Abstract
In photography, changing the f-number by factors of $\sqrt{2}$ is said to change image brightness by halves. Starting from the proportionality of image illuminance to the square of the sine of the image-side half-angle, this paper counts in which way of measuring that halving is exact, how far it misses when the measure is changed, and how small an f-number can be. No new theorem or law is claimed.
First, measuring the half-angle by its sine, illuminance falls exactly by half per stop: setting the f-number to half the reciprocal of the sine of the half-angle, illuminance is inversely proportional to the square of the f-number, and over $11$ steps of $\sqrt{2}$ the ratio departed from $2$ by only $4.4\times10^{-16}$. The ratio between F1 and F16 is $\mathbf{256.000000}$ (Section 2).
Second, at f-number $0.5$ light arrives from the whole image-side hemisphere: the half-angle is then $90$ degrees and the illuminance $\pi$ ($3.141593$); a smaller f-number would need a sine above $1$ and cannot be made in air (Section 2).
Third, measuring the half-angle by its tangent misses on the bright side: the ratio of illuminance measured by tangent to that measured by sine is $0.500000$ at f-number $0.5$, $\mathbf{0.800000}$ at $1$, $0.941176$ at $2$ and $0.999024$ at $16$, and the ratio between F1 and F16 was $\mathbf{205.000000}$ (Section 3).
Fourth, illuminance looks not at the solid angle but at the projected solid angle: the ratio of projected solid angle to the solid angle of the exit cone is $0.500000$ at f-number $0.5$, $0.933013$ at $1$ and $0.984123$ at $2$ (Section 4).
Fifth, the separator is whether the half-angle of the aperture is measured by its sine or by its tangent: measured by the sine, the square law in the f-number is exact and the lower limit of the f-number is $0.5$. Measured by the tangent, the values approach the same on the dark side but depart from the square law the brighter the setting (Section 5).
Keywords: f-number, image illuminance, sine condition, numerical aperture, projected solid angle, photographic lens.
Scope (what is not claimed)
No new theorem or law is claimed: the proportionality of image illuminance to the squared sine of the image-side half-angle, the sine condition, the relation between working f-number and numerical aperture, and projected solid angle are all standard. Only the axis is treated: fall-off of brightness toward the edge of the frame, vignetting, transmittance and reflection losses are not included. No lens is designed: aberrations are not computed; whether the sine condition is satisfied is only set as two ways of measuring. Real lenses are not discussed: f-number markings of products and which lens is brighter are not treated. Air only: the lower limit is stated for an image-side medium of refractive index $1$. Relation to earlier papers: Paper 358 showed that the depth of depth of field is a convention, not a measurement. Here, for brightness, it is counted in which way of measuring the same f-number becomes exact. Paper 757 showed that which of wavelength, numerical aperture and factor buys the resolution decides what the depth of focus pays. Here the same sine as in the numerical aperture is counted as deciding the square law of brightness in the f-number and its lower limit of $0.5$. Paper 918 showed that antenna gain is a ratio measuring how the $4\pi$ of solid angle is shared out. Here it is counted that image illuminance looks not at solid angle but at projected solid angle, and reaches the hemisphere’s projected solid angle $\pi$ at f-number $0.5$. What is added: confirming that illuminance measured by the sine halves exactly per step of $\sqrt{2}$, showing that illuminance reaches $\pi$ at the lower limit of f-number $0.5$, setting down the departures when measured by the tangent, finding the ratio of projected to plain solid angle, and placing the separator on “sine, or tangent”.
1. Notation and the Question
Setting: imaging a surface of luminance $L$, image illuminance can be written with the image-side half-angle $\theta'$ as follows (transmittance $1$). Two ways of relating the f-number $N$ to the half-angle, by sine and by tangent, are set side by side:
$$E=\pi L\sin^2\theta',\qquad \sin\theta'=\frac{1}{2N}\ \Rightarrow\ E=\frac{\pi L}{4N^2},\qquad \tan\theta'=\frac{1}{2N}\ \Rightarrow\ E=\frac{\pi L}{1+4N^2}$$Measuring by sine corresponds to a lens satisfying the sine condition, and by tangent to the geometry of a lens with a flat principal plane. The check sets $L=1$ and varies the f-number from $0.5$ to $16$ in steps of $\sqrt{2}$.
Question: in which way of measuring is the halving of brightness with the square of the f-number exact, and how small can the f-number be?
2. Measuring by the Sine
Computation 2.1: illuminance and half-angle for each f-number.
| f-number | illuminance by sine | illuminance by tangent | tangent over sine | half-angle by sine (deg) | half-angle by tangent (deg) |
|---|---|---|---|---|---|
| $0.500000$ | $3.141593$ | $1.570796$ | $0.500000$ | $90.0000$ | $45.0000$ |
| $0.707107$ | $1.570796$ | $1.047198$ | $0.666667$ | $45.0000$ | $35.2644$ |
| $1.000000$ | $0.785398$ | $0.628319$ | $0.800000$ | $30.0000$ | $26.5651$ |
| $1.414214$ | $0.392699$ | $0.349066$ | $0.888889$ | $20.7048$ | $19.4712$ |
| $2.000000$ | $0.196350$ | $0.184800$ | $0.941176$ | $14.4775$ | $14.0362$ |
| $2.828427$ | $0.098175$ | $0.095200$ | $0.969697$ | $10.1821$ | $10.0250$ |
| $4.000000$ | $0.049087$ | $0.048332$ | $0.984615$ | $7.1808$ | $7.1250$ |
| $5.656854$ | $0.024544$ | $0.024353$ | $0.992248$ | $5.0709$ | $5.0512$ |
| $8.000000$ | $0.012272$ | $0.012224$ | $0.996109$ | $3.5833$ | $3.5763$ |
| $11.313708$ | $0.006136$ | $0.006124$ | $0.998051$ | $2.5330$ | $2.5305$ |
| $16.000000$ | $0.003068$ | $0.003065$ | $0.999024$ | $1.7908$ | $1.7899$ |
Illuminance measured by sine departed from a ratio of $2$ per step of $\sqrt{2}$ by at most $4.4\times10^{-16}$, and the ratio between F1 and F16 was $\mathbf{256.000000}$. The illuminance at f-number $0.5$ matched $\pi$ with difference $0$.
Observation 2.2 measured by the sine, the square law in the f-number is exact and the lower limit is $0.5$. At f-number $0.5$ the half-angle becomes $90$ degrees and light arrives from the whole image-side hemisphere: a smaller f-number asks for a sine above $1$ and cannot be made on the image side in air.
3. Measuring by the Tangent
Computation 3.1: the ratio of illuminance by tangent to illuminance by sine is $4N^2/(1+4N^2)$, matching $\mathbf{0.800000}$ at f-number $1$ with difference $0$. The ratio of illuminance between F1 and F16 was $\mathbf{205.000000}$.
Observation 3.2 measured by the tangent, the brighter the setting, the further from the square law. At f-number $16$ the ratio is $0.999024$ and the two measures are nearly the same, but at $1$ it is $0.8$ and at $0.5$ a half: measured by the tangent, however small the f-number, the half-angle never reaches $90$ degrees and illuminance never reaches $\pi$.
4. Solid Angle and Projected Solid Angle
Computation 4.1: solid angle and projected solid angle of the exit cone (half-angle by sine).
| f-number | solid angle | projected solid angle | projected over solid angle |
|---|---|---|---|
| $0.5$ | $6.283185$ | $3.141593$ | $0.500000$ |
| $1.0$ | $0.841787$ | $0.785398$ | $0.933013$ |
| $2.0$ | $0.199517$ | $0.196350$ | $0.984123$ |
Observation 4.2 illuminance looks not at solid angle but at projected solid angle. The cone at f-number $0.5$ is a hemisphere of solid angle $2\pi$, but light arriving obliquely strikes the image plane obliquely and is weakened, so what counts for illuminance is the projected solid angle $\pi$: the larger the f-number and the narrower the cone, the closer the two angles.
5. The Separator
Observation 5.1 the separator is whether the half-angle of the aperture is measured by its sine or by its tangent. Measured by the sine, illuminance is exactly inversely proportional to the square of the f-number, the lower limit of the f-number is $0.5$, and there illuminance becomes $\pi$: measured by the tangent, the ratio between F1 and F16 becomes $205$ rather than $256$, departing more on the bright side. On the dark side the difference between the two measures vanishes.
Observation 5.2 what is not said. Computing aberrations, brightness at the edge of the frame, transmittance and the markings of real lenses are not treated. What is shown is that the square law in the f-number is exact when measured by the sine, and that then the f-number in air cannot go below $0.5$.
6. Roll Call and Closing
- $E=\pi L\sin^2\theta'$ (Section 1): the squared sine.
- ratio $2$ per step of $\sqrt{2}$; $\mathbf{256.000000}$ between F1 and F16 (Computation 2.1): exact halving.
- illuminance $\pi$ at f-number $0.5$ (Computation 2.1): whole hemisphere, lower limit.
- tangent over sine $\mathbf{0.800000}$ at f-number $1$ (Computation 3.1): departs on the bright side.
- by tangent, F1 over F16 is $\mathbf{205.000000}$ (Computation 3.1): not $256$.
- projected over solid angle $0.500000$, $0.933013$, $0.984123$ (Computation 4.1): projected solid angle counts.
- sine, or tangent (Observation 5.1): the separator; Papers 358, 757, 918.
Closing: Brightness follows the square of the f-number when the half-angle is measured by its sine, and in air the f-number cannot go below $0.5$. Measured by the sine, illuminance halved exactly per step of $\sqrt{2}$, and the ratio between F1 and F16 was $256$: at f-number $0.5$ the half-angle is $90$ degrees and illuminance reaches the hemisphere’s projected solid angle $\pi$; a smaller f-number cannot be made in air. Measured by the tangent, illuminance at f-number $1$ is $0.8$ times as large, and the ratio between F1 and F16 was $205$: the separator is whether the half-angle of the aperture is measured by its sine or by its tangent. Placed among the earlier papers: against the depth side of the f-number shown to be a convention in Paper 358, the exactness of its brightness side was counted; the same sine as the numerical aperture of Paper 757 was shown to set the lower limit; and projected solid angle was set beside the sharing of solid angle in Paper 918. To be honest: nothing is said about which lens is better. In which way of measuring the square law is exact, and no further.
References
- Born, M., & Wolf, E. (1999). Principles of Optics. Cambridge University Press.
- Kingslake, R. (1978). Lens Design Fundamentals. Academic Press.
- Yamagishi, Y. (2026). The “Depth” of Depth of Field Is a Convention, Not a Measurement [Paper 358]. YOI: https://www.yamagishiyuuki.com/yoi/358; DOI: 10.5281/zenodo.22697216
- Yamagishi, Y. (2026). Which of Three Quantities Buys the Resolution Decides What the Depth of Focus Pays [Paper 757]. YOI: https://www.yamagishiyuuki.com/yoi/757; DOI: 10.5281/zenodo.23015169
- Yamagishi, Y. (2026). Antenna Gain Averages to $1$ over All Directions, and the Factor That Turns Beamwidth into Gain Depends on the Power in the Skirts [Paper 918]. YOI: https://www.yamagishiyuuki.com/yoi/918
Reference
Yamagishi, Y. (2026). Brightness Follows the Square of the F-Number When the Half-Angle Is Measured by Its Sine, and in Air the F-Number Cannot Go Below $0.5$ [Paper 923]. YOI: https://www.yamagishiyuuki.com/yoi/923
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 923. This paper is a preprint and has not been peer reviewed.