> For the complete documentation index, see [llms.txt](https://simplemetalens.gitbook.io/docs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://simplemetalens.gitbook.io/docs/smd-pro/tutorial/lens-characterization/relative-illumination.md).

# Relative Illumination

## Definition

Relative illumination (RI) is a normalized irradiance of fields at the image plane. RI is influenced by vignetting, roll-off, and aberrations. For an object at infinity, RI can be obtained from the irradiance at the focal plane. For such objects, let us normalize the irradiance of off-axis fields to the irradiance of the on-axis field at the focal plane:

$$
\begin{equation}        RI(\theta) = { dI\_{f,\theta} \over dI\_{f,o}} = {I\_{i,\theta} \cdot \eta\_\theta  \over  I\_{i,o} \cdot \eta\_o} = { (I\_{i,o}  \cdot \kappa\_s) \cdot \eta\_\theta \over I\_{i,o} \cdot \eta\_o }  = {\kappa\_s \cdot \eta\_\theta \over \eta\_o}               \end{equation}
$$

$$\theta$$:  field angle on the object side (*θ* = 0 for the on-axis field*)* 

$$dI\_{f,\theta}$$: irradiance element of the field at *θ* at focal plane

$$dI\_{f,o}$$: irradiance element of the on-axis field at focal plane

$$I\_{i,\theta}$$: total irradiance of the field at *θ* at entrance pupil

$$I\_{i,o}$$: total irradiance of the on-axis field at entrance pupil

$$\kappa\_s$$:  a constant given by  $$I\_{i,\theta} / I\_{i,o}$$. 

<p align="center">For an <a data-footnote-ref href="#user-content-fn-1">isotropic source</a> <span class="math">\kappa_s = cos^3(\theta)</span> </p>

<p align="center">For a Lambertian source <span class="math">\kappa_s = cos^4(\theta)</span></p>

$$\eta\_\theta$$: a constant given by $$dI\_{f,\theta} / I\_{i,\theta}$$

$$\eta\_o$$: a constant given by $$dI\_{f,o} / I\_{i,o}$$

## How to Compute Relative Illumination (RI)

{% stepper %}
{% step %}

### Set field angle and run simulation (SIMULATE - FOCUSING)

Use the entry field 'Incidence Angle' to set the field angle. Under Optional Settings, if needed, adjust the values of 'I<sub>xy</sub> xy-range' and/or 'Distortion' to properly locate the center of the focal spot in the simulation domain of I<sub>xy.</sub>&#x20;
{% endstep %}

{% step %}

### Evaluate focusing efficiency (ANALYZE - FOCUSING)

Under 'Focal Area' in the toolbar at the bottom of the window, select the shape of the focal area to be used. Then, select 'Custom' in the option menu and enter a value in the entry field to define the size of your focal area (See [Focusing Efficiency](/docs/smd-pro/topical-discussions/focusing-efficiency.md) for details about how focusing efficiency is calculated). <mark style="color:$warning;">Use the same focal area shape and size for all field angles</mark>. Make sure the focal area is within the simulated area of I<sub>xy</sub>; otherwise, go back to SIMULATE tap and adjust the values of I<sub>xy</sub> xy-range and/or Distortion and simulate again. From the results of I<sub>xy</sub>, identify the y-axis position of the center of the focal spot to compute focusing efficiency.
{% endstep %}

{% step %}

### Write down results

Write down focusing efficiency and field angle. These values will be needed to compute RI based on Eq. (1) above.
{% endstep %}

{% step %}

### Change field angle and repeat Steps 1 - 3

{% endstep %}

{% step %}

### Compute RI

Use Eq. (1) to compute RI for each field angle.
{% endstep %}
{% endstepper %}

## RI Computation Example

The lens structure from [1. Build a Metalens](/docs/smd-pro/tutorial/basic-workflow/1.-build-a-metalens.md) is used for this RI computation example. Below are shown parameters used at two field angles (*θ):* 0 and 30°. A circular lens aperture with a diameter of 50 µm was used for all fields. To represent a pixel of a camera sensor, a square focal area is used in focusing efficiency calculation. The side length of the square focal area is chosen to be 8 µm to represent a few pixels assuming a typical pixel size of \~4 µm.&#x20;

<details>

<summary>1. On-axis field (<em>θ</em> = 0)</summary>

<div align="center"><figure><img src="/files/7wAgWbL0PiAHUl9d6nIK" alt=""><figcaption><p>Focusing simulation parameters (<em>θ</em> = 0).</p></figcaption></figure></div>

<figure><img src="/files/1Q220VCnY3PhtyNkxHom" alt=""><figcaption><p>I<sub>xy</sub> and focusing efficiency calculation (<em>θ</em> = 0).</p></figcaption></figure>

</details>

<details>

<summary>2. Off-axis field (<em>θ</em> = 30°)</summary>

<figure><img src="/files/2ujMQWHbObmMBY02tA5t" alt=""><figcaption><p>Focusing simulation parameters (<em>θ</em> = 30°).</p></figcaption></figure>

<figure><img src="/files/zuBHXktrxF8H5QUcza4t" alt=""><figcaption><p>I<sub>xy</sub> and focusing efficiency calculation (<em>θ</em> = 30°).</p></figcaption></figure>

</details>

The table below shows parameters used for the RI computation for two different sources at six field angles. The RI results are also plotted below.

<table><thead><tr><th>Field angle (θ) (°))</th><th>Distortion (%)</th><th>y-position of focal spot center (µm)</th><th>Focusing efficiency (η)</th><th width="121.5999755859375">RI (isotropic source)</th><th>RI (Lambertian source)</th></tr></thead><tbody><tr><td>0</td><td>0</td><td>0</td><td>0.724</td><td>1</td><td>1</td></tr><tr><td>10</td><td>0</td><td>9.34</td><td>0.718</td><td>0.947</td><td>0.933</td></tr><tr><td>20</td><td>0</td><td>18.2</td><td>0.585</td><td>0.670</td><td>0.630</td></tr><tr><td>30</td><td>-8</td><td>26.7</td><td>0.421</td><td>0.378</td><td>0.327</td></tr><tr><td>40</td><td>-18</td><td>34.9</td><td>0.303</td><td>0.188</td><td>0.144</td></tr><tr><td>50</td><td>-29</td><td>42.8</td><td>0.225</td><td>0.083</td><td>0.053</td></tr><tr><td>60</td><td>-43</td><td>50</td><td>0.166</td><td>0.029</td><td>0.014</td></tr></tbody></table>

<figure><img src="/files/okPRGUC5JsFWsDvykDRX" alt="" width="563"><figcaption></figcaption></figure>

[^1]: An isotropic source emits the equal amount of radiation at all angles: the radiant flux (or radiated power) per unit solid angle is constant at all viewing angles.


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