How to compute the Brewster angle: discovered by David Brewster (Scotland 1781-1868) Edinburgh University 1815 observing inclined glass windows eliminating reflections at particular incident angle. Fundamental formula: tan(theta_B) = n2/n1, where theta_B Brewster angle (degrees or radians), n1 incident medium refractive index, n2 refracted medium refractive index. At theta_B reflected light is 100% s-polarized (TE transverse electric, perpendicular to incidence plane), completely eliminating p-polarized component (TM transverse magnetic). Examples: air-water n2/n1=1.333 -> theta_B=53.06°, air-BK7 glass n=1.5168 -> theta_B=56.57°, air-diamond n=2.417 -> theta_B=67.53°.

Essential tool for: polarized sunglasses (Polaroid Cambridge MA invented Edwin Land 1929 - now Bausch&Lomb Rochester NY, Ray-Ban EssilorLuxottica global leader, Oakley Foothill Ranch CA, Maui Jim Peoria IL, Costa Del Mar Daytona Beach FL marine fishing), photography (CPL polarizer filters Hoya Tokyo Japan, B+W Schneider Bad Kreuznach Germany, Marumi Tokyo, Tiffen Hauppauge NY USA), astronomical telescopes (eliminate sky reflections night atmospheric Rayleigh scattering), CD/DVD/Blu-ray laser (Brewster reflection minimizes losses), laser cavities (Brewster windows window CO2 He-Ne lasers Coherent Spectra-Physics), photovoltaic industry (solar cell anti-reflection coating ARC Brewster angle Si-air SunPower First Solar Tempe AZ), polarimetry chemistry (sucrose industry sugarmeters optical rotation).

Brewster angle and Fresnel formulas

Brewster's law (1815): tan(theta_B) = n2/n1. Equivalently: theta_B = atan(n2/n1). For n1=1 (air) and n2 refracted medium: theta_B=atan(n2). At theta_B Snell-refraction: n1*sin(theta_B)=n2*sin(theta_t), and theta_B+theta_t=90° (reflected and refracted rays perpendicular).

Fresnel equations (Augustin-Jean Fresnel France 1823): electric field reflection coefficients for s-polarization (TE) and p-polarization (TM). For s polarization: r_s = (n1*cos(theta_i) - n2*cos(theta_t))/(n1*cos(theta_i) + n2*cos(theta_t)). For p polarization: r_p = (n2*cos(theta_i) - n1*cos(theta_t))/(n2*cos(theta_i) + n1*cos(theta_t)). Intensity reflectance: R_s = r_s^2, R_p = r_p^2. At theta_B: r_p = 0 -> R_p = 0 (TM total elimination). Transmittance: T = 1 - R for each polarization.

Critical angle total internal reflection: for n1>n2 (e.g. water-air): sin(theta_c) = n2/n1, theta_c = asin(n2/n1). Water-air: theta_c=asin(1/1.333)=48.75°. BK7 glass-air: theta_c=asin(1/1.5168)=41.23°. Diamond-air: theta_c=asin(1/2.417)=24.4° (low critical angle = high brilliance). Application: optical fibers cladding-core index difference, snorkeling underwater view "Snell's window" circle of light, gemology diamond brilliant cut 57-58 facets maximum brilliance.

Refractive index and dispersion: n(lambda) varies with wavelength. Sellmeier equation: n^2(lambda) = 1 + sum(B_i*lambda^2/(lambda^2-C_i)). BK7 standard optical glass: B1=1.03961212, B2=0.231792344, B3=1.01046945, C1=0.00600069867, C2=0.0200179144, C3=103.560653 um^2. Abbe number V_d = (n_d - 1)/(n_F - n_C) measures dispersion: V_d>50 crown glass low dispersion, V_d<50 flint high chromatic dispersion.

Stokes parameters polarization: vector [I, Q, U, V] completely describes light polarization. I total intensity, Q diff polarizations 0°/90°, U diff 45°/135°, V circular component. Poincare sphere geometric representation. Mueller matrix 4x4 decomposition optical components model (polarizers, quarter-wave half-wave plates, depolarizers). Software: Polaris Optics CodeV ASAP Synopsys Mountain View CA USA.

History and context

David Brewster (Scotland 1781-1868) Edinburgh optician founder Royal Society Edinburgh 1815 published "On the Laws Which Regulate the Polarisation of Light by Reflexion from Transparent Bodies" Philosophical Transactions Royal Society 105:125 experimentally observing reflected light from inclined glass at particular angle is fully polarized. His observations: 1813 angle of polarization. Inventor kaleidoscope 1816 Edinburgh patent, Brewster stereoscope 1849. Augustin-Jean Fresnel (France 1788-1827) Polytechnique Paris derived 1823 Fresnel equations electric field reflection/refraction coefficients for s/p polarization, had already proposed 1818 wave theory of light beating Newton corpuscular theory. James Clerk Maxwell (Scotland 1831-1879) 1865 electromagnetic unification Maxwell equations light transverse EM wave. Sir George Stokes (Great Britain 1819-1903) Cambridge 1852 introduced Stokes parameters [I,Q,U,V] partial polarization description. Henri Poincare (France 1854-1912) Sorbonne 1892 Poincare sphere geometric polarization representation. Hans Mueller (USA 1900-1965) MIT 1943 Mueller matrices 4x4 polarization propagation through optical elements. R Clark Jones (USA 1916-2004) Polaroid Corp 1941 Jones matrices 2x2 fully polarized light. Edwin Land (USA 1909-1991) MIT-Polaroid Corp Cambridge MA 1929 invented J-Sheet polarizing film polyvinyl alcohol stretched oriented eliminating reflections, founded Polaroid Corporation 1937 now Bausch+Lomb Rochester NY.

Global polarized sunglasses lens industry: EssilorLuxottica Agordo Belluno Italy global leader 100 mln pairs/year (Ray-Ban Persol Oakley Vogue), Safilo Padova Italy eyewear premium (Carrera Polaroid sunglasses), Marcolin Longarone Belluno (Tom Ford Tod's eyewear), Bausch+Lomb Rochester NY USA, Polaroid Eyewear, Maui Jim Peoria IL USA, Costa Del Mar Daytona Beach FL marine fishing, Smith Optics Sun Valley ID USA. Photographic CPL polarizer filters: Hoya Filter Hoya Corp Tokyo Japan global leader, B+W Filter Schneider Optical Bad Kreuznach Germany premium, Marumi Filter Tokyo, Tiffen Filter Hauppauge NY USA, Heliopan Munich Germany, Lee Filters Andover UK cinema. Optical glass industry: Schott AG Mainz Germany (1884) global optical glass leader BK7 SF11, Ohara Sagamihara Japan, CDGM Chengdu China, Sumita Tokyo Japan, Hoya Optical Tokyo, Corning Sullivan Park Corning NY USA Gorilla Glass smartphone leader, Saint-Gobain Paris France. Photovoltaic industry: SunPower San Jose CA, First Solar Tempe AZ, Tesla Solar Buffalo NY, Q CELLS Hanwha Bitterfeld-Wolfen Germany, JinkoSolar China leader. Anti-reflection coating ARC: Schott AG, Edmund Optics Barrington NJ USA, Thorlabs Newton NJ optics distributor major. USA: National Institute Standards Technology NIST Gaithersburg MD precision optical standards, MIT Microsystems Technology Laboratory Cambridge MA, Caltech Pasadena Optical Sciences.

Universities: MIT 6.1610 Optics, Caltech Ph10, Stanford EE142 Optoelectronic Devices, UC Berkeley EE234, Tufts Boston MA, Princeton EE321, Rochester Institute Technology Imaging Science Rochester NY USA Polaroid roots. Astrophysics: Hubble Space Telescope NASA-ESA optical Wide Field Camera 3 polarimetry filters, JWST James Webb Space Telescope NIRCam imaging.

Refractive indices and Brewster angle table

• Vacuum: n=1 exact definition
• Air 0°C 1 atm 589 nm: n=1.000293, theta_B(from vacuum)=45.01°
• Water 20°C 589 nm: n=1.333, theta_B(air-water)=53.06°
• Ice 0°C: n=1.31, theta_B=52.64°
• Ethanol 20°C: n=1.361, theta_B=53.68°
• Glycerol 20°C: n=1.473, theta_B=55.82°
• Olive oil: n=1.467, theta_B=55.71°
• Crown standard glass: n=1.52, theta_B=56.66°
• Schott BK7 optical glass: n=1.5168, theta_B=56.57°
• Fused silica SiO2: n=1.458, theta_B=55.55°
• Crystal quartz: n=1.544, theta_B=57.06°
• Flint dispersive glass: n=1.62, theta_B=58.32°
• Acrylic PMMA: n=1.49, theta_B=56.12°
• Polystyrene: n=1.59, theta_B=57.82°
• Polycarbonate PC: n=1.585, theta_B=57.72°
• Ruby corundum Al2O3: n=1.77, theta_B=60.52°
• Sapphire corundum Al2O3: n=1.77, theta_B=60.52°
• Diamond 589 nm: n=2.417, theta_B=67.53°
• Silicon Si IR 1550 nm: n=3.48, theta_B=73.99°
• Germanium Ge IR: n=4.0, theta_B=75.96°
• GaAs IR: n=3.30, theta_B=73.15°
• Human cornea: n=1.376, theta_B=53.98°
• Critical angle water-air: theta_c=48.75° (total internal reflection)
• Critical angle BK7 glass-air: theta_c=41.23°
• Critical angle diamond-air: theta_c=24.4° (brilliance)

Detailed numerical examples

Example 1 - Air-water interface (lake mirror). Air n1=1.0003, water n2=1.333 (20°C 589 nm). theta_B=atan(1.333/1.0003)=atan(1.333)=53.06°. Sun at 53° height (37° elevation) reflected lake is 100% s-polarized (horizontal). Polarized sunglasses horizontally rejecting eliminate glare. Refracted angle at Brewster: theta_t=90-53.06=36.94°. R_s at Brewster: 7.98% reflected, R_p=0%. Application: photography landscapes lakes seas low sun, CPL polarizer Hoya Tokyo reduces glare. USA fishing Costa Del Mar Florida polarized lenses standard.

Example 2 - Air-Schott BK7 glass optics. Air n1=1, Schott BK7 glass n2=1.5168 (589 nm sodium D-line). theta_B=atan(1.5168)=56.57°. R_s at Brewster=12.1%, R_p=0%, R_total=6.05%. Application: laser cavities Brewster windows He-Ne 632.8 nm, Ar+ 488 nm, CO2 10.6 um, Czerny-Turner spectrometers Andor Belfast UK, Princeton Instruments Trenton NJ USA. Schott AG Mainz Germany global BK7 SF11 optics leader.

Example 3 - Air-diamond gemology. Air n1=1, diamond n2=2.417 (589 nm). theta_B=atan(2.417)=67.53°. Critical angle total internal reflection diamond-air: theta_c=asin(1/2.417)=24.4° (low). Diamond brilliance: brilliant cut 57-58 facets crown-pavilion 34.5°-40.75° angles exploit theta_c=24.4° for multiple total internal reflection. Application: gemology GIA Gemological Institute America Carlsbad CA, IGI International Gemological Institute Antwerp Belgium, De Beers Antwerp-London diamond industry global.

Example 4 - Water-air critical angle total internal reflection. Diver view from water n1=1.333 toward air n2=1. For theta_inc>theta_c=48.75°: total internal reflection 100%. For theta_inc=50° (incidence) >48.75°: sin_t=(1.333/1)*sin(50°)=1.021>1 NO refraction! All R=100% T=0%. Snell's window: cone aperture half-angle theta_c=48.75° view surface above water, beyond reflected lake bottom. Application: underwater photography Nikon Subacquea Olympus underwater housings, snorkeling diving experience Maui HI Caribbean.

Example 5 - Air-Si silicon IR detector telecom 1550 nm. Air n1=1, silicon Si n2=3.48 at 1550 nm (telecom C-band optical fiber). theta_B=atan(3.48)=73.99°. R_s at Brewster=27.1%, R_p=0%, R_total=13.55%. Application: silicon IR detector 1550 nm fiber optics, anti-reflection coating ARC SiO2/Si3N4 multilayer reduce reflection 0.1% solar cell SunPower First Solar Q CELLS, photodetector InGaAs Hamamatsu Tokyo. Telecom backbone 1550 nm Cisco San Jose CA, Juniper Sunnyvale CA, Nokia Espoo Finland.

Limits and considerations

1. Brewster's law valid only for transparent dielectrics (real n). Metals complex n require generalized Fresnel equations.
2. Refractive index varies with wavelength (dispersion): Sellmeier equation n(lambda). Chromatic Brewster 5-10° span visible.
3. Polarization requires polarized light or balanced unpolarized: circular light s/p equal.
4. Macroscopic phenomenon: rough surfaces relative to lambda lose polarization coherence.
5. Fresnel equations assume monochromatic plane wave: short fs laser pulses require time-frequency analysis.
6. Birefringent materials (calcite quartz LiNbO3): double refraction, two ordinary/extraordinary Brewsters.
7. Absorbing materials (metals colored glass): extinction coefficient k added complex n=n+i*k.
8. Critical angle requires n1>n2: air to glass NO theta_c, glass to air YES theta_c.
9. Circular/elliptical polarization: Brewster maintains s-pol but p-pol Fresnel coefficients vary phase.
10. Measurement precision: optical goniometers Newport Irvine CA, Optosigma Tokyo precision ±0.01° typical.

Common errors

1. Confusing n1/n2 vs n2/n1: tan(theta_B)=n2/n1 (refracted/incident), NOT inverse.
2. Computing Brewster angle without specifying wavelength: dispersion varies 5-10°.
3. Confusing s-polarization (TE) with p-polarization (TM): at Brewster R_p=0, R_s residual.
4. Assuming theta_c for n1<n2: critical angle exists only internal reflection.
5. Using n_air=1 exact: air n=1.0003 (0.03% correction precision measurements).
6. Neglecting temperature: dn/dT glass ~10^-6 per °C, water -10^-4 per °C.
7. Assuming Brewster for metals: complex n, reflection does not vanish, only minimum.
8. Confusing Snell's window 48.75° with Brewster water-air 53.06°: different phenomena.
9. Computing R wrong formula: R = ((n2-n1)/(n2+n1))^2 valid ONLY normal incidence 0°.
10. Neglecting initial light polarization: laser polarized vs natural light different applications.

Numbered applications

1. Polarized sunglasses: Maui Jim Peoria IL, Costa Del Mar Daytona Beach FL marine fishing, Bausch+Lomb Rochester NY, Smith Optics Sun Valley ID.
2. CPL photographic filters: Hoya Tokyo Japan global leader, B+W Schneider Bad Kreuznach Germany, Tiffen Hauppauge NY USA premium.
3. Industrial optical glass: Schott AG Mainz Germany (1884) BK7 SF11 leader, Ohara Sagamihara Japan, Hoya Optical Tokyo.
4. Smartphone Gorilla Glass: Corning Sullivan Park NY USA leader globally, Schott Mainz, Asahi AGC Tokyo Japan competitors.
5. Laser cavities: Brewster windows He-Ne CO2 Ar+ lasers, Coherent Santa Clara CA, Spectra-Physics Newport Irvine CA.
6. Photovoltaic ARC: SunPower San Jose CA, First Solar Tempe AZ, Q CELLS Hanwha Germany, JinkoSolar China leader.
7. Diamond gemology: GIA Gemological Institute Carlsbad CA leader, IGI Antwerp Belgium, De Beers London-Antwerp diamond industry.
8. Telecom 1550 nm: Cisco San Jose CA, Juniper Sunnyvale CA, Nokia Espoo Finland, fiber Corning fused silica.
9. Astronomy NASA Hubble JWST: HST Wide Field Camera 3 polarimetry filters, JWST NIRCam imaging California USA.
10. Educational teaching: MIT 6.1610 Optics, Caltech Ph10, Stanford EE142, UC Berkeley EE234, Rochester Institute Imaging Science.

Support

For technical questions or reports: info@calcoi.com. Educational tool for preliminary optics. For professional engineering Brewster D 1815 'On the Laws Which Regulate the Polarisation of Light' Phil Trans R Soc 105:125, Fresnel AJ 1823 'Memoire sur la loi des modifications que la reflexion imprime a la lumiere polarisee', Born M Wolf E 'Principles of Optics' 1999 Cambridge, Hecht E 'Optics' 2017 Pearson, Saleh BEA Teich MC 'Fundamentals of Photonics' 2007 Wiley.

Updated July 2026