The Science of Photography 
Text and images Copyright © 1998, 1999 John A. Lind




The simplest camera lens (other than a pinhole) consists of a barrel containing a single lens element.  Nearly all camera lenses have three or more elements made of glass and an aperture ring to adjust the diameter of an internal diaphragm.  This diaphragm controls how much light passes through the lens to the film in the camera body.  The great majority for 35mm and Medium Format have a second ring to focus the lens although some very old bodies may perform focusing in the body itself.  Each of these are topics within this section.  On some cameras with leaf shutters, the shutter is mounted inside the lens also.  The leaf shutter is discussed in the Shutter Section.

This section is presented as an overview of lenses, the basics of their features, and how they work.  It is intended for the photographer.  It is not intended to be an in-depth, detailed discussion of modern photography lenses for a camera designer.  Advanced optics is a subject that easily fills college level Physics textbooks.  Consequently there are some simplifications and generalizations to facilitate understanding basic principles.  Some of the equations presented are practical approximations more than sufficient for applied field use and require only Secondary School Basic Algebra.  The slightly more accurate, definitive versions of them are more complex and require advanced college level mathematics.

Lens Elements

A lens with a single piece of glass, such as a magnifying glass or the lens in a pair of glasses, is called a "simple" lens.  Still photography lens designs have contained more than one piece of glass for over 100 years and are called "compound" lenses.  The object is to have all wavelengths of light from a single point converge to a single point on the film when that point is in focus.  If it is out of focus, all the wavelengths need to converge into a circle of confusion (see definition of this below) equally so the same color results throughout the circle.  A lens that can do this properly is called "achromatic."  Inability to do this results in "fringing," typically visible in extreme enlargement with bands of color near an edge of high contrast between two different colors.  Another very important objective for rectilinear lenses (see defnition of this below) is to have straight lines in space appear as straight lines on the film.  There are other types of aberration (coma, astigmatism, etc.) but these two are the most important.

A lens element is a single piece of glass within the lens.  There are different types of glass used for their light transmission properties and how they refract (bend) light.  Different minerals can be added when the glass is being made that change its transmission and refractive properties.  Two or more different types of glass can be ground into elements with the front surface of one exactly matching the rear surface of another.  These can then be cemented together making a "group" within the lens.  The boundary between the different glass types, since they refract light differently, will also refract light at a specific angle.  A single element not cemented to any other is considered a group also.  The diagram in this subsection has six elements in four groups.  The four rear elements are joined together in pairs to form two groups.

Lens coatings are often added to reduce reflection and improve light transmission at an air-glass or glass-glass interface (decreasing one increases the other).  A single coating is typically centered within the visible spectrum giving off a single greenish or yellowish reflection if viewed at the correct angle.  Since a coating only affects a band of wavelengths less than the entire visible spectrum, more expensive lenses have multiple coatings.  Each coating will reflect a different color.  The visible spectrum is divided into segments with each one typically centered within its segment.  Sometimes it is unnecessary to coat all the elements to adequately reduce reflection.

Of historical note is lens coatings were noted to improve lens performance circa 1900, however these coatings were natural deposits on very old lenses as they aged.  The first practical coatings (applied using vacuum deposition) were developed by Carl Zeiss in the 1930's.  Held as a state secret by the German government, they did not appear on camera lenses for the general public until mid-way through World War II.  Widespread use of coated camera lenses did not occur until after World War II.  Multiple coatings followed and their use became more common during the 1970's.

Focal Length

To keep it uncluttered, the figure above uses a simple double-convex lens instead of the typical compound type used for a camera.  Further, it is assumed to be a theoretical "thin" lens; one without any appreciable thickness.  In this case the "front lens node" and "rear lens node" are located inside the single lens element at the same point.  The meaning of the lens nodes will be developed further with a simple "thick lens" under the section about lens nodes.  There are equivalent locations for a compound lens with multiple elements and groups.  The solid lines show "ray paths" of light passing through the lens.  The horizontal dashed line is the lens axis.  The focal length of a lens, f, is the distance behind the rear lens node at which all light ray paths parallel to the lens axis converge (they converge with the lens axis also).  This is also the "infinity" focus distance.


The following are definitions of the various dimensions labeled in the diagram and will be used throughout the remaining discussion about lenses:

f    focal length
O    subject size
I    image size
u    distance from subject to front lens node
v    distance from rear lens node to image

The focal length, f, is the distance between the rear lens node and the point behind the lens at which ray paths parallel to the lens axis prior to entering the lens intersect with the lens axis after leaving the lens.  One such parallel ray path is the top one depicted in the diagram.  The following is a fundamental equation in optics for focal length:

As already explained, if a lens is focused at "infinity," all the ray paths are parallel to the lens axis, and will converge there at a distance equal to the focal length of the lens.  The first equation supports this.  At an "infinite" distance, the first term goes to zero leaving the focal length, f, equal to the distance from the rear lens node to the the image, v.  If a subject distance, u, is known, solving for the image distance, v, can give (or at least estimate) the length of lens tube or bellows required for macro photography requiring extremely close focusing.  Remember that these distances are between subject and front lens node, and film plane and rear lens node, not the front lens ring or lens mounting flange.  For practical use, the apparent location of the diaphragm as viewed from the front of the lens (more on this under the "Nodes") can be used to estimate the location of the rear lens node.  Remember also that units of distance must be the same.  If subject distance is measured in inches, then the focal length and all other distances must be in inches (an inch is exactly 25.4mm).

A second basic set of equations give the image magnification, M, which is the ratio of the image size to the subject size:

If the image (on the film negative or transparency itself) is smaller than the subject, then the magnification is less than one.  If the image is larger than the subject, the magnification is greater than one.

A special condition exists when the subject and image are the same size giving a magnification of one.  Using the first equation, setting u equal to v (by substituting it with v), and solving for v gives:

This means the image distance is twice the focal length behind the rear lens node.  Doing the same with u gives a similar result.  The subject must be twice the focal length in front of the front lens node.  Similar mathematics can be performed with a desired magnification to determine distances required for half size, twice size, etc., subject images on film.  The relationships expressed by these equations are important when precision of the image magnification is required for photometrics.


In the above diagrams:
NPf = Front Nodal Plane NPr = Rear Nodal Plane PPf = Front Principal Plane PPr = Rear Principal Plane
Nf = Front Nodal Point Nr = Rear Nodal Point Pf = Front Principal Point Pr = Rear Principal Point
Solid lines depict actual light paths; dashed lines show the geometry that can be used to find where a ray will emerge.

In reality, lens elements are "thick."  They have a finite thickness.  A lens can be defined in its entirety by its six "cardinal points."  These are the front and rear "focal points," the front and rear "nodes," and the front and rear "principle points."  If a lens has the same material, with the same index of refraction, in front and behind the lens, then the location of the front node is the same as the front principle point, and the location of the rear node is the same as the rear principle point.  In the case of practical lenses used in nearly all photography, the substance on each side of the lens is air.  In discussion of photographic lenses, the terminology of principle points and nodes is often intermixed, because they are at the same locations, even though they are two separate definitions.

An example of an exception is the meniscus lens, used in some microscopes, in which the objective is immersed in a fluid.  For that type of lens, the index of refraction is different at the front and rear surface, and therefore the nodes and principal points will be in different locations along the lens axis.

The left diagram shows the nodal planes and points:
If a light "ray" is aimed at the front nodal point, it will emerge from the lens as if it were traveling through the rear nodal point.

The right diagram shows the principal planes and points:
If a light "ray" enters parallel to the lens axis, it will emerge aimed at the rear focal point.  If a light "ray" passing through the front focal point enters the lens, it will emerge parallel to the lens axis.

Photographic lenses are not only thick, they are compound lenses with a number of elements.  In a compound lens, not only does each individual element have its own focal, nodal and principal points, the compound lens as a whole has them too.


[under construction]


[under construction]

Depth of Field

When a lens is focused to a specific distance, only objects at that distance are in sharpest possible focus.  Anything at any other distance will be at something less than sharpest possible focus.  Because we cannot immediately perceive this degradation with the naked eye, there is a region both farther and closer than this exact distance that appear to be in sharp focus. This is the "Depth of Field" or DOF.

Circle of Confusion

A point theoretically in exact focus produces a point on the film.  Anything at less than exact focus distance will theoretically produce a circle with fuzzy edges on the film.  Many lenses produce something less than a perfect fuzzy circle because of their design (tradeoffs for other desirable traits) or perhaps even damage.  At other than the exact focus distance, all the light "rays" from a point converge is in front of the film if it is farther than focus distance, or behind the film if it is closer than focus distance.  On the film itself, the fuzzy edge circle is called the "Circle of Confusion."  How big this circle can be before the average human eye detects it is no longer a point is the largest acceptable circle of confusion and is the primary parameter that determines DOF.

The largest acceptable circle of confusion is a subjective measure based on largest "practical" print size at a "normal" viewing distance and has been extensively studied.  The generally accepted values for its size on the film is based on film format.  Larger film formats do not need to be enlarged as much as smaller film formats.  For small format 35mm, values range from 0.025mm to 0.033mm and 0.030mm is commonly used.  The smaller, more conservative 0.025mm allows a combination of greater enlargement at a smaller viewing disatance.  What appears to be a greater DOF in a small proof print can create a surprise when enlarged to 8x10 or especially 11x14 when viewed at the same distance as the small proof print; the DOF narrows some with enlargment.  This tutorial uses 0.025mm for small format 35mm; example calculations will be more conservative compared to other sites.  Commonly used maximum circle of confusion diameters for small, medium and large formats:

Max. CoC
 35mm   0.025mm 
 6x4.5 cm   0.050mm 
 6x6 cm   0.060mm 
 6x7 cm   0.065mm 
 6x9 cm   0.075mm 
 4x5 in.   0.150mm 
 5x7 in.   0.200mm 
 8x10 in.   0.300mm

Hyperfocal Distance

[NOTE:  When using the equations presented, remember units of linear measure must be the same!  Conversion of everything to millimeters is recommended.  The inch is defined as exactly 25.4mm.]
The hyperfocal distance is the focus distance for a given focal length, aperture and largest acceptable circle of confusion at which "infinity" is barely in focus.  Stated another way, this is the shortest focus distance at which "infinity" will still have the appearance of being in focus (just barely).  The front of the DOF begins at half the hyperfocal distance which will appear to be just as barely in focus as "infinity" does.  A unique property of the hyperfocal distance is it has the greatest (deepest) possible DOF for a given focal length, aperture and largest acceptable circle of confusion.  This property can be used for creative purposes with scenes having great depth and a desire for as much of that depth as possible to appear in focus.

There are three basic equations for computing the DOF:

  Hyperfocal Distance
f   Lens Focal Length
A   Lens Aperture Setting
c   Maximum Acceptable Circle of Confusion
S   Lens Focus Distance
Sn   Near Depth of Field Limit
Sf   Far Depth of Field Limit

The first finds the hyperfocal distance.  Once the hyperfocal distance is computed, the near and far end of the DOF for any other focus distance can be calculated using the second and third equations respectively.

Field of View

[under construction]


[still under construction]

A rectilinear lens captures how we perceive the world around us and maps flat planes in space to the film.  In doing so it preserves angles and perspectives of angles.  Straight lines in space theoretically map to straight lines on film regardless of location or orientation in space.  No lens is perfect in this regard and deviation from the theoretical mapping is called barrel or pincushion distortion.    For most lenses under most conditions this may be measurable on a bench with magnified photographs of a special lens test chart, but it is not noticeable in the images.  With a few lenses this distortion is enough to be noticeable under certain conditions with high contrast straight lines, especially if there are many parallel ones, not on the lens axis (not running through the center of the image).  While a rectilinear lens maps angles correctly (and perspectives of them) it does not map areas correctly.  This is most noticeable for objects near the edge of an image, especially with wide angle lenses (more details about wide angle lenses are below).


The standard or "normal" focal length is one that approximates the perspective and, to a lesser extent the field of view (not including peripheral vision), of the human eye.  The accepted standards spanning different sizes of film formats is a focal length approximately the same as the diagonal of the film plane, or a diagonal angle of view between 50° and 55°.  In small format 35mm the 24 X 36mm diagonal is about 43mm.  Lenses with focal lengths between 40mm and 55mm are considered "normal" with 50mm overwhelmingly the standard length.  In medium format for 645 (6 x 4.5cm) and 6 x 6cm, the diagonals are 7.5cm and about 8.5cm respectively.  The overwhelming majority of standard lenses for them today are 80mm.  There are still some 75mm lenses made, a length much more commonly used in the past (circa 1930-1960).  The following table shows film format, the diagional, and most popular focal length with its diagonal angle of view for subminiature, small, medium and large formats (length of diagonal and diagonal angle of view are rounded off):
Film Format Diagonal Focal Length Angle of View
110 (13 X 17mm) 21.4mm  26mm  45°
35mm ½-frame (18 X 24mm) 30mm  30mm 53°
APS (16.7 X 30.2mm) 35mm  35mm 52°
35mm (24 X 36mm) 43mm  50mm  47°
4 X 4cm (127) 57mm  60mm  50°
645 (6 X 4.5cm) 75mm  80mm  50°
6 X 6cm 85mm  80mm  56°
6 X 7cm 92mm  90mm  54°
6 X 9cm (70mm) 108mm  100mm  57°
4 X 5 inches 6.4 inches  150mm / 6 inch  56°
8 X 10 inches 12.8 inches  300mm/12 inch  56°
1Also very popular was 28mm. The Olympus PEN F super speed lenses were 40mm, but most of the other PEN cameras were 28, 30 or 32mm.
2APS does not have a clearly defined focal length for standard lenses.  Most APS cameras are fixed focus wide angle or a zoom lens centered around 35mm.
Standard lenses for all the film formats are in or very close to this range with the exception of 110 and 35mm which have a slightly narrower view.  6x6cm is almost always cropped to a 4x5 proof and a rectangular aspect ratio standard print size (8x10, 11x14, or 16x20) reducing its angle of view in the print.

Wide Angle

[under construction]


[under construction]

Spherical (Fisheye)

[still under construction]

A spherical, or fisheye, lens is often said to produce a distorted image.  In reality a theoretical fisheye does not.  Any distortions are the product of tradeoffs or flaws of a less than perfect lens design, or possibly damage to the lens.  A spherical lens does map space to a flat image differently than we perceive the world around us.  A perfect fisheye preserves areas, but does not preserve angles and perspectives of them.  It maps spherical planes in space to a flat plane of film.  Objects, especially those near the edge, appear "warped" as a consequence.  Next time you see an atlas (most often found in the reference section of a library), look for maps of the North and South Poles.  These are typically "Equidistant Polar Projections" and will look much like a fisheye image because it maps a spherical plane in space to a flat map page just as a fisheye lens does.  Not all straight lines in space are warped into curves in a fisheye image.  Any straight running through the center (lens axis) of the image will be straight in the image.  However, they are the only straight lines in space that will be straight in the image.


[under construction]

Full Frame

[under construction]