Integral Targets

In OptiLayer, integral targets are represented as finite sums and can be considered as approximations of an integral expression with the help of rectangle rule.

As in conventional targets it is possible to specify

• Angle of incidence (Angle);
• Spectral characteristic (Char);
• Value of the integral (Target);
• Tolerances for target values (Tol);
• Qualifiers for the target values in the case of range targets  (Q).

Fig. 1. Specification of integral target. The integral target window consists of Targets and Integral Weights panels.

In the example (Fig. 1), transmittance (TA, no polarization) is integrated with three commonly used spectral weight functions. It is required that the integral values are to be more than 60%.

Fig. 2. Spectral distributions of the target weights can be chosen from a pop-up list in the Integral Weights tab.

There are three predefined commonly used spectral weights distributions:

Light T,R,A – normalized relative spectral distribution of illuminant D65 multiplied by the spectral luminous efficiency.

Solar direct T,R,A – normalized relative spectral distribution of global solar radiation.

UV T,R,A – normalized relative spectral distribution of the UV part of the global solar radiation.

Switching between different distributions on the left panel, you can observe different spectral weights distributions.

Up to 128 different spectral weight functions \(W(\lambda)\) can be defined and used in OptiLayer.

Fig. 3. Spectral distributions of the target weights are plotted in Preview panel of the Integral Weight tab.

• In any external software (for example, Excel, Origin) and imported to OptiLayer (Copy-Paste), Fig. 4
• Directly in OptiLayer: create a new spreadsheet (‘+’ button, Fig. 5), specify a number of spectral points and specify wavelength grid/weights via F6 tool.

Fig. 4. Example of user-defined integral weights.

Fig. 5. Adding “My integral” to the available integrals.

Default names are F1, F2, etc. Of course, you can rename the distributions for your convenience.

Specifying this unit function in Integral Target window (Fig. 1), you will get averaged values in the specified spectral range over the specified number of spectral points.

\[ MF_{int}^2=\frac 1M\sum\limits_{i=1}M\left[\frac{F^{(i)}-\hat{F}^{(i)}}{\Delta F^{(i)}}\right]^2, \]

where \(M\) is the number of different integrals in the resulting target (Fig. 1), \(F^{(i)}\) are target values,  \(\Delta F^{(i)}\) are corresponding tolerances (Fig. 1).

Fig. 6. Normalized and Source/Data check boxes are unchecked.

\[ F=\int\limits_{\lambda_d}^{\lambda_u} W(\lambda) C(\lambda)d\lambda,\]

where \(\lambda_d\) and \(\lambda_u\) are boundaries of the wavelength interval of interest, \(W(\lambda)\) is a given weight function, \(C(\lambda)\) is a spectral characteristic of the coating.

\[ F=\frac{\int\limits_{\lambda_d}^{\lambda_u} W(\lambda)C(\lambda)d\lambda}{\int\limits_{\lambda_d}^{\lambda_u} W(\lambda) d\lambda},\]

where \(\lambda_d\) and \(\lambda_u\) are boundaries of the wavelength interval of interest, \(W(\lambda)\) is a given weight function, \(C(\lambda)\) is a spectral characteristic of a coating.

Fig. 7. An integral characteristic will be normalized with respect to the target weights, source and detector are not taken into account.

Fig. 8. An integral characteristic will be not normalized with respect to the target weights, source and detector are taken into account.

\[ F=\int\limits_{\lambda_d}^{\lambda_u} W(\lambda)D(\lambda) S(\lambda) C(\lambda)d\lambda,\]

where \(\lambda_d\) and \(\lambda_u\) are boundaries of the wavelength interval of interest, \(W(\lambda)\) is a given weight function, \(C(\lambda)\) is a spectral characteristic of a coating, \(D(\lambda)\) and \(S(\lambda)\) are spectral distributions of the detector and light source, respectively.

\[ F=\frac{\int\limits_{\lambda_d}^{\lambda_u} W(\lambda)D(\lambda) S(\lambda) C(\lambda)d\lambda}{\int\limits_{\lambda_d}^{\lambda_u} W(\lambda) S(\lambda) D(\lambda)d\lambda},\]

where \(\lambda_d\) and \(\lambda_u\) are boundaries of the wavelength interval of interest, \(W(\lambda)\) is a given weight function, \(S(\lambda)\) is a spectral characteristic of a coating, \(D(\lambda)\) and \(S(\lambda)\) are spectral distributions of the detector and light source, respectively.

Fig. 9. An integral characteristic will be normalized with respect to the target weights, source and detector are taken into account.