Polarizers and Extinction Ratio

A polarizer is a device that preferentially transmits one polarization component, producing light that is predominantly polarized along a chosen axis. Malus's law describes the ideal version — perfect transmission along the axis, perfect blocking across it — but no real polarizer is ideal, and the gap between the two is exactly what an optical engineer has to budget. A real linear polarizer transmits a large fraction $k_1$ of the component along its axis and a small but nonzero fraction $k_2$ of the component across it. The single number that captures its quality is the extinction ratio $k_1/k_2$ : how much more of the wanted polarization gets through than the unwanted one. A cheap sheet polarizer reaches $\sim 10^3$ – $10^4$ ; a good crystal prism reaches $10^5$ – $10^6$ . That one leak parameter $k_2$ rewrites several ideal results. Crossed polarizers no longer go perfectly black — they floor at a residual $k_1 k_2$ of the input. Light passed by a single polarizer is not perfectly polarized — its degree of polarization is $(k_1 - k_2)/(k_1 + k_2)$ , a hair under one. And the rotating-analyzer curve, ideally $\cos^2\theta$ diving to zero, instead rides on a small pedestal set by $k_2$ . This article builds the two-transmittance law, defines the extinction ratio and its decibel form, works out what a real crossed pair actually transmits, and surveys the four physical mechanisms — selective absorption, wire grids, birefringent walk-off, and reflection near Brewster's angle — that real polarizers use to separate the two components in the first place.

Key takeaways

A real linear polarizer obeys the two-transmittance law $I = I_0(k_1\cos^2\theta + k_2\sin^2\theta)$ : principal transmittance $k_1 \le 1$ along the axis, minor transmittance $k_2 > 0$ across it.
Extinction ratio $\mathrm{ER} = k_1/k_2$ is the figure of merit; in decibels, $\mathrm{ER}_{\mathrm{dB}} = 10\log_{10}(k_1/k_2)$ .
Crossed real polarizers do not reach zero: a pair transmits $T_\perp = k_1 k_2$ ; a parallel pair transmits $T_\parallel = \tfrac12(k_1^2 + k_2^2)$ .
Unpolarized light through one real polarizer emerges with degree of polarization $(k_1 - k_2)/(k_1 + k_2)$ — just short of fully polarized.
Common polarizer technologies exploit four mechanisms: dichroic absorption (Polaroid), reflection from wire grids, birefringent walk-off (Nicol, Glan–Thompson), and polarization-selective reflection near Brewster's angle.
Key idea: one leak number $k_2$ turns the ideal Malus polarizer into a real device — it sets the extinction ratio, the crossed-pair floor $k_1 k_2$ , and the sub-unity degree of polarization.

The core insight

An ideal polarizer is defined by $\{k_1 = 1, k_2 = 0\}$ ; a real one by a small nonzero minor transmittance $k_2$ . Many practical departures from the ideal Malus-law picture can be captured by the nonzero orthogonal transmittance $k_2$ , which sets the residual transmission, degree of polarization, and extinction ratio $k_1/k_2$ in this two-transmittance model.

Where this lives today

Extinction ratio is the number on every polarizer datasheet, and it decides which device an application can use. A liquid-crystal display uses two crossed sheet polarizers around a voltage-controlled cell; in the dark state, the residual transmission is limited in part by the polarizers' finite $k_2$ leakage, which contributes to the achievable contrast ratio. Optical isolators — the components that suppress back-reflections into a laser — pair a Faraday rotator with polarizers, and the polarizers' extinction ratio contributes directly to the achievable isolation. Wire-grid polarizers, fabricated by nanolithography, now sit on camera sensors as per-pixel polarization filters for polarimetric imaging that sees glare, stress, and surface orientation invisible to ordinary cameras. And in precision metrology — ellipsometry, gravitational-wave interferometer input optics, quantum-optics state preparation — Glan–Thompson and Glan–Taylor calcite prisms can provide extinction ratios of $10^{5}$ – $10^{6}$ , making them high-purity polarization references for precision measurements.

The physical picture

Picture unpolarized light hitting a polarizer as two independent, equal streams: one oscillating along the transmission axis, one across it. An ideal polarizer passes the first stream untouched and annihilates the second. A real polarizer is leakier — it passes a large fraction $k_1$ of the wanted stream and lets a small fraction $k_2$ of the unwanted stream slip through. Everything about polarizer quality is the size of that unwanted leak: the smaller $k_2$ , the darker the "off" state and the purer the output. The most sensitive place to see the leak is between crossed polarizers, where the ideal answer is exactly zero. Rotate a real analyzer from parallel to crossed and the transmitted intensity does not vanish — it bottoms out at a small floor. Plotting that on a logarithmic scale makes the extinction ratio jump out: the curve falls from $k_1$ near the top to a floor at $k_2$ , and the ratio of those two levels is the extinction ratio. That is the punchline figure below.

Figure

Real vs ideal polarizer on a log intensity scale: transmission falls from k₁ ≈ 0.8 to a leak floor k₂ = 10⁻⁴ instead of diving to zero — the ratio of the two levels is the extinction ratio, 8000 (39 dB)

Why does the leak exist at all? Because separating two perpendicular field components perfectly is hard, and each physical mechanism does it imperfectly in its own way. There are four broad families, sketched below: dichroic materials absorb one component (Polaroid sheet); wire grids reflect the component along the wires and pass the other; birefringent prisms send the two components off along different ray paths and throw one away (Nicol, Glan–Thompson); and Brewster-angle reflection transmits almost only the parallel component. Each buys a different trade of extinction ratio, acceptance angle, bandwidth, and cost.

Figure

Four ways to build a linear polarizer: dichroic absorption of one component, a wire grid reflecting the parallel field, birefringent walk-off separating the two rays, and Brewster-angle reflection passing mostly p-polarization

The core equations

An ideal polarizer keeps the field component along its axis and kills the orthogonal one — Malus's law. A real polarizer keeps most of the first and a little of the second, so it needs two transmittance parameters instead of one.

I = I_0\big(k_1\cos^2\theta + k_2\sin^2\theta\big) \qquad (1)

where: $I_0$ — incident intensity of a linearly polarized beam; $\theta$ — angle between the incident polarization and the transmission axis; $k_1$ — principal transmittance (the fraction passed when aligned, $k_1 \le 1$ ); $k_2$ — minor transmittance (the fraction passed when crossed, $k_2 \ge 0$ ). Setting $k_1 = 1, k_2 = 0$ recovers Malus's $I_0\cos^2\theta$ . The device figure of merit is the ratio of the two transmittances,

\mathrm{ER} = \frac{k_1}{k_2}, \qquad \mathrm{ER}_{\mathrm{dB}} = 10\log_{10}\!\frac{k_1}{k_2} \qquad (2)

where: $\mathrm{ER}$ — extinction ratio (dimensionless); the decibel form is standard in datasheets. A ratio of $10^3$ is $30$ dB, $10^4$ is $40$ dB, $10^6$ is $60$ dB — each factor of ten adds $10$ dB. Bigger is better: it is how completely the polarizer rejects the unwanted polarization. Because $\cos^2\theta = \tfrac12(1+\cos2\theta)$ , equation (1) is a pedestal plus a modulation,

T = \frac{I}{I_0} = \tfrac12(k_1 + k_2) + \tfrac12(k_1 - k_2)\cos 2\theta \qquad (3)

where: the constant $\tfrac12(k_1+k_2)$ is the mean transmittance and the modulation amplitude is $\tfrac12(k_1-k_2)$ . The fractional depth of the ripple, $(k_1-k_2)/(k_1+k_2)$ , is exactly the degree of polarization the polarizer can impose — the rotating-analyzer signal never reaches zero because $k_2 \neq 0$ .

Derivation AWhat a real crossed pair actually transmits

The goal is to compute the transmittance of two identical real polarizers, first parallel and then crossed, starting from unpolarized input — and to show that crossed sheets floor at $k_1 k_2$ rather than at zero. The strategy is to follow the two equal, independent field streams of unpolarized light (one along the first axis, one across it) through each sheet, applying $k_1$ to whichever component is along a sheet's axis and $k_2$ to whichever is across it. Intensities add because the streams are incoherent.

Step 1. Unpolarized input, intensity $1$ , splits into two equal incoherent streams of $\tfrac12$ each: one along the first polarizer's axis $\hat{\mathbf a}_1$ , one across it. Why two equal streams: "unpolarized" means no preferred direction, which decomposes into any orthogonal pair with equal power — choosing $\hat{\mathbf a}_1$ and its perpendicular makes the first sheet trivial to apply.

Step 2. After polarizer 1: the along- $\hat{\mathbf a}_1$ stream is attenuated by $k_1$ to $k_1/2$ ; the across stream by $k_2$ to $k_2/2$ . The beam is now partially polarized, with the larger transmitted component along $\hat{\mathbf a}_1$ . Why both survive: a real sheet does not fully kill the orthogonal component — that surviving $k_2/2$ is the seed of every leak downstream.

Step 3. Parallel case, polarizer 2 with axis $\hat{\mathbf a}_1$ : it passes $k_1$ of the aligned $k_1/2$ and $k_2$ of the crossed $k_2/2$ , giving $T_\parallel = \tfrac12(k_1^2 + k_2^2)$ . Why the squares: each surviving stream passes through a same-orientation transmittance twice, once per sheet.

Step 4. Crossed case, polarizer 2 with axis $\perp\hat{\mathbf a}_1$ : the component transmitted by the first polarizer along $\hat{\mathbf a}_1$ , with intensity $k_1/2$ , is attenuated by $k_2$ , while the orthogonal component with intensity $k_2/2$ is attenuated by $k_1$ , giving $T_\perp=\frac{k_2}{2}k_1+\frac{k_1}{2}k_2=k_1k_2$ . Why the floor is $k_1 k_2$ and not $0$ : the leak of the second sheet acts on the strong survivor of the first — the product of one big and one small transmittance, symmetric in the two sheets. *Check: for $(k_1,k_2)=(0.80,10^{-4})$ , the formulas give $T_\parallel = \tfrac12(k_1^2+k_2^2)$ and $T_\perp = k_1 k_2 = 8.0\times10^{-5}$ , consistent with the step-by-step derivation above.*

Go deeper · Derivation B · 1 minThe degree of polarization a real polarizer can impose

Derivation BThe degree of polarization a real polarizer can impose

The goal is to find how completely a single real polarizer polarizes unpolarized light — its output degree of polarization — and to show it equals $(k_1-k_2)/(k_1+k_2)$ , just under one for a good sheet. The strategy is to write the output as two orthogonal incoherent intensities (the surviving along-axis and across-axis streams), then apply the definition of degree of polarization as the contrast of those two intensities, which is also the depth of the rotating-analyzer ripple in equation (3).

Step 1. From the previous derivation, one real polarizer on unpolarized input leaves $I_\parallel = k_1/2$ along the axis and $I_\perp = k_2/2$ across it, incoherent. Why these are the extremes: an analyzer downstream reads maximum when aligned with the strong stream and minimum when aligned with the weak one — those are $I_\parallel$ and $I_\perp$ .

Step 2. The degree of polarization is the normalized contrast $P = \dfrac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}} = \dfrac{k_1/2 - k_2/2}{k_1/2 + k_2/2} = \dfrac{k_1 - k_2}{k_1 + k_2}$ . Why this equals the ripple depth: equation (3) has mean $\tfrac12(k_1+k_2)$ and amplitude $\tfrac12(k_1-k_2)$ , whose ratio is the same expression — the analyzer literally traces out the degree of polarization.

Step 3. For $k_1 = 0.80, k_2 = 10^{-4}$ : $P = (0.80 - 0.0001)/(0.80 + 0.0001) = 0.99975$ . As $k_2 \to 0$ , $P \to 1$ : an ideal polarizer produces perfectly polarized light. Why it never quite reaches one: the residual leak always injects a sliver of the orthogonal state, so real "polarized" light carries a small unpolarized admixture. *Check: for $(k_1,k_2)=(0.80,10^{-4})$ , the formula gives $P = (k_1-k_2)/(k_1+k_2) = 0.99975$ , and as $k_2\to0$ the degree of polarization approaches $1$ , as expected for an ideal polarizer.*

Limiting cases

Case	Result	Meaning
Ideal polarizer	$k_1 = 1, k_2 = 0$	Recovers Malus $I_0\cos^2\theta$ ; crossed pair $= 0$ ; $P = 1$ .
Aligned ( $\theta = 0$ )	$I = k_1 I_0$	Principal transmittance; the best the device does.
Crossed ( $\theta = 90°$ )	$I = k_2 I_0$	Minor transmittance — the leak, not zero.
Two crossed sheets	$T_\perp = k_1 k_2$	The realistic "dark state" of an LCD or crossed-polarizer null.
Unpolarized in, one sheet	$I = \tfrac12(k_1+k_2)I_0$ , $P = \tfrac{k_1-k_2}{k_1+k_2}$	Halves and (nearly) polarizes.
High-ER prism	$k_2 \to 10^{-6}$	$\mathrm{ER} = 60$ dB; Glan–Thompson / calcite regime.

Common trapA high principal transmittance does not mean a good polarizer — the extinction ratio is set by the leak k₂

It is tempting to rate a polarizer by how bright the "on" state is, i.e. by $k_1$ . But two polarizers with the same $k_1 = 0.8$ can differ by orders of magnitude in quality, because quality lives in the minor transmittance $k_2$ : the extinction ratio is $k_1/k_2$ , and it is $k_2$ that sets the darkness of the crossed state, the contrast of a display, and the purity of the output. A sheet with $k_2 = 10^{-3}$ ( $\mathrm{ER} = 800$ ) and one with $k_2 = 10^{-6}$ ( $\mathrm{ER} = 8\times10^{5}$ ) look identical when aligned and differ by a factor of a thousand when crossed. Always quote — and read — the extinction ratio, not just the transmission.

Worked examples

1 · Rating a sheet polarizer and its crossed pair

A Polaroid sheet has principal transmittance $k_1 = 0.80$ and minor transmittance $k_2 = 1.0\times10^{-4}$ . Find its extinction ratio in plain numbers and in decibels, and the transmittance of two such sheets crossed.

Step 1. Extinction ratio: $\mathrm{ER} = k_1/k_2 = 0.80/10^{-4} = \boldsymbol{8000}$ .

Step 2. In decibels: $\mathrm{ER}_{\mathrm{dB}} = 10\log_{10}(8000) = \boldsymbol{39.0\ \mathrm{dB}}$ .

Step 3. Crossed pair (unpolarized input): $T_\perp = k_1 k_2 = 0.80\times10^{-4} = \boldsymbol{8.0\times10^{-5}}$ — so a "dark" crossed pair still leaks eight parts in a hundred thousand, which is why a good LCD needs a much higher-ER polarizer to render true black.

2 · A crystal prism and the purity of its output

A Glan–Thompson calcite prism has an extinction ratio of $1\times10^{5}$ . Express it in decibels, and find the degree of polarization it imposes on unpolarized light, taking $k_1 = 0.90$ so that $k_2 = k_1/\mathrm{ER}$ .

Step 1. Decibels: $\mathrm{ER}_{\mathrm{dB}} = 10\log_{10}(10^{5}) = \boldsymbol{50\ \mathrm{dB}}$ — an order of magnitude better rejection than the sheet, in dB terms.

Step 2. Minor transmittance: $k_2 = k_1/\mathrm{ER} = 0.90/10^{5} = 9.0\times10^{-6}$ .

Step 3. Degree of polarization: $P = (k_1 - k_2)/(k_1 + k_2) = (0.90 - 9\times10^{-6})/(0.90 + 9\times10^{-6}) = \boldsymbol{0.99998}$ — as close to perfectly polarized as tabletop optics gets, which is why prism polarizers are chosen for precision nulling and polarimetry.

Physics of the portrait

The "linear polarizer" is less a single invention than a family of tricks for the same job: peel a chosen electric-field direction out of a beam. For two centuries the only good option was a chunk of birefringent crystal — Iceland spar, cut and cemented so that the two refracted rays part company and one can be blocked. The Nicol prism (1828) made practical crystal-prism polarizers possible, and the Glan and Glan–Thompson designs subsequently refined the approach for high-extinction applications. The revolution was making polarizers cheap and large: a thin plastic sheet you can cut with scissors. That came from a Harvard undergraduate who aligned tiny dichroic crystals — and later iodine-stained polymer chains — so that one field component is absorbed and the other passes. His name became the product's: Polaroid.

Origins · 1 minlinear polarizer and how this was discovered

linear polarizer

Nicol prism 1828 · Land's Polaroid sheet 1929–1938 · four mechanisms, one job — pull a single field direction out of the light

The linear polarizer is where materials physics meets Malus's law. A birefringent prism (Nicol, 1828; later Glan–Thompson) exploits a calcite crystal's two refractive indices to send the ordinary and extraordinary rays along different paths, then discards one — giving extinction ratios of $10^5$ – $10^6$ but over a narrow acceptance angle. A dichroic sheet absorbs one component: Edwin Land, as a Harvard undergraduate around 1929, aligned needle-like herapathite microcrystals in a polymer (J-sheet), then in 1938 invented H-sheet — stretched polyvinyl alcohol stained with iodine, whose aligned polymer chains conduct and thus absorb the field component along them. H-sheet is the Polaroid in every pair of polarized sunglasses and most LCDs. A wire grid does the same by reflection rather than absorption: metal wires spaced finer than the wavelength short out the parallel field and pass the perpendicular one — Hertz demonstrated it with radio waves in 1888, and evaporated-metal grids now polarize the infrared. A Brewster-angle pile transmits almost pure p-polarization at $\tan\theta_B = n_2/n_1$ (its own article). Same output state, four physical routes, four different trades of extinction, aperture, bandwidth, and price.

History

1669

— Bartholin discovers double refraction in Iceland spar, the crystal that makes prism polarizers possible

1828

— Nicol cements two calcite prisms into the first practical polarizing prism

1852

— Stokes formalizes partial polarization, giving the degree of polarization its measure

1888

— Hertz polarizes radio waves with a grid of parallel wires — the first wire-grid polarizer

1929

— Land patents J-sheet, the first synthetic dichroic sheet polarizer, as a Harvard undergraduate

1938

— Land invents H-sheet, iodine-stained stretched polyvinyl alcohol — the modern Polaroid

Going further

Texts. Hecht, Optics, 5th ed., ch. 8 (polarization; dichroism, birefringent and reflection polarizers). Saleh & Teich, Fundamentals of Photonics, 3rd ed., ch. 6. Bennett, Polarizers (Handbook of Optics) for extinction-ratio metrology. Collett, Polarized Light, for the Mueller-matrix treatment of imperfect polarizers.

Provenance (base-corpus facts carried forward, re-verified). The two-transmittance law $I = I_0(k_1\cos^2\theta + k_2\sin^2\theta)$ , the extinction ratio $k_1/k_2$ , and its reduction to Malus's law as $k_1\to1, k_2\to0$ follow Saleh & Teich ch. 6 and Hecht ch. 8. Recomputed from scratch: the crossed-pair transmittance $T_\perp = k_1 k_2$ and parallel-pair $T_\parallel = \tfrac12(k_1^2+k_2^2)$ from the incoherent two-stream model; the decibel conversion $10\log_{10}(k_1/k_2)$ ; and the output degree of polarization $(k_1-k_2)/(k_1+k_2)$ . Worked numbers use $(k_1,k_2)=(0.80,10^{-4})$ for a sheet and $\mathrm{ER}=10^5$ for a prism. History note: the Polaroid dates are distinct and must not be conflated — Land's J-sheet patent is 1929, and H-sheet (the modern iodine-PVA Polaroid) is 1938.

Builds on: Malus's Law · Polarization of Light · Degree of Polarization. Leads to: Birefringence (Nicol and Glan prisms) · Brewster's Angle (reflective polarizers) · Jones Calculus (polarizers as projection matrices) · Optical Isolators.

What is the difference between the principal transmittance k₁ and the extinction ratio?

$k_1$ is how much of the wanted polarization gets through — a brightness number, typically $0.4$ – $0.9$ for sheet polarizers (a single sheet can't exceed $0.5$ for unpolarized input, since half the power is in the blocked component). The extinction ratio $k_1/k_2$ is a quality number: how much more of the wanted than the unwanted polarization passes. Two sheets can share the same $k_1$ but differ a thousandfold in extinction ratio. Rejection, contrast, and output purity all track the extinction ratio, so it — not $k_1$ — is the headline spec.

Why don't crossed real polarizers give perfect darkness?

Because each sheet leaks. The first polarizer leaves a small orthogonal component (fraction $k_2$ of that stream survived), and the second polarizer passes a small fraction $k_2$ of the strong component that reaches it. Following both streams gives a crossed transmittance $T_\perp = k_1 k_2$ , not zero — for a sheet with $k_1 = 0.8, k_2 = 10^{-4}$ that is $8\times10^{-5}$ of the input. To get darker you need a smaller $k_2$ , i.e. a higher extinction ratio, which is exactly why LCD contrast and optical-isolator isolation are polarizer-quality problems.

How do I convert an extinction ratio to decibels, and why bother?

Take ten times the base-ten logarithm: $\mathrm{ER}_{\mathrm{dB}} = 10\log_{10}(k_1/k_2)$ . So $10^3 \to 30$ dB, $10^4 \to 40$ dB, $10^5 \to 50$ dB, $10^6 \to 60$ dB — every decade adds 10 dB. Decibels are used because extinction ratios span many orders of magnitude and because isolation budgets add: cascading two $30$ -dB polarizers with a rotator can give $60$ dB of isolation, a sum that is awkward with bare ratios but natural in dB.

Which polarizer should I use — sheet, prism, or wire grid?

It depends on the trade. A dichroic sheet (Polaroid) is cheap, large-area, wide-angle, and low-extinction ( $10^3$ – $10^4$ ); perfect for displays and sunglasses. A birefringent prism (Glan–Thompson, Glan–Taylor) gives the highest extinction ( $10^5$ – $10^6$ ) and handles high laser power, but over a small aperture and acceptance angle, and it is expensive. A wire grid is broadband (especially in the infrared), works at large apertures and steep angles, reflects rather than absorbs the rejected light (useful for recycling), but has moderate extinction. Match the extinction ratio, aperture, bandwidth, power handling, and cost to the job.

Is a polarizer that "reflects" the rejected light better than one that absorbs it?

Neither is universally better; it changes where the rejected energy goes. A dichroic sheet absorbs the blocked component, turning it into heat — a problem at high laser power, where the sheet can burn. A wire grid or a birefringent prism redirects the blocked component (reflection or walk-off), so it can be dumped safely or even reused. For power handling and thermal management, reflect/redirect wins; for compact, cheap, wide-angle polarizing where the rejected power is small, absorb is fine.

Does the degree of polarization of the output depend on the input?

For unpolarized input, one real polarizer produces output with degree of polarization $(k_1-k_2)/(k_1+k_2)$ , set entirely by the device. For polarized input already aligned near the axis, the output is even purer (little orthogonal component is present to begin with). The device's degree-of-polarization figure is the worst case — the purity it can impose on completely unpolarized light — and it is the number that matters when you use a polarizer to define a clean reference state from an arbitrary beam.