Goal
Calculate transmitted intensity and the ideal transmission fraction for linearly polarized light passing through an analyzer at a stated angle.
Malus's Law Polarized Intensity
Calculate transmitted intensity and the ideal transmission fraction for linearly polarized light passing through an analyzer at a stated angle.
Key facts
- What it does
- Calculate transmitted intensity and the ideal transmission fraction for linearly polarized light passing through an analyzer at a stated angle.
- Formula
Malus's law is I = I0 cos(theta)^2, where theta is the angle in degrees between the incoming linear-polarization axis and the ideal analyzer axis; transmission fraction is cos(theta)^2.- You enter
- Incoming linearly polarized intensity · Polarizer-axis angle
- Worked example
- An incoming intensity of 100 W/m^2 at 30 degrees gives 75 W/m^2 transmitted and a 0.75 ideal transmission fraction.
A clearer path to an answer
From your question to a useful result
This page keeps the calculation transparent: define the goal, enter the matching values, inspect the method, and decide what the result means in your situation.
Inputs
Incoming linearly polarized intensity · Polarizer-axis angle
Method
Malus's law is I = I0 cos(theta)^2, where theta is the angle in degrees between the incoming linear-polarization axis and the ideal analyzer axis; transmission fraction is cos(theta)^2.
Next step
Calculate, review the assumptions below, then compare a related tool when the decision needs more context.
Malus's Law Polarized Intensity
Calculate transmitted intensity and the ideal transmission fraction for linearly polarized light passing through an analyzer at a stated angle.
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Result
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01
Inputs (2)
- Incoming linearly polarized intensity Ready
- Polarizer-axis angle Ready
02
Formula
Malus's law is I = I0 cos(theta)^2, where theta is the angle in degrees between the incoming linear-polarization axis and the ideal analyzer axis; transmission fraction is cos(theta)^2.
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Result
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Formula, assumptions, and example
Formula: Malus's law is I = I0 cos(theta)^2, where theta is the angle in degrees between the incoming linear-polarization axis and the ideal analyzer axis; transmission fraction is cos(theta)^2.
This calculator applies Malus's law to a nonnegative incoming linearly polarized intensity and an analyzer angle from 0 to 90 degrees. It returns transmitted intensity in W/m^2 and the dimensionless ideal transmission fraction, including the angular factor when the incoming intensity is zero.
- Incoming intensity is a finite nonnegative magnitude in watts per square metre and is already linearly polarized.
- The angle is a finite degree measure between the polarization and analyzer transmission axes, restricted to the first-quarter range 0 through 90 degrees.
- The analyzer is ideal, with no absorption beyond polarization selection, scattering, reflection, wavelength dependence, or detector response.
- Partial polarization, coherence details, beam geometry, optical alignment, laser classification, and safety advice are outside the calculation.
Worked example: An incoming intensity of 100 W/m^2 at 30 degrees gives 75 W/m^2 transmitted and a 0.75 ideal transmission fraction.
Displayed input contract
- Incoming linearly polarized intensity · minimum 0 · maximum 1000000000000
- Polarizer-axis angle · minimum 0 · maximum 90
The displayed limits are checked before the handler runs. Model-specific domain checks may also reject impossible or non-finite inputs.
Methodology: This calculator follows the WorldCalculate input, formula, precision, and boundary policy. Read the official methodology.
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Answer-first guide
How to use the Malus's Law Polarized Intensity for a real question
Calculate transmitted intensity and the ideal transmission fraction for linearly polarized light passing through an analyzer at a stated angle. Start with one clearly defined goal, enter values in the units shown, and keep the result attached to the assumptions below.
What this answers
This tool is useful when your question includes Malus law, polarization, analyzer angle. It returns the outputs declared in the calculator contract rather than a live quote, approval, diagnosis, or professional sign-off.
What you enter
Incoming linearly polarized intensity · Polarizer-axis angle. Keep the same time period, unit system, and currency wherever the form requires comparable values.
How to check it
Run the worked example first, compare its output with the page's example, then change one input at a time. This makes an unexpected result easier to trace to a unit, boundary, or assumption.
Three checks before you rely on the answer
- Match the question. Confirm that the result means the quantity you need, not a similar-sounding percentage, balance, rate, or estimate.
- Match the inputs. Use the requested units and period, and read each hint before replacing the example values with your own.
- Read the boundary. Review the assumptions and limits. Incoming intensity is a finite nonnegative magnitude in watts per square metre and is already linearly polarized.
Need a wider view? Browse Science Calculators or compare the related tools below. The WorldCalculate methodology explains how formulas, examples, limits, and revisions are reviewed.
How to use the Malus's Law Polarized Intensity
- Enter Incoming linearly polarized intensity — Finite nonnegative intensity incident on the ideal analyzer. (W/m^2).
- Enter Polarizer-axis angle — Finite angle from 0 through 90 degrees between transmission axes. (degrees).
- Choose Calculate and read the result panel.
- Use Download PDF or Download Word to save a result sheet.
Formula
Malus's law is I = I0 cos(theta)^2, where theta is the angle in degrees between the incoming linear-polarization axis and the ideal analyzer axis; transmission fraction is cos(theta)^2.
This calculator applies Malus's law to a nonnegative incoming linearly polarized intensity and an analyzer angle from 0 to 90 degrees. It returns transmitted intensity in W/m^2 and the dimensionless ideal transmission fraction, including the angular factor when the incoming intensity is zero.
Worked example
An incoming intensity of 100 W/m^2 at 30 degrees gives 75 W/m^2 transmitted and a 0.75 ideal transmission fraction.
Assumptions and limits
- Incoming intensity is a finite nonnegative magnitude in watts per square metre and is already linearly polarized.
- The angle is a finite degree measure between the polarization and analyzer transmission axes, restricted to the first-quarter range 0 through 90 degrees.
- The analyzer is ideal, with no absorption beyond polarization selection, scattering, reflection, wavelength dependence, or detector response.
- Partial polarization, coherence details, beam geometry, optical alignment, laser classification, and safety advice are outside the calculation.
Who uses this calculator?
- Optics students learning polarization intensity
- Science learners practicing degree-to-radian trigonometry
- Teachers demonstrating cosine-squared transmission
When is it useful?
- Calculate intensity after an ideal analyzer at a known relative angle.
- Find the ideal transmission fraction for a polarization setup.
- Compare parallel, intermediate, and crossed-axis limiting cases.
Context and background
The model-first approach to science
Science calculators define a system, choose an equation, apply units and constants, and show the substitution. Effects outside that model remain outside the result.
Introductory science problem solving builds from measured quantities and idealized relationships. Those models are valuable for learning and first-pass estimates, while experiments and engineering decisions need additional evidence.
Research and review
How this guide was researched
Researched by Hassan ALRowaie, Founder and editorial researcher at WorldCalculate.
This guide follows the live calculator's declared inputs, formula, worked example, assumptions, validation boundaries, and source-backed methodology. The review date describes editorial review of the calculator explanation; it is not a promise that external facts or rates remain current.
Malus's law describes how an ideal analyzer transmits linearly polarized light when its axis is at an angle to the incoming polarization direction. This calculator accepts a nonnegative incoming intensity in W/m^2 and an angle in degrees from 0 through 90. It converts the angle for the trigonometric calculation, evaluates I = I0 cos(theta)^2, and reports both transmitted intensity and the dimensionless transmission fraction. The fraction is the ideal angular factor even when the incoming intensity is zero. The page is an ideal polarization equation check, not an optical alignment diagnosis, detector model, laser classification, or safety instruction. The guide explains the exact fields, endpoints, example, validation, and limits.
The polarization question answered here
The calculator answers a focused optics question: how much of an incoming linearly polarized intensity remains after an ideal analyzer whose transmission axis is separated by a stated angle? The input intensity supplies the scale, and the angle supplies the cosine-squared factor. The handler returns the resulting intensity and the factor used to obtain it. It does not identify the source, measure the polarization state, inspect an optical bench, or infer whether the entered angle was measured from a particular drawing. The geometric convention must be established before entry.
The word ideal is part of the contract. A real polarizer or analyzer can absorb, reflect, scatter, or transmit imperfectly, and a source can be partially polarized. Those effects can make observed intensity differ from the value produced by the simple relation. The page deliberately reports the textbook result under declared conditions rather than adding a guessed efficiency. This makes the output useful for equation checks while preventing a clean number from being mistaken for a complete optical-system prediction.
- Inputs are incoming intensity and relative axis angle.
- Output includes intensity and an ideal fraction.
- The angle is the separation of transmission axes.
- Real optical losses are outside the relation.
What linearly polarized intensity means
Linear polarization describes an electric-field oscillation whose transverse direction is organized around one axis for the beam being considered. The initialIntensity field is the intensity arriving at the analyzer before the ideal selection step. It is entered as a nonnegative magnitude in watts per square metre. The calculator does not ask for a phase, wavelength, beam profile, polarization ellipse, or source power, so it cannot reconstruct the full electromagnetic field. The surrounding problem must establish that the incoming light meets the linear-polarization assumption.
Intensity is a power-per-area quantity, not the amplitude of an electric field. Malus's law is often derived by resolving the field amplitude along the analyzer axis and then squaring to obtain an intensity factor. Entering an amplitude in the intensity field would apply the cosine-squared relation to the wrong physical quantity. The label and unit are intended to prevent that substitution. If a source reports total optical power instead, an area definition is needed before a W/m^2 intensity can be entered.
- The input is intensity, not field amplitude.
- The unit is W/m^2.
- The incoming state is assumed linearly polarized.
- Wavelength and beam area are not inferred.
The analyzer angle
The angle field is the relative angle between the incoming polarization axis and the analyzer transmission axis. Zero degrees means the axes are parallel in the selected convention, while 90 degrees means they are crossed. Angles between those endpoints describe intermediate projections. The catalog restricts the input to 0 through 90 degrees because that interval covers the distinct magnitude behavior needed here. The handler does not accept a negative angle or silently reduce an angle outside the range modulo 180 degrees; doing so would hide the stated geometry.
The angle is entered in degrees for readability, but JavaScript's trigonometric functions use radians. The engine converts angle times pi divided by 180 before evaluating cosine. Exact endpoint branches return a factor of one at zero degrees and zero at 90 degrees, avoiding tiny floating-point residue at the crossed-axis boundary. This explicit conversion and endpoint handling are part of the numerical contract, not a claim that an optical setup can align axes perfectly.
- Zero degrees represents parallel axes.
- Ninety degrees represents crossed axes.
- The accepted range is inclusive from 0 to 90 degrees.
- Degree input is converted to radians internally.
Deriving the cosine-squared factor
The electric-field component along the analyzer axis is proportional to cos(theta). Intensity is proportional to the square of the field amplitude, so the transmitted intensity becomes I = I0 cos(theta)^2. Dividing by the incoming intensity gives the ideal transmission fraction cos(theta)^2 when the incoming intensity is positive. The handler calculates the factor directly, then multiplies by initialIntensity. This order keeps the zero-intensity case well-defined as a model factor instead of attempting an undefined 0 divided by 0 operation.
The fraction is dimensionless and lies between zero and one over the supported angle range. At an intermediate angle it reports the portion selected by the ideal projection. It does not include a separate polarizer efficiency because no such field is present. If a real device has an efficiency less than one, that factor belongs to a different, explicitly defined model and should not be hidden inside the incoming intensity or the angle.
- Field projection contributes cos(theta).
- Intensity produces the squared factor.
- I/I0 is cos(theta)^2 for positive I0.
- No unentered device-efficiency factor is applied.
The 100 W/m^2 and 30 degree example
The catalog example enters 100 W/m^2 and 30 degrees. The cosine of 30 degrees is about 0.8660254, and its square is 0.75. Multiplying the incoming intensity by that factor gives 75 W/m^2. The second result is 0.75, which makes the percentage interpretation easy: the ideal analyzer transmits 75 percent of the incoming intensity under this model. The handler calculates the same result using degree conversion rather than relying on a hard-coded table of trigonometric values.
This example is an arithmetic and geometry check. It does not claim that a physical analyzer has no absorption or that a beam meter will display exactly 75 W/m^2. A real report should state the source polarization, the definition of the relative angle, the measurement plane, and any known optical losses. The calculator intentionally leaves those details out so the core cosine-squared relation remains inspectable and reusable in a classroom problem.
- Incoming intensity is 100 W/m^2.
- Relative angle is 30 degrees.
- Ideal transmission fraction is 0.75.
- Ideal transmitted intensity is 75 W/m^2.
Parallel and crossed-axis endpoints
At zero degrees, cos(theta)^2 equals one, so the transmitted intensity equals the incoming intensity and the fraction is one. At 90 degrees, the factor is zero, so the ideal transmitted intensity is zero and the fraction is zero. These endpoints are useful implementation tests because they express the geometry without approximation. The handler makes them exact branches, which avoids a tiny residual caused by representing pi and the cosine of a right angle in floating-point arithmetic.
The crossed-axis zero is an ideal polarization result, not evidence that every real device blocks every photon. Leakage, imperfect extinction, scattering, detector background, and partially polarized input can all produce a nonzero observation. Similarly, a real parallel-axis arrangement can lose intensity before or inside the analyzer. The endpoint results should therefore be labeled ideal. The page reports the mathematical factor for the stated model and does not diagnose why an instrument differs.
- Zero degrees returns factor 1 and unchanged intensity.
- 90 degrees returns factor 0 and zero ideal intensity.
- Endpoint values are normalized exactly.
- Real leakage and loss are not modeled.
Zero incoming intensity
The intensity field permits zero. In that case the transmitted intensity is zero for every allowed angle because the multiplier I0 is zero. The transmission fraction remains the cosine-squared angular factor, since it describes what the ideal analyzer would transmit from a nonzero linearly polarized input with the same geometry. This distinction avoids a division-by-zero definition while still providing a useful result for the angle. The note states this convention so a zero input is not confused with an experimentally measured ratio.
A zero result does not reveal why no light arrived. It could represent a mathematical baseline, an obscured beam, an unavailable source, or a deliberately chosen test. The calculator does not inspect a detector or decide whether zero is a meaningful observation. If a measured ratio is needed, the incoming and outgoing readings and their uncertainties should be analyzed outside the form. Here the factor is a model output, and the intensity multiplication is the only energy-scale operation performed.
- Zero input intensity gives zero transmitted intensity.
- The fraction still reports the ideal angular factor.
- No 0/0 division is attempted.
- The page does not explain a real zero detector reading.
Catalog bounds and finite validation
The catalog accepts incoming intensity from 0 through 1,000,000,000,000 W/m^2 and angle from 0 through 90 degrees. The default of 100 W/m^2 and 30 degrees produces a finite nontrivial example. The intensity bounds prevent unbounded display scale, while the angle bounds define the first-quarter magnitude domain. They are conservative software limits rather than a statement about the strongest safe beam, the full range of laboratory instruments, or the maximum possible angle in a different polarization convention.
The pure handler repeats the type, finiteness, and range checks. It rejects numeric strings, missing values, NaN, either infinity, negative intensity, and angles outside the inclusive range. It finite-checks the transmission factor and the multiplied intensity. A value just beyond a bound is rejected instead of being folded into the range or clipped. This preserves the visitor's geometry and makes errors visible to a caller that bypasses the browser form.
- Intensity bounds are 0 to 1e12 W/m^2.
- Angle bounds are 0 to 90 degrees.
- All inputs and outputs must be finite.
- Invalid values are rejected rather than normalized into range.
Units and angle preparation
The intensity unit W/m^2 identifies a flux density. If a source gives total optical power, divide by a defined illuminated area before entering the value, and preserve that area assumption in the record. The angle is a pure geometric measure displayed in degrees. The cosine function itself uses radians, so the conversion is theta_rad = angle_deg pi/180. A unit label does not perform either conversion automatically. The handler receives numeric values that are already consistent with the catalog contract.
A useful dimensional check is that cos(theta)^2 has no unit, so multiplying it by W/m^2 leaves W/m^2. The fraction should never be reported in watts, and the intensity should not be described as a percentage. Conversely, multiplying the fraction by 100 is a presentation choice that this engine does not use for the result row. Keeping the rows separate helps a reader distinguish the physical intensity scale from the dimensionless polarization factor.
- Convert total power to intensity only with a stated area.
- Convert degrees to radians for trigonometry.
- The transmission factor is dimensionless.
- Intensity remains measured in W/m^2.
Ideal analyzer and polarization limits
Malus's law in this form assumes that the incident light is fully linearly polarized and that the analyzer is ideal. If the input is partially polarized, an unpolarized component can contribute a different baseline behavior. If the analyzer has finite extinction, absorption, reflection, or wavelength-dependent transmission, a device factor is needed. The calculator does not ask for a degree of polarization or an efficiency because those are not implied by I0 and theta. Adding them without a new contract would make the displayed formula appear more general than it is.
Coherence and phase can matter when multiple optical paths or additional elements are involved. Beam divergence, aperture clipping, detector response, and spatial variation can also affect a measured power density. The page does not model those effects. Its ideal result is still a valid algebraic reference when the surrounding problem has already established a fully linearly polarized beam and ideal analyzer. The note is meant to preserve that model boundary next to the number, not to diminish the usefulness of the relation.
- Full linear polarization is assumed.
- Analyzer loss and extinction are excluded.
- Partial polarization needs a different model.
- Multiple paths and detector behavior are not represented.
Comparing angles without overinterpreting
For a fixed incoming intensity, the transmitted result follows the cosine-squared curve. Small changes near zero degrees can leave the factor close to one, while changes near 90 degrees can strongly reduce it. Comparing 0, 30, 60, and 90 degrees is a simple way to see the progression from full ideal transmission to complete ideal rejection. The comparison is mathematical and assumes the same incoming polarization, analyzer, beam area, and measurement definition for every case.
Changing the incoming intensity scales the transmitted intensity linearly but does not change the transmission fraction. Changing the angle changes the fraction but does not change the input scale. This separation helps identify a common mistake in which a lower output is attributed to polarization when the source intensity also changed. The page cannot decide whether two measurements are comparable. It supplies the factor for one stated pair of inputs and leaves experimental normalization outside the calculator.
- At fixed I0, angle controls the factor.
- At fixed angle, intensity scales the output linearly.
- The fraction is independent of input intensity.
- Comparisons need matching beam and measurement definitions.
Optical safety and device decisions are excluded
A transmitted intensity number is not a laser hazard classification, exposure limit, eye-safety decision, or recommendation for viewing a beam. The calculator does not know wavelength, pulse duration, beam diameter, access time, enclosure, reflections, or the applicable safety standard. Even a low ideal output can be unsafe under another wavelength or exposure condition, and a high input can be attenuated by a real device in ways this model does not describe. Those questions require dedicated information and qualified review.
The page also does not select a polarizer, specify an optical coating, design an instrument, or certify an alignment. It provides a deterministic projection calculation. If the result is used in a real optical system, carry the source and analyzer specifications, losses, uncertainty, and safety controls into the larger analysis. Never infer operational permission from a finite mathematical output or from the fact that the angle lies within the catalog range.
- No wavelength or pulse exposure is modeled.
- No laser or optical safety decision is made.
- No analyzer or coating is selected.
- A finite result is not permission to operate a beam.
How to report a Malus-law result
A reproducible report should state incoming linearly polarized intensity in W/m^2, the relative angle in degrees, the degree-to-radian conversion, and I = I0 cos(theta)^2. Include both the transmitted intensity and the dimensionless fraction. At zero input intensity, label the fraction as the ideal angular factor rather than an observed quotient. At the endpoint angles, state whether the result is the exact ideal parallel or crossed-axis case. These details keep units, geometry, and interpretation aligned.
The report should repeat that the analyzer is ideal and that partial polarization, device losses, scattering, detector response, beam geometry, and safety are not modeled. If an observed intensity is being compared, preserve the input measurement, calibration, area, wavelength, and uncertainty outside the calculator. The honest conclusion is that the cosine-squared projection was evaluated for the entered pair. It is not that a particular optical device has been approved or that a measured beam must match the ideal number.
- Record I0, angle, units, and axis definition.
- Show the degree conversion and cosine-squared step.
- Keep fraction and intensity as separate outputs.
- Attach real-device and safety context elsewhere.
Final scope checklist
Before accepting the result, confirm that the input is an intensity rather than an amplitude, that it is nonnegative and finite, and that the angle is between 0 and 90 degrees. Check the endpoint behavior, the 30-degree example if useful, and the unit path from dimensionless fraction to W/m^2. Verify that a zero input returns zero transmitted intensity without treating the fraction as a measured ratio. These checks validate the implementation and the model's declared geometry.
The final statement should remain narrow: ideal Malus's law was evaluated for an incoming linearly polarized intensity and a relative analyzer angle. No partial-polarization correction, device loss, detector interpretation, optical alignment, or safety advice was produced. If the intended question includes one of those features, define that additional model separately. The strength of this page is a clear cosine-squared result with no hidden claims about a real optical system.
- Confirm intensity, angle, and units.
- Check exact parallel and crossed-axis cases.
- Treat zero-input fraction as a model factor.
- Keep optical-device and safety decisions outside the formula.
Measured intensity versus the ideal factor
A measured transmission ratio can be compared with the cosine-squared factor only after the incoming and outgoing readings use the same area, wavelength, detector response, and timing definition. Background subtraction, aperture changes, alignment drift, and source fluctuations can affect the observed ratio. The calculator has no fields for those effects. It supplies the factor predicted by the ideal analyzer geometry, so a difference between a measurement and the output is a prompt to inspect the experimental contract rather than an automatic proof that the trigonometric relation failed.
The distinction is also important when several polarizing elements are placed in sequence. The output of one element can become the input to the next, but each stage needs its own axis relationship and device behavior. This page evaluates one incoming intensity and one analyzer angle only. It does not multiply a chain of unknown elements, infer a polarization angle from an observed intensity, or fit a source. Use the result as one clearly labeled projection step in any larger optical analysis.
- Compare measurements only with matching definitions.
- Background and detector effects are external data.
- Sequential analyzers need separate stage contracts.
- The page does not fit angles or infer source polarization.
Uncertainty in an intensity comparison
The handler returns the central arithmetic value for the entered intensity and angle. It does not propagate uncertainty in either quantity. Near an angle where the cosine-squared curve changes rapidly, an angle interval can produce a visible range of transmitted intensities; near a flat part of the curve, the same angle interval can have a smaller first-order effect. A measured input may also include calibration and background uncertainty. Those intervals should be carried through a separate analysis rather than inferred from the number of displayed digits.
This boundary does not make the ideal fraction useless. It gives a reference against which a carefully defined measurement can be compared. To make that comparison meaningful, state whether the input is a mean, peak, integrated, or background-corrected intensity and whether the angle is an alignment reading or a nominal setting. The calculator accepts one scalar pair and reports one scalar pair of outputs. It does not choose a statistical method, confidence interval, or acceptance threshold for an optical experiment.
- The outputs are central values without propagated intervals.
- Angle sensitivity varies across the cosine-squared curve.
- Input calibration and background uncertainty are external.
- No measurement acceptance threshold is inferred.
Frequently asked questions
What is the Malus's Law Polarized Intensity?
Calculate transmitted intensity and the ideal transmission fraction for linearly polarized light passing through an analyzer at a stated angle.
What is the formula for the Malus's Law Polarized Intensity?
Malus's law is I = I0 cos(theta)^2, where theta is the angle in degrees between the incoming linear-polarization axis and the ideal analyzer axis; transmission fraction is cos(theta)^2. This calculator applies Malus's law to a nonnegative incoming linearly polarized intensity and an analyzer angle from 0 to 90 degrees. It returns transmitted intensity in W/m^2 and the dimensionless ideal transmission fraction, including the angular factor when the incoming intensity is zero.
What do I need to use this calculator?
Enter Incoming linearly polarized intensity, Polarizer-axis angle, then choose Calculate.
What are the limits of this calculator?
Incoming intensity is a finite nonnegative magnitude in watts per square metre and is already linearly polarized. The angle is a finite degree measure between the polarization and analyzer transmission axes, restricted to the first-quarter range 0 through 90 degrees. The analyzer is ideal, with no absorption beyond polarization selection, scattering, reflection, wavelength dependence, or detector response. Partial polarization, coherence details, beam geometry, optical alignment, laser classification, and safety advice are outside the calculation.
Methodology
This calculator is part of the WorldCalculate library. Its formula, example, assumptions, input bounds, and output formatting follow the official methodology.
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