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Transmittance

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Earth's atmospheric transmittance over 1 nautical mile sea level path (infrared region[1]). Because of the natural radiation of the hot atmosphere, the intensity of radiation is different from the transmitted part.
Transmittance of ruby in optical and near-IR spectra. Note the two broad blue and green absorption bands and one narrow absorption band on the wavelength of 694 nm, which is the wavelength of the ruby laser.

Electromagnetic radiation can be affected in several ways by the medium in which it propagates.  It can be scattered, absorbed, and reflected and refracted at discontinuities in the medium.  This page is an overview of the last 3. The transmittance of a material and any surfaces is its effectiveness in transmitting radiant energy; the fraction of the initial (incident) radiation which propagates to a location of interest (often an observation location). This may be described by the transmission coefficient.

Surface Transmittance

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Hemispherical transmittance

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Hemispherical transmittance of a surface, denoted T, is defined as[2]

where

  • Φet is the radiant flux transmitted by that surface into the hemisphere on the opposite side from the incident radiation;
  • Φei is the radiant flux received by that surface.

Hemispheric transmittance may be calculated as an integral over the directional transmittance described below.

Spectral hemispherical transmittance

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Spectral hemispherical transmittance in frequency and spectral hemispherical transmittance in wavelength of a surface, denoted Tν and Tλ respectively, are defined as[2]

where

  • Φe,νt is the spectral radiant flux in frequency transmitted by that surface into the hemisphere on the opposite side from the incident radiation;
  • Φe,νi is the spectral radiant flux in frequency received by that surface;
  • Φe,λt is the spectral radiant flux in wavelength transmitted by that surface into the hemisphere on the opposite side from the incident radiation;
  • Φe,λi is the spectral radiant flux in wavelength received by that surface.

Directional transmittance

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Directional transmittance of a surface, denoted TΩ, is defined as[2]

where

  • Le,Ωt is the radiance transmitted by that surface into the solid angle Ω;
  • Le,Ωi is the radiance received by that surface.

Spectral directional transmittance

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Spectral directional transmittance in frequency and spectral directional transmittance in wavelength of a surface, denoted Tν,Ω and Tλ,Ω respectively, are defined as[2]

where

Luminous transmittance

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In the field of photometry (optics), the luminous transmittance of a filter is a measure of the amount of luminous flux or intensity transmitted by an optical filter. It is generally defined in terms of a standard illuminant (e.g. Illuminant A, Iluminant C, or Illuminant E). The luminous transmittance with respect to the standard illuminant is defined as:

where:

  • is the spectral radiant flux or intensity of the standard illuminant (unspecified magnitude).
  • is the spectral transmittance of the filter
  • is the luminous efficiency function

The luminous transmittance is independent of the magnitude of the flux or intensity of the standard illuminant used to measure it, and is a dimensionless quantity.

Internal Transmittance

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Optical Depth

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By definition, internal transmittance is related to optical depth and to absorbance as

where

  • τ is the optical depth;
  • A is the absorbance.

Beer–Lambert law

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Main article: Beer–Lambert law

The Beer–Lambert law states that, for N attenuating species in the material sample,

where

Attenuation cross section and molar attenuation coefficient are related by

and number density and amount concentration by

where NA is the Avogadro constant.

In case of uniform attenuation, these relations become[3]

Cases of non-uniform attenuation occur in atmospheric science applications and radiation shielding theory for instance.

Other radiometric coefficients

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Radiometry coefficients
Quantity SI units Notes
Name Sym.
Hemispherical emissivity ε — Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity εν
ελ 		— Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity εΩ — Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity εΩ,ν
εΩ,λ 		— Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance A — Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance".
Spectral hemispherical absorptance Aν
Aλ 		— Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance AΩ — Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance AΩ,ν
AΩ,λ 		— Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance".
Hemispherical reflectance R — Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance Rν
Rλ 		— Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance RΩ — Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance RΩ,ν
RΩ,λ 		— Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance T — Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance Tν
Tλ 		— Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance TΩ — Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance TΩ,ν
TΩ,λ 		— Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficient μ m−1 Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficient μν
μλ 		m−1 Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Directional attenuation coefficient μΩ m−1 Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral directional attenuation coefficient μΩ,ν
μΩ,λ 		m−1 Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Transmittance

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Transmittance is the fraction of incident electromagnetic radiation, such as light, that passes through a material without being absorbed or reflected, defined as the ratio of the transmitted intensity ItI_t to the incident intensity I0I_0, expressed as T=It/I0T = I_t / I_0, where TT ranges from 0 (opaque) to 1 (fully transparent) or equivalently as a percentage.[1] This optical property is wavelength-dependent and characterizes the transparency of materials like glass, liquids, and biological tissues across the ultraviolet, visible, and infrared spectra.[2] In spectroscopy and analytical chemistry, transmittance is central to the Beer-Lambert law, which relates it to absorbance A=−log⁡10(T)=ϵclA = -\log_{10}(T) = \epsilon c l, where ϵ\epsilon is the molar absorptivity, cc is the concentration of the absorbing species, and ll is the path length through the sample, enabling quantitative determination of solute concentrations in solutions.[3] Transmittance measurements are typically performed using spectrophotometers, which compare the intensity of light before and after passing through the sample at specific wavelengths to assess material properties or chemical compositions.[4] Transmittance plays a critical role in applications ranging from optical engineering, where it guides the design of lenses, filters, and windows for high transmission in the visible range (400–800 nm), to photovoltaics and biomedical imaging, where it influences energy efficiency and light penetration in encapsulants or tissues.[2][5] Factors affecting transmittance include material thickness, surface reflections, scattering, and absorption bands, with advanced models accounting for these in complex media like the atmosphere or turbid liquids.[6]

Basic Principles

Definition

Transmittance is a fundamental concept in optics and radiometry, defined as the fraction of incident radiant flux that is successfully transmitted through a sample, surface, or medium. It quantifies the effectiveness of the material or interface in allowing radiant energy to pass without being reflected or absorbed. This ratio is dimensionless and ranges from 0 (complete opacity) to 1 (perfect transparency).[7] In the context of radiometry, radiant flux Φe\Phi_e represents the total power carried by electromagnetic radiation, measured in watts (W), encompassing the energy emitted, transferred, or received over time. The incident radiant flux Φei\Phi_e^i is the power approaching the sample, while the transmitted radiant flux Φet\Phi_e^t is the power emerging on the other side. The transmittance TT is mathematically expressed as
T=ΦetΦei, T = \frac{\Phi_e^t}{\Phi_e^i},
where both fluxes are evaluated under the same conditions.[8][9] While transmittance specifically denotes the power or intensity ratio in classical optics, the term transmission coefficient is sometimes used interchangeably but more precisely refers to the amplitude transmission factor in wave optics, with the transmittance being its squared magnitude adjusted for refractive indices and angles. In quantum mechanics, the transmission coefficient often describes the probability of transmission for particles or photons. For non-scattering media, transmittance relates to other optical properties through energy conservation, where T+R+A=1T + R + A = 1, with RR as reflectance and AA as absorptance.[10][7]

Mathematical Representation

Transmittance is conventionally denoted by the symbol $ T $, representing the fraction of incident radiant energy that is transmitted through a material or optical element. The radiant flux, denoted $ \Phi_e ,quantifiesthetotalpowerofelectromagneticradiation,measuredinwatts(W).Subscriptsareusedtodistinguishincidentflux(, quantifies the total power of electromagnetic radiation, measured in watts (W). Subscripts are used to distinguish incident flux ( \Phi_{e,i} ),transmittedflux(), transmitted flux ( \Phi_{e,t} ),andreflectedflux(), and reflected flux ( \Phi_{e,r} $).[11][12] The general mathematical expression for transmittance is given by the ratio of transmitted radiant flux to incident radiant flux:
T=Φe,tΦe,i T = \frac{\Phi_{e,t}}{\Phi_{e,i}}
This formulation applies to the total flux crossing a surface or beam. For directional cases, such as collimated beams, the flux may be expressed per unit solid angle in watts per steradian (W/sr). Transmittance is inherently dimensionless, as it is a ratio of like quantities.[11][12] Extensions of this definition account for other radiometric quantities relevant to specific geometries. For planar illumination, transmittance can be defined in terms of irradiance $ E $ (radiant flux per unit area, in W/m²), where $ T = E_t / E_i $. Similarly, for imaging or extended sources, it relates to radiance $ L $ (radiant flux per unit area per unit solid angle, in W/m²/sr), yielding $ T = L_t / L_i $. These forms assume uniform distribution and conservation along the propagation path.[13][14] The standard formulation assumes negligible scattering within the medium, such that transmitted light consists primarily of directly propagated radiation without diffuse redirection. It typically applies to incoherent light sources, though for coherent illumination (e.g., lasers), interference effects in thin films can modulate transmittance beyond simple absorption. In ideal measurements, the setup isolates transmitted flux without external contributions like ambient light; real measurements may require corrections for scattering or multiple reflections to approximate these conditions.[15][12] Energy conservation in optical systems relates transmittance to complementary quantities: reflectance $ R = \Phi_{e,r} / \Phi_{e,i} $ and absorptance $ A = (\Phi_{e,i} - \Phi_{e,t} - \Phi_{e,r}) / \Phi_{e,i} $. For non-scattering, opaque samples where transmission occurs minimally but is considered, the flux balance yields $ T + R + A = 1 $, ensuring all incident energy is accounted for via transmission, reflection, or absorption. This relation derives directly from the principle that incident flux equals the sum of outgoing (transmitted and reflected) and absorbed components.[13]

Surface Transmittance

Hemispherical Transmittance

Hemispherical transmittance quantifies the fraction of radiant flux that passes through a surface when the incident illumination originates from all directions within a hemisphere, integrating over the projected solid angle to account for the total energy transfer. It is defined as the ratio of the total transmitted radiant flux Φet\Phi_e^t to the total incident radiant flux Φei\Phi_e^i across the hemisphere, assuming uniform or diffuse incidence conditions.[11] In general, for non-uniform radiance distributions, it is expressed as
T=∫hemisphereLt(θ,ϕ)cos⁡θ dΩ∫hemisphereLi(θ,ϕ)cos⁡θ dΩ, T = \frac{\int_{\text{hemisphere}} L^t(\theta, \phi) \cos \theta \, d\Omega}{\int_{\text{hemisphere}} L^i(\theta, \phi) \cos \theta \, d\Omega},
where LtL^t and LiL^i are the transmitted and incident radiances, respectively, θ\theta is the polar angle from the surface normal, ϕ\phi is the azimuthal angle, and dΩ=sin⁡θ dθ dϕd\Omega = \sin \theta \, d\theta \, d\phi is the differential solid angle.[8] This measure is particularly relevant for surface interactions in optics, focusing on the overall energy passage without resolving specific angles. Hemispherical transmittance is commonly applied to opaque surfaces exhibiting partial transmission through interfaces or to thin films where reflection at boundaries dominates the optical behavior, such as in coatings or layered materials. In these scenarios, the value represents the net flux transmission after accounting for interfacial losses, aiding in the design of optical components like mirrors or filters. Key factors influencing hemispherical transmittance include surface roughness, which scatters incident light and alters the reflectance distribution, thereby reducing specular reflection and potentially increasing diffuse transmission in low-absorption materials.[16] Additionally, mismatches in refractive index between the incident medium and the surface lead to reflection governed by the Fresnel equations, where the power transmittance decreases with larger index differences, particularly at oblique angles.[10] For example, a clear soda-lime glass pane under solar illumination exhibits a hemispherical transmittance of approximately 92%, reflecting the high passage of broadband solar flux through the material while minimal absorption and reflection occur at the air-glass interfaces.[17]

Directional Transmittance

Directional transmittance quantifies the transmission of radiant energy through an interface or medium for light incident and transmitted along specific directions, emphasizing the angular dependence that arises due to refraction and reflection at boundaries. It is particularly relevant in scenarios involving ray tracing or directional light propagation, where the overall energy transfer cannot be captured by angle-integrated measures. Unlike broader transmittance definitions, directional transmittance focuses on the ratio of radiance or irradiance in precisely defined incident and transmitted directions, governed by the geometry of the interface.[12] Mathematically, the directional transmittance is expressed as
T(θi,ϕi;θt,ϕt)=Lt(θt,ϕt)Li(θi,ϕi), T(\theta_i, \phi_i; \theta_t, \phi_t) = \frac{L^t(\theta_t, \phi_t)}{L^i(\theta_i, \phi_i)},
where $ L^i(\theta_i, \phi_i) $ is the incident radiance from polar angle $ \theta_i $ and azimuthal angle $ \phi_i $, and $ L^t(\theta_t, \phi_t) $ is the transmitted radiance in the corresponding transmitted directions $ \theta_t $ and $ \phi_t $. The transmitted angles are related to the incident angles via Snell's law: $ n_1 \sin \theta_i = n_2 \sin \theta_t $, where $ n_1 $ and $ n_2 $ are the refractive indices of the incident and transmitting media, respectively; the azimuthal angle typically remains unchanged ($ \phi_t = \phi_i $) for planar interfaces. This formulation assumes conservation of energy and applies to non-scattering, non-absorbing conditions at the boundary.[18][19] The angular dependence of directional transmittance is primarily determined by the Fresnel coefficients, which describe the amplitude reflection and transmission at the interface for s- (perpendicular) and p- (parallel) polarized light. For oblique incidence, the power transmittance $ T $ incorporates these coefficients along with the cosine factors from the Poynting vector: $ T = \frac{n_2 \cos \theta_t}{n_1 \cos \theta_i} |t|^2 $, where $ t $ is the amplitude transmission coefficient. At normal incidence ($ \theta_i = 0 $), the polarization distinction vanishes, and the transmittance simplifies to $ T = \frac{4 n_1 n_2}{(n_1 + n_2)^2} $ for non-absorbing media, representing the maximum transmission efficiency without angular complications.[19][20] For oblique angles, transmittance varies significantly with polarization and incidence angle, peaking at the Brewster angle where p-polarized light experiences zero reflection ($ R_p = 0 $), yielding $ T_p = 1 $ in ideal non-absorbing dielectrics; the Brewster angle is given by $ \theta_B = \tan^{-1}(n_2 / n_1) $. In contrast, s-polarized light shows decreasing transmittance with increasing angle, approaching total internal reflection beyond the critical angle if $ n_2 < n_1 $. This angular sensitivity is crucial for understanding phenomena like glare reduction in polarized sunglasses.[19] A representative example is the transmission of polarized light through a dielectric interface, such as unpolarized light from air ($ n_1 = 1 )incidentonglass() incident on glass ( n_2 = 1.5 $) at the Brewster angle of approximately 56.3°. Here, the p-component transmits fully without reflection, while the s-component partially reflects, resulting in transmitted light that is partially p-polarized and directionally refracted according to Snell's law. For normal incidence on the same interface, $ T \approx 0.96 $, illustrating the reduced losses compared to oblique cases.[19]

Internal Transmittance

Optical Depth

Optical depth, denoted as τ, serves as a logarithmic measure of the attenuation experienced by light propagating through a medium, specifically defined as the negative natural logarithm of the internal transmittance $ T_i $, where $ T_i $ represents the fraction of incident radiant power transmitted through the bulk material excluding losses due to surface reflection or refraction:
τ=−ln⁡(Ti). \tau = -\ln(T_i).
This quantity quantifies the cumulative effect of interactions that remove photons from the beam along the propagation path.[21][22] In its general formulation for inhomogeneous media, the optical depth is expressed as the line integral of the local attenuation coefficient along the path $ s $:
τ=∫α(s) ds, \tau = \int \alpha(s) \, ds,
where $ \alpha(s) $ is the attenuation coefficient at position $ s $, which encompasses both absorption (photon energy conversion to heat or chemical energy) and scattering (photon redirection out of the beam). The distinction between these contributions arises in the Napierian scale (natural logarithm base, typical for optical depth) versus the decadic scale (common logarithm base, used in spectrophotometry), with the former emphasizing exponential decay directly tied to transmittance.[22][23] Optical depth is dimensionless, often expressed in nepers (Np), reflecting its origin in the natural logarithm; equivalently, it can be converted to decibels for certain applications as $ \tau_{db} = 10 \log_{10}(1/T_i) $, yielding approximately 4.343 times the value in nepers. Absorption dominates in scenarios with high material opacity, while scattering prevails in turbid media like aerosols, altering the effective $ \alpha $ based on particle size and wavelength relative to the scattering regime. A practical example occurs in Earth's atmosphere, where the optical depth for sunlight varies from about 0.1 in clear conditions (dominated by molecular Rayleigh scattering) to 10 or more in heavily polluted or dusty environments, depending on wavelength and aerosol loading; for visible light around 500 nm, typical values range from 0.1 to 1 under varying weather.[24][25]

Beer-Lambert Law

The Beer-Lambert law serves as the foundational model for calculating internal transmittance in dilute, homogeneous media, assuming no scattering or other losses beyond absorption. It expresses the internal transmittance $ T_i $ as $ T_i = e^{-\alpha \ell} $, where $ \alpha $ is the linear absorption coefficient (in units of inverse length) and $ \ell $ is the path length through the medium. An equivalent form, commonly used in chemical analysis, is $ T_i = 10^{-\epsilon c \ell} $, where $ \epsilon $ is the molar absorptivity (in liters per mole per centimeter), $ c $ is the molar concentration of the absorbing species, and the base-10 logarithm relates to the definition of absorbance $ A = -\log_{10} T_i = \epsilon c \ell $. This law quantifies how absorbed radiant power leads to exponential decay of transmitted intensity along the optical path.[26][27] The law's development traces back to Johann Heinrich Lambert's 1760 work Photometria, which established that light intensity diminishes exponentially with distance in an absorbing medium, independent of the incident angle for diffuse sources. August Beer extended this in 1852 by incorporating the effect of solute concentration in solutions, demonstrating that absorption scales linearly with both path length and concentration for dilute systems. Beer's contribution, published in Annalen der Physik und Chemie, merged Lambert's geometric optics insight with chemical principles, forming the modern Beer-Lambert law. This optical depth $ \tau = \alpha \ell $ directly corresponds to the exponent in the transmittance formula for uniform media.[26][28] The derivation starts from the infinitesimal attenuation of radiant flux $ \Phi_e $ due to absorption along a differential path $ ds $:
dΦeds=−αΦe. \frac{d\Phi_e}{ds} = -\alpha \Phi_e.
Separating variables and integrating from $ s = 0 $ (incident flux $ \Phi_e(0) $) to $ s = \ell $ (transmitted flux $ \Phi_e(\ell) $) gives
ln⁡(Φe(ℓ)Φe(0))=−αℓ, \ln \left( \frac{\Phi_e(\ell)}{\Phi_e(0)} \right) = -\alpha \ell,
yielding the exponential form $ T_i = \frac{\Phi_e(\ell)}{\Phi_e(0)} = e^{-\alpha \ell} $. For the molar version, $ \alpha = \epsilon c \ln 10 $, leading to the decadic expression after substitution. This assumes monochromatic light and a homogeneous medium where absorption is the sole interaction.[26][27] While powerful for quantitative analysis, the Beer-Lambert law has limitations and applies strictly to dilute, non-scattering media. It breaks down at high concentrations (typically above 0.01 M) due to non-linear effects like intermolecular interactions that alter the effective absorptivity $ \epsilon $, causing deviations from the predicted straight-line calibration. Scattering, such as in turbid solutions, redirects light paths and violates the assumption of unidirectional propagation, while fluorescence re-emits absorbed energy, reducing net attenuation. Validity requires thin samples where $ \alpha \ell \ll 1 $ (absorbance $ A < 1 $) to minimize higher-order effects like stray light or polychromatic deviations in spectrometers.[29][27] A practical example is spectrophotometric analysis of dye solutions, where the concentration of a colored analyte like Congo red is determined by measuring absorbance at its peak wavelength (around 500 nm) in a 1 cm cuvette; with known $ \epsilon \approx 4 \times 10^4 $ L mol⁻¹ cm⁻¹, the law allows solving for $ c = A / (\epsilon \ell) $ from observed $ A $, enabling quality control in textile or pharmaceutical industries.[26]

Spectral and Luminous Transmittance

Spectral Transmittance

Spectral transmittance describes the fraction of incident radiant flux that is transmitted through a material or interface as a function of wavelength λ\lambda, denoted as T(λ)T(\lambda). It is formally defined as the ratio of the differential spectral transmitted radiant flux dΦet(λ)d\Phi_e^t(\lambda) to the differential spectral incident radiant flux dΦei(λ)d\Phi_e^i(\lambda), expressed as
T(λ)=dΦet(λ)dΦei(λ). T(\lambda) = \frac{d\Phi_e^t(\lambda)}{d\Phi_e^i(\lambda)}.
[30] This extends the general transmittance concept by differentiating the radiant fluxes with respect to wavelength, allowing analysis of wavelength-dependent transmission behavior. An equivalent formulation exists in terms of frequency ν\nu, where T(ν)=dΦet(ν)/dΦei(ν)T(\nu) = d\Phi_e^t(\nu) / d\Phi_e^i(\nu), though the wavelength form is more common in optical spectroscopy.[31] Spectral transmittance applies to both surface and internal transmission, capturing variations due to reflection, scattering, or absorption that differ across the electromagnetic spectrum. For internal spectral transmittance within a bulk material, the formulation arises from the wavelength-dependent absorption along the path, given by
T(λ)=e−∫α(λ) ds, T(\lambda) = e^{-\int \alpha(\lambda) \, ds},
[32] where α(λ)\alpha(\lambda) is the spectral absorption coefficient and the integral is over the path length ss. This exponential decay reflects how materials selectively attenuate light at wavelengths corresponding to molecular or atomic transitions. Key features of spectral transmittance include absorption bands, regions of low T(λ)T(\lambda) where energy is absorbed by electronic excitations (common in ultraviolet) or vibrational modes (prominent in infrared), and transparency windows, intervals of high T(λ)T(\lambda) approaching unity where minimal interaction occurs. For instance, soda-lime glass exhibits near-complete transmittance (>90%) in the visible range (400–700 nm) due to a transparency window matching human vision, but drops sharply to opacity in the ultraviolet (below ~300 nm) from electronic absorption and in the near-infrared (beyond ~2.5 μ\mum) from vibrational overtones.[33] Spectral transmittance is measured using a spectrophotometer incorporating a monochromator to isolate narrow wavelength bands from a broadband source, such as a deuterium or tungsten-halogen lamp, and detect the transmitted intensity relative to a reference beam.[34] The monochromator disperses light via a diffraction grating or prism, enabling sequential scanning across the spectrum (typically 200–2500 nm for UV-Vis-NIR instruments), with detectors like photodiodes or photomultipliers quantifying the flux ratio at each λ\lambda. This yields a transmittance spectrum directly convertible to an absorption spectrum via A(λ)=−log⁡10T(λ)A(\lambda) = -\log_{10} T(\lambda), facilitating identification of material composition through characteristic peaks.[7] A representative example is water, which displays high spectral transmittance (>99% for thin layers) throughout the visible range (400–700 nm) owing to low absorption in this window, enabling its clarity for optical applications. However, transmittance falls dramatically in the near-infrared due to absorption bands from O-H vibrations, for example, dropping below 50% around 1200 nm for a 1 cm path length from combination modes, and approaching zero beyond 1400 nm from the first overtone of O-H stretching.[35] This wavelength selectivity underscores water's role in biological and environmental optics, where visible transmission supports photosynthesis while IR opacity limits thermal imaging through aqueous media.

Luminous Transmittance

Luminous transmittance, denoted as τv\tau_v, quantifies the fraction of incident luminous flux that passes through a material, weighted according to the human eye's sensitivity in the visible spectrum under photopic conditions. It is defined by the formula τv=∫V(λ)T(λ)S(λ) dλ∫V(λ)S(λ) dλ\tau_v = \frac{\int V(\lambda) T(\lambda) S(\lambda) \, d\lambda}{\int V(\lambda) S(\lambda) \, d\lambda}, where V(λ)V(\lambda) is the CIE photopic luminosity function, T(λ)T(\lambda) is the spectral transmittance of the material, and S(λ)S(\lambda) is the spectral power distribution of the light source; the constant Km=683K_m = 683 lm/W represents the maximum luminous efficacy at 555 nm but cancels out in the ratio, yielding a dimensionless value tied to luminous flux in lumens.[36][37][11] Unlike radiometric transmittance, which treats all wavelengths equally, luminous transmittance accounts for the eye's peak sensitivity at 555 nm via V(λ)V(\lambda), emphasizing green light while reducing the contribution from red and blue regions; in low-light scotopic conditions, a separate function V′(λ)V'(\lambda) with peak sensitivity at 507 nm applies, though photopic weighting dominates for typical daylight applications.[38][39] This perceptual weighting ensures τv\tau_v reflects visible brightness rather than total energy, making it essential in photometry for assessing transparency in human vision contexts. In practical applications, luminous transmittance is critical for protective eyewear and vehicle glazing, where standards mandate minimum values to balance visibility and safety. For sunglasses, ANSI Z80.3 specifies categories based on τv\tau_v ranges, such as 8–18% for general outdoor use to reduce glare while maintaining color perception.[40] Automotive windshields must achieve τv>70%\tau_v > 70\% per Federal Motor Vehicle Safety Standard No. 205 to ensure driver visibility under normal conditions.[41] A representative example is neutral density filters, which exhibit nearly flat T(λ)T(\lambda) across the visible spectrum, allowing τv\tau_v to approximate the uniform attenuation factor (e.g., an optical density of 0.3 corresponds to τv≈50%\tau_v \approx 50\%) without spectral bias, ideal for photography and machine vision where consistent light reduction is needed.[42]

Related Quantities and Applications

Other Radiometric Coefficients

In radiometry, transmittance is complemented by reflectance and absorptance, which together describe the interaction of radiant flux with a material. Reflectance $ R $ is defined as the ratio of the reflected radiant flux $ \Phi_e^r $ to the incident radiant flux $ \Phi_e^i $, expressed as $ R = \frac{\Phi_e^r}{\Phi_e^i} $. Absorptance $ a $ is similarly the ratio of the absorbed radiant flux $ \Phi_e^a $ to the incident radiant flux, given by $ a = \frac{\Phi_e^a}{\Phi_e^i} $. For non-scattering media, conservation of energy requires that the sum of transmittance $ T $, reflectance $ R $, and absorptance $ a $ equals unity: $ T + R + a = 1 $. These coefficients exhibit various forms depending on geometry and wavelength. Hemispherical reflectance accounts for reflection integrated over the hemisphere, while directional reflectance specifies the incident and reflected directions. Spectral variants, such as $ R_\lambda $ or $ T_\lambda $, describe wavelength-dependent behavior. The table below summarizes key radiometric coefficients related to transmittance.
CoefficientSymbolDefinitionCommon Variants
Transmittance$ T $$ \frac{\Phi_e^t}{\Phi_e^i} $Hemispherical $ T $, directional $ T $, spectral $ T_\lambda $
Reflectance$ R $$ \frac{\Phi_e^r}{\Phi_e^i} $Hemispherical $ R $, directional-hemispherical $ R $, spectral $ R_\lambda $
Absorptance$ a $$ \frac{\Phi_e^a}{\Phi_e^i} $Spectral $ a_\lambda $
Emissivity$ \varepsilon $Ratio of emitted radiance to blackbody radianceSpectral $ \varepsilon_\lambda = a_\lambda $ by Kirchhoff's law
By Kirchhoff's law of thermal radiation, the emissivity $ \varepsilon $ of a surface in thermal equilibrium equals its absorptance $ a $ at each wavelength, $ \varepsilon(\lambda) = a(\lambda) $. This relation links emission properties to absorption and underpins applications in thermal radiation analysis. A related quantity is decadic absorbance, defined as $ A = -\log_{10}(T) $, which quantifies absorption on a logarithmic scale and is particularly useful for volumes where attenuation occurs gradually along the path length, as opposed to surfaces where interactions are primarily at the interface. For surfaces, absorptance directly follows from the energy balance, while in volumes, it integrates over the material thickness, often excluding surface reflections in internal transmittance assessments. These radiometric coefficients were standardized in the 20th century through efforts by the International Commission on Illumination (CIE), which provided foundational definitions for reflectance, transmittance, and absorptance to ensure consistent measurement and application in optics. For example, in an opaque polished metal like silver under visible light, transmittance $ T \approx 0 $, reflectance $ R \approx 0.95 $, and absorptance $ a \approx 0.05 $.

Practical Applications

Transmittance measurements are essential in various fields and are typically performed using spectrophotometers to determine spectral transmittance $ T(\lambda) $, integrating spheres for hemispherical transmittance, and laser-based systems for directional transmittance.[43] The ASTM E903 standard specifies procedures for measuring solar absorptance, reflectance, and transmittance of materials with spectrophotometers equipped with integrating spheres, ensuring accurate hemispherical assessments across wavelengths from 300 to 2500 nm.[43] These methods account for diffuse scattering and provide reliable data for material characterization.[44] In optics, transmittance is critical for designing lenses and filters, where high values ensure minimal light loss and optimal image quality in cameras and eyeglasses.[45] Atmospheric applications involve modeling solar radiation, with clear-sky transmittance often approximated at τ≈0.7\tau \approx 0.7 for broadband solar flux, aiding predictions of energy availability for solar energy systems.[46] In materials science, transmittance testing evaluates polymers for transparency and durability, such as in packaging or displays, by quantifying light passage through thin films.[47] Biomedically, transmittance principles underpin tissue oximetry, where the Beer-Lambert law is applied to assess oxygen saturation in blood via light absorption at specific wavelengths.[48] Modern applications in photovoltaics leverage anti-reflective coatings to achieve transmittance exceeding 95%, enhancing solar cell efficiency by reducing reflection losses at the glass surface.[49] In laser optics, ruby lasers operating at 694 nm require optimized transmittance in the gain medium to maximize pulse energy output, with coatings minimizing losses for applications in holography and medical procedures.[50] Environmental monitoring uses transmittance data to track ozone layer thickness, where increased τ\tau in the UV range signals depletion, leading to higher UV exposure risks.[51] As of 2025, monitoring indicates the ozone layer is recovering, projected to return to 1980 levels by around 2066, gradually reducing UV transmittance in the UV range.[52] Challenges in transmittance measurement include correcting for scattering effects, which can inflate apparent absorption, and accounting for temperature influences on the absorption coefficient α(λ)\alpha(\lambda), potentially altering results by several percent in variable conditions.[53][54] Software tools like HITRAN provide high-resolution spectral databases for simulating transmittance in gaseous media, supporting accurate modeling of atmospheric and molecular interactions.[55] A practical example is windshield testing in automotive regulations, such as in the European Union, where visible luminous transmittance τv\tau_v must exceed 75% to ensure driver visibility and safety compliance.[56]

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