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A foot-lambert or footlambert (fL, sometimes fl or ft-L) is a unit of luminance in United States customary units and some other unit systems. A foot-lambert equals 1/π or 0.3183 candela per square foot, or 3.426 candela per square meter (the corresponding SI unit). The foot-lambert is named after Johann Heinrich Lambert (1728–1777), a Swiss-German mathematician, physicist and astronomer. It is rarely used by electrical and lighting engineers, who prefer the candela per square foot or candela per square meter units.

The luminance of a perfect Lambertian diffuse reflecting surface in foot-lamberts is equal to the incident illuminance in foot-candles. For real diffuse reflectors, the ratio of luminance to illuminance in these units is roughly equal to the reflectance of the surface. Mathematically,

,

where

is the luminance, in foot-lamberts,
is the illuminance, in foot-candles, and
is the reflectivity, expressed as a fractional number (for example, a grey card with 18% reflectivity would have ).

The foot-lambert is used in the motion picture industry for measuring the luminance of images on a projection screen. The Society of Motion Picture and Television Engineers (SMPTE) recommended, in SMPTE 196M, a screen luminance of 16 foot-lamberts for commercial movie theaters, when measured "open-gate" (i.e. with no film in the projector). (Typical base density of 0.05 yields peak white of about 14 fL.) The current revision of SMPTE 196M specifies 55 candela per square meter (nits).

The foot-lambert is also used in the flight simulation industry to measure the highlight brightness of visual display systems. The minimum required highlight brightness varies based on the type and level of Flight Simulation Training Device (FSTD), but is generally 3–6 foot-lamberts for most devices qualified under Federal Aviation Administration (FAA) or Joint Aviation Authorities (JAA) regulations.[1][2][3]

Military specifications for illuminated switches, panels, and displays, such as MIL-PRF-22885 and SAE AS7788, also require luminance measurements in foot-lamberts. Luminance levels can vary from hundreds of foot-lamberts for sunlight readable switch displays per MIL-PRF-22885 to only a few foot-lamberts in night conditions for panels in accordance with SAE AS7788.

Units of luminance
cd/m2 (SI unit)
≡ nit ≡ lm/m2/sr 		stilb (sb) (CGS unit)
≡ cd/cm2 		apostilb (asb)
≡ blondel 		bril skot (sk) lambert (L) foot-lambert (fL)
= 1 ⁄ π cd/ft2
1 cd/m2 = 1 10−4 π
≈ 3.142 		107 π
≈ 3.142×107 		103 π
≈ 3.142×103 		10−4 π
≈ 3.142×10−4 		0.30482 π
≈ 0.2919
1 sb = 104 1 104 π
≈ 3.142×104 		1011 π
≈ 3.142×1011 		107 π
≈ 3.142×107 		π
≈ 3.142 		30.482 π
≈ 2919
1 asb = 1 ⁄ π
≈ 0.3183 		10−4 ⁄ π
≈ 3.183×10−5 		1 107 103 10−4 0.30482
≈ 0.09290
1 bril = 10−7 ⁄ π
≈ 3.183×10−8 		10−11 ⁄ π
≈ 3.183×10−12 		10−7 1 10−4 10−11 0.30482×10−7
≈ 9.290×10−9
1 sk = 10−3 ⁄ π
≈ 3.183×10−4 		10−7 ⁄ π
≈ 3.183×10−8 		10−3 104 1 10−7 0.30482×10−3
≈ 9.290×10−5
1 L = 104 ⁄ π
≈ 3183 		1 ⁄ π
≈ 0.3183 		104 1011 107 1 0.30482×104
≈ 929.0
1 fL = 1 ⁄ 0.30482 ⁄ π
≈ 3.426 		1 ⁄ 30.482 ⁄ π
≈ 3.426×10−4 		1 ⁄ 0.30482
≈ 10.76 		107 ⁄ 0.30482
≈ 1.076×108 		103 ⁄ 0.30482
≈ 1.076×104 		10−4 ⁄ 0.30482
≈ 1.076×10−3 		1

See also

[edit]

Other units of luminance:

SI photometry quantities
Quantity Unit Dimension
[nb 1] 		Notes
Name Symbol[nb 2] Name Symbol
Luminous energy Qv[nb 3] lumen second lm⋅s T⋅J The lumen second is sometimes called the talbot.
Luminous flux, luminous power Φv[nb 3] lumen (= candela steradian) lm (= cd⋅sr) J Luminous energy per unit time
Luminous intensity Iv candela (= lumen per steradian) cd (= lm/sr) J Luminous flux per unit solid angle
Luminance Lv candela per square metre cd/m2 (= lm/(sr⋅m2)) L−2⋅J Luminous flux per unit solid angle per unit projected source area. The candela per square metre is sometimes called the nit.
Illuminance Ev lux (= lumen per square metre) lx (= lm/m2) L−2⋅J Luminous flux incident on a surface
Luminous exitance, luminous emittance Mv lumen per square metre lm/m2 L−2⋅J Luminous flux emitted from a surface
Luminous exposure Hv lux second lx⋅s L−2⋅T⋅J Time-integrated illuminance
Luminous energy density ωv lumen second per cubic metre lm⋅s/m3 L−3⋅T⋅J
Luminous efficacy (of radiation) K lumen per watt lm/W M−1⋅L−2⋅T3⋅J Ratio of luminous flux to radiant flux
Luminous efficacy (of a source) η[nb 3] lumen per watt lm/W M−1⋅L−2⋅T3⋅J Ratio of luminous flux to power consumption
Luminous efficiency, luminous coefficient V 1 Luminous efficacy normalized by the maximum possible efficacy
See also:
  1. ^ The symbols in this column denote dimensions; "L", "T" and "J" are for length, time and luminous intensity respectively, not the symbols for the units litre, tesla and joule.
  2. ^ Standards organizations recommend that photometric quantities be denoted with a subscript "v" (for "visual") to avoid confusion with radiometric or photon quantities. For example: USA Standard Letter Symbols for Illuminating Engineering USAS Z7.1-1967, Y10.18-1967
  3. ^ a b c Alternative symbols sometimes seen: W for luminous energy, P or F for luminous flux, and ρ for luminous efficacy of a source.

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Foot-lambert

View on Grokipedia
from Grokipedia
The foot-lambert (symbol: fL) is a unit of luminance in the English system of photometry, defined as the luminance produced by a perfectly diffusing surface uniformly illuminated at one foot-candle (1 lumen per square foot), which equates to 1/π candela per square foot (approximately 0.3183 cd/ft²). This unit corresponds to approximately 3.426 candela per square meter (cd/m²) in the International System of Units (SI). Equivalent to one lumen per square foot for a perfectly diffusing reflector under ideal conditions, the foot-lambert quantifies the brightness of surfaces as perceived by the human eye. Historically rooted in mid-20th-century photometric practices, the foot-lambert derives from the lambert (a centimeter-based unit equal to 1/π cd/cm²) but is scaled for imperial measurements, with widespread adoption in the United States since around 1948 for applications in and visibility. It gained prominence in illumination engineering, where a surface illuminated at one yields a luminance of one foot-lambert for a perfect diffuser, facilitating straightforward conversions in non-SI contexts. Although authoritative bodies like the National Institute of Standards and Technology (NIST) now discourage its use in favor of the SI per square meter due to the global shift toward metric standards since the 1979 redefinition of the , the foot-lambert persists in legacy systems and specific industries. In practical applications, the foot-lambert is commonly employed to measure screen in cinema projection and display technologies, where typical values range from 10 to 20 fL for optimal viewing conditions, such as 14 fL in controlled evaluations of laser-illuminated projectors to assess image quality and speckle effects. For instance, cinema screens are often calibrated to achieve 16 fL or higher to ensure visibility in darkened auditoriums, balancing with factors like viewer and content type. Its equivalence to nits (cd/m²) allows conversions like 200 fL equaling about 685 nits, aiding in mixed-unit environments such as street lighting and visibility assessments. Calibration of meters using the foot-lambert typically involves blackbody sources at around 2856 , with NIST providing traceable standards at uncertainties below 1% for reliable measurements.

Definition

Formal Definition

The foot-lambert (symbolized as fL or fl) is a unit of , which quantifies the photometric brightness of a surface or source in the US customary system of units. It is precisely defined as the produced by a perfectly diffusing (Lambertian) surface that emits or reflects a total of 1 lumen per , with the flux distributed uniformly over the hemisphere above the surface. This definition yields the mathematical expression 1 fL=1π1 \, \mathrm{fL} = \frac{1}{\pi} candela per square foot (cd/ft2\mathrm{cd/ft^2}), or approximately 0.31831 cd/ft20.31831 \, \mathrm{cd/ft^2}. The relation arises from the fundamental photometric derivation for luminance LL of a Lambertian surface, given by L=IAL = \frac{I}{A} where II is the luminous intensity in candela and AA is the area in square feet, adjusted by the hemispherical emission factor such that the exitance (flux per unit area) equals πL\pi L.

Physical Interpretation

The foot-lambert (fL) serves as a unit of in photometry, quantifying the perceived brightness of a surface resulting from the interaction of incident light with that surface. It specifically represents the luminance achieved by a perfectly diffusing (Lambertian) surface when uniformly illuminated by 1 of , where the surface reflects or transmits light equally in all directions following . This unit links directly to human visual perception by measuring luminous intensity per unit projected area in a manner that aligns with the eye's sensitivity to direction and intensity, particularly under photopic (daylight) conditions where the retina's cells dominate sensation. Unlike , which describes incoming flux per unit area on a surface, in foot-lamberts characterizes the outgoing emanating from the surface toward the observer, thus better capturing the subjective experience of . For instance, a uniformly reflecting screen receiving 1 lumen per of total distributed over the hemispherical would appear 1 fL bright to an observer, illustrating how the unit integrates surface properties and light distribution to model perceived surface glow. This interpretation stems from the formal definition of 1 fL as approximately 1/π per , emphasizing its photometric foundation.

Unit Equivalents and Conversions

SI Unit Equivalents

The foot-lambert (fL) is equivalent to approximately 3.426 / in SI units, derived from its as 1/π per (/ft²). To perform the conversion, begin with the base : 1 fL = \frac{1}{\pi} /ft², where \pi ≈ 3.1415926535 yields approximately 0.31831 /ft². Next, account for the area unit difference, as 1 ft² = 0.09290304 exactly (or equivalently, 1 = 10.76391041671 ft²). Thus, multiply the /ft² value by the number of square feet per square meter: \frac{1}{\pi} \times 10.76391041671 ≈ 3.426259 /. This process ensures precise bridging between imperial and metric systems for measurements. The following table provides conversions for common foot-lambert values, including those relevant to display and projection standards:
Foot-lamberts (fL) per square meter (cd/m²)
13.426
1034.26
1654.82
These conversions are essential for international standardization in scientific, , and photometric applications, as the SI unit (cd/m²) is the globally adopted measure maintained by bodies like NIST to ensure consistency across borders.

Relation to Other Legacy Units

The foot-lambert (fL) is a legacy unit of in the imperial system, analogous to the lambert (L), a CGS unit defined as the luminance of a perfectly diffusing surface with an intensity of 1/π candela per square centimeter, equivalent to approximately 3183 cd/m². Due to the area scaling between square centimeters and square feet (1 ft² ≈ 929.0304 cm²), the direct conversion is 1 fL = 0.001076391 lambert, or conversely, 1 lambert ≈ 929.0304 fL. This relationship arises because both units incorporate the 1/π factor for Lambertian surfaces but differ in base area units, with the foot-lambert tailored for U.S. practices that favored foot-based measurements in and projection systems. Another related CGS unit is the apostilb (asb), also known as the blondel, defined as the luminance of a perfectly diffusing surface emitting 1/π candela per square meter, or approximately 0.3183 cd/m², making it 0.0001 lambert. The foot-lambert converts to about 10.76391 apostilbs, reflecting the larger square foot area relative to the square meter (1 ft² ≈ 0.092903 m²). In the CGS system, the stilb (sb), defined simply as 1 cd/cm² or 10,000 cd/m², serves as a brighter reference without the 1/π diffusion factor, equating to approximately π lamberts or 2918.64 fL. While the foot-lambert measures (brightness of a surface), it is sometimes contrasted with the (fc), an imperial unit of (light incident on a surface, in lumens per ), highlighting the distinction in photometric quantities despite shared imperial roots. The foot-lambert emerged in U.S. to provide a consistent imperial counterpart for calculations in applications like screen , mirroring the foot-candle's role in .
UnitSymbolEquivalent in fLSI Equivalent (cd/m²)
LambertL1 L ≈ 929.0304 fL≈ 3183.10
Apostilbasb1 asb ≈ 0.092903 fL≈ 0.3183
Stilbsb1 sb ≈ 2918.64 fL10,000
Foot-lambertfL1 fL≈ 3.426

Historical Development

Origins in Photometry

The roots of the foot-lambert lie in 19th-century advancements in photometry, which built upon Johann Heinrich Lambert's seminal 1760 publication Photometria sive de mensura et gradibus luminis, colorum et umbrae. In this work, Lambert formulated the cosine law of emission, stating that the radiant intensity observed from an ideal diffusely emitting surface is proportional to the cosine of the angle between the observer's and the surface normal. This principle provided a foundational framework for quantifying light emission and reflection, influencing subsequent efforts to define standardized units for and related quantities. These developments culminated in the proposal of the lambert as a unit of within the centimeter-gram-second (cgs) system, to measure the of emitting or reflecting surfaces in a manner consistent with . The lambert represented one lumen per square centimeter divided by π, emphasizing the unit's ties to hemispherical light distribution from diffuse sources. As photometric practices expanded in the late , particularly with the rise of electric , the need for compatible units in non-metric systems became evident, especially in engineering contexts outside . The foot-lambert emerged between 1920 and 1925 as an imperial counterpart to the lambert, adapting it to foot-based measurements for in English-unit systems prevalent in American lighting . This unit was developed alongside the , a measure of formalized around 1909–1910 by the Illuminating Engineering Society to quantify light falling on a surface in lumens per . The foot-lambert maintained conceptual consistency with the , such that for a perfectly diffusing reflector, the in foot-lamberts equals the incident in foot-candles, facilitating unified calculations in customary units. Post-World War I, the foot-lambert gained traction in burgeoning applications like automobile headlight design and indoor illumination standards, where precise photometric evaluation supported innovations in safety and architectural amid rapid . These fields demanded reliable, practical units to assess beam patterns and , aligning with the era's emphasis on standardized practices in the U.S.

Standardization Efforts

The Illuminating Engineering Society (IES), founded in 1906, played a pivotal role in adopting the foot-lambert as a standard unit for measurement in its technical handbooks and reports during the 1920s and 1930s, promoting its use for consistent reporting in illumination and photometric applications. By the early , this adoption was formalized through IES efforts, culminating in the 1941 IES nomenclature report approved by the American Standards Association as ASA Z7.1-1942, which integrated the foot-lambert into official photometric terminology for . In the international arena, the International Commission on Illumination (CIE) addressed photometric standardization in its 1948 recommendations, redefining the candela as the luminous intensity of 1/60 cm² of a blackbody radiator at the temperature of freezing platinum (approximately 60 cd/cm²), which established the precise equivalence of 1 foot-lambert to 1/π cd/ft² in US practices despite growing advocacy for metric units. This definition aligned with earlier 1930s US standards, where the foot-lambert was fixed at 1/π cd/ft² to reflect the luminance of a perfect diffuser under 1 foot-candle illumination, ensuring compatibility with English-unit photometry. Key milestones included the foot-lambert's prominent role in the 1947 IES Lighting Handbook, published under ANSI/IES auspices, which incorporated it into comprehensive standards for interior, exterior, and industrial lighting calculations to enhance and efficiency. During , the unit appeared in specifications for assessment in night operations and equipment design, supporting photometric evaluations in low-light environments. Postwar evolution involved minor refinements in the 1950s by IES and NIST to harmonize with the 1948 candela redefinition, adjusting calibration methods for greater precision while maintaining the foot-lambert's core value; the unit continued in widespread US engineering use until the 1960s push for SI adoption.

Applications

Projection Systems

The foot-lambert serves as a key metric for screen brightness in cinema and theater projection systems, where controlled ambient lighting is essential for image quality. The Society of Motion Picture and Television Engineers (SMPTE) recommends a luminance of 16 foot-lamberts for commercial theater screens to achieve optimal contrast and visibility in darkened environments. This standard, outlined in SMPTE 196M, equates to approximately 55 candela per square meter, facilitating international comparisons while maintaining the legacy unit's role in U.S.-centric specifications. This practice arose amid advancements in arc lamp technology and screen materials, allowing precise calibration of projector output against screen reflectance. In projection setups, foot-lamberts are calculated based on projector lumens, screen gain, and area, using the formula: fL=projector lumens×screen gainscreen area in ft2×π\text{fL} = \frac{\text{projector lumens} \times \text{screen gain}}{\text{screen area in ft}^2 \times \pi} This equation accounts for the lambertian diffusion of light on the screen, where the π factor derives from the unit's definition as luminance per steradian. Screen gain, typically ranging from 1.0 for matte white surfaces to higher values for directional materials, amplifies brightness but must balance uniformity. The unit persists in modern digital cinema through Digital Cinema Initiatives (DCI) standards, which reference 14 foot-lamberts (±3 foot-lamberts) as the target for screen in 2K and 4K systems, ensuring compatibility with legacy film workflows. In home theater applications, projectors are designed to achieve 12–22 foot-lamberts on typical screens, aligning with SMPTE guidelines for immersive viewing in light-controlled rooms. However, issues like hotspotting—where higher-gain screens create brighter central areas due to —can distort uniform foot-lambert measurements, reducing off-axis viewing quality and necessitating low-gain alternatives for even illumination.

Illumination Engineering

In illumination , foot-lamberts serve as a key metric for assessing surface in static lighting designs, ensuring visual comfort by controlling levels to minimize while supporting task performance in interiors such as offices and factories. The Illuminating Engineering Society (IES) guidelines emphasize maintaining luminance ratios where task surfaces exhibit 3 to 10 times the of surrounding areas, with recommended levels for workspace surfaces typically ranging from 90 to 150 foot-lamberts to achieve near-maximum and reduce . For example, in office desk areas, surface luminance targets of around 100 foot-lamberts on work planes, combined with lower surrounding levels (e.g., 30-50 foot-lamberts on walls), promote uniform adaptation and comfort during prolonged tasks like reading or , as derived from reflectance-adjusted calculations in early standards. For and static displays, foot-lamberts quantify the emitted or reflected essential for in varied ambient conditions, guiding designs for billboards, LED panels, and highway markers. Preferred levels for illuminated signs fall between 10 and 20 foot-lamberts to balance visibility and discomfort , particularly in low-ambient rural settings, while applications for signs often target 100 foot-lamberts or higher to ensure against bright backgrounds. These specifications help engineers select materials and illumination sources that maintain contrast without excessive veiling reflections, enhancing safety for motorists. Measurement techniques in the field rely on photometers calibrated directly in foot-lamberts to evaluate and sources during installation and testing. These instruments, often cosine-corrected for accurate angular response, quantify veiling reflections— that reduces contrast on tasks—with tolerable levels capped at 18 foot-lamberts to preserve visual performance in engineered environments. In calculations, foot-lambert values integrate with models to predict discomfort, allowing adjustments like diffuser placement for optimal uniformity. A notable early occurred in factory , where foot-lambert metrics standardized evaluations during the shift to fluorescent systems. In , implementing 100 foot-lamberts for drawing-in tasks improved worker accuracy and speed by enhancing contrast sensitivity, as documented in IES evaluations linking uniform to reduced errors and higher output in industrial settings. This approach, outlined in the 1947 IES Lighting Handbook, influenced widespread retrofits in U.S. factories, correlating brightness-controlled environments with up to 20% gains in visual task efficiency.

Modern Context

Shift to SI Units

The adoption of the (SI) in 1960 by the 11th General Conference on Weights and Measures (CGPM), with support from the (ISO) and the (IEC), established the (cd) as the base unit for , promoting the candela per square meter (cd/m²) as the coherent derived unit for in photometry to ensure international consistency. This shift aimed to unify measurements across scientific and engineering disciplines, replacing disparate legacy units like the foot-lambert (fL) with a decimal-based system better suited for global collaboration and precision. The foot-lambert is equivalent to 3.426 cd/m². In the , the of 1975 formalized a voluntary national policy to coordinate and encourage the transition to SI units across industries, including illumination , without imposing mandates that could disrupt established practices. This created the Metric Board to oversee , highlighting rationales such as enhanced compatibility in and reduced conversion errors in technical specifications. The Commission Internationale de l'Éclairage (CIE) reinforced this global movement through its 1970s publications on physical photometry, recommending SI units like cd/m² over legacy ones such as the foot-lambert to promote scientific consistency and in lighting standards worldwide. These reports emphasized the benefits for cross-border and , where non-SI units could introduce inconsistencies in exchange. Despite these efforts, the transition encountered challenges, particularly in U.S. industries reliant on long-standing tools and conventions calibrated to , leading to gradual rather than immediate adoption. The Illuminating Engineering Society (IES) addressed this by adopting a policy in 1979 to prioritize SI units and updating its handbooks throughout the to incorporate dual-unit presentations, facilitating partial conversions in professional standards. By the , the foot-lambert saw diminished application in formal , publications, and international standards, though it persisted in certain legacy software interfaces and specifications due to compatibility needs. This decline reflected broader success in embedding SI units within photometric practices, aligning with global norms while acknowledging practical hurdles in full replacement.

Persistent Usage Areas

Despite the widespread adoption of SI units, the foot-lambert persists in specific U.S.-centric industries, particularly , where (FAA) specifications for and signs require an average of 10 to 30 foot-lamberts (fL) for visibility and identifiability up to 800 feet. This unit also remains standard in flight for calibrating visual display systems, with instruments typically measured in foot-lamberts or equivalent nits (1 fL = 3.426 nits) to ensure realistic highlight brightness. regulations, such as those from the General Civil Aviation Authority, further reference foot-lamberts in defining requirements for Level A flight simulators. In software and measurement tools, the foot-lambert continues for compatibility with older U.S. standards, as seen in engineering simulation programs like SPEOS Pro, which quantifies in foot-lamberts alongside per square meter. Photometers, such as the Gould-Bass DLM-1500 Digital , directly output measurements in foot-lamberts for ranges of 380-700 nanometers, supporting legacy in photometric applications. Niche applications in North American high-end (AV) integration and systems maintain foot-lambert usage, often alongside nits (cd/m²), for specifying screen in client projects and reviews. For instance, buyer's guides and evaluations reference foot-lamberts to assess , targeting 16-30 fL for optimal theater viewing. This practice aids compatibility with traditional motion picture standards while interpreting modern equipment outputs. Although a gradual phase-out is anticipated with SI unit dominance, the foot-lambert retains value for analyzing historical photometric data in these domains.

References

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  48. https://www.projectorcentral.com/home_theater_projector_buyers_guide-Powell.htm
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