Malus Law Calculator

What This Calculator Does

The Malus Law Calculator helps you quickly determine the intensity of polarized light transmitted through a polarizing filter at a specified angle. By inputting the initial light intensity and the angle between the light’s polarization direction and the filter’s axis, you can instantly find the transmitted intensity, transmission coefficient, and percentage transmitted. This tool is ideal for anyone looking to understand or apply Malus’s Law in practical optics, physics studies, laboratory work, or general curiosity about polarized light.

How to Use This Calculator

1. Enter the Initial Intensity (I₀): Type in the intensity of the incoming polarized light before it passes through the polarizing filter. This value is typically measured in units such as watts per square meter (W/m²) or lumens, depending on your context.
2. Specify the Angle (θ): Input the angle in degrees between the polarization direction of the incident light and the axis of the polarizing filter. This angle should be a value between 0° and 90°.
3. Review the Results: The calculator will automatically display the transmitted intensity (I), the transmission coefficient (how much light passes through relative to the initial intensity), and the percentage of the original intensity that is transmitted.
4. Interpret the Outputs: Use the output values to analyze how much of the original light passes through the filter and how the angle affects transmission.
5. Adjust Inputs as Needed: Experiment with different initial intensities and angles to explore how polarization affects light transmission in various scenarios.

Definitions of Key Terms

Initial Intensity (I₀)

The power per unit area of the polarized light before it passes through the polarizing filter. It represents the starting brightness or energy of the light beam and is usually measured in watts per square meter (W/m²).

Angle (θ)

The angle formed between the direction of polarization of the incoming light and the axis of the polarizing filter. This angle directly influences how much light is transmitted according to Malus’s Law.

Transmitted Intensity (I)

The intensity of light that successfully passes through the polarizing filter. It is calculated using Malus’s Law and is always less than or equal to the initial intensity.

Transmission Coefficient

The ratio of transmitted intensity to the initial intensity (I/I₀). This value, always between 0 and 1, indicates the fraction of the original light that passes through the filter at the given angle.

Percentage Transmitted

This is the transmission coefficient expressed as a percentage. It tells you, in percentage terms, how much of the original light’s intensity remains after passing through the polarizing filter.

Calculation Methodology

The Malus Law Calculator uses the well-established Malus’s Law from optics to determine how much of the initial polarized light passes through a polarizing filter at a particular angle. The core formula is as follows:

Transmitted Intensity (I) = Initial Intensity (I₀) × cos²(θ)

Transmission Coefficient = I / I₀

Percentage Transmitted = (I / I₀) × 100

Where:
I₀ = Initial intensity of incident polarized light
θ = Angle (in degrees) between the light's polarization direction and the filter's axis
I = Intensity of light transmitted through the filter

The fundamental principle is that as the angle between the light’s polarization and the filter increases, the transmitted intensity decreases according to the square of the cosine of the angle. This relationship allows you to predict and understand how much light will pass through polarized filters in various settings.

Practical Scenarios

• Photography and Polarizing Filters: A photographer wants to understand how rotating a polarizing filter on the camera lens at different angles affects the brightness of the captured image, especially when shooting through reflective surfaces or dealing with glare.
• Physics Lab Experiments: A student in an optics laboratory is measuring the intensity of polarized laser light after it passes through a rotating polarizer to verify Malus’s Law experimentally.
• LCD Display Design: An engineer is assessing how much light is transmitted through stacked polarizing sheets in a liquid crystal display (LCD) at various angles to optimize display brightness.
• Quality Control in Sunglasses Manufacturing: A manufacturer uses the calculator to check the effectiveness of polarized sunglasses by measuring the reduction in light intensity at different filter orientations.

Advanced Tips & Best Practices

• Double-Check Angle Measurements: Always ensure the angle you enter is between the polarization direction of the incoming light and the transmission axis of the filter. Incorrect angles can lead to inaccurate results.
• Use Consistent Units: Keep the units of initial intensity consistent throughout your calculations. The calculator assumes I₀ and I are in the same units, so mixing units like W/m² and lumens may provide misleading results.
• Explore Edge Cases: Try entering angles of 0° and 90° to see the extremes: at 0°, all light passes through; at 90°, no light passes through. This helps build intuition for how Malus’s Law operates.
• Apply in Multi-Filter Systems: For setups with more than one polarizing filter, apply Malus’s Law successively to each filter with the appropriate angle to model compound effects.
• Verify with Experimental Data: If you are using the calculator for lab work or product testing, compare the calculated values with actual measurements to spot calibration issues or experimental errors.

Frequently Asked Questions (Optional)

Does the calculator work with unpolarized light?

Malus’s Law applies strictly to completely polarized light. For unpolarized light, only half of the intensity is transmitted through a single polarizer, regardless of the angle. For such cases, apply an initial reduction factor of 0.5 before using the calculator for subsequent polarizers.

What happens if I enter an angle outside 0° to 90°?

The calculator expects angles between 0° and 90°, as these represent the physical range for polarization analysis. Angles outside this range may produce mathematically correct but physically irrelevant results, as the cosine squared function repeats every 180°.

Can I use the calculator for multiple filters?

Yes. For multiple polarizing filters, apply the calculator sequentially. Use the transmitted intensity from the first filter as the initial intensity for the next, and set the angle for each filter relative to the previous filter’s axis.