DPanswers

A primer on sensor sizes (crop factor)

What effect do the sensor size have on images

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Table of Contents

• Introduction
• Field of View
• Perspective
• Depth of Field
• Diffraction
• F-Number
• Hand Holding
• Macro Ratio
• DOF Examples

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Introduction

The negative of a 135-format film (aka. 35 mm film) has an image area that is 36 mm by 24 mm. Most digital cameras have sensors that are smaller than this (figure 1). The amount you need to multiply the diagonal of the digital sensor with to match the diagonal of 35 mm film (aka. FX or 135-format) is usually called the digital sensor's “crop factor”¹⁾.

There exists digital cameras where the sensor is roughly the same size as the 135-format film negative, for example the Canon EOS 5D II, the Nikon D700, and the Sony Alpha A-900. We say that these cameras have a crop factor of 1.0x, and refer to them as cameras with FX-format sensors.

The sensors in most DSLR bodies from Nikon, Fujifilm, Pentax and Sony have a crop factor of 1.5x. We refer to them as cameras with DX-format sensors.

Canon use sensors that have a crop factor of 1.0x, 1.3x and 1.6x. Sigma's DSLR bodies have a crop factor of 1.7x. Cameras built according to the Four-Thirds standard (Olympus, Panasonic and some Leicas) have a crop factor of 2.0x. Digital consumer compacts have sensors with an even larger crop factor. For instance, the 2/3" type CCD used in a number of popular digital compacts has a crop factor of 3.9x, and a 1/1.8" type CCD has a crop factor of 4.8x.

The crop factor is a physical property of the image sensor. Some people become confused by the crop because they have observed that there exists DSLR lenses that designed to be used on bodies with cropped sensors (e.g. the lenses from Nikon carrying the “DX” designation, or the lenses from Canon carrying the “EF-S” designation). However, the only significance of such designations is that the lens is designed with a reduced image circle that makes it unsuitable for use on a camera with an FX-format sensor. The focal length of such lenses are the same as the focal length of lenses designed for the FX-format. In other words, the Nikkor 35 mm DX (designed for a DX-format sensor) and the Nikkor 35 mm AF-D (designed for an FX-format sensor) will cover exactly the same field of view when placed on a camera with a DX-sized sensor such as the Nikon D3100. In both cases, the field of view will be 44°. This is what most people will call a “normal” field of view.

Below I discuss the effects of the crop factor in terms of field of view, perspective, depth of field, diffraction, f-number, hand holding, and finally in terms of macro ratio.

Field of View

The field of view (FOV) of a lens is the angular cone extending from the focal plane of the camera into space. The FOV depends on two things, the diagonal of the sensor (D) and the focal length (f) of the lens. The formula below shows how to compute the diagonal FOV (the answer will be in degrees or radians, depending upon what unit the atan function puts out).

FOV = 2 x atan (D / 2 / f)
(D = diagonal of the sensor, f = focal length of the lens)

The table below show the results of computing the diagonal FOV for three different lenses on three different sensor sizes. The first lens is a 135 mm lens used on a camera with an FX-format sensor (i.e. a crop factor of 1.0x), the second lens is a 90 mm lens used on a camera with an DX-format sensor (i.e. a crop factor of 1.5x), and the third lens is a 28 mm lens used on a camera with a 1/1.8" type CCD sensor (i.e. a crop factor of 4.8x).

Sensor	Crop f.	D (mm)	f (mm)	FOV (rad.)	FOV (deg.)
FX/135-format	1.0x	43.3	135	0.318	18.2
DX-format	1.5x	28.4	90	0.313	17.9
1/1.8" type CCD	4.8x	8.9	28	0.316	18.1

The table shows that all three lenses in this case will produce about the same field of view: 18°. Lenses with a more narrow FOV of view than 34° is generally regarded as telephoto lenses. I.e.:given a small enough sensor, a 28 mm lens is a telephoto lens.

Some photographers, however, have worked for a long time with 135-format film and are used to thinking about FOV in terms of focal lengths of lenses attached to 35 mm film cameras instead of degrees. For them, the crop factor is a convenient way of computing the 35 mm film “equivalent” or “effective” focal length from the real focal length. Those photographers recommend that you multiply the real focal length with the crop factor for whatever type of camera you are using to get the 35 mm film “equivalent” or “effective” focal length.

Example: Given that you use a camera with sensor that is DX-format and a 90 mm lens, the FX/35 mm film “equivalent” focal length of this combination is:

90 mm x 1.5 = 135 mm.

Note, however, that the real focal length of the lens does not change. The crop factor multiplicator is just a convenient device to help a photographer that is used to working with 35 mm film to visualize the FOV when using a certain lens on a digital camera with an unfamiliar sensor size.

An alternative way of looking at this is to start with the common names used for different classes of lenses that provides the photographer with a different perspective (i.e. superwide, wide, normal, short tele, tele and supertele), and how these classes of lenses maps onto specific focal lengths depending upon sensor size. This is done in the two tables below, based upon a suggestion from Audun Sjøseth.

The table shows the focal length (in millimeters) for for a number of common DSLR formats. The rows labeled “FOV” shows the field of view in degrees for the focal lengths listed. The top FOV row shows diagonal field of view, and the bottom FOV shows horizontal field of view for sensors with a 3:2 aspect ratio (i.e. all except Four-Thirds, which has a 4:3 aspect ratio). You can use the table to figure out the focal lengths to look for if you want a specific class of lens, given the sensor size of the camera you want to use the lens on.

Format	Crop f.	Superwide		Wide		Normal			Short tele		Tele		Supertele
FOV (diag.):		114°	84°	75°	63°	57°	47°	37°	34°	18°	16°	6.2°	5.0°	3.1°
FX:	1.0x	14.0	24	28	35	40	50	65	70	135	150	400	500	800
Canon (1.3x):	1.3x	10.8	18	22	27	31	38	50	54	104	115	308	385	615
DX:	1.5x	9.3	16	19	23	27	33	43	47	90	100	267	333	533
Canon (1.6x):	1.6x	8.8	15	18	22	25	31	41	44	84	94	250	313	500
Sigma:	1.7x	8.2	14	16	21	24	29	38	41	79	88	235	294	471
Four-Thirds:	2.0x	7.0	12	14	18	20	25	33	35	68	75	200	250	400
FOV (hor.):		104°	74°	65°	54°	48°	40°	31°	29°	15°	14°	5.2°	4.1°	2.6°

When using the table, you will need to match focal length to the nearest focal length that is actually available. E.g. if you want a “normal” fixed focal length lens, matching the popular 50 mm on a camera with an FX-format sensor on a camera with a DX-format sensor, the table tells you that an exact match is 33 mm. However, a 33 mm lens does not exist. In this case, you should pick a 35 mm or a 30 mm.

Let's say that you want to buy a superwide class lens. You can see from the table that for a camera with an FX-format sensor, you need to find a lens where the focal length is 24 mm or shorter. If your camera uses a DX-format sensor, you need to use a lens that is 16 mm or shorter for the same superwide effect.

This means that a lens such as the Nikkor 14-24 mm will be clearly classified as superwide for use on a camera with a FX-format sensor, but only marginally so for a camera with a DX-format sensor. If we use this Nikkor with an adapter on a camera with a Four-Thirds sensor, it will be classified as wide-to-normal (not superwide).

We can also see from the table that for a camera with a DX-format sensor, you would need a lens with a zoom range 9.3 mm to 16 mm to match the FOV of the Nikkor 14-24 mm when used on a FX-format sensor. Unfortunately, such a lens does not exist for the DX-format, so one must use something like the Sigma 10-20 mm instead. This lens, however, does not go as wide as 114° on the wide end. On a camera with a DX-format sensor, f=10 mm gives a diagonal FOV=110°.

While there exists lenses with zoom ranges to goes from wide to telephoto (e.g. the Canon EF 28-300mm f/3.5-5.6 L IS USM and the Nikon 18-200mm f/3.5-5.6G IF-ED AF-S DX VR), the compromises that is inherent in such designs make many photographers build a collection of quality zooms that covers the range from superwide (FOV ≈114°) to telephoto (FOV ≈12°) spread over three lenses. The table below shows the typical zoom lenses in such a collection. The table also shows the diagonal FOV at both ends of the range.

Format	Crop f.	FOV	Superwide zoom	Standard zoom	Telephoto zoom	FOV
FX:	1.0x	114°	14-24 mm	24-70 mm	70-200 mm	12.4°
DX:	1.5x	110°	10-20 mm	17-55 mm	50-135 mm	12.0°
Canon (1.6x):	1.6x	107°	10-22 mm	17-55 mm	50-135 mm	11.4°
Four-Thirds:	2.0x	114°	7-14 mm	14-35 mm	35-100 mm	12.4°

Currently, users of cameras with FX-format sensors have access to original zooms with a maximum aperture of f/2.8 in the FOV-range from superwide) to telephoto. For users of smaller sensors, third party lenses must be used to cover at least part of this range if one want wide or fast zooms.

Perspective

Some people believe that “perspective” in photography is determined by the focal length of the lens used. They will argue that cer­tain focal lengths have a certain, inherent perspective or “look” that will be present irrespective of the size of the sensor. In other words, they will argue that a 135 mm lens will always give the “compressed” look one gets from using a telephoto lens, and a 28 mm lens will always give the “expanded” look of a wide-angle lens.

This is wrong. There is nothing special about the focal length. Perspective in a photograph is deter­mined by one thing: the position of the photographer relative to the scene. (To be precise, perspective is dermined by the distance from the front nodal point to the scene).

The reason people believe that a specific focal length effect a certain look is because the narrow FOV of a telephoto lens forces the photographer to take up a position far away from the scene, and the large FOV of a wide-angle lens allows the photographer to move closer. It is these differences in position, and not the focal length as such, that effect the difference in perspective.

It follows that as long as you stay in the same postion, you get the same perspective no matter what focal length lens you put on the camera.

Below is a demonstration of this. It shows two photographs of the same scene. The first is taken with a 135 mm lens, the second with a 28 mm lens. The reason they show the same perspective is because the 28 mm lens is used on a camera with a much smaller sensor. The second photograph can be considered to be a 7.18 mm by 5.32 mm crop from the middle of a 36 mm by 24 mm film frame.

Depth of Field

One of the things that seems to confuse a lot of photographers, is something called “depth of field“. In this section of the essay, I shall try to drill down on how depth of field behave when subjected to variations in sensor size and other parameters.

For starters, consider the following two statements:

1. You will get a more shallow DOF on a camera with an FX-format sensor than on a camera with a DX-format sensor if you photograph the same scene from the same point using both formats and change the lens to one with a longer focal length when using FX-format to keep the FOV constant – and also keep all other parameters (aperture, print size, distance to subject) constant.
2. You will get a deeper DOF on a camera with an FX-format sensor than on a camera with a DX-format sensor if you photograph the same scene from the same point using both formats and you change the FOV by using the same lens and the same focal length on both bodies – and also keep all other parameters (aperture, print size, distance to subject) constant.

True or false? You will find the answer at the end of this section. But before you look up the answer, try to work this out for yourself.

DOF Defintion

According to Leslie Stroebel et al, Basic Photographic Materials and Processes, 2nd ed., Focal Press, 2000, depth of field (DOF) is defined as the range of object distances within which objects are imaged with acceptable sharpness.

Of course, only one single plane will be in perfect focus. But because humans do not have perfect eyesight, and because lenses and sensors do not have infinite resolution, there is a region surrounding the point in perfect focus that most viewers will deem to be in focus. An old rule of thumb says the region of acceptable sharpness extends from 1/3 in front of perfect focus plane to 2/3 behind it. However, this rule of thumb is only strictly true at 1/3 of the hyperfocal distance. In the macro region, the perfect focus plane is in the middle of the region of acceptable sharpness, and when the lens focused at the hyperfocal distance; then region of acceptable sharpness extends from 1/2 the focusing distance to infinity.

Most of the confusion surrounding the term DOF in photographic circles stems from a failure to understand what DOF is, and is not.

DOF is not an objective, intrinsic or inherent property of a film negative or a digital image file. DOF is mainly determined by magnification. A postcard sized and poster sized print of the same negative or file will show different DOF if both are viewed from the same distance. The smaller print will appear to have a greater DOF. DOF is a property of an photographic image that depends upon, and varies with, the context of presentation, including such things as the physical size of the printed image, the distance of the observer relative to the printed image, and even the visual acuity of the observer. The DOF numbers one can find in manufacturers DOF-tables or engraved upon lenses are not universal and ubiquitous. They are just guidelines for what the DOF will be like, based upon assumptions about the how the DOF in an average print will be perceived when it is viewed by an average observer from an average distance.

DOF-shift with identical FOV

The measure for DOF most people are interested in when moving between equipment with different sensor sizes, is the shift in DOF that happens at identical FOV. As explained in the FOV-section, to maintain FOV we need to change the focal length by the amout indicated by the crop factor. For instance, a f=90 mm lens gives us a FOV of 27° on a camera with a FX-format sensor. To get the same 27° FOV on a camera with a DX-format sensor (crop factor 1.5x), we must use a lens with 90 mm/1.5x = 60 mm focal length. The shift in DOF is also determined by the crop factor. To get the same DOF on both cameras, we need to open up the aperture on the camera with the smaller sensor to the f-number we get when we divide the f-number on the camera with the FX-format sensor with the crop factor. Example: If we use f=90 mm, f/4.0 on the camera with the FX-format sensor, we need to open up to 4.0/1.5 = f/2.67 on the f=60 mm lens fitted to the body with the DX-format sensor. to get an identical DOF. This constitutes a shift of 1.2 EV. The table below shows the aperture shift needed to maintain DOF, comparing a number of popular sensor sizes to the FX-format.

Shift in DOF between FX-format and other sensor sizes, same FOV.

Type	Crop	Ø (mm)	WxH (mm)	Examples	Shift (EV)
1/1.8"	4.8x	8.9	7.2x5.3	Canon G5	4.5
2/3"	3.9x	11.0	8.8x6.6	Canon Pro 1	4.0
4/3	2.0x	21.8	17.4x13.1	Olympus E-1	2.0
?	1.7x	24.9	20.7x13.8	Sigma SD14	1.6
?	1.6x	27.0	22.5x15.0	Canon EOS 30D	1.3
DX	1.5x	28.4	23.7x15.6	Nikon D90	1.2
APS-C	1.4x	30.1	25.1x16.7	Nikon Pronea 6i, Canon EOS IX	1.0
APS-P	1.4x	31.7	30.2x9.5	Nikon Pronea 6i, Canon EOS IX	0.9
?	1.3x	34.3	28.7x18.7	Canon EOS 1D Mk III	0.7
APS-H	1.3x	34.5	30.2x16.7	Nikon Pronea 6i, Canon EOS IX	0.7
FX	1.0x	43.3	36.0x24.0	Canon EOS 5D, Nikon D700	0.0

When you examine the table above, you should see why it is so difficult to get a shal­low DOF on compact digicams with small sensors. The shift on a camera with a 1/1.8" type sensor is 4.5 EV. This means that a Canon Powerhot G5, at its maximum aperture f/2.0 will have the same DOF as a camera with an FX-format sensor have at f/9.6, given the same FOV.

What this means is that there is no way to duplicate both the FOV and the very shallow DOF that you get when you use a 85 mm f/1.4 portrait lens at full bore on a camera with a FX-format sensor, on a camera with a DX-format sensor. In theory, you should get the same effect by using a 57 mm f/0.9 lens – but no such lens exist.

The converse, however, is not true. As R.N. Clark explains in his article The Depth-of-Field Myth and Digital Cameras, the very deep DOF of small-sensor cameras can also be obtained by using a camera with a larger sensor, just by upping the ISO and stop­ping down the lens. Because of the better noise characteristics of large sensors, and because small sensor cameras are more diffraction limited than large sensor cameras, we find that when making the images from two cameras with different sized sensors, and arranging for them being identical in terms of resolution, FOV, exposure time, and signal-to-noise ratio, we find that the DOF is also identical. Example: To reproduce the deep DOF of a Canon Powershot G5 (1/1.8" type sensor) stopped down to f/8.0 (it's minimum aperture), we must stop down the aperture of a DSLR with a FX-format sensor a further 4.5 stops. To maintain the same shutter speed, we must also increase the ISO on the DSLR by 4.5 stops. I.e.: if we use ISO 100 and f/8.0 with the small sensor, we must use ISO 2300 and f/38 on the camera with the FX-format sensor. This will result in both images having identical DOF, and very similar image quality. If one does the math, it nicely works out that one get the same amount of diffraction, and the same amount of noise, in both images.

DOF formula

If we want to calculate numbers for the DOF, there exists a number of different models that let us do this, given certain variables, such as aperture, focal length, subject distance, and imager size. They all have one thing in common: They recognize that DOF is a subjective property depending upon context.

Common for all models, however, is that they have a notion about tolerable amount of blur in an image. “Blur” that is smaller than this tolerance will be perceived by an human observer as being “sharp”. Th perception of DOF is, in other words, caused by blur that is so small that it can not be detected by a human observer.

German lens manufacturer Carl Zeiss had this to say about the tolerable amount of blur in their newsletter Camera Lens News No. 1 (1997):

A certain amount of blur is supposed to be tolerable. According to international standards the degree of blur tolerable is defined as 1/1000th of the camera format diagonal, as the normally satisfactory value. With 35 mm format and its 43 mm diagonal only 1/1500th is deemed tolerable, resulting in 43 mm/1500 [= 0.029 mm] ≈ 0.030 mm = 30 µm of blur.

The tolerable amount of blur blur projected onto the film or sensor plane is often referred to as the Circle of Confusion or CoC. Understanding the CoC is essential to understand how DOF works. As Zeiss explaims, it can be computed by dividing the length of the diagonal of the negative or the camera's digital sensor with a constant. Vice versa, if we know the diagonal and the CoC, we can calculate the value of the constant. In this article, I will refer to this constant as the z-constant in honour of Zeiss.

CoC = D / z-constant
z-constant = D / CoC

In the quote above, Zeiss says that they think CoC should be set equal to 1/1500th of the diagonal, which (rounded) gives a CoC equal to 30 µm for a 43 mm diagonal capture. Zeiss does not say why they think that 30 µm is a suitable value for the CoC.

As noted in introduction, these numbers are derived from assumptions about what constitutes an “average” print and viewer. To cut a long story short: From experiments, Zeiss knows that a persion of average eyesight in good light is capable of resolving 5 lines per millimeter or 200 µm. If this detail exists on a a 30 cm print that is enlarged from a 43 mm capture (i.e. a ≈ 7x enlargement) the detail measures 200 µm/7 ≈ 29 µm on the original imager. In the quoted portion from Camera Lens Use, this number is just rounded up to become 30 µm.

Some online DOF-calculators uses a z-constant set equal to 1730, and Canon's online DOF-calculator appearently uses 1443. However, in this article we shall stick to Zeiss' original suggestion and use 1500.

If we set the z-constant equal to 1500, we find that CoC for FX/135-format film (43.3 mm diagonal) is 29 µm, that the CoC for a DX-format digital sensor (28.4 mm diagonal) is 19 µm, and that the CoC for a 1/1.8" type digital sensor (8.9 mm diagonal) it is 6 µm.

The formula below shows how to calculate the DOF for any focal length, aperture subject distance and sensor size. It is based upon an approximate gaussian model of an optical system, taken from Allen R. Greenleaf: Photographic Optics, MacMillan, New York, 1950, pp. 25-27. (Notation: f is the focal length, d is the subject distance, CoC is the circle of confusion, and N is the aperture expressed as an f-number.)

This simple model of an optical system assumes that singular points are infinitely small, that all lenses are perfect and symmetrical, that diffraction and airy disks do not exist, and that cross-talk, sensor pixel pitch, bayer interpolation and film grain does not interfere with resolution.

While by no means totally accurate, the simple gaussian model will suffice for the purposes of this essay, which is to predict how sensors of different physical sizes will impact upon the DOF of a photograph. For digital sen­sors, it should be noted that this model assumes that the CoC is larger than the sampling limit according to the Nyquist theorem (i.e. larger than twice the photosite pitch) . This contraint is, however, satisfied for all modern digital cameras with a pixel count of 6 Mpx or more.

As can be easily seen, this is a hyperbolic function that converges towards zero in the macro region. To be precise, it is zero when the subject distance (d) is equal to the focal length (f). It converges towards infinity as the distance (d) increases. The value of the distance d when the function becomes infinty is known as the hyperfocal distance (H), and can be computed as follows:

H = f²/(CoC x N)

Also note that at smaller apertures, resolution is limited by diffraction and not by the CoC. All the computations below are done for aperture equal to f/2, which is not diffraction limited, even for the smallest sensors (10 mm diagonal) plotted. However, a sensor of that size will become diffraction limited already at f/4 in a 10 Mpx camera.

DOF plots

Below is five plots that is intended to demonstrate how depth of field (DOF) changes with various parameters. Aperture is kept constant in all five plots.

All the calculations used to generate the plots below is based upon the gaussian model of the optical system presented above. I have used a z-constant of 1500 to compute the CoC.

Figure 4 shows how DOF changes when we vary the sensor dimension, but keep the field of view (FOV) constant. The focal length is at all times kept is equal the sensor diagonal (this translates into a so-called “normal” lens). Aperture is kept constant at f/2.0 and distance to scene is also constant at 1 meter.

Figure 4 below clearly shows that given the same field of view, and distance to scene, the wider focal length that must be used with a small sensor size (f=10 mm) will result in a deeper DOF than a lens that must be used with a large diagonal sensor (f=45 mm).

If we, instead, keep the focal length constant, but vary the sensor size, there is a strict linear relation between sensor diameter and DOF. I.e. a larger sensor also results in a larger DOF (given that we keep all other factors: focal length, aperture, print size and distance to scene, constant). This relationship for a f=23 mm lens at f/2.0, with distance to scene 1 meter, is plotted in figure 5 below.

Note that if we keep the focal length constant, DOF actually decreases with smaller sensor diagonals. The reason digital compacts has large DOFs is not because a smaller sensor gives a larger DOF, but because the very short focal lengths of the lenses these compact cameras are equipped with, gives you a large DOF.

Too see how the models used for figures 4 and 5 behave in reality, see my DOF examples page. It contains some images that demonstrates the effect of sensor size or film size on depth of field.

Figure 6 is a 3D plot combining the data from figure 4 and 5. I.e. it shows how the DOF changes when both the focal length and the sensor diagonal vary. As in the previous two figures, the vertical Y-axis shows the DOF. The X-axis shows the focal length (as in figure 3), and the sensor diagonal is now plotted indpendent of focal length along the Z-axis.

Figure 7 shows how DOF changes when we keep the focal length constant and maintain constant coverage of our main subject (i.e. we change the camera-subject distance to make sure that our main subject always fills the same percentage of the frame) while varying the sensor size. In this simulation, we use a lens with a constant focal length equal to 50 mm and a constant aperture of f/2.0. In this particular simulation, subject distance vary from 4.7 m (10 mm sensor) to 1.1 m (45 mm sensor). The simulation shows that given the same coverage, the longer distance that must be used with a small sensor size (10 mm) will result in a deeper DOF than at the distance that must be used with a large sensor (45 mm).

Michael Reichmann has posted an article on his Luminous Landscape website where he argues that if you the change focal length from wide angle to telephoto, and at the same time change camera to subject distance so that the main subject fills the same percentage of the frame, then at any given aperture all lenses will give the same depth of field. His article comes with a set of photographs where he attempts to visualize this.

Figure 8 below is a simulation of Reichmann's experiment where I've set up my mathematical DOF-model so that a main subject measuring 34 cm (Reichmann's Gremlin) is always filling 23 % of the frame by automatically changing subject distance when the focal length changes. I've also set a constant aperture equal to f/5.6 on a (simulated) camera with an FX-format sensor. The model indicates that the subject distance must be less than a meter when f=17 mm and almost 14 meters from it when f=400 mm.

The result is shown in figure 8 and shows that Reichmann is almost right. But at the wide angle end, the DOF is not constant as he claims. Instead, it changes rapidly. His experiment is unfortunately badly set up, and he also makes the mistake of comparing apples and oranges. For some reason, he sets up his scene so that this main reference target (the hand puppet) is no longer visible when he goes wider than 50 mm. To remedy this, he takes a crop from his 17 mm shot (showing the distant tower), and enlarges the cropped portion much more than the other photographs in his comparison set. Since we know that the degree of enlargement is one of the factors that determines the perceived DOF, it doesn't make sense to compare this enlarged crop to other photos that has not been subject to the same cropping and enlargement.

My model reports a DOF spanning 1400 mm when f=17 mm;, and this is reduced to half, 700 mm, when f=26 mm;. But from f=100 mm to f=400 mm, it only drops by 5 % (from 397 mm to 378 mm).

Answers

At the beginning of this section, I presented two statements and asked: True or false? Here they are again with the correct answer filled in.

1. You will get a more shallow DOF on a camera with an FX-format sensor than on a camera with a DX-format sensor if you photograph the same scene from the same point using both formats and change the lens to one with a longer focal length when using FX-format to keep the FOV constant – and also keep all other parameters (aperture, print size, distance to subject) constant. True.
2. You will get a deeper DOF on a camera with an FX-format sensor than on a camera with a DX-format sensor if you photograph the same scene from the same point using both formats and you change the FOV by using the same lens and the same focal length on both bodies – and also keep all other parameters (aperture, print size, distance to subject) constant. True.

Further reading

For more about DOF, see these articles:

• Bob Atkins: Depth of Field and the Small-Sensor Digital Cameras
• Canon Europe: Tools: Canon depth-of-field calculator
• Don Fleming: DOFMaster Depth of Field Calculator
• Gisle Hannemyr: DOF examples
• Sean McHugh: Tutorials: Depth of Field
• Norman Koren: Depth of field and diffraction
• Paul van Walree: Depth of field

Diffraction

Diffraction limits the resolving power of an optical system at small apertures. Why this happens is beyond the scope on this article. For an explanation, see the tutorial on diffraction in Sean McHugh's Cambridge in Colour. [Note: The diffraction calculator at the end of the tutorial has a bug that will give you the wrong results for cameras with more than 3 Mpx if you tick the checkbox marked “Set Circle of Confusion = Twice Pixel Size?”. Ticking this box will make the CoC a function of the pixel pitch for all sensors. While this is the correct thing to do for a camera with less than 3 Mpx, it will (with the presets used by McHugh), give bogus results for cameras with roughly more than 3 megapixels.]

When uniform light passes through an aperture, it is diffracted and creates a pattern of light and dark regions known as an Airy disc. The angular resolution of the system, which is important in astronomy when one is photographing point light sources, is limited by the Rayleigh criterion, which is met when two Airy discs from two point light sources are so close together that they overlap.

In digital photography, we say that an optical system is diffraction limited when the size of the Airy disc is larger than half the CoC, or larger than about 3/4 of the pixel pitch²⁾ (whichever is largest). The CoC depends on the diagonal of the sensor (D). The size of the Airy disc (AD-size) depends upon the wavelength of light (λ), and the aperture (N) of the optical system, and a constant (1.22) derived from a calculation of the position of the first dark ring surrounding the central Airy disc of the diffraction pattern. The formulas to compute the AD-size (derived from the articles linked to above) and CoC, are:

AD-size = 1.22 x λ x N
CoC = D / z-constant
(λ = wavelength of light, D = diagonal of the sensor)

Setting CoC equal to twice the AD-size and solving with respect to aperture (N) gives us a function to compute the aperture where the systems becomes differaction limited for different sensor sizes (assuming that about 3/4 of the sensor's pixel pitch is smaller than this).

N = CoC / (2 x 1.22 x λ)

Note that the sensor diagonal (D) used to compute Coc and wavelength of light should be expressed in the same units (e.g. nm).

In the table below, I've computed the aperture/f-number (N) where this happens for a number of different sensor sizes, for light at wavelengths 400 nm (blue), 550 nm (green) and 700 nm (red). Since the eye's resolution is mainly determined by green, the numbers for the column with the heading “λ=550nm” is the most relevant. The z-constant used to compute the CoC is set equal to 1500.

Aperture where the system becomes diffraction limited

Format	Crop f.	D (mm)	λ=400nm	λ=550nm	λ=700nm	Example
1/2.7"	6.6x	6.6	f/4.5	f/3.3	f/2.6	Nikon 775
1/1.8"	4.8x	8.9	f/6.1	f/4.4	f/3.5	Canon G5
1/1.7"	4.6x	9.5	f/6.5	f/4.7	f/3.7	Canon G9
2/3"	3.9x	11.0	f/7.5	f/5.5	f/4.3	Canon Pro 1
4/3	2.0x	21.8	f/14.9	f/10.8	f/8.5	Olympus E-1
?	1.7x	24.9	f/17.0	f/12.4	f/9.7	Sigma SD14
?	1.6x	27.0	f/18.5	f/13.4	f/10.6	Canon EOS 40D
DX	1.5x	28.4	f/19.4	f/14.1	f/11.1	Nikon D90
APS-C	1.4x	30.6	f/20.9	f/15.2	f/12.0	Nikon Pronea 6i
?	1.3x	34.5	f/23.5	f/17.1	f/13.5	Canon EOS 1D
FX	1.0x	43.3	f/29.6	f/21.5	f/16.9	Nikon D3

Looking at this table, it becomes clear why the smallest aperture available on the Canon Powershot G5 is f/8. In the green channel (550 nm), the G5 becomes diffraction limited at f/4.4, so even at f/5.6, the image is degraded by diffraction.

F-number

As noted above in the section discussing field of view (FOV) the real focal length of a lens is a fundamental property of its design and does not change if the size of sensor, and therefore the FOV, changes.

The f-number is a numerical desigation for a lens' aperture. The f-number is derived from the lens' real focal length. It expresses the diameter of the entrance pupil of the lens as a fraction of the real focal length of the lens. The “f” in “f-number” is actually shorthand for focal length. For example, f/4.0 represents a entrance pupil diameter that is one-quarter of the lens' focal length. If the lens has f=100 mm, f/4.0 designates an entrance pupil diameter equal to 100/4=25 mm.

It follows that the size of the sensor does not affect the f-number. In other words, a lens that is f/4.0 on a camera with an FX-format sensor will also be f/4.0 on a camera with an sensor that is a different size.

Hand Holding

The so-called “focal length reciprocal rule” is an old rule of thumb that is known by almost every photographer using 135-format film. The “rule” says that to avoid blur from camera shake, you should use a shutter speed equal to the reciprocal of the focal length (or faster). For example, if you are shooting with a 100 mm lens, for hand held shots you should use a shutter speed of at least 1/100th of a second.

For digital, to get identical results with different sensor sizes, you need to take the crop factor into account. In other words, if you belive the focal length reciprocal rule is a good guideline, you must multiply the focal length with the camera's crop factor if you shoot with a camera that has a sensor smaller than FX-format. I.e. an updated version of the old rule of thumb looks like this:

min. shutter speed = 1/(focal length x crop factor)

Example: Let's say we are using a 100 mm lens on a camera with a DX-format sensor. Our slowest stutter speed for hand held shots is 1/(100 x 1.5) = 1/150th second.

The reason you need to figure in the crop factor is that an image projected onto a physical DX-format sensor need to be magnified more than an image projected onto a FX-format sensor for any given print size. As part of the process, camera shake blur is also magnified, and the increased magnification must be compensated for by using a faster shutter speed.

It should be pointed out that the focal length reciprocal rule for hand-holding is not a hard, scientific fact. It is just a rule-of-thumb that some people claim works reasonable well if the photographer has average steady hands, and the resulting average sized print is viewed by a person with average eyesight from a normal viewing distance (e.g. a print sized up to 17 x 25 cm viewed from a distance of around 30 cm). If your hands are less steady, and/or you use a crop from the total frame, and/or you print larger than 25 cm, you may need to use a higher shutter speed than indicated by the reciprocal rule to get a sharp print – and vice versa.

Not everyone agree with the rule. The rule indicates that with a normal lens, 1/50 second is safe for hand holding. Ansel Adams wrote:

Tests I conducted some years ago, photographing leafless trees against the sky, indicated that, using a normal lens with a hand-held camera, the slowest shutter speed that ensured maximum sharpness was 1/250 second. I found that even with firm body support image sharpness was noticeably degraded at 1/125 second, a speed that many photographers consider safe for hand-holding a camera with normal lens.
— Ansel Adams, The Camera, p. 116.

Adams' observation are confirmed by a test conducted in 2008 by Erwin Puts. He photographed a Siemens star with three top class bodies (Nikon, Leica and Olympus) from a distance of two meters. Each body was fitted with very good lens with a FOV equal to around 100 mm on a body with a FX-format sensor. According to the “focal length reciprocal rule”, 1/100 should be a safe shutter speed for hand holding. However, Puts found that hand-held shots at much shorter shutter speeds did not have the same sharpness as shots taken on a tripod. He concluded:

Overall one might say that a shutter speed below 1/200 does not do justice to the potential image quality in all three cases.
&mdash Edwin Puts, Vibration Reduction compared with handholding.

Another of Puts' findings that is worth mentioning is that VR actually works:

The Nikon offers in-lens vibration reduction and at 1/100 the results are very close to what you get on tripod!! Only the very fine details get lost in the process, but a contrast of 20% at 1200 lp/ph is an excellent performance. This result holds for shutter speeds as low as 1/15. One may say that VR at 1/15 is much better than non-VR at 1/180. For number aficionados this implies an improvement of 3.5 to 4 stops.
— ibid.

Without VR, I think the old “focal length reciprocal rule” is of dubious value. Without any form of support, the minimum shutter speed to use for hand held shots should be:

min. shutter speed = 1/(focal length x crop factor x 2)

But the again, as Magnum photographer Henri Cartier-Bresson once said: “Sharpness is a bourgeois concept.”.

Macro Ratio

In macro photography, the term macro ratio (or magnification ratio) denotes the ratio between real life dimensions and how those dimensions are projected on to the sensor.

A macro ratio of 1:1 means that the projection of an object on to the sensor will have the same dimensions as the actual object (i.e. life size). A macro ratio of 1:2 means half life size, and a macro ratio of 3:1 will magnify objects to three times life size, and so on.

The macro ratio for a specific lens is constant and not a function of the sensor's physical size. A 1:1 macro lens on a 135-format film camera or a DSLR with a FX-format sensor will still be a 1:1 macro lens when you put it on a DSLR with a smaller sensor.

All a macro ratio of 1:1 tells you, is that 1 millimeter in “real life” will measure 1 millimeter when projected on to the camera's sensor. This does not means that the acquisition will be the same if you use a DSLR with a FX-format sensor and one with a DX-format sized sensor. If you, for example, were to photograph a ruler at 1:1, you would capture 36 mm of it along the long side if your camera has a FX-format sensor, but only 24 mm of it if you use a camera with a DX-format sensor.

In 35 mm film photography, a lens is not regarded as a “real” macro lens unless its macro ratio is at least 1:1³⁾. The image of a ruler photo­graphed at 1:1 on 135-format film will show a 36 mm long section of it across the long side of the negative frame. How­ever, if one used a Four-Thirds camera (2.0x crop factor) to photograph the same ruler, a macro factor of 1:2 would be suffiscient to capture 36 mm of the ruler across the long side. This has lead to some people insisting that on the Four-Thirds system, a macro ratio equal to 1:2 is “equivalent” to 1:1 on a film camera or a digital camera with a FX-format sensor, and that it is proper to use the term “macro” to designate Four-Thirds lenses with a macro ratio equal to 1:2, or more.

IMHO, the notion of an “equivalent” something (e.g. macro factor, focal length, etc.) when discussing the effects of using sensors of different sizes should be avoided, Whatever pedagogical qualities such as an approach may have, there is also a lot of evidence that this approach leads to a lot of confusion among those mistaking equivalence for actuality. It is much better to use the actual value – and to educate oneself and others to understand what this entails.

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1) A brief note on terminology. Some people dislike the term “crop factor” and insist on using another term, such as “lens factor” or “lens focal length conversion factor”. I think those terms are misleading because they suggest that the lens' focal lens somehow is “converted” to some other focal length, and - as I shall show - it is not. The only thing that happens is that the parts of the image circle that is outside the sensor is cropped, and that makes “crop factor” the most appropriate term.

2) Strictly speaking, for a Bayer camera the true value lies somewhere between half the pixel pitch and half the diagonal of one RGGB block, which is equal to the pixel pitch. The exact value depends on the quality of Bayer demosaicing algorithm used, but 3/4 of the pixel pitch may be a reasonable approximation.

3) Except by marketing people. Marketing people use the word “macro” to describe any lens that has a minimum focusing distance less than 50 cm.

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