Advertisement

Journal of Optics

, Volume 47, Issue 1, pp 65–74 | Cite as

Making aspherical lens by thin film deposition coated under vacuum

Research Article
  • 44 Downloads

Abstract

An aspherical lens is made by a thin-film coating technique. A special mask is placed between the evaporation source and the substrate that is to be coated as an aspheric. The design principle of the mask is completely described. The final surface is tested by an interferometric double path auto-collimation measuring method. Less than 0.05λ rms of a wave (632.8 nm) of aberrations is achieved without much trial and error. The static mask is described and analysed by simulations of the film thickness theory, based on the assumption that atoms and molecules emitted by the source travel in straight lines to the substrates, which rotate about the central axis during films deposition. The design method can be practically used for all optical coatings produced by vacuum deposition. The solution of a mask shape is unique for a stationary set of coating parameters. A set of coating parameters include the height of the apex of the dome above the source, the curvature radius of the dome, the distance from the source to the rotation axis of the dome, the emissive characteristics of the source which modify the cosine law of the surface source, the form and area of the source. The static mask designed by this method can be used for correction optical element’s aberrations. Dioxide silicon thin film was evaporated with 6 µm thickness on BK7 optical glass and fused quartz optical glass for apherization refractive optical elements. The SiO2 film was tested for stability and environmental durability, it achieved good sticking and passed the test without peeling or cracking.

Keywords

Mask Film thickness Optical coating Vacuum deposition Vacuum aspherization Aspherical lens Coating technique 

Introduction

Aspheric surfaces are usually made by a subtractive process, that is, the surface figure is corrected to be aberration free by grinding and polishing. This requires skilled operators and high technique machine. Diamond-turning techniques with high-precision CNC machines have been used to fabricate aspherics. This makes things easier, but is good only for IR optics because of residual surface roughness. Aspherics made by ion milling have been studied by Pozdimek et al. [1]. However, this requires a well-controlled ion beam, both in position and in angle of incidence.

Alternatively, additive processes have been proposed, for example, making aspherics by adding material instead of removing material. Strong and Gaviola [2] corrected spherics by controlled deposition of aluminum, but found that the coating began to show a bloom and scatter light when the thickness of the aluminum became greater than 2 µm. Schulz [3] carefully arranged the source geometry and the substrate placement to make off-axis parabolic Fresnel mirrors by evaporation of aluminum. Aspherical refracting surfaces have been coated by Schulz [4]. He used lithium fluoride and found that films began to peel off when the thickness became greater than 5 µm. Dobrowolski and Weinstein [5] and Weinstein and Dobrowolski [6] substituted zinc sulfide to make aspherical surfaces that did not peel off or crack and that exhibited negligible optical loss in the visible range shorter than 500 nm. Kurdock and Austin [7] summarized the additive processes used to make aspherics.

In this paper, we describe how to make aspherical lens additively by deposition silicon dioxide on fused quartz and BK7 optical glasses with thickness about 5.8 µm in vacuum evaporation.

Theory

Surface correction

The conventional way to make an aspheric is to start from a spherical substrate. Therefore, we need to know what the distribution of layer thickness is on a spherical substrate from a directed surface source before making any correction. Let m be the total mass of material emitted from the source in the upper direction and µ be the film density. Then the thickness of film, t, at a given point P of radius S on the substrate Sub, as shown in Fig. 1, is t [8].
$$dM = \frac{m}{{\frac{2\pi }{n + 1}}}\cos^{n} \theta dw$$
$$t = \frac{m}{\mu \pi }\frac{\cos \varPhi \cos \theta }{{r^{{{\prime }2}} }},$$
(1)
where:
$$r^{{{\prime }2}} = h^{2} + S^{2} + r^{2} - 2Sr\cos\uppsi$$
$$R_{s}^{2} = R^{2} + r^{{{\prime }2}} - 2Rr^{{\prime }} \cos \theta$$
Also, \(R_{s}^{2} = r^{2} + \left( {h_{0} - R} \right)^{2}\), therefore
$$\cos \theta = \frac{{h^{2} + S^{2} - h_{0}^{2} + 2Rh_{0} - 2Sr\cos \psi }}{{2Rr^{{\prime }} }}$$
$$\cos \varPhi = \frac{h}{{r^{{\prime }} }},$$
Fig. 1

Geometry of the vapor flow in the surface source [9]

Substituting cos θ and cos Φ into Eq. (1), we have
$$t = \frac{m}{\pi \mu }\frac{{h\left( {h^{2} + S^{2} - h_{0}^{2} + 2Rh_{0} - 2Sr\cos \psi } \right)}}{{2R\left( {h^{2} + S^{2} + r^{2} - 2Sr\cos \psi } \right)^{2} }},$$
(2)

In order to have enough material to evaporate without any further feeding during evaporation, we use an electron-beam source.

In this case, Eq. (2) must be modified, since the distribution of evaporants is no longer a simple cosine form. Instead, it has the form cos n Φ, where n varies from 1.0 to 3.0, depending on the power of the electron beams. Note that n = 0 corresponds to a point source, and n = 1 corresponds to a simple directed surface source. Figure 2 shows the vapor distribution of various cosine exponents. Assume that the total mass m is deposited on a substrate according to the distribution of cos n Φ then the integration over the upper sphere is
$$\int\limits_{0}^{2\pi } {\int\limits_{0}^{{\frac{\pi }{2}}} {\frac{{\cos^{n} \varPhi }}{{r^{{{\prime }2}} }}dA = \int\limits_{0}^{2\pi } {\int\limits_{0}^{{\frac{\pi }{2}}} {\frac{{\cos^{n} \varPhi }}{{r^{{{\prime }2}} }}r^{{{\prime }2}} \sin \varPhi d\varPhi d\alpha = \frac{2\pi }{n + 1}.} } } }$$
Fig. 2

Geometry of evaporation: r′ is the distance from the source to an arbitrary point P, which is at a height h above the source. h0 is the height of the apex of the substrate above the source. R is the curvature radius of the substrate. r is the distance from the source to the rotation axis. R S is the distance from the source to the center of curvature of the substrate [9]

Therefore, Eq. (2) is corrected as
$$t_{n} = \frac{m}{{\frac{2\pi }{n + 1}\mu }}\frac{{h^{n} \left( {h^{2} + S^{2} - h_{0}^{2} + 2Rh_{0} - 2Sr\cos \psi } \right)}}{{2R\left( {h^{2} + S^{2} + r^{2} - 2Sr\cos \psi } \right)^{{\frac{3 + n}{2}}} }}.$$
(3)
When the spherical substrate is rotating about the central axis during evaporation, the layer thickness at point P on the ring of radius S.
$$t_{n} = \frac{m}{{\frac{2\pi }{n + 1}\mu }}\frac{1}{2\pi } \times \int\limits_{0}^{2\pi } {\frac{{h^{n} \left( {h^{2} + S^{2} - h_{0}^{2} + 2Rh_{0} - 2Sr\cos \psi } \right)}}{{2R\left( {h^{2} + S^{2} + r^{2} - 2Sr\cos \psi } \right)^{{\frac{3 + n}{2}}} }}.}$$
(4)
Figure 3 shows the vapor distribution of various cosine exponents.
Fig. 3

Vapor distribution of various cosine exponents. The dotted line indicates the diameter of the substrate to be coated. The radius and height are counted from the center of the source surface [10]

Figure 4 shows the thickness distribution for a spherical substrate with curvature radius R = 330-mm and at the height from the electron-beam source, h 0 = 600 mm, with diameter 540 mm and r = 170 mm, where t0 is the film thickness at the center of the substrate.
Fig. 4

Thickness distribution t on a spherical substrate with curvature radius R = 330 mm for a cos n Φ evaporation source at r = 170 mm and the substrate height from evaporation source h0 = 600 mm. n is given as a parameter for each curve

In a real case, the evaporation source has an area A. Therefore the real film thickness tn is
$$t_{n} = \frac{m}{{\frac{2\pi }{n + 1}\mu }}\frac{1}{2\pi }\frac{1}{A} \times \int\limits_{area} {\int\limits_{0}^{2\pi } {\frac{{h^{n} \left( {h^{2} + S^{2} - h_{0}^{2} + 2Rh_{0} - 2Sr\cos \psi } \right)}}{{2R\left( {h^{2} + S^{2} + r^{2} - 2Sr\cos \psi } \right)^{{\frac{3 + n}{2}}} }} \times d\psi dA.} }$$
(5)
Let us take conic surface with parameters (R: radius at the vertex, k: conic constant) as examples of aspherics. Then, as shown in Fig. 5, we can make conics aspherics from spherical surfaces. Spherical surface Zs and conic surface Z c can be written as
$$Z_{s} = R - \left( {R^{2} - S^{2} } \right)^{{\frac{1}{2}}} = R - R\left[ {1 - \left( {\frac{S}{R}} \right)^{2} } \right]^{{\frac{1}{2}}} \approx \left( {\frac{{S^{2} }}{2R}} \right) + \left( {\frac{{S^{4} }}{{8R^{3} }}} \right),\quad for\;R \gg S,$$
$$Z_{c} = \frac{{CS^{2} }}{{1 + \left[ {1 - \left( {k + 1} \right)C^{2} S^{2} } \right]^{{\frac{1}{2}}} }}$$
$$Z_{c} = \left( {\frac{{S^{2} }}{2R}} \right) + \left( {\frac{{\left( {1 + k} \right)S^{4} }}{{8R^{3} }}} \right) = \left( {\frac{{S^{2} }}{2R}} \right) + \left( {\frac{{S^{4} }}{{8R^{3} }}} \right)\left( {k + 1} \right),$$
$$Z_{c} = \tau + \left( {\frac{{S^{2} }}{2R}} \right) + \left( {\frac{{S^{4} }}{{8R^{3} }}} \right)\left( {k + 1} \right),$$
where τ is the distance between the two surfaces on the Z axis. For a subtractive process, τ = 0. For an additive process, as we are performing here, τ ≠ 0. Since s = s0 at the edge of the substrate, the two surfaces are coincident, so we have
$$\tau + \left( {\frac{{S_{0}^{2} }}{2R}} \right) + \left( {\frac{{S_{0}^{4} }}{{8R^{3} }}} \right)\left( {k + 1} \right) = \left( {\frac{{S_{0}^{2} }}{2R}} \right) + \left( {\frac{{S_{0}^{4} }}{{8R^{3} }}} \right),$$
i.e.,
$$\tau = - k\left( {\frac{{S_{0}^{4} }}{{8R^{3} }}} \right).$$
(6)
Fig. 5

Convex Conics can be made from spherical surfaces. a spherics; b conics (subtractive); c conics (additive)

Therefore, the difference between the two surfaces can be expressed as
$$\begin{aligned} D\left( S \right) = &\, Z_{c} - Z_{s} = - k\left( {\frac{{S_{0}^{4} }}{{8R^{3} }}} \right) + k\left( {\frac{{S^{4} }}{{8R^{3} }}} \right), \\ = & - k\left( {\frac{{S_{0}^{4} }}{{8R^{3} }}} \right)\left[ {1 - \left( {\frac{S}{{S_{0} }}} \right)^{4} } \right], \\ = & \left( {\frac{{W_{040} }}{2}} \right)\left[ {1 - \left( {\frac{S}{{S_{0} }}} \right)^{4} } \right], \\ \end{aligned}$$
(7)
where
$$W_{040} = - k\left( {\frac{{S_{0}^{4} }}{{4R^{3} }}} \right).$$
(8)

Mask shape

When a mask is placed in front of the substrate, the distribution of film thickness will change. Let the solid section in Fig. 6 represent the mask just beneath the substrate. If \(M\left( s \right) = \widehat{EF}\) then the relation between the thickness with the mask, tm and the thickness without the mask, tn, for a given S is
$$t_{m} \left( S \right) = t_{n} \left( S \right)\left[ {1 - \frac{M}{2\pi S}} \right]$$
Fig. 6

Mask in front of substrate. The mask is represented by the solid section [10]

This must be equal to the target thickness, which is D(S) in Eq. (7). Therefore the size of the mask at any given S can be plotted according to the following equation:
$$M\left( S \right) = 2\pi S\left[ {1 - \frac{D\left( S \right)}{{t_{n} \left( S \right)}}} \right],$$
(9)
where tn and D can be calculated from Eq. (5) and Eq. (7), respectively. Since we cannot place the mask in contact with the substrate, the correct mask is smaller than that described by Eq. (9). Let the heights of the substrate and of the mask from the evaporation source be h 0, and h m , respectively; then, by the method of view projection, we know that the mask profile is
$$M\left( S \right) = \frac{{h_{m} }}{{h_{s} }}2\pi S\left[ {1 - \frac{D\left( S \right)}{{t_{n} \left( S \right)}}} \right],$$
(10)

Experiments

The substrate spherical surface was measured by DPAC using Fizeau interferometer referred to an aspherical wave; the Fig. 7 shows the spherical surface before correction by simulation with zemax and by measuring.
Fig. 7

a Interferogram of the spherical surface referred to aspherical wave from simulation. b Interferogram of the spherical surface referred to aspherical wave from measuring. Both figures show large number of unreservable fringes that cannot be measured practically

Table 1 listed a comparison between the Zernike fringe coefficients for both surfaces (spheric and aspheric). The Zernike coefficients were obtained by simulation with Zemax. Table 2 listed the same comparison for both surfaces by measuring (DPAC method) after correction (after coating).
Table 1

Zernik fringe coefficients by simulation

Term

Z(ρ, φ)

Spherical surface

Aspherical surface

1

1

35.77097455

−0.03355851

2

ρcosφ

0.0

0.0

3

ρsinφ

0.0

0.0

4

2 − 1

50.78951699

−0.00016943

5

ρ2cos2φ

0.0

0.0

6

ρ2sin2φ

0.0

0.0

7

(3ρ2 − 2)ρcosφ

0.0

0.0

8

(3ρ2 − 2)ρsinφ

0.0

0.0

9

4 − 6ρ2 + 1

13.27083670

0.00265766

10

ρ3cos3φ

0.0

0.0

11

ρ3sin3φ

0.0

0.0

12

(4ρ2 − 3) ρ2cos2φ

0.0

0.0

13

(4ρ2 − 3) ρ2sin2φ

0.0

0.0

14

(10ρ4 − 12ρ2 + 3) ρcosφ

0.0

0.0

15

(10ρ4 − 12ρ2 + 3) ρsinφ

0.0

0.0

16

20ρ6 − 30ρ4 + 12ρ2 − 1

−1.59056224

−0.02685179

17

ρ4cos4φ

0.0

0.0

18

ρ4sin4φ

0.0

0.0

19

(5ρ2 − 4) ρ3cos3φ

0.0

0.0

20

(5ρ2 − 4) ρ3sin3φ

0.0

0.0

21

(15ρ4 − 20ρ2 + 6) ρ2cos2φ

0.0

0.0

22

(15ρ4 − 20ρ2 + 6) ρ2sin2φ

0.0

0.0

23

(35ρ6 − 60ρ4 + 30ρ2 − 4) ρcosφ

0.0

0.0

24

(35ρ6 − 60ρ4 + 30ρ2 − 4) ρsinφ

0.0

0.0

25

70ρ8 − 140ρ6 + 90ρ4 − 20ρ2 + 1

0.14410723

0.00354235

26

ρ5cos5φ

0.0

0.0

27

ρ5sin5φ

0.0

0.0

28

(6ρ2 − 5) ρ4cos4φ

0.0

0.0

29

(6ρ2 − 5) ρ4sin4φ

0.0

0.0

30

(21ρ4 − 30ρ2 + 10) ρ3cos3φ

0.0

0.0

31

(21ρ4 − 30ρ2 + 10) ρ3sin3φ

0.0

0.0

32

(56ρ6 − 105ρ4 + 60ρ2 − 10)ρ2cos2φ

0.0

0.0

33

(56ρ6 − 105ρ4 + 60ρ2 − 10)ρ2sin2φ

0.0

0.0

34

(126ρ8 − 280ρ6 + 210ρ4 − 60ρ2 + 5)ρcosφ

0.0

0.0

35

(126ρ8 − 280ρ6 + 210ρ4 − 60ρ2 + 5)ρsinφ

0.0

0.0

36

252ρ10 − 630ρ8 + 560ρ6 − 210ρ4 + 30ρ2 − 1

−0.01198950

−0.00031330

37

924ρ12 − 2772ρ10 + 3150ρ8 − 1680ρ6 + 420ρ4 − 42ρ2 + 1

0.00096630

0.00002259

Table 2

Zernike fringe coefficients by measuring

Term

N

RMS fit

Aspherical surface

Spherical surface

Plane

2

0.063

0.312, 1.508

Cannot be measured practically with aspherical wave a cause of unreservable fringes

Sphere

3

0.058

0.315, 1.513, 0.025

4th order

8

0.045

0.309, 1.499, 0.010, 0.017, −0.050, −0.007, −0.017, −0.018

6th order

15

0.028

0.298, 1.476, −0.004, 0.047, −0.070, −0.023, −0.045, −0.026, 0.044, 0.031, 0.024, −0.031, −0.026, 0.023, 0.012

8th order

24

0.017

0.314, 1.503, 0.018, 0.019, −0.037, 0.001, −0.011, 0.006, 0.015, 0.014, 0.006, 0.004, −0.004, 0.043, 0.023, −0.010, −0.029, −0.023, −0.011, −0.003, 0.045, 0.024, −0.010, −0.004

Complete

36

0.009

0.250, 1.170, −0.079, 0.204, −0.096, −0.083, −0.427, −0.058, 0.229, 0.368, 0.106, −0.034, −0.050, −0.157, 0.025, −0.101, 0.024, 0.082, 0.161, −0.007, 0.038, 0.018, 0.005, −0.003, −0.165, −0.127, 0.010, −0.011, −0.012, −0.011, 0.000, −0.020, −0.012, 0.013, 0.001, 0.000

We notice here that there is only spherical aberration because the simulation is performed only for zero field.

We notice here that there is no Zernike coefficients for the spherical surface a cause of the unreservable fringes.

Table 3 listed Sidel aberration coefficients in waves for the spheric surface by simulation with DPAC method. Since the field is zero, the aberrations could be shown is only spherical aberration.
Table 3

Sidel aberration coefficients in waves by simulation for spherical surface

Sidel aberration coefficients in waves

Aberration

0.0

Tilt

0.0

Focus

0.0

Astigmatism

0.0

Coma

−53.309687

Spherical

Table 4 listed Sidel aberration coefficients from fringe centers for the aspheric surface measured with DPAC method.
Table 4

Sidel aberration coefficients from fringe centers by measuring for the aspheric surface

Magnitude waves

Angle

Aberration

1.530

78.4

Tilt

0.020

Focus

0.106

−35.6

Astigmatism

0.056

−113.3

Coma

−0.110

Spherical

Coatings were deposited with a 61-cm bell chamber coater. The coating system was pumped to a base pressure of 5 × 10−6 mbar. The coater is provided with two pumps, first one is a mechanical rotary pump, Edwards type E2M40, it is fitted to provide rough pumping until 8 × 10−2 mbar, second one is an Edwards E09K diffusion pump can provide vacuum until 4 × 10−7 mbar, which is achieved using liquid nitrogen cooled trap. The pressure inside the bell chamber could be measured with two Pirani and Penning gauges to measure and monitor the pressure in the chamber during the process cycle. The substrate rotation speed is adjusted up to 60 rpm; it was fixed in our experiment at 25 rpm. The temperature of the substrate during deposition is 200 °C.

Figure 8 shows the schematic drawing of the setup for making conic lenses. Substrate sub is a spherical surface and is placed 600 mm above electron-beam source E. The diameter of the substrate is 50 mm, and its curvature radius is 57.28 mm. The n for cosn θ is 1. M is the mask, which is made according to Eq. (10) and is set 20 mm below the substrate holder. The substrate is rotated during evaporation.
Fig. 8

Schematic drawing of the coater for making aspherical lenses. M is the mask, sub is the substrate, W is the window, source is the source of the evaporated material

Film thicknesses are controlled by a quartz crystal monitor.

After deposition, the aberrations of the lens were measured by double pass auto-collimation method (DPAC) [11] using Fizeau interferometer λ = 632.8 nm and the generated interference pattern shown in Fig. 9.
Fig. 9

Measuring the aspheric surface by DPAC using Fizeau interferometer

The sticking of the coated film is tested by adhesive tape, which is pressed firmly against the film and quickly removed, there is no coating removed from the surface. The coating was tested for its environmental durability: for humidity, the coated surface was exposure to an atmosphere of 50 °C and 95% relative humidity; there is no peeling in the surface, the coated surface was durable for acidic solutions, but it failed when it was exposure to the salty fog for 24 h. The test was made according to the MIL-C-675C. The coating was tested also for thermal durability, the part was put in an atmosphere of −62 and 71 °C for 2 h at each temperature (the rate of temperature change ≤2.2 °C), the film was not affected and there is no peeling or cracking, this test was done according to the MIL-C-48497A.

Results and discussions

The coating has been tested for its stability and environmental durability. The SiO2 film of 6 µm thickness was stable and not bloom, and there is no peeling or cracking.

Film of 6 µm thickness could achieve with this technique whose apparatus is available. The film of this thickness was stable, may for thicker film there is no stability. Other research will be done for achieving films thicker than 6 µm, we expected that 12 µm could be achieved with the present technique.

Less than one-eighth wavelength spherical surface was used as a starting substrate whose aberration (or deviation from a perfect spherical surface) can affect the design of coating mask. The substrate holder prevented a 2-mm-wide ring around the edge of the substrate from being coated. The film thickness at the center of the substrate was 5.816 µm. The deposition rate dioxide silicon is 0.55 nm/s (1.98 µm/h) so the coating time was about 3 h.

Figure 10a shows the interferogram given by simulation with Zemax program using the DPAC method for the aspherical surface of the lens.
Fig. 10

Interferograms for aspherical surface referred to aspherical waves by DPAC method. a Interferogram of the aspherical surface referred to aspherical wave from simulation. b Interferogram of the aspherical surface referred to aspherical wave from measuring

Figure 10b shows the interferogram of the coated surface tested by the DPAC method indicated in Fig. 9. This value of rms aberration is about 0.03λ, P–V is about 0.20λ. Compared with traditional grinding and polishing methods for making aspherics, coating with a mask is much simpler and faster. We did it without much trial and error, and the coatings are easily reproducible, provided that the spot of the electron beam is kept in good shape.

Figure 11 shows 3D plot of the phase map of the aspheric surface coated and measured by Fizeau interferometer; which is mentioned in the Fig. 10b.
Fig. 11

3D plot of phase map of aspheric surface mentioned in the Fig. 10b

Depositing a thinner film for the aspherical mirror is also possible. In our example of S0 = 50 mm, we want an aspherical surface mirror with a vertex radius of R = 137.483 mm and conic constant of k = −0.071547; then a starting spherical surface with radius R′ = 137.8189 mm must be used, and the largest thickness of film is 5.9 μm. The mask shape designed according to Eqs. (5), (10), and (12) for this case is shown in Fig. 12. Assume that R is the vertex radius of a desired aspheric mirror (the dashed curve as shown in Fig. 13). Its sagitta is \(Z = \left( {\frac{{S^{2} }}{2R}} \right) + \left( {\frac{{S^{4} }}{{8R^{3} }}} \right)\left( {k + 1} \right)\). For a spherical surface with radius R′ different from R, its sagitta is \(Z^{{\prime }} = R^{{\prime }} - \sqrt {R^{{{\prime }2}} - S^{2} }\). In order to make sure that the substrate sphere and the aspheric coincide at the edge,
$$R^{{\prime }} - \sqrt {R^{{{\prime }2}} - S_{0}^{2} } = \left( {\frac{{S_{0}^{2} }}{2R}} \right) + \left( {\frac{{S_{0}^{4} }}{{8R^{3} }}} \right)\left( {k + 1} \right).$$
Fig. 12

Mask for making the aspheric mirror

Fig. 13

On the left, aspheric mirror with radius R and a sphere with radius R′, on the right, departure of the aspheric lens from the spherical surface used as substrate

Assume that the sagitta of the aspheric surface at S0 is Z0, so, The relationship between R and R′ is
$$R^{{\prime }} = \frac{{S_{0}^{2} }}{4R}\left( {1 + \frac{{S_{0}^{2} }}{{4R^{2} }}\left( {k + 1} \right)} \right) + \frac{R}{{1 + \frac{{S_{0}^{2} }}{{4R^{2} }}\left( {k + 1} \right)}}.$$
$$R^{{\prime }} = \frac{{Z_{0}^{2} + S_{0}^{2} }}{{2Z_{0} }}.$$
(11)
Therefore the radius of the spherical surface, R′, which we prepare for making the aspheric mirror, can be found from Eq. (11). Thus the film thickness that must be deposited is
$$D\left( S \right) = Z - Z^{{\prime }}$$
$$D\left( S \right) = \frac{{S^{2} }}{2}\left( {\frac{1}{R} - \frac{1}{{R^{{\prime }} }}} \right) + \frac{{S^{4} }}{8}\left( {\frac{k + 1}{{R^{3} }} - \frac{1}{{R^{{{\prime }3}} }}} \right).$$
(12)
By Differentiating D with respect to S, we then find that the largest film thickness will be at
$$S = \sqrt {\frac{{2\left( {\frac{1}{{R^{{\prime }} }} - \frac{1}{R}} \right)}}{{\frac{k + 1}{{R^{3} }} - \frac{1}{{R^{{{\prime }3}} }}}}} .$$

In our example of S 0 = 50 mm, we want an aspherical surface mirror with a vertex radius of R = 137.483 mm and conic constant of k = −0.071547; then a starting spherical surface with radius R′ = 137.8189 mm must be used, and the largest thickness of film is 5.9 µm. The mask shape designed according to Eqs. (5), (10), and (12) for this case is shown in Fig. 11.

An aluminum film of 5.9 µm thickness could be achieved with good stability, for aspherizing a mirror surface. The deposition rate of aluminum is 2 nm/s (7.2 µm/h). The aluminum film has been tested for environmental durability; it could pass the test successfully. The tests were performed according to the MIL-C-675C and MIL-C-48497A the same as the SiO2 film.

The mask can be different if the spherical substrate which be coated is different. For example, if we assume the conic surface coincide the spherical substrate at the apex, so the maximum deviation will be at the edge of the substrate as shown in Fig. 14. The parameters of the conic surface are R = 57.28, k = −0.074, Φ = 38 mm. The radius of the spherical substrate is R = 57.28. The previous conic surface is defined with next equation:
$$Z_{c} = \frac{{S^{2} }}{{57.28\left[ {1 + \left( {1 - 0.926\frac{{S^{2} }}{{\left( {57.28} \right)^{2} }}} \right)^{{\frac{1}{2}}} } \right]}}$$
Fig. 14

a is a spherical substrate. c is a conic surface coincide with spherical one at the vertex

Since R = R′, the film thickness distribution that must be deposited is
$$D\left( S \right) = k\frac{{S^{4} }}{{8R^{3} }}.$$
(13)
Figure 15 shows the departure of the aspherical surface of the lens from the spherical ones.
Fig. 15

On the left, departure of the aspheric surface of the lens from the best fit sphere; on the right, departure of the aspheric lens from spherical surface used as substrate

Figure 16 shows the distribution of the film thickness on the convex spherical substrate.
Fig. 16

Thickness distribution t/t 0 on a convex spherical substrate with curvature radius R = 57.28 and diameter = 38 mm. The parameter of the evaporator are radius of the dome R = 330 mm for a cos n Φ evaporation source with n = 1 at r = 170 mm and the substrate height from evaporation source h0 = 600 mm

The mask shape designed according to Eqs. (5), (10), and (13) for aspherizing this convex spherical surface is shown in Fig. 17.
Fig. 17

Mask for making the conic lenses in Fig. 14

Conclusions

An aspherical lens can be made by a thin-film coating technique with a specially designed mask. A fused quartz and BK7 optical glasses are coated with silicon dioxide with about 6-µm thickness film. An aspherical lens with 0.03λ rms aberration of the wavelength (λ = 632.8 nm) has been achieved without much trial and error. In addition, an aspherical mirror can be made by coating aluminum thin film with thickness about 6 µm. The SiO2 and aluminum films were stable with good sticking; they could pass durability environmental without peeling or cracking. But they failed when exposure the salty fog test. The method described in this paper is better preferred over conventional grinding and polishing methods for aspheric fabrication wherein the aspheric departure is of the order 6 µm. This method needs only some calculations for designing the appropriate mask and knowing the specification of the used coater. The calculations and designing could be performed by computer. It is expected that a film of 12 µm thickness could be achieved with this technique. This will be my future work. The effect of the part rotation with the mask need some studies could be performed in the future.

References

  1. 1.
    O. Podzimek, L.H. Beckmann, J.A. Brok, Fabrication of aspheric germanium-lens surfaces by ion milling, in Proceedings of SPIE 0656, Contemporary Optical Instrument Design, Fabrication, and Testing, ed. by H.J.F. Leo Beckmann, J. David Briers, P.R. Yoder, vol 656. doi: 10.1117/12.938469
  2. 2.
    J. Strong, E. Gaviola, On the figuring and correcting of mirrors by controlled deposition of aluminum. J. Opt. Soc. Am. 26(4), 153–162 (1936)ADSCrossRefGoogle Scholar
  3. 3.
    L.G. Schulz, Making Fresnel off-axis parabolic mirrors by the evaporation technique. J. Opt. Soc. Am. 37(5), 349–354 (1947)ADSCrossRefGoogle Scholar
  4. 4.
    L.G. Schulz, Preparation of aspherical refracting optical surfaces by an evaporation technique. J. Opt. Soc. Am. 38(5), 432–441 (1948)ADSCrossRefGoogle Scholar
  5. 5.
    J.A. Dobrowolski, W. Weinstein, Optical aspherizing by vacuum evaporation. Nat. (Lond.) 175(4458), 646–647 (1955)ADSCrossRefGoogle Scholar
  6. 6.
    W. Weinstein, J.A. Dobrowolski, Production of aspheric surfaces by vacuum deposition. Astronomical Optics and Related Subjects, in Proceedings of a symposium held 19-22 April, 1955 at the University of Manchester, ed. by Z. Kopal (North Holland Publishing Company, Amsterdam, 1956), p. 360Google Scholar
  7. 7.
    J.R. Kurdock, R.R. Austin, Correction of optical elements by the addition of evaporated films. Phys. Thin Films 10, 261–308 (1978)Google Scholar
  8. 8.
    H.A. Macleod, Thin Film Optical Filters, 4th edn. (Macmillan, Tucson, 2010), pp. 595–602Google Scholar
  9. 9.
    C.-C. Jaing, Designs of masks in thickness uniformity, in 5th International Symposium on Advanced Optical Manufacturing and Testing Technologies: Advanced Optical Manufacturing Technologies, p. 76551QGoogle Scholar
  10. 10.
    C.-C. Lee, D. Wan, C. Jaing, C. Chu, Making aspherical mirrors by thin-film deposition. J. Opt. Soc. Am. 32(28), 5535–5540 (1993)Google Scholar
  11. 11.
    R. ter Horst, R. Stuik, Manufacturing and Testing of a Convex Aspheric Mirror for ASSIST, in Proceedings of SPIE 8450, Modern Technologies in Space- and Ground-based Telescopes and Instrumentation II, p. 84504XGoogle Scholar

Copyright information

© The Optical Society of India 2017

Authors and Affiliations

  1. 1.Higher Institute for Applied Sciences and TechnologyDamascusSyria

Personalised recommendations