An intuitionistic method is proposed to design shadow masks to achieve thickness profile control for evaporation coating processes. The proposed method is based on the concept of the shadow matrix, which is a matrix that contains coefficients that build quantitive relations between shape parameters of masks and shadow quantities of substrate directly. By using the shadow matrix, shape parameters of shadow masks could be derived simply by solving a matrix equation. Verification experiments were performed on a special case where coating materials have different condensation characteristics. By using the designed mask pair with complementary shapes, thickness uniformities of better than 98% are demonstrated for MgF_{2} (_{3} (

Thickness uniformity control of optical coatings is of great importance for precision optical applications [

We develop an approach to solve the two problems. By appropriate partitioning of the substrate and the mask, a shadow matrix could be introduced. The shadow matrix contains coefficients that build quantitive relations between shape parameters of mask partitions and shadow quantities of substrate partitions one by one. By using this method, a mask pair with complementary shapes could be designed for specified surface figures to get the control of uniformities for two materials with different condensation parameters. Verification experiments of the proposed method were made on a convex spherical substrate with the radius curvature of 200 mm by using the planetary rotation fixture. Thickness uniformities of better than 98% were achieved for both MgF_{2} (_{3} (

The design of shadow masks begins with the deposition rate that is expressed as [

The most challenging problem on the design of shadow masks is how to establish quantitative relation between portions shadowed on each position of the substrate and shape parameters of the mask. Here we propose a shadow matrix to solve the problem. The shadow matrix is a matrix that corresponds to a given partition arrangement that divides the substrate and the mask into several partitions, respectively, as illustrated by the example in Figure

Partition arrangement of the mask and the substrate.

In this partition arrangement, the substrate could be divided into

According the definitions above, the shape of the shadow mask could be derived easily by the following routine: first, calculate the normalized thickness distribution vector (

Then the mask shape parameter vector (

In case two materials (high and low refractive index materials) used in the coating process have different

Therefore, shape parameters could be derived easily by solving equation set (

The shadow mask design and verification procedure were performed by a coating plant (SYRUSpro 1110 DUV by Leybold Optics). The 1100 mm diameter evaporation chamber contains a planetary rotation fixture carrying four substrate holders, and the distance between each spin rotation axis and the revolution axis is 300 mm. MgF_{2} and LaF_{3} coatings were deposited by molybdenum boats onto 25 mm diameter DUV grade fused silica substrates. Substrates were distributed along the axis on a convex spherical jig with the radius curvature of 200 mm. To determine thickness distributions, single layer MgF_{2} and LaF_{3} coatings were deposited onto substrates. The vertical distance between evaporation sources and the substrate holder is 700 mm. Planes that contain shadow masks are 10 mm below the lowest point of the spherical jig. Transmissions of coatings were measured with spectrophotometer (Lambda 1050 by Perkin Elmer) and layer thickness was derived by fitting transmission spectrum with the commercial thin film software Optilayer.

We wrote a computation code to calculate coating thickness distributions and shadow matrix in (

Determination of emission parameters and condensation parameters is the perquisition to the design of shadow masks. These parameters are acquired by fitting thickness distributions without shadow masks, as shown in Figure

Measured and fitted results of normalized thickness distributions for MgF_{2} and LaF_{3} coatings.

In our experiment, values of emission parameters and condensation parameters were determined to be _{2} and _{3} by fitting thickness distributions without shadow masks.

By using these parameters, shadow matrix and vectors needed to derive shape parameters in (

Designed and measured uniformity distributions of MgF_{2} and LaF_{3} single layers, together with shadow masks shown in the inserted figure.

By using the mask pair with complementary shapes, thickness distributions of both MgF_{2} and LaF_{3} were well tuned simultaneously. Measured thickness uniformities were better than 98%, which were in good agreement with the designed ones. To verify optical performances under this thickness control, additional test was performed by designing and depositing antireflection (AR) coatings. Transmission spectrum of both side AR coated samples are illustrated in Figure

Transmissions of both side AR coated samples on a convex surface, measured at positions shown in the inset.

The central wavelength of AR coatings for transmission spectrum is in good agreement with the designed wavelength of 193 nm, and transmission difference measured at 193 nm of all samples is less than 0.1%. These results show that uniform coating thickness yields uniform optical performances. Meanwhile, there are still some differences among these samples, when comparing their overall transmission spectrum on the wavelength region far from the design wavelength. This is due to their refractive index difference caused by the difference of incidence angle range of coating particles along the substrate axis [

We have proposed an approach method to design shadow masks based on the concept of the shadow matrix. By solving an equation set that contains four shadow matrix, thickness distributions adjustment for two coating materials with different condensation parameters could be achieved simultaneously. By using this method, the design of shadow masks becomes a rather simple and straightforward routine. We have applied this method to a convex spherical substrate with the diameter of 280 mm and the radius curvature of 200 mm with the coating material combination of MgF_{2} (_{3} (