Relation of F‑numbers to the Aperture

F‑numbers and the aperture have an inverse relationship

Hi there, my name is Paul, and this is Expo­sure Ther­a­py. In this video, I’ll explain the rea­son for the inverse numer­i­cal rela­tion­ship between f‑numbers and the aper­ture. This rela­tion­ship is a wide­spread point of con­fu­sion for many begin­ner pho­tog­ra­phers, who regard it as irra­tional or need­less­ly com­plex. My goal is to dis­pel the mys­tery around f‑numbers and demon­strate why they’re a per­fect­ly rea­son­able method for express­ing how the aper­ture affects exposure.

Under­stand­ing the rela­tion­ship between pic­ture bright­ness and both the shut­ter speed and ISO is straight­for­ward for stu­dents learn­ing the basics of pho­tog­ra­phy. Shut­ter speed is expressed numer­i­cal­ly in time units, with the most com­mon being frac­tions of a sec­ond; longer dura­tions result in brighter pic­tures, and short­er dura­tions result in dark­er pic­tures. ISO is also expressed numer­i­cal­ly; big­ger num­bers pro­duce brighter pho­tos, and small­er num­bers make dark­er pho­tos. 

In both cas­es, the rela­tion­ship between the set­ting and its effect on pic­ture bright­ness is easy to under­stand because there’s a pos­i­tive cor­re­la­tion, and they move in tan­dem. For exam­ple, when you dou­ble the expo­sure dura­tion, it dou­bles the bright­ness; when you halve the ISO, it halves the bright­ness. It’s a sim­ple rela­tion­ship that stu­dents in my pho­tog­ra­phy work­shops grasp with ease. 

The inverse relationship between f‑numbers and the aperture is confusing for many beginners

Unfor­tu­nate­ly, the rela­tion­ship between f‑numbers, aper­ture size, and pic­ture bright­ness is not as imme­di­ate­ly intu­itive. Begin­ners are con­fused by the neg­a­tive (or inverse) rela­tion­ship between f‑numbers and aper­ture size. In addi­tion, they have a hard time under­stand­ing why big­ger f‑numbers rep­re­sent small­er aper­tures that reduce bright­ness, and small­er f‑numbers define larg­er aper­tures that increase bright­ness. 

The best way to address this is by start­ing with the basics. Inside every inter­change­able lens is a ring of over­lap­ping blades col­lec­tive­ly known as an iris diaphragm or iris. Expand­ing or con­tract­ing the blades adjusts the open­ing in the cen­tre of the iris, called the aper­ture. 

Introducing the Entrance Pupil

When you hold a lens up and look at the aper­ture, what you’re see­ing is tech­ni­cal­ly called the “entrance pupil.” The entrance pupil is the opti­cal image of the phys­i­cal aper­ture as seen through the front of the lens. This dis­tinc­tion mat­ters because when you look at the front of a lens, you see the aper­ture through mul­ti­ple lay­ers of glass that affect its mag­ni­fi­ca­tion and per­ceived loca­tion in space com­pared to the phys­i­cal open­ing in the iris. For the sake of sim­plic­i­ty, I’ll use “aper­ture” when refer­ring to both the set­ting and the phys­i­cal open­ing and “entrance pupil” in ref­er­ence to dimensions.

Chang­ing the size of the aper­ture adjusts the inten­si­ty of light pass­ing through the lens. Increas­ing the aperture’s size allows more light to pass through the lens, increas­ing expo­sure and cre­at­ing a brighter pic­ture. Con­verse­ly, decreas­ing the aperture’s size reduces how much light pass­es through the lens, reduc­ing expo­sure and result­ing in a dark­er pho­to. 

Why are apertures expressed using f‑numbers?

We express aper­ture val­ues using f‑numbers and not as the mea­sured size of the entrance pupil, such as its diam­e­ter, radius, or area, because it neglects the essen­tial role of focal length. This can be demon­strat­ed with a thought exercise.

Let’s pre­tend we have two lens­es attached to iden­ti­cal cam­eras: one lens is 50 mm and the oth­er is 100 mm, and both have entrance pupils with 25 mm diam­e­ters. Since their entrance pupils are iden­ti­cal in size, an equal amount of light enters each lens. How­ev­er, because the focal length of the 100 mm lens is twice that of the 50 mm lens, the light pass­ing through it has to trav­el twice the dis­tance to reach its camera’s image sen­sor, which pro­duces a dark­er image. 

Reduc­tion in bright­ness occurs because light has the prop­er­ty of spread­ing out as it recedes from its source, and from the per­spec­tive of your camera’s image sen­sor, this source is the point inside the lens from which focal length is mea­sured. This trait of light to dif­fuse out­wards is described by the Inverse Square Law, which states that inten­si­ty is inverse­ly pro­por­tion­al to the square of the dis­tance. In this exam­ple, the inverse square law informs us that the 100 mm lens expos­es its camera’s image sen­sor to 1/4 the light com­pared to the 50 mm lens because it’s twice as long. This occurs because one over two squared equals one-quarter.

The 100 mm lens can pro­vide an expo­sure equal to its 50 mm coun­ter­part by open­ing its aper­ture to col­lect four times more light, assum­ing its aper­ture can open that much. Since aper­tures are rough­ly cir­cu­lar, we can deter­mine how big they should be by cal­cu­lat­ing the area of a cir­cle. An entrance pupil with a 25 mm diam­e­ter has an area of about 491 mm^2. The 100 mm lens would need an entrance pupil with an area of 1,964 mm^2, which is formed by a cir­cle with a 50 mm diam­e­ter. Sim­ple, right?

F‑numbers express ratios

For­tu­nate­ly, pho­tog­ra­phers don’t need to per­form such cal­cu­la­tions to take pic­tures! That’s because hid­den with­in these num­bers is a straight­for­ward rela­tion­ship. For exam­ple, notice how the expo­sure pro­duced by the 50 mm lens with a 25 mm entrance pupil is iden­ti­cal to the 100 mm lens with a 50 mm entrance pupil. This is because in both cas­es, the ratio of the focal length to the entrance pupil diam­e­ter is 2:1. 

This is pre­cise­ly why the f‑number is some­times called the f‑ratio. The f‑number express­es a ratio of the lens focal length to the diam­e­ter of the entrance pupil, and it’s defined by the equa­tion N=ƒ/D. Thus, the f‑number equals the focal length divid­ed by the entrance pupil diam­e­ter. It can also be mod­i­fied to solve for the entrance pupil diam­e­ter using the equa­tion D=ƒ/N. Thus, the entrance pupil diam­e­ter equals the focal length divid­ed by the f‑number. 

These equa­tions demon­strate that choos­ing the same f‑number on a lens of any focal length will result in the same amount of light pass­ing through the lens. They also explain the inverse rela­tion­ship between f‑numbers and expo­sure. For a giv­en focal length, as the aperture’s size increas­es, the ratio decreas­es, and vice ver­sa. 

A 50 mm lens set to ƒ/4 will have an entrance pupil diam­e­ter of 12.5 mm—because 50 divid­ed by 12.5 equals 4. A 24 mm lens set to ƒ/8 will have an entrance pupil diam­e­ter of 3 mm. Some lens­es can open to ƒ1.0, in which case the entrance pupil diam­e­ter and focal length are equal. 

The f‑number scale

The stan­dard f‑number scale is: 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, and so on. The dif­fer­ence in expo­sure between adja­cent num­bers is one stop, which means that it either dou­bles or halves the amount of light pass­ing through the lens depend­ing on whether you’re open­ing or clos­ing the aper­ture. How­ev­er, the numer­ic sequence grows by a fac­tor of about 1.4 or shrinks by a fac­tor of about 0.7. 

Most pho­tog­ra­phers sim­ply com­mit the stan­dard f‑number scale to mem­o­ry. How­ev­er, if you’re hav­ing trou­ble, a more straight­for­ward method is to remem­ber just the first two numbers—1 and 1.4—because the rest of the scale is an iter­a­tion of dou­bling each in alter­nat­ing order. The next f‑number is always dou­ble the pre­vi­ous one. So the num­ber after ƒ/1.4 is dou­ble of ƒ/1, which is ƒ2. Like­wise, the num­ber after ƒ/2 is dou­ble of ƒ/1.4, which is ƒ/2.8.  And on and on it goes.

Last­ly, dou­bling the f‑number, such as chang­ing it from ƒ/2.8 to ƒ/5.6, reduces pic­ture bright­ness by one-quar­ter. And con­verse­ly, halv­ing the f‑number, such as adjust­ing from ƒ/8 to ƒ/4, increas­es pic­ture bright­ness four times. 

I hope this helped you under­stand the inverse numer­i­cal rela­tion­ship between f‑numbers and their effect on the aper­ture. If you have requests for future top­ics, let me know in the com­ments, and I’ll address them in future videos. In the mean­time, you can learn more about pho­tog­ra­phy on ExposureTherapy.ca. See you next time.

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