$
• Analogously, ( ) the refractive index for the extraordinary ray (engl. Extraordinary ray ), and is referred to as  n ao , n e , n e or designated.${\ displaystyle n _ {\ gamma}}$${\ displaystyle \ neq n _ {\ alpha}}$${\ displaystyle n _ {\ parallel}}$

### For partially crystalline and amorphous materials

Relationship between refractive index and density for silicate and borosilicate glasses

In the case of partially crystalline or amorphous materials, the atomic structure also has a significant influence on the refractive index. As a rule, the refractive index of silicate , lead silicate and borosilicate glasses increases with their density .

Despite this general trend, the relationship between the index of refraction and density is not always linear, and exceptions occur, as shown in the diagram:

• a relatively large refractive index and a small density can be obtained with glasses, the light metal oxides such as Li 2 O or MgO containing
• the opposite is achieved with glasses containing PbO and BaO .

## Negative indices of refraction

### history

In 1968, the Soviet physicist Wiktor Wesselago described the strange behavior of materials with a negative refractive index: "If the production were successful, one could use them to manufacture lenses whose resolution would be far better than that of lenses made from ordinary optical materials".

In 1999, Sir John Pendry proposed a design for metamaterials with negative refractive index for microwaves, which was realized shortly afterwards.

2003, a group to Yong Zhang discovered in Colorado that crystals of yttrium - vanadate (YVO 4 ), a compound of yttrium, vanadium and oxygen , a negative refractive index for light waves of a large even without further processing frequency domain have. The crystal consists of two nested crystal lattices with symmetrical optical axes. The negative refraction occurs only in a certain angular range of the angle of incidence . In future experiments , the researchers want to test other suspected properties of negative refraction - such as the reversal of the Doppler effect and Cherenkov radiation .

In 2007, Vladimir Shalaev and his colleagues from Purdue University presented a metamaterial with a negative refractive index for radiation in the near infrared range .

In 2007, physicists led by Ulf Leonhardt from the University of St Andrews , using metamaterial with a negative refractive index ("left-handed material") succeeded in reversing the so-called Casimir effect (reverse Casimir effect, also known as quantum levitation). This opens up the future perspective of (almost) smooth nanotechnology .

### Lenses not limited by diffraction

In 2000, John Pendry showed that a material with a negative refractive index can be used to produce a lens whose resolution is not limited by the diffraction limit. A limiting condition is that the lens must be in the near field of the object so that the evanescent wave has not yet died down too much. For visible light this means a distance of about <1 µm. A few years later, researchers working with Xiang Zhang at Berkeley University succeeded in building a microscope with a resolution of one sixth the wavelength of the light used.

## literature

• Michael Bass: Handbook of Optics Volume 1. Optical Techniques and Design: . 2nd Edition. Mcgraw-Hill Professional, 1994, ISBN 0-07-047740-X .
• Martin Roß-Messemer: The smallest angle in sight . In: innovation . No. 10 , 2001, p. 22–23 ( PDF; 705 kB, archived on Nov. 9, 2012 ( memento of November 9, 2012 in the Internet Archive ) [accessed on June 20, 2016]).
• Schott Glass (Ed.): Optical Glass Properties. 2000 (product catalog; refractive indices of various types of glass). PDF; 257 kB.

Wiktionary: refractive index  - explanations of meanings, word origins, synonyms, translations
Commons : Refraction  - collection of images, videos, and audio files

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