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.

Numerical apertureunit

Are you curious about how microscope objectives capture finer object structures to produce higher-resolution images? This foundational knowledge article on Numerical Aperture and Light Cone Geometry will give you a sound understanding of the light gathering ability of microscope objectives and how it is expressed through the numerical aperture (NA). An interactive tutorial allows you to visualize changes in the illumination cone as you vary NA values. You will also learn about the role of the refractive index and the limitations of the maximum achievable NA values.

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.

Numerical apertureof microscope

Join the ranks of world-leading microscopists with our expert training courses. Whether you're in academia or industry, a biologist, materials scientist, or somewhere in between, our training courses will help you unlock the full potential of your microscopy skills.

The light gathering ability of a microscope objective is quantitatively expressed in terms of the numerical aperture (NA). The objective’s NA is a measure of its ability to capture image-forming light rays: Higher NA values allow increasingly oblique rays (representing finer object structures) to enter the front lens of the objective, producing a higher-resolution image with greater specimen detail. This interactive tutorial demonstrates the change in numerical aperture light cones displayed by a microscope objective with corresponding changes in numerical aperture. The angular aperture value corresponding to a given NA-value is also depicted here.

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.

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.

n is the refractive index of the media in the object space (between the cover glass and the objective’s front lens) and θ is half the full angular aperture. The value of n varies between 1.0 for air and 1.58 for most immersion media used in optical microscopy. The angular aperture, which varies with the objective focal length, is the maximum angle of image-forming light rays diffracted by the specimen that the front lens of the objective can capture when the specimen is in focus. As the objective focal length decreases, the maximum angle between the specimen and the outer diameter of the objective front lens increases, causing a proportional increase in the angular aperture. From the above equation, it is obvious that the NA increases with both the angular aperture and the refractive index of the imaging medium.

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.

Numerical aperture nacalculator

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.

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.

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.

Numerical apertureformula

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.

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.

Numerical apertureof optical fiber

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.

Numerical Aperturecalculator

Please note that in some cases, you may be prompted to enter your login credentials again to access certain areas or resources within our training platform.

Theoretically, the maximum angular aperture achievable with a dry (air) microscope objective would be 180 degrees, resulting in a value of 90 degrees for the half angle used in the NA equation. The sine of 90 degrees is one, indicating that the numerical aperture is limited not only by the angular aperture but also by the refractive index of the imaging medium. Most microscope objectives are designed to operate with air (refractive index= 1.0) as the imaging medium between the cover glass and the front lens of the objective. This yields a theoretical maximum NA of 1.00. For practical reasons (available working distance), the highest desirable value for the NA of a dry objective is 0.95 (the half angle of the aperture is approximately 72 degrees). Immersion objectives achieve much higher NAs at the expense of free working distance and spherical aberration sensitivity.

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.

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.

Numerical apertureof lens

The tutorial displays a schematic drawing of a microscope objective. The actual angular aperture of the light cone and the corresponding NA value are indicated in the tutorial window. To operate the tutorial, use the Numerical Aperture slider to change the NA value from low (left) to high (right). As you vary the numerical aperture value with the slider, the size and shape of the illumination cone entering the objective’s front lens is altered. The adjustable NA for this tutorial is 0.03 to 0.95. The approximate objective magnification has also been assigned to each NA value.

Numerical apertureof objective lens

The brightness and resolution of an image formed by an objective at a given magnification increases with its NA value, respectively the diameter of the angular aperture (the angle of the light cone collected by the objective). Light rays emanating from the specimen pass through air (or a liquid-based immersion medium) located between the cover glass and the objective’s front lens. The angular aperture is expressed as the angle between the microscope’s optical axis and the direction of the most oblique light rays captured by the objective (see the tutorial figure). Mathematically, the NA is expressed as:

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.

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.

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.

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?

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.

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.