Rugate mirror

Rugate filters made from porous silicon carbide .

A rugate mirror or rugate filter is a dielectric mirror that selectively reflects a certain wavelength range of light . This effect is achieved through a periodic, constant change in the refractive index, depending on the thickness of the mirror. A certain proportion of the wavelength of light cannot propagate in the rugate mirror and is reflected. The word rugate mirror is derived from corrugated structures that are found in nature and also selectively reflect certain wavelengths of light. An example of this are the wings of the morpho butterfly .

Characteristics

Refractive index profiles of a Bragg and Rugate mirror with maximum reflectivity at 700 nm.

Reflection spectra of a Bragg and Rugate mirror with normal incidence of light with maximum reflectivity at 700 nm.

With rugate mirrors, the refractive index varies periodically and continuously as a function of the thickness of the mirror. This is similar to the Bragg mirror with the difference that the refractive index profile is discontinuous there. The comparison of the refractive index profile of a Rugate and Bragg mirror is illustrated in the graph on the right. In the case of the Bragg mirror, the discontinuous transitions are responsible for the reflection of incident light, while in the case of Rugate mirrors, incident light is reflected at every position. According to the Fresnel equations , however, the reflection coefficient is greatest where the greatest change in the refractive index takes place. With the Rugate mirror, these are the turning points in the refractive index profile. With the help of the theory of the Bragg mirror, it is possible to calculate the wavelength at which the reflection of a Rugate mirror is greatest. For an alternating sequence in the Bragg mirror, the maximum of the reflection at one wavelength is: ${\ displaystyle \ lambda _ {0}}$

{\ displaystyle n_ {L} d_ {L} + n_ {H} d_ {H} = {\ frac {\ lambda _ {0}} {2}}}

In this equation and stand for the high and low refractive index in the Bragg mirror during and the respective physical thicknesses of these layers. For the more general case that the refractive index changes continuously, the previous equation can be rewritten: ${\ displaystyle n_ {L}}$ ${\ displaystyle n_ {H}}$ ${\ displaystyle d_ {L}}$ ${\ displaystyle d_ {H}}$

{\ displaystyle {\ frac {\ int _ {0} ^ {d} {n (x)} dx} {d}} d = \ left \ langle n \ right \ rangle d = {\ frac {\ lambda _ { 0}} {2}}}

On the left is the integral over the refractive index over a period of the refractive index profile divided by the period length . This term corresponds to the mean value of the refractive index profile. You can convince yourself of the correctness of this equation by solving the integral for a discrete refractive index profile and inserting the period of a Bragg mirror . ${\ displaystyle n (x)}$ ${\ displaystyle d}$ ${\ displaystyle d = d_ {H} + d_ {L}}$

The image on the right shows the reflection spectra calculated using the matrix transfer method for the refractive index profiles of a Bragg and Rugate mirror shown at the beginning. It can be seen that both mirrors have their maximum reflectivity at 700 nm, the rugate mirror having a smaller bandwidth. For this reason, rugate mirrors are often used as optical notch filters . You can also see a smaller peak in the spectrum of the rugate filter . This peak is not present in the spectrum of the Bragg mirror because its discrete layer system leads to destructive interference at this wavelength. Bragg mirrors, however, have secondary maxima at wavelengths of which can be undesirable if you only want to filter out a certain wavelength as far as possible. Rugate mirrors are better suited here, as the sinusoidal refractive index profile has anti-reflective properties similar to those of black silicon . As a result, the secondary maxima are reduced in their intensity. ${\ displaystyle \ lambda _ {0} / 2}$ ${\ displaystyle \ lambda _ {0} / (2n-1)}$

Manufacturing

Rugate mirrors can be made using sputtering and chemical vapor deposition . A particular challenge is the realization of the constant 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 these separation processes. Another possibility for the production of Rugate mirrors is electrochemical porosification of silicon . The current density during the etching process is chosen so that the resulting porosity and thus the refractive index varies sinusoidally with the layer thickness.

Individual evidence

↑ Markus Leitgeb, Christopher Zellner, Michael Schneider, Ulrich Schmid: Porous single crystalline 4H silicon carbide rugate mirrors . In: APL Materials . 5, No. 10, 2017, p. 106106. doi : 10.1063 / 1.5001876 .
^ HA Macleod: Thin-Film Optical Filters . 3. Edition. Institute of Physics Publishing, Bristol / Philadelphia 2001, ISBN 0-7503-0688-2 (first edition: 1986).
↑ Radislav Potyrailo, Ravi K. Bonam, John G. Hartley, Timothy A. Starkey, Peter Vukusic, Milana Vasudev, Timothy Bunning, Rajesh R. Naik, Zhexiong Tang, Manuel A. Palacios, Michael Larsen, Laurie A. Le Tarte, James C. Grande, Sheng Zhong, Tao Deng: Towards outperforming conventional sensor arrays with fabricated individual photonic vapor sensors inspired by Morpho butterflies . In: Nature Communications . 6, 2014, p. 7959. doi : 10.1038 / ncomms8959 .
↑ ^a ^b Olaf Stenzel: Optical Coatings . 3. Edition. Springer Science & Business, Heidelberg 2014, ISBN 978-3-642-54063-9 (first edition: 2014).
↑ Bartzsch, H., Lange, S., Frach, P., & Goedicke, K .: Graded refractive index layer systems for antireflective coatings and rugate filters deposited by reactive pulse magnetron sputtering. In: Surface and Coatings Technology . tape 180 , 2004, p. 616-620 , doi : 10.1016 / j.surfcoat.2003.10.105 .
↑ Lim, S., Ryu, JH, Wager, JF, & Plant, TK: Rugate filters grown by plasma-enhanced chemical vapor deposition. In: Thin Solid Films . tape 245 , no. 1-2 , 1994, pp. 141-145 , doi : 10.1016 / 0040-6090 (94) 90889-3 .
↑ Claudia Pacholski: Photonic Crystal Sensors Based on Porous Silicon . In: Sensors . tape 13 , no. 4 , April 9, 2013, p. 4694-4713 , doi : 10.3390 / s130404694 .

[1] Markus Leitgeb, Christopher Zellner, Michael Schneider, Ulrich Schmid: Porous single crystalline 4H silicon carbide rugate mirrors . In: APL Materials . 5, No. 10, 2017, p. 106106. doi : 10.1063 / 1.5001876 .

[Macleod-2] HA Macleod: Thin-Film Optical Filters . 3. Edition. Institute of Physics Publishing, Bristol / Philadelphia 2001, ISBN 0-7503-0688-2 (first edition: 1986).

[3] Radislav Potyrailo, Ravi K. Bonam, John G. Hartley, Timothy A. Starkey, Peter Vukusic, Milana Vasudev, Timothy Bunning, Rajesh R. Naik, Zhexiong Tang, Manuel A. Palacios, Michael Larsen, Laurie A. Le Tarte, James C. Grande, Sheng Zhong, Tao Deng: Towards outperforming conventional sensor arrays with fabricated individual photonic vapor sensors inspired by Morpho butterflies . In: Nature Communications . 6, 2014, p. 7959. doi : 10.1038 / ncomms8959 .

[stenzel-4] Olaf Stenzel: Optical Coatings . 3. Edition. Springer Science & Business, Heidelberg 2014, ISBN 978-3-642-54063-9 (first edition: 2014).

[5] Bartzsch, H., Lange, S., Frach, P., & Goedicke, K .: Graded refractive index layer systems for antireflective coatings and rugate filters deposited by reactive pulse magnetron sputtering. In: Surface and Coatings Technology . tape 180 , 2004, p. 616-620 , doi : 10.1016 / j.surfcoat.2003.10.105 .

[6] Lim, S., Ryu, JH, Wager, JF, & Plant, TK: Rugate filters grown by plasma-enhanced chemical vapor deposition. In: Thin Solid Films . tape 245 , no. 1-2 , 1994, pp. 141-145 , doi : 10.1016 / 0040-6090 (94) 90889-3 .

[7] Claudia Pacholski: Photonic Crystal Sensors Based on Porous Silicon . In: Sensors . tape 13 , no. 4 , April 9, 2013, p. 4694-4713 , doi : 10.3390 / s130404694 .