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# About

About

Merit functions are criteria used to optimize the optical system currently under study. During an optimization sequence, the algorithm tends to modify the system variables defined in the "variable" table so that the value of each merit function tends towards a setpoint value specified by the user.

Each merit function "f_{i}" is defined by its setpoint value "f_{0i}" (**value** columns) and its weight "w_{i}" (**weight** columns). The global merit function is given by the formula:

"Φ = ∑ w_{i}^{2}×(f_{i}-f_{0i})^{2}".

During an optimization cycle, the variables (defined in the "variables" table) are modified to minimize the global merit function "Φ".

merit functions are constantly evolving. Those currently proposed are grouped in three categories as follows:

-**Spot diagram** (merit functions related to the spot size on the observation surface and for a given field),

-**Paraxial** (merit functions related to paraxial parameters),

-**Chief ray** (merit functions related to the chief ray position/orientation after a given surface and for a given field).

Only the merit functions selected in the **active** check box column are taken into account during an optimization sequence.

**Spot diagram**

Each "spot diagram-fieldX" function corresponds to the RMS value of the spot diagram diameter on the observation surface for the field number **X** from which the rays are launched (the number **X** specifying the field is the same as that defined in the table "fields" of the entry page "aperture and fields").

"spot diagram-fieldX" is specified by its setpoint value (generally taken equal to "0" because an optimization is most often performed to minimize the size of a spot diagram), the wavelength(s) to be taken into account (**wave** column), as well as the weight (**weight** column). The number to be selected and specifying a wavelength is the same as that defined in the "input/wavelengths" page.

For each "spot diagram-fieldX" function, only numbers corresponding to active wavelengths are listed in the **wave** column. If "all" is chosen, all active wavelengths are considered for the related spot size calculation during optimization. Note that active wavelengths can be changed in the "inputs/wavelengths" page.

Vignetting is automatically taken into account in the calculation of a "spot diagram-fieldX" function. Indeed, the merit function can be significantly decreased if some rays are blocked when variables are modified during an optimization sequence. This precaution minimizes the risk of false optimization due to an increasing number of vignetted rays that may unadequately decrease the spot size in the observation surface.

**Paraxial**

**Paraxial** groups two merit functions : "magnification" and "efl". Both are defined at the active wavelength corresponding to the smallest number of the active wavelengths specified in the "input/wavelengths" page.

"Magnification" corresponds to the transverse magnification of the system. It is related to sum of the difference between the magnification of the system calculated in two ways (starting from the object and considering the observation surface as conjugated with the object one) and the targetted value. Therefore, it allows to obtain the conjugation between the object surface and the observation surface with the targetted magnification provided that distance from the object to the first surface as well as the distance from the last surface to the observation surface are both variables.

"efl" corresponds to the effective focal length of the system.

Both are defined at the active wavelength corresponding to the smallest number of the active wavelengths specified in the "input/wavelengths" page.

**Single ray**

Each "ray angle - field X" function corresponds to the angle of a single ray after surface number **surf.** with the "Z" axis of surface number **ref. surf**. The ray is launched from field number **X** with the normalized pupil coordinates **Pnx** and **Pny** (respectively along the "X" and "Y" axis of the aperture) at the active wavelength number **wave**.

Each "ray shift - fieldX" function is related to the impact of a ray on surface number **surf.**. The ray is launched from field number **X** with the normalized pupil coordinates **Pnx** and **Pny** (respectively along the "X" and "Y" axis of the aperture) and at the active wavelength number **wave**. The function gives the distance from the projection of the aforementioned impact on the "XY" plane of surface number **ref. surf** to its aperture center (if any). This aperture should have been previously defined by its "X" and "Y" coordinates (repectively **X dec** and **Y dec**). If no aperture is defined, the function gives the distance to the coordinate center of surface number **ref. surf** .

**opt. path diff.**

Each "OPD - field X" function corresponds to the optical path difference between two rays at different wavelengths (**ref. wave** and **wave**) launched from field number **X** and transmitted up to the observation surface. Each ray has the same normalized pupil coordinates **Pnx** and **Pny** (respectively along the "X" and "Y" axis of the aperture).