Datasheet Parameters

The key parameters used in the following calculations are taken from the LED’s official datasheet: CSL1501RW1

• Radiation Intensity: Maximum of 3.4 \(\frac{\text{mW}}{\text{sr}}\) at 30 mA
• Horizontal Emission Angle: 140°
• Vertical Emission Angle: 160°

The LED emits 3.4 milliwatts of optical power per steradian² when operated at a forward current of 30 mA. Since the LED is limited to 2.4 mA due to the board design, and the datasheet indicates that the radiation intensity decreases linearly, we can calculate the adjusted radiation intensity as follows:

\[ I_{\text{op}} = \frac{3.4 \ \frac{\text{mW}}{\text{sr}} \times 2.4 \ \text{mA}}{30 \ \text{mA}} = 0.273 \ \frac{\text{mW}}{\text{sr}} \]

² A steradian is a unit used to measure angles in three-dimensional space, similar to how a radian measures angles in a circle.

Solid Angle

In the context of light sources like LEDs, the solid angle describes the portion of space into which the light is emitted. A larger solid angle means the light is spread over a wider area, while a smaller solid angle indicates a more focused beam. For reference, the total solid angle surrounding a point in all directions (a full sphere) is 4π sr, which is approximately 12.57 sr.

Since LEDs do not emit light in a perfect cone in just one direction but rather in more of an elliptical pattern, the usual formula for calculating the solid angle based on a circular cone is not entirely accurate. Instead, it’s better to approximate the emission using an elliptical model that takes into account the different horizontal and vertical beam angles. This can be done using the formula:

\[ \Omega = 4 \arcsin \left( \sin \left( \frac{\theta_x}{2} \right) \cdot \sin \left( \frac{\theta_y}{2} \right) \right) \]

where $\theta_x$ and $\theta_y$ are the horizontal and vertical beam angles, respectively, given in radians.

It’s important to note, however, that this formula is still an approximation. In reality, the light distribution of LEDs is not perfectly rectangular as the formula might suggest, but rather elliptical or somewhat smoother in shape. The formula deliberately simplifies the geometry for ease of calculation, but despite this, it provides a sufficiently accurate estimation of the solid angle.

When the beam angles from the datasheet are applied in the formula, the resulting calculation yields a solid angle of approximately 4.73 steradians:

\[ \Omega = 4 \arcsin \left( \sin \left( \frac{160^\circ \cdot \pi}{360} \right) \cdot \sin \left( \frac{140^\circ \cdot \pi}{360} \right) \right) = 4.73 \,\text{sr} \]

Illuminated Area

To calculate the area illuminated by each individual LED, the Oosterom-Strackee formula can be used after a suitable transformation. This formula has already been applied in combination with the horizontal and vertical beam angles to estimate the solid angle (steradian) covered by the LED.

However, by rearranging the formula, it can also be used to calculate the projected area of the light cone at a given distance. Specifically, it allows us to estimate the size of the illuminated spot, which is formed by the horizontal and vertical spread of the LED beam.

\[ \text{Width} = 2 \cdot h \cdot \tan\left(\frac{\theta_x}{2}\right) \quad ; \quad \text{Height} = 2 \cdot h \cdot \tan\left(\frac{\theta_y}{2}\right) \]

\[ A = \text{Width} \cdot \text{Height} = 4 \cdot h^2 \cdot \tan\left(\frac{\theta_x}{2}\right) \cdot \tan\left(\frac{\theta_y}{2}\right) \]

where $\theta_x$ and $\theta_y$ are the horizontal and vertical beam angles, respectively, given in radians, and $h$ is the distance from the LED to the illuminated surface.

Using the beam angles from the datasheet and a distance of 1 cm in the formula, the calculated illuminated area is approximately 63.33 cm².

\[ A = 4 \cdot (1\,\text{cm})^2 \cdot \tan\left(\frac{140^\circ \cdot \pi}{360}\right) \cdot \tan\left(\frac{160^\circ \cdot \pi}{360}\right) = 63.33 \,\text{cm}^2 \]

Total Power

Since the datasheet only specifies the LED’s intensity in \(\frac{\text{mW}}{\text{sr}}\), meaning the power emitted per unit solid angle, but not the total emitted power, the actual power emitted by the LED must be calculated by multiplying this value by the LED’s total solid angle. This yields the total power in milliwatts emitted over the LED’s full emission angle, which is necessary to determine the LED’s power over a specific area (Irradiance) in the next step.

\[ P = I_{\text{op}} \times \Omega = 0.273\,\text{mW/sr} \times 4.73\,\text{sr} = 1.29\,\text{mW} \]

Irradiance

Irradiance is the measure of the radiant power received by a surface per unit area. It quantifies how much energy falls onto a given area. In simpler terms, it tells you how „intense“ the light is on a surface. To calculate it, the total radiant power incident on the surface is divided by the surface area.

Using the previously calculated values, the irradiance from 7 LEDs at a distance of 1 cm can be expressed as:

\[ E = \frac{7 \times P}{A} = \frac{7 \times 1.29\,\text{mW}}{63.33\,\text{cm}^2} = 0.14\,\text{mW/cm}^2 = 1.4\,\text{W/m}^2 \]

Radiance

The previously calculated values focused on irradiance, which relates the LED’s emitted power to the illuminated area on a surface at a given distance. While irradiance is important for evaluating the overall light intensity received by a surface, the EN 62471 standard—particularly regarding the Retinal Thermal Hazard Exposure Limit under weak visual stimulus—requires consideration of radiance as well.

Radiance describes the radiant power emitted by the source per unit projected emitting area and per unit solid angle. This makes it a key parameter for assessing potential retinal hazards, as it captures not only how much power is emitted, but also how concentrated the light is in specific directions. Unlike irradiance, which depends on the receiving surface, radiance is intrinsic to the source’s emission characteristics and better reflects the risk of thermal damage to the retina from focused or directional light.

Radiance can be calculated using the formal definition:

\[ L = \frac{d^2\Phi}{\cos(\beta) \cdot dA \cdot d\Omega} \quad \left[\frac{\text{W}}{\text{m}^2 \cdot \text{sr}}\right] \]

This formula describes how much radiant power \( d^2\Phi \) is emitted from a projected area element \( dA \) of the source into a given solid angle \( d\Omega \), with \( \beta \) being the angle between the surface normal and the emission direction. Importantly, the area \( dA \) refers to the source’s emitting surface—not the illuminated target—and the cosine term \( \cos(\beta) \) accounts for the projection of the surface in the emission direction.

To simplify the calculation, we assume \( \beta = 0^\circ \), meaning the observation direction is perpendicular to the emitting surface. This represents the worst-case scenario, as the radiance is maximized when the viewing angle aligns directly with the surface normal, resulting in \( \cos(\beta) = 1 \):

\[ L = \frac{d^2 \Phi}{dA_{\mathrm{led}} \cdot d\Omega} = \frac{P}{A_{\mathrm{led}} \cdot \Omega} = \frac{1{,}29\,\mathrm{mW}}{0{,}5\,\mathrm{mm}^2 \cdot 4{,}73\,\mathrm{sr}} \approx 545 \, \frac{\mathrm{W}}{\mathrm{m}^2 \cdot \mathrm{sr}} \]

Radiance Limit set by EN 62471

The radiance limit set by EN 62471 is not a fixed value; instead, it depends on the angular subtense of the light source, measured in radians. The angular subtense describes how large the light source appears to the observer — basically, the angle subtended by the source at the eye.

To calculate the limit allowed by EN 62471 for exposures longer than 10 seconds, you simply divide 6000 by the angular subtense $a$ (in radians):

\[ L_{\text{limit}} = \frac{6000}{a} \quad \left[\frac{\text{W}}{\text{m}^2 \cdot \text{sr}}\right] \]

To calculate the angular subtense \( a \), both the physical size of the light source and its distance from the observer’s eye must be known. The angular subtense in radians is defined as the angle subtended by the source, based on its characteristic dimension \( d \) (e.g., diameter or side length) and the viewing distance \( r \). For the 0402 LEDs used here, where \( d = 1\,\text{mm} \) and \( r = 1\,\text{cm} \), the angular subtense is:

\[
a = \frac{d}{r} = \frac{1\,\text{mm}}{10\,\text{mm}} = 0.1\,\text{rad}
\]

This results in a limit for exposure durations longer than 10 seconds of:

\[
L_{\text{limit}} = \frac{6000}{a} = \frac{6000}{0.1} = 60{,}000\,\frac{\text{W}}{\text{m}^2 \cdot \text{sr}}
\]

Compliance Check

The calculated irradiance for 7 LEDs at a distance of 1 cm is approximately 1.4 \(\frac{\text{W}}{\text{m}^2}\) (0.14 \(\frac{\text{mW}}{\text{cm}^2}\)). This value is about 70 times lower than the safety irradiance limit of 100 \(\frac{\text{W}}{\text{m}^2}\) (10 \(\frac{\text{mW}}{\text{cm}^2}\)) for prolonged exposure, meaning, based on the calculations, it is well within safe limits.

I also observed similarly low irradiance values when measuring the LEDs.

The radiance of the LED was calculated to be about 545 \(\frac{\text{W}}{\text{m}^2 \cdot \text{sr}}\). According to the EN 62471 standard, the radiance limit for the given angular subtense of 0.1 radians is 60,000 \(\frac{\text{W}}{\text{m}^2 \cdot \text{sr}}\). This means the LED’s radiance is roughly 110 times lower than the safety threshold.

In conclusion, both irradiance and radiance are far below the EN 62471 safety limits.