Two ideal crossed polarizers block the beam, yet adding a third at an intermediate angle can let some light through. The extra filter still absorbs energy; it changes the polarization of what it transmits. Following the beam one element at a time resolves the surprise.
This chapter develops single and repeated projections, explains the half-intensity rule and its limits, and connects reflection geometry to Brewster polarization.
What you will learn to do
- Distinguish linear, circular and unpolarized light conceptually.
- Derive Malus’s law from field projection.
- Apply the unpolarized half rule only at the appropriate stage.
- Calculate a sequence of rotated polarizers.
- Interpret analyzer extrema under a declared partial-polarization model.
- Derive Brewster’s angle and identify reflected s polarization.
- Relate glare suppression to the actual incidence plane.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Polarization, analyzer chains and Brewster reflection Learning goals: - Distinguish linear, circular and unpolarized light conceptually. - Derive Malus’s law from field projection. - Apply the unpolarized half rule only at the appropriate stage. - Calculate a sequence of rotated polarizers. - Interpret analyzer extrema under a declared partial-polarization model. - Derive Brewster’s angle and identify reflected s polarization. - Relate glare suppression to the actual incidence plane. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
What distinguishes changing a light wave’s polarization from changing its propagation direction?
Compare your reasoning
Polarization specifies the orientation and behaviour of its transverse electric field. The propagation direction tells where the wave travels. Rotating an analyzer need not redirect the beam.
Revisit the explanation →01 / Polarization, analyzer chains and Brewster reflection
Polarization describes the electric field’s transverse motion
A travelling electromagnetic plane wave has an electric field perpendicular to its propagation direction. There are two independent transverse directions in which that field can vary. Polarization describes the pattern traced by the field vector at a fixed position as time passes.
For linear polarization, the field oscillates along one fixed transverse line. Circular polarization has a vector rotating with constant magnitude; elliptical polarization is the more general fixed ellipse. These are ordered field relationships, not random orientations.
Unpolarized light has no preferred transverse orientation in its relevant statistical average. Its field fluctuations cannot be described as one fixed linearly polarized wave with a stable axis. Ordinary light can also be partially polarized.
A linear polarizer transmits the field component along its transmission axis and rejects the perpendicular component in the ideal model. The emerging transmitted beam is linearly polarized along that axis. We use ideal lossless transmission of the allowed component, while rejected power leaves the transmitted beam.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Polarization, analyzer chains and Brewster reflection Focus: Polarization describes the electric field’s transverse motion Learning goals: - Distinguish linear, circular and unpolarized light conceptually. - Derive Malus’s law from field projection. - Apply the unpolarized half rule only at the appropriate stage. - Calculate a sequence of rotated polarizers. - Interpret analyzer extrema under a declared partial-polarization model. - Derive Brewster’s angle and identify reflected s polarization. - Relate glare suppression to the actual incidence plane. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
02 / Polarization, analyzer chains and Brewster reflection
Project the field amplitude, then square to get intensity
Suppose linearly polarized light reaches an ideal linear analyzer. If its electric-field direction makes angle θ with the transmission axis, the transmitted field amplitude is E0 cos θ along that axis.
Intensity is proportional to the squared field amplitude, so the transmitted intensity is Iin cos²θ. This is Malus’s law. The incident intensity here is the intensity immediately before this analyzer, not necessarily the original source intensity before earlier filters.
Parallel axes give full transmission of the allowed input component in the ideal model; crossed axes give zero. Rotating an axis by 180° leaves the same transmission line and therefore the same intensity. A negative cosine changes the field phase but cannot create negative intensity.
The cosine applies to the relative angle between the incoming linear polarization and the current analyzer. In a chain, each polarizer changes the outgoing polarization direction, so the next angle is measured from that new direction.
Ideal linear analyzer with linearly polarized input. θ is the relative angle to its transmission axis.
WORKED EXAMPLE
Do not use the amplitude projection as an intensity ratio
A linearly polarized beam of intensity 80 W/m² reaches an ideal analyzer at 60° to its transmission axis. Find the transmitted intensity.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Polarization, analyzer chains and Brewster reflection
Focus: Project the field amplitude, then square to get intensity
Learning goals:
- Distinguish linear, circular and unpolarized light conceptually.
- Derive Malus’s law from field projection.
- Apply the unpolarized half rule only at the appropriate stage.
- Calculate a sequence of rotated polarizers.
- Interpret analyzer extrema under a declared partial-polarization model.
- Derive Brewster’s angle and identify reflected s polarization.
- Relate glare suppression to the actual incidence plane.
Equations in this section (LaTeX):
E_{\mathrm{out}}=E_{\mathrm{in}}\cos\theta,\qquad I_{\mathrm{out}}=I_{\mathrm{in}}\cos^2\theta
Meaning: Ideal linear analyzer with linearly polarized input. θ is the relative angle to its transmission axis.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.03 / Polarization, analyzer chains and Brewster reflection
The first polarizer transmits half of an ideal unpolarized beam
For an unpolarized incident beam, all transverse directions are statistically equivalent. Averaging the squared projection onto any chosen linear axis gives one half, so an ideal linear polarizer transmits I0/2.
The transmitted beam is then linearly polarized. A second analyzer at relative angle θ transmits another cos²θ fraction, giving I0 cos²θ/2. Applying a factor one half at every stage would ignore the polarization created by the first filter.
This half-intensity result is an ideal model. Real polarizers can have absorption along the allowed axis, leakage through the blocked axis and wavelength dependence. A measured setup may therefore transmit less than half of an initially unpolarized beam.
If the input is specified as a mixture of an unpolarized contribution Iu and an independent linearly polarized contribution Il, with their intensities adding incoherently, an analyzer at angle θ to the linear component transmits Iu/2 + Il cos²θ. State that mixture model before inferring component fractions.
After the first filter, the input to the second is linearly polarized. Use the relative axes, not an arbitrary laboratory angle.
Compare analyzer rotation for two different incident polarization states
- Linear input
- 1
- Unpolarized input
- 0.5
WORKED EXAMPLE
Handle the first filter and analyzer separately
An unpolarized beam has intensity 120 W/m². It passes through two ideal linear polarizers whose axes differ by 30°. Find the final intensity.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Polarization, analyzer chains and Brewster reflection
Focus: The first polarizer transmits half of an ideal unpolarized beam
Learning goals:
- Distinguish linear, circular and unpolarized light conceptually.
- Derive Malus’s law from field projection.
- Apply the unpolarized half rule only at the appropriate stage.
- Calculate a sequence of rotated polarizers.
- Interpret analyzer extrema under a declared partial-polarization model.
- Derive Brewster’s angle and identify reflected s polarization.
- Relate glare suppression to the actual incidence plane.
Equations in this section (LaTeX):
I_1=\frac{I_0}{2},\qquad I_2=\frac{I_0}{2}\cos^2\theta\quad\text{for unpolarized input followed by two ideal filters}
Meaning: After the first filter, the input to the second is linearly polarized. Use the relative axes, not an arbitrary laboratory angle.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.04 / Polarization, analyzer chains and Brewster reflection
An intermediate axis changes what reaches the final filter
Two crossed ideal polarizers transmit no light from the linear output of the first. Inserting a third polarizer between them can make the final transmitted intensity nonzero because the middle filter changes the direction of the transmitted field.
Let the first and last axes be 0° and 90°, with the middle at θ. For unpolarized original intensity I0, the output is (I0/2) cos²θ cos²(90° − θ) = (I0/8) sin²2θ.
The largest transmission occurs at θ = 45°, giving I0/8. This does not mean adding a passive filter creates energy. The original last filter rejected the first filter’s entire output; the inserted filter presents a different polarization to it, while still rejecting some incident energy itself.
The order of nonparallel polarizers matters for the transmitted intensity of a specified input. Treat each stage as a new projection, and keep an intensity ledger so that every loss is applied to the current beam.
WORKED EXAMPLE
Calculate the chain from left to right
An unpolarized 160 W/m² beam passes through ideal polarizer axes at 0°, 45° and 90°, in that order. Find the intensity after each stage.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Polarization, analyzer chains and Brewster reflection Focus: An intermediate axis changes what reaches the final filter Learning goals: - Distinguish linear, circular and unpolarized light conceptually. - Derive Malus’s law from field projection. - Apply the unpolarized half rule only at the appropriate stage. - Calculate a sequence of rotated polarizers. - Interpret analyzer extrema under a declared partial-polarization model. - Derive Brewster’s angle and identify reflected s polarization. - Relate glare suppression to the actual incidence plane. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
05 / Polarization, analyzer chains and Brewster reflection
Analyzer maxima and minima can quantify a specified mixture
For the incoherent mixture model Iout(θ) = Iu/2 + Il cos²θ, the maximum is Iu/2 + Il and the minimum is Iu/2. Their difference isolates the linearly polarized contribution, and twice the minimum gives the unpolarized contribution.
The fractional linear polarization in this particular model is Il/(Iu + Il) = (Imax − Imin)/(Imax + Imin). The denominator reconstructs the total input intensity rather than the maximum transmitted intensity alone.
This interpretation depends on the assumed mixture. A general elliptical or circular state needs more polarization measurements, such as a retarder combined with an analyzer, to identify its complete state.
In an exam, a statement that the light is partially polarized may imply a particular model, but the safest reasoning is to state it. A single rotating linear analyzer does not measure every aspect of an arbitrary polarization state.
WORKED EXAMPLE
Recover components only under a declared model
A beam is specified as an incoherent mixture of unpolarized light and one linearly polarized component. An ideal rotating analyzer gives maximum 70 W/m² and minimum 20 W/m². Find Iu, Il and the linearly polarized fraction.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Polarization, analyzer chains and Brewster reflection Focus: Analyzer maxima and minima can quantify a specified mixture Learning goals: - Distinguish linear, circular and unpolarized light conceptually. - Derive Malus’s law from field projection. - Apply the unpolarized half rule only at the appropriate stage. - Calculate a sequence of rotated polarizers. - Interpret analyzer extrema under a declared partial-polarization model. - Derive Brewster’s angle and identify reflected s polarization. - Relate glare suppression to the actual incidence plane. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
06 / Polarization, analyzer chains and Brewster reflection
Reflection treats the two polarization directions differently
The plane of incidence contains the incident ray and the surface normal. Resolve the electric field into a component perpendicular to this plane, called s polarization, and a component within it, called p polarization.
At a boundary between ideal transparent isotropic nonmagnetic dielectric media, the p-polarized reflection coefficient vanishes at a particular incidence angle, Brewster’s angle. Unpolarized incident light then produces reflected light polarized perpendicular to the plane of incidence.
At this angle the reflected and transmitted ray directions are perpendicular. Since reflection angle equals incidence angle, the refraction angle r satisfies θB + r = 90°. Combining that geometry with Snell’s law gives tan θB = n2/n1.
This is a polarization-selective absence of reflection, not total internal reflection. The transmitted beam is generally partially polarized for unpolarized incidence, and the reflected beam still contains the s component. Absorbing materials and more general optical media require a fuller reflection model.
Ideal transparent isotropic nonmagnetic dielectric interface. Incidence is from n1 toward n2 and angles are measured from the normal.
WORKED EXAMPLE
Find the reflected polarization and refracted direction
Unpolarized light in air reaches a transparent dielectric of index 1.50 at Brewster incidence. Find the incidence and refraction angles and the reflected polarization direction.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Polarization, analyzer chains and Brewster reflection
Focus: Reflection treats the two polarization directions differently
Learning goals:
- Distinguish linear, circular and unpolarized light conceptually.
- Derive Malus’s law from field projection.
- Apply the unpolarized half rule only at the appropriate stage.
- Calculate a sequence of rotated polarizers.
- Interpret analyzer extrema under a declared partial-polarization model.
- Derive Brewster’s angle and identify reflected s polarization.
- Relate glare suppression to the actual incidence plane.
Equations in this section (LaTeX):
\tan\theta_B=\frac{n_2}{n_1},\qquad\theta_B+r=90^\circ
Meaning: Ideal transparent isotropic nonmagnetic dielectric interface. Incidence is from n1 toward n2 and angles are measured from the normal.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.07 / Polarization, analyzer chains and Brewster reflection
Connect a polarization axis to the actual reflection geometry
For reflection from a horizontal surface in a vertical plane of incidence, the s direction is horizontal. Near Brewster incidence, much of the reflected glare is polarized along that horizontal direction.
A linear filter with vertical transmission axis can reduce that component because the unwanted field projects weakly onto its allowed axis. The phrase blocks glare is therefore shorthand for a particular field and surface geometry.
Rotating the reflecting surface or viewing plane changes the s and p directions in the laboratory. Identify the incidence plane before assigning horizontal or vertical labels. The general rule is perpendicular or parallel to that plane.
A polarizer can also reduce desired light, and real surfaces may scatter or depolarize it. The ideal geometry explains the mechanism while leaving room for nonideal contrast and transmission.
WORKED EXAMPLE
Determine which analyzer axis suppresses the reflected component
A reflected beam at ideal Brewster incidence has its plane of incidence vertical. Which linear-analyzer transmission direction suppresses the reflected beam in the ideal model?
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Polarization, analyzer chains and Brewster reflection Focus: Connect a polarization axis to the actual reflection geometry Learning goals: - Distinguish linear, circular and unpolarized light conceptually. - Derive Malus’s law from field projection. - Apply the unpolarized half rule only at the appropriate stage. - Calculate a sequence of rotated polarizers. - Interpret analyzer extrema under a declared partial-polarization model. - Derive Brewster’s angle and identify reflected s polarization. - Relate glare suppression to the actual incidence plane. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BRING THE IDEAS TOGETHER
What should stay with you
- A linear polarizer selects a field component; intensity follows its squared amplitude.
- Malus’s law uses the relative angle and the intensity immediately before each analyzer.
- Unpolarized input loses half its intensity through one ideal linear polarizer.
- A middle polarizer changes the polarization reaching a crossed final filter.
- A rotating linear analyzer alone cannot identify every polarization state.
- At ideal Brewster incidence, the reflected beam from unpolarized input is polarized perpendicular to the incidence plane.
- Brewster transmission and total internal reflection are different optical conditions.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
A linearly polarized 100 W/m² beam meets an ideal analyzer at 60°. What intensity emerges?
Answer in W/m².
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SELF CHECK 2
An unpolarized beam of intensity 80 W/m² passes through ideal axes 0°, 45° and 90° in that order. What final intensity emerges?
Answer in W/m².
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SELF CHECK 3
At a transparent nonmagnetic interface, tan of the Brewster angle is 1.40 for incidence from air of index 1. What destination index follows?
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SELF CHECK 4
For the stated unpolarized-plus-linear incoherent mixture, an analyzer minimum is 12 W/m². What is the unpolarized input contribution?
Answer in W/m².
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SELF CHECK 5
After unpolarized light passes through one ideal linear polarizer, how should a second rotated filter be treated?
- Always multiply intensity by one half again
- Apply cos² of its angle relative to the first filter’s axis
- Add the two filter angles to the source intensity
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SELF CHECK 6
At ideal Brewster incidence from unpolarized light, the reflected beam is polarized:
- Within the plane of incidence
- Perpendicular to the plane of incidence
- Along the propagation direction
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SELF CHECK 7
An ideal rotating linear analyzer gives constant transmitted intensity. Does this alone prove the input is unpolarized?
- Yes, uniquely
- No; circular polarization also gives an angle-independent result
- Yes, if the wavelength is visible
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APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
Use the index ratio at the interfaceIntermediate · Hints and guided reasoningAn analyzer transmits a projectionIntermediate · Hints and guided reasoningA third polarizer between crossed axesIntermediate · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
Why can you not apply only the angle between the first and last filters when a third polarizer lies between them?
Compare your reasoning
Each ideal polarizer transmits a component and sets the polarization direction for the next stage. Apply the intensity factor at each step; the middle filter changes the state arriving at the last one.
Revisit the explanation →A few days later: change the situation
For an ideal first polarizer, why does unpolarized incident light lose half its average intensity rather than follow one fixed cos² angle?
Compare your reasoning
There is no single fixed incident polarization angle. Averaging the squared projection over uniformly distributed orientations gives one half. Subsequent ideal analyzers act on the transmitted polarized beam.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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