CA1298724C - Reflectivity mirrors and transversely variable transmission apertures with an azimuthal symmetry - Google Patents

Abstract

ABSTRACT OF THE DISCLOSURE
A variable reflectivity mirror or aperture, comprising an AR-coated transparent substrate and a thin shaped layer of a high-refractive-indexed dielectric on the substrate to provide a reflective surface is formed by depositing thin layer of predetermined radial profile of a high-refractive-index dielectric onto the substrate.

Description

129~7~L

The present invention relates to improved mirrors or apertures with a graded reflectivity profile and a method for making same.

BACKGROUND OF THE INVENTION
Mirrors with graded reflectivity profiles can be used as couplers, spatial filters or apodizers to reduce the detrimental effects of edge diffraction in optical systems and, more specifically, such mirrors and transmission apertures can be used in optical resonators to help control the optical and spectral qualities of lasers or externally of the laser resonator to shape the light beam or for any other purposes.
There are various known techniques for fabricating these mirrors and all present some drawbacks. In essence, the problem is to fabricate transversely variable reflectivity mirrors and transversely variable transmission apertures which do not significantly absorb the light at the working wavelength. Some known methods are discussed hereinbelow.
Absorbing Mirrors and/or Apertures Mirrors or apertures with radially varying reflectivity or transmission have been fabricated by means of evaporated metal or absorptive coatings, exposed and developed photographic films, absorptive filters with radially tapered density, or shaping of an absorptive material. It has been found that all variable reflectivity mirrors or variable transmission apertures based on such schemes absorb some of the power and, hence, both reduce laser efficiency and are damage prone in high power lasers.
Tape~ed-Groove n~th Diffraction Gratings Variable reflectivity mirrors can be made by etching a diffraction grating into an otherwise high reflectivity surface with a constant grating period and, hence constant diffraction angle, but with a groove depth and, hence diffraction efficiency, which varies from the center to the edge of the mirror. This approach requires sophisticated techniques, such as ion beam milling, to radially vary the groove depth of the grating in a controlled manner and, accordingly, the method is very expensive.

~' Radiallv Varying Birefringent Mirrors or Apertures In accordance with this technique, a birefringent element or an optically active element with radially varying strength or thickness is placed inside a laser cavity. The polarization of a wave passing through the element will be rotated or converted from linear to elliptical by an amount which varies with radius from the center of the element. A polarization sensitive coupling element is used to extract the rotated polarization component with a strength which varies radially at the desired rate. This technique requires at least two optical elements (a birefringent element and a polarizer) which are rather expensive. Furthermore, at long wavelengths such as 10um, for example, known birefringent elements have a low damage threshold and, consequently, this approach is not considered suitable for high power lasers.
lS Newton's Ring Reflectors A radially varying reflectivity and/or transmission apertures can be obtained by the combination of a flat surface and a curved surface placed in contact with or in close proximity to one another. This approach also requires two optical elements and very precise position-ing of the elements. Accordingly, it is also relatively expensive andawkward to use.
It will be seen, therefore, that all of the foregoing techniques have severe limitations and that, therefore, there is a need for a better method of making mirrors or apertures with a graded reflectivity profile.

SUMMARY OF TNE INVENTION
In accordance with the present invention, it is proposed to make transversely variable reflectivity mirrors and transversely variable transmission apertures by a method based on the principle that the effective reflectivity of a thin layer of dielectric depends on its thickness and its index of refraction. Thus, the proposed variable reflectlvity mirrors and variable transmission apertures are made of i) a substrate which is transparent at the wavelengths at which the device is intended to be used and ii) a layer of non-absorbing dielectric whose thickness is varied in order to obtain the required reflectivity or transmission profile.
A

lZ9~

Anti-reflection coatings or other uniform thin dielectric layers can be added to increase either the minimum or maximum reflectivity values or both. Two or more graded coatings inserted between uniform coatings can also be deposited to increase the reflectivity range.
Two methods are provided to produce the profiled dielectric layer. One method comprises controlling the thickness of the dielectric by placing a mask with an appropriately shaped aperture in front of a substrate during a deposition process and rotating either the mask or the substrate or both about the axis of symmetry during the deposition process.
The second method consists of placing a variable aperture iris in front of the substrate and varying the aperture diameter during deposition in order to control the relative deposition time and conse-quently the thickness as a function of radius.
Broadly, the present invention provides a method of forming transversely variable reflectivity mirrors and transversely variable transmission apertures with a symmetry of revolution, comprises depositing a thin layer of predetermined radial profile of a high-refractive-index dielectric onto a transparent substrate.
In accordance with another aspect of the present invention, there is provided a variable reflectivity mirror or aperture, comprising an AR-coated transparent substrate and a thin shaped layer of a high-refractive-indexed dielectric deposited on to provide a reflective surface.

BRIEF DESCRIPTION OF THE DRAWINGS
These and other features of the invention will become more apparent from the following description in which reference is made to the appended drawings, wherein:
S FIGURE lA illustrates the basic mirror configuration constructed in accordance with the present invention;
FIGURE lB illustrates the periodic dependence of R on layer thickness ~;
FIGURE 2A illustrates the thickness profile required to produce a Gaussian mirror;
FIGURE 2B illustrates the reflectivity curve of the Gaussian mirror of FIGURE 2A;
FIGURE 3 illustrates the phase shift distortion introduced in the reflected (FIGURE 3A) and transmitted (FIGURE 3B) wave fronts when a plane wave is incident on a mirror with a Gaussian reflectivity profile made of a shaped layer of germanium deposited on perfectly (dotted line) and imperfectly (solid line) matched AR coating-substrate combinations;
FIGURE 4A shows the reflectivity profile obtained when the layer is 33 thicker than the layer calculated to yield a Gaussian reflec-tivity profile (dashed line) for the air-Ge-air configuration;
FIGURE 4B illustrates the manner in which the variation of the half-reflectivity (HWHM) radius Pl/2 normalized to the HWHM design value Pl/2 varies linearly with ~e2/e2;
FIGURFS 5A AND 5B illustrate the method in accordance with a preferred embodiment of forming a profiled layer on a transparent substrate;
FIGURES 6A, B snd C illustrate the contour lines of the aperture of a mask used during deposition;
FIGURF 7 is a graph illustrating the measured reflectivity and transmissivity curves of a Gaussian mirror prototype;
FIGURE 8A is a layout of a 10~m interferometer; and FIGUR~ 8B is a graph showing relative phase shift versus radius.

lZ~37~4 DESCRIPTION OF PREFERRED EMBODIMENTS
The essential feature of the present invention comprises depositing a thin shaped layer of a dielectric onto a transparent substrate and yields reflectivity (or transmissivity) profiles which continuously vary with the radial distance. Mirrors (or filters) of any diameter and with any azimuthally symmetric transfer functions can be obtained. High damage thresholds are obtained because all of the non-reflected light is transmitted but not absorbed. The proposed design can be applied to any wavelength with appropriate materials.
FIGURE lA illustrates the basic mirror configuration. It essentially consists of a shaped thin layer 10 of a high-refractive-index dielectric deposited on an AR-coated transparent substrate 12 to provide a reflective surface 14. When the reflection from the second surface 16 is neglected, the theory of multiple beam interference inside a thin dielectric film provides an expression for the reflectivity as a function of the film thickness ez:

Ur 2 ¦rl2 + re exp [-i4~n2e2/ o]l R(e2) - _ _ q (1) Ui 1 + rl2 req exp [-i4~n2e2/ O]
with the thickness of the AR layer of index n3 equal to ~0/4n3 and n7 - (n~2/n,n4) (2) ~q n2 + (n3 /nln4) Subscripts 1, 2, 3 and 4 refer to the incidence medium, shaped layer, AR-coating and substrate materials, respectively;
~0 is the design wavelength in vacuum;
nl, the refractive index of medium i; and rlJ - (nl-nJ)/(nl+nJ), the Fresnel reflection coefficient at the interface between media i and ~.
FIGURE lB illustrates the periodic dependence of R on the layer thickness. From Equation (1), it will be seen that the thickness variation required to yield the reflectivity profile R(p) (and the transmissivity profile T(p)-l-R(p)) is given by:
A

1~91~3'7;~:~

n2~2(P) 1 -1 1 (1 - rl2)(1 - req ) 2 2 0 4~[2rl2r [ 1 - R(p) 12 eq ]]
(3) where p is the radial distance. With perfectly matched AR-coatings (n3 ~ (nln4) 5), the maximal reflectivity RmaX is achieved when ~2 = ~0/4n2 and is equal to (2rl2)2/(l+rl22)2. As RmaX increases with the refractive index n2, high-index dielectrics are desirable; Germanium (n=4.0) appears a good candidate for use at ~ lO~m giving RmaX = 0.78. The dashed curve in FIGURE lB (obtained assuming an AR coating with n3=1.5 (nln4) 5) shows that, in the more practical case of imperfectly matched coatings, RmaX is smaller than the value of the ideal case and Rmin no longer vanishes. This residual reflectivity constitutes a limitation of the present invention which may be reduced by using multilayer AR-coatings.
FIGURE 2A illustrates the thickness profile required to produce a Gaussian mirror for which the reflectivity dependence on the radial distance is given by:

R(p)--Rmaxexp~-p /*int) ~ (4) where Wlnt is the radius at reflectivity Rmaxe-l~ It is seen that imperfect matching of the AR coating (n331.5~nln4) 5) results in a truncation of the dielectric layer which causes the residual reflectivity at the mirror edge (FIGURE 2B).
Phase front distortion. At different radial positions, the light which is reflected or transmitted travels through different optical path lengths. Assuming a perfectly plane substrate, the radial phase variation of the reflected light is given by:

~(P) ~ ~/'2(P) ~ g~2() + (~alr(P) ~ (5) where ~2(P) represents the phase shift as determined on the dielectric surface:

~298~

1 rimaginary (Ui/ Ur)l ~2(P) = tan l real (ui/ur) ~ (6) with ur/ul given by Eq.(l). From FIGURE 1, it is seen that:

~air(P) - 2k [~2(P) ~ ~2()]. (7) where the factor of 2 takes into account the double passage through Zair The evaluation of the transmitted wave front distortion ~(p) is similarly calculated. It is given by:

~(P) = ~2() ~ ~2(P) ~ ~8ir(P), (8) where ~air(P) and ~2(P) are the free air and dielectric contributions, respectively, with ~8ir=k[~2(p)-~2(0)]. The value of ~2(P) is given by a relation similar to Equation (5) in which the ratio of the transmitted to incident amplitude is ut/ui-l+ur/ui.
FIGURE 3 illustrates the phase shift distortion introduced in the reflected (FIGURE 3A) and transmitted (FIGURE 3B) wave fronts when a plane wave is incident on a mirror with a Gaussian reflectivity profile made of a shaped layer of germanium deposited on perfectly (dotted line) and imperfectly (solid line) matched AR coating-substrate combinations. With a perfectly matched AR coating-substrate combination, the maximum phase variations introduced by the germanium profiled layer amounts to ~~o/6 and ~3~o/16 for the reflected and transmitted wave fronts, respectively. With an imperfect AR coating (air-Ge-ZnS-Ge, for example), a moderate phase jump (~~o/6) occurs near the profiled layer boundary where the reflectivity is very low. In that case, it is not expected that such small phase distortions will significantly deteriorate the reflected and transmitted light properties.
Sensitivity to Design Parameters. The design is based on a variable thickness single layer deposited on a transparent substrate.
The layer thickness is adapted for use at a design wavelength ~0.
Operation at a different wavelength or an error on the layer thickness will modify the transfer functions of the mirror.

~A

129~37Z~

To evaluate the sensitivity of the mirror characteristics to the design parameters, calculations of the reflectivity profiles have been done assuming perfectly matched AR coatings but a layer thickness different from the required value. As an example, FIGURE 4A shows the reflectivity profile obtained when the layer is 33~ thicker than the layer calculated to yield a Gaussian reflectivity profile (dashed line) for the air-Ge-air configuration. The Gaussian shape is approximately maintained but the width of the reflectivity profile is modified.
Operations with thicker and thinner dielectric result in broader and narrower reflectivity profiles, respectively. The variation of the half-reflectivity (HWHM) radius Pl/2 normalized to the HWHM design value Pl/2 varies linearly with ~2 (see FIGURE 4B). Consequently, the graded mirror performances are not expected to be very sensitive to the design parameters.
FIGURE 5A illustrates the technique used to obtain the profiled layer. The layer thickness is controlled by placing a mask 20 with a appropriately shaped deposition aperture 22 in front of a turning substrate 12 during the deposition. This provides a continuous variation of the reflectivity with the radial position and can then provide any azimuthally symmetric transfer function. To avoid shadow effects, the mask must be placed as close as possible to the substrate and the edge of the aperture, bevelled as shown in FIGURE 5B.
Mask 20 is in the form of a slab of any rigid material. An aperture with a contour calculated to yield the desired thickness profile is cut in the slab. This contour is determined by relating the full angle ~(p) of the free aperture to the calculated thickness ~2(P) at the radial position p:

~(p) _ ~(o)~7(p) (9) where ~2() is the required thickness at the center of the mirror. This relation gives the fraction of the rotation period during which the evaporated dielectric can reach a given point on a circumference located at a radius p. As shown in FIGURE 6A, the contour lines of the aperture are determined by the coordinates x(p) - pcos[~(p)/2] and 40 y(p) ~ +psin[~(p)/2]. At the radial position Pm~ the thickness equals 1~87~4 g zero and so does the aperture angle.
In Equation (9), the normalization angle ~(0) at the center of the mirror can take any arbitrary value between 0 and 2~. For ~<~(0)<2~, the aperture has a heart form as shown in FIGURE 6B. As ~(0) increases, the cutting of the sharp central point becomes fas-tidious. For small values oE ~(0) the aperture is very narrow and the deposition time necessary to reach the required central thickness is increased. A normalization angle ~(0)=~ (FIGURE 6C) constitutes a good compromise between the deposition time and the difficulty in cutting the aperture in the slab.
One-lobe masks as shown in FIGURES 6A and 6B can yield a thickness profile with a hole in the center if the rotation axis of the substrate falls in the dark region of the masking slab. The possibility of using a mask with two identical lobes, as shown in FIGURE 6C, has been considered. The blunting of the sharp central points gives a small region in the mask that is always open during deposition and, thus, reduces the alignment constraints:. The reflectivity profile will then exhibit a small flat at its center.
Moreover, the two-lobe mask shown in FIGURE 6C requires the same deposition time as a one-lobe mask with ~(0)-2~, but it provides an advantage for the cutting operation in that the contours of the aperture are identical in the four quadrants.
Two-lobe masks have been used to fabricate many prototypes of mirrorswith Gaussian reflectivity profiles of different diameters (12.5 and 5.0 cm), with different substrate materials (Ge and NaCl) and for use at different wavelengths (10.6, 3.7 and 2.7 ~m). The Ge and NaCl substrates were AR coated with single layers of ZnS and NaF, respectively.
Reflectivity and Transmissivity Profiles. To verify the above design, the radial reflectivity and transmissivity profiles of the 5-cm diameter flat Ge prototype of Gaussian reflectivity mirror have been measured with a cw CO2 probing laser. By focusing with a lens, spatial resolution better than 0.5 mm was possible without damaging the mirror surface. The experimental results are shown in FIGURE 7, where the dashed curve represents a Gaussian of the same peak height and width as the experimental curve. It will be seen that the reflectivity profile follows very closely the Gaussian shape. As expected, the A

~Z~ 72~

reflectivity (R) and transmissivity (T) curves are complementary, i.e.
T+R~l everywhere.
Phase front distortions. FIGURE 8A illustrates a 10-~m inter-ferometer which was used to measure the radial phase variations in the wave fronts reflected and transmitted by the Gaussian mirror prototype.
An acoustooptic modulator split, into two parts, the output beam of a dither stabilizer cw CO2 laser. The deviated beam, which was upshifted in frequency by 40 MHz, was used as a reference and mixed to the probing beam on a fast HgCdTe photodiode. Optical phase variations in either path shifted the phase of the beating signal at the output of the IR detector. Comparison of this signal with a reference rf signal in a phase detector allowed a precise measurement of any phase variation introduced in the probing beam path. The curve of the radial phase variations was obtained by transversely translating the prototype lS across the probing beam. For the transmitted wave front measurements, the prototype was placed before the beam splitter which was rotated by 90 as indicated by the dashed line in FIGURE 8A.
FIGURE 8B represents the measured radial phase variation of the transmitted and reflected signals. Both curves exhibit variations very similar to the predicted behaviour (see FIGURE 3). The reference curves have been obtained wlth flat optical elements and are representative of the sensitivity and stability of the arrangement.
For transmitted light, the maximum phase shift produced by the Ge profiled layer amounts to ~0.22~2, in good agreement with the calculated value 3~o/16. The reflected wave front presents a phase ~ump of ~~0/4 near the reflectivity profile border. This shift is higher than the ~o/6 predicted value for that position. However, the ~~o/10 phase variation introduced in the center of the mirror is close to the expected ~o/8 phase shift. These data appear reasonably close to the theoretical values.
It will be seen therefore that graded reflectivity mirrors can be made advantageously by employing deposition techniques. All of the non-reflected light is transmitted when non-absorptive materials are used. The turning mask technique yields reflectivity and transmis-sivity profiles which vary continuously with the radial position. The same can be applied to any wavelength with appropriate materials. The ~Z9~

method could also be used to fabricate smooth radially varying phase filters.
Prototypes of Gaussian reflectivity mirrors have been made on NaCl and Ge substrates for use at about 3.7 and 10 ~m. The full characterization of a prototype have confirmed the theoretical calculations. Experiments in a 50-cm long TEA C02 laser indicate that the Ge-ZnS-Ge configuration has a damage threshold close to that of standard coated germanium couplers (>1 J/cm2).

A

Claims (20)

1. A method of forming transversely variable reflectivity mirrors and transversely variable transmission apertures with a symmetry of revolution, comprising the steps of:
depositing a thin layer of predetermined radial profile of a high-refractive-index dielectric onto a transparent substrate.

2. A method as defined in claim 1, said depositing step further including:
preparing a transparent substrate;
placing a mask having a deposition aperture of predetermined contour in front of said substrate;
depositing a thin layer of said dielectric with a thickness which varies radially with a symmetry of revolution on by rotating either said mask or said substrate or both.

3. A method as defined in claim 1, said depositing step further including:
preparing a transparent substrate;
placing a mask having a variable deposition aperture over a surface of said substrate; and depositing a thin layer of said dielectric with a thickness which varies radially with a symmetry of revolution while varying said aperture.

4. A method as defined in claim 2, further including the step of placing said mask in close proximity to said substrate.

5. A method as defined in claim 3, further including the step of placing said mask in close proximity to said substrate.

6. A method as defined in claim 2, said deposition aperture having a bevelled edge.

7. A method as defined in claim 3, said variable deposition aperture having a bevelled edge.

8. A method as defined in claim 2, said aperture being formed with a contour determined by relating the full angle .theta.(p) of said free aperture to the calculated thickness .epsilon.2(P) at the radial position p:

(9) where .epsilon.2(0) is the required thickness at the center of the mirror or transmission aperture.

9. A method as defined in claim 8, said deposition aperture having contour lines determined by the co-ordinates:

x(p) = pcos[.theta.(p)/2]; and y(p) = +psin[.theta.(p)/2].

10. A method as defined in claim 8, wherein the normalization angle .theta.(0) at the center of said substrate being selected from any arbitrary value between 0 and 2.pi..

11. A method as defined in claim 10, wherein the normalization angle .theta.(0)-.pi..

12. A method as defined in claim 1, said substrate including multilayer AR-coatings.

13. A method of forming transversely variable reflectivity mirrors and transversely variable transmission apertures with a symmetry of revolution, comprising the steps of:
preparing a AR-coated transparent substrate;
placing a mask having a deposition aperture of predetermined contour in close proximity to a surface of said substrate, said aperture having a bevelled edge or being very thin and being formed with a contour determined by relating the full angle .theta.(p) of the free aperture to the calculated thickness .epsilon.2(p) at the radial position p:

(9) where .epsilon.2(0) is the required thickness at the center of the mirror or transmission aperture; and depositing a thin layer of said dielectric with a thickness which varies radially with a symmetry of revolution by rotating either said mask or said substrate or both.

14. A method as defined in claim 13, wherein the contour lines of said deposition aperture are determined by the coordinates:

x(p) - pcos[.theta.(p)/2]; and y(p) - +psin[.theta.(p)/2].

15. A method as defined in claim 14, wherein the normalization angle .theta.(0) at the center of said substrate being selected from any arbitrary value between .theta. and 2.pi..

16. A method as defined in claim 15, wherein the normalization angle .theta.(0)-.pi..

17. A method of forming transversely variable reflectivity mirrors and transversely variable transmission apertures with a symmetry of revolution, comprising the steps of:
preparing a AR-coated transparent substrate;
placing a mask having a variable deposition aperture having a bevelled edge in close proximity to a surface of said substrate; and depositing a thin layer of said dielectric with a thickness which radially varies with a symmetry of revolution while varying said aperture.

18. A variable reflectivity mirror or aperture, comprising an AR-coated transparent substrate and a thin shaped layer of a high-refractive-indexed dielectric deposited on said substrate to provide a reflective surface having a reflectivity or transmissivity profile which varies continuously with radial distance from an axis extending through said substrate.

19. A variable reflectivity mirror or aperture as defined in claim 18, said shaped layer having a thickness profile given by:

...(3) where (2) and the thickness of the layer of index n3 is .lambda.0/4n3 and p is the radial distance, Subscripts 1, 2, 3 and 4 refer to the incidence medium, shaped layer, AR-coating and substrate materials, respectively;
.lambda.o is the design wavelength in vacuum;
ni is the refractive index of medium i;
rij - (ni-nj)/(ni+nj), the Fresnel reflection coefficient at the interface between media i and j;
.epsilon.2 is the layer thickness.

20. A variable reflectivity mirror or aperture as defined in claim 18 or 19, said substrate including multilayer AR-coatings.