
A rugate filter, also known as a gradient-index filter, is an
optical filter
An optical filter is a device that selectively transmits light of different wavelengths, usually implemented as a glass plane or plastic device in the optical path, which are either dyed in the bulk or have interference coatings. The optic ...
based on a
dielectric mirror
A dielectric mirror, also known as a Bragg mirror, is a type of mirror composed of multiple thin layers of dielectric material, typically deposited on a substrate of glass or some other optical material. By careful choice of the type and thickne ...
that selectively reflects specific wavelength ranges of light. This effect is achieved by a periodic, continuous change of the
refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, o ...
of the
dielectric coating
A dielectric mirror, also known as a Bragg mirror, is a type of mirror composed of multiple thin layers of dielectric material, typically deposited on a substrate of glass or some other optical material. By careful choice of the type and thickne ...
. The word "rugate" is derived from corrugated structures found in nature, which also selectively reflect certain wavelength ranges of light, for example the wings of the Morpho butterfly.
Characteristics

In rugate filters the refractive index varies periodically and continuously as a function of the depth of the mirror coating. This is similar to
Bragg mirrors with the difference that the refractive index profile of a Bragg mirror is discontinuous. The refractive index profiles of a Rugate and a Bragg mirror are shown in the graph on the right. In Bragg mirrors, the discontinuous transitions are responsible for reflection of incident light, whereas in rugate filters, incident light is reflected throughout the thickness of the coating. According to the
Fresnel equations
The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media. They were deduced by Augustin-Jean Fr ...
, however, the
reflection coefficient
In physics and electrical engineering the reflection coefficient is a parameter that describes how much of a wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the amplitude of the reflected w ...
is greatest where the greatest change in refractive index occurs. For rugate filters, these are the
inflection point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...
s in the refractive index profile. The theory of the Bragg mirror leads to a calculation of the wavelength at which the reflection of a rugate filter is greatest. For an alternating sequence in the Bragg mirror, the maximum reflection at a wavelength
is:
:
In this equation
and
stand for the high and low refractive indices of the Bragg mirror while
and
are the respective thicknesses of these layers. For the more general case that the refractive index changes continuously, the previous equation can be rewritten as:
:
On the left hand side is the integral over the refractive index over one period of the refractive index profile
divided by the period length
. This term corresponds to the mean value of the refractive index profile.
As a
sanity check
A sanity check or sanity test is a basic test to quickly evaluate whether a claim or the result of a calculation can possibly be true. It is a simple check to see if the produced material is rational (that the material's creator was thinking ration ...
for the correctness of this equation, one can solve the integral for a discrete refractive index profile and substitute the period of a Bragg mirror
.
The figure on the right shows the reflection spectra calculated by the
transfer-matrix method
In statistical mechanics, the transfer-matrix method is a mathematical technique which is used to write the partition function into a simpler form. It was introduced in 1941 by Hans Kramers and Gregory Wannier. In many one dimensional latti ...
for the refractive index profiles of a Bragg and Rugate filter. It can be seen that both mirrors have their maximum reflectivity at 700 nm, whereas the rugate filter has a lower bandwidth. For this reason rugate filters are often used as optical
notch filters. Furthermore, one can see a smaller peak in the spectrum of the rugate filter at
. This peak is not present in the spectrum of the Bragg mirror because of its discrete layer system, which causes destructive interference at this wavelength. However, Bragg mirrors have secondary maxima at wavelengths of
, which may be undesirable if you only want to filter out a certain wavelength. Rugate filters are better suited for this purpose because the sinusoidal refractive index profile has
anti-reflection properties similar to those of
black silicon. This reduces the intensity of the secondary maxima.
Production
Rugate filters can be produced by
sputtering
In physics, sputtering is a phenomenon in which microscopic particles of a solid material are ejected from its surface, after the material is itself bombarded by energetic particles of a plasma or gas. It occurs naturally in outer space, and ...
and
chemical vapor deposition
Chemical vapor deposition (CVD) is a vacuum deposition method used to produce high quality, and high-performance, solid materials. The process is often used in the semiconductor industry to produce thin films.
In typical CVD, the wafer (subst ...
. A special challenge is the creation of the continuous refractive index profile. To achieve this, the chemical composition of the mirror must also change continuously as a function of the layer thickness. This can be achieved by continuously changing the gas composition during the deposition process. Another possibility for the production of rugate filters is electrochemical
porosification of silicon. Here, the current density during the etching process is selected so that the resulting porosity and thus the refractive index varies sinusoidally with the layer thickness.
[{{cite journal , vauthors = Pacholski C , title = Photonic crystal sensors based on porous silicon , journal = Sensors , volume = 13 , issue = 4 , pages = 4694–713 , date = April 2013 , pmid = 23571671 , pmc = 3673107 , doi = 10.3390/s130404694 , bibcode = 2013Senso..13.4694P , doi-access = free ]
References
Optical filters
Electrodynamics
Physical optics