Image sensor format

Three possible depth-of-field comparisons between formats are discussed, applying the formulae derived in the article on depth of field. The depths of field of the three cameras may be the same, or different in either order, depending on what is held constant in the comparison.

Considering a picture with the same subject distance and angle of view for two different formats:

${\displaystyle {\frac {\mathrm {DOF} _{2}}{\mathrm {DOF} _{1}}}\approx {\frac {d_{1}}{d_{2}}}}$

so the DOFs are in inverse proportion to the absolute aperture diameters ${\displaystyle d_{1}}$ and ${\displaystyle d_{2}}$.

Using the same absolute aperture diameter for both formats with the "same picture" criterion (equal angle of view, magnified to same final size) yields the same depth of field. It is equivalent to adjusting the f-number inversely in proportion to crop factor – a smaller f-number for smaller sensors (this also means that, when holding the shutter speed fixed, the exposure is changed by the adjustment of the f-number required to equalise depth of field. But the aperture area is held constant, so sensors of all sizes receive the same total amount of light energy from the subject. The smaller sensor is then operating at a lower ISO setting, by the square of the crop factor). This condition of equal field of view, equal depth of field, equal aperture diameter, and equal exposure time is known as "equivalence".[1]

And, we might compare the depth of field of sensors receiving the same photometric exposure – the f-number is fixed instead of the aperture diameter – the sensors are operating at the same ISO setting in that case, but the smaller sensor is receiving less total light, by the area ratio. The ratio of depths of field is then

${\displaystyle {\frac {\mathrm {DOF} _{2}}{\mathrm {DOF} _{1}}}\approx {\frac {l_{1}}{l_{2}}}}$

where ${\displaystyle l_{1}}$ and ${\displaystyle l_{2}}$ are the characteristic dimensions of the format, and thus ${\displaystyle l_{1}/l_{2}}$ is the relative crop factor between the sensors. It is this result that gives rise to the common opinion that small sensors yield greater depth of field than large ones.

An alternative is to consider the depth of field given by the same lens in conjunction with different sized sensors (changing the angle of view). The change in depth of field is brought about by the requirement for a different degree of enlargement to achieve the same final image size. In this case the ratio of depths of field becomes

${\displaystyle {\frac {\mathrm {DOF} _{2}}{\mathrm {DOF} _{1}}}\approx {\frac {l_{2}}{l_{1}}}}$.

In practice, if applying a lens with a fixed focal length and a fixed aperture and made for an image circle to meet the requirements for a large sensor is to be adapted, without changing its physical properties, to smaller sensor sizes neither the depth of field nor the light gathering ${\displaystyle \mathrm {lx=\,{\frac {lm}{m^{2}}}} }$ will change.

Discounting pixel response non-uniformity (PRNU) and dark noise variation, which are not intrinsically sensor-size dependent, the noises in an image sensor are shot noise, read noise, and dark noise. The overall signal to noise ratio of a sensor (SNR), expressed as signal electrons relative to rms noise in electrons, observed at the scale of a single pixel, assuming shot noise from Poisson distribution of signal electrons and dark electrons, is

${\displaystyle \mathrm {SNR} ={\frac {PQ_{e}t}{\sqrt {\left({\sqrt {PQ_{e}t}}\right)^{2}+\left({\sqrt {Dt}}\right)^{2}+N_{r}^{2}}}}={\frac {PQ_{e}t}{\sqrt {PQ_{e}t+Dt+N_{r}^{2}}}}}$

where ${\displaystyle P}$ is the incident photon flux (photons per second in the area of a pixel), ${\displaystyle Q_{e}}$ is the quantum efficiency, ${\displaystyle t}$ is the exposure time, ${\displaystyle D}$ is the pixel dark current in electrons per second and ${\displaystyle N_{r}}$ is the pixel read noise in electrons rms.[2]

Each of these noises has a different dependency on sensor size.

The resolution of all optical systems is limited by diffraction. One way of considering the effect that diffraction has on cameras using different sized sensors is to consider the modulation transfer function (MTF). Diffraction is one of the factors that contribute to the overall system MTF. Other factors are typically the MTFs of the lens, anti-aliasing filter and sensor sampling window.[7] The spatial cut-off frequency due to diffraction through a lens aperture is

${\displaystyle \xi _{\mathrm {cutoff} }={\frac {1}{\lambda N}}}$

where λ is the wavelength of the light passing through the system and N is the f-number of the lens. If that aperture is circular, as are (approximately) most photographic apertures, then the MTF is given by

${\displaystyle \mathrm {MTF} \left({\frac {\xi }{\xi _{\mathrm {cutoff} }}}\right)={\frac {2}{\pi }}\left\{\cos ^{-1}\left({\frac {\xi }{\xi _{\mathrm {cutoff} }}}\right)-\left({\frac {\xi }{\xi _{\mathrm {cutoff} }}}\right)\left[1-\left({\frac {\xi }{\xi _{\mathrm {cutoff} }}}\right)^{2}\right]^{\frac {1}{2}}\right\}}$

for ${\displaystyle \xi <\xi _{\mathrm {cutoff} }}$ and ${\displaystyle 0}$ for ${\displaystyle \xi \geq \xi _{\mathrm {cutoff} }}$[8] The diffraction based factor of the system MTF will therefore scale according to ${\displaystyle \xi _{\mathrm {cutoff} }}$ and in turn according to ${\displaystyle 1/N}$ (for the same light wavelength).

In considering the effect of sensor size, and its effect on the final image, the different magnification required to obtain the same size image for viewing must be accounted for, resulting in an additional scale factor of ${\displaystyle 1/{C}}$ where ${\displaystyle {C}}$ is the relative crop factor, making the overall scale factor ${\displaystyle 1/(NC)}$. Considering the three cases above:

For the 'same picture' conditions, same angle of view, subject distance and depth of field, then the F-numbers are in the ratio ${\displaystyle 1/C}$, so the scale factor for the diffraction MTF is 1, leading to the conclusion that the diffraction MTF at a given depth of field is independent of sensor size.

In both the 'same photometric exposure' and 'same lens' conditions, the F-number is not changed, and thus the spatial cutoff and resultant MTF on the sensor is unchanged, leaving the MTF in the viewed image to be scaled as the magnification, or inversely as the crop factor.

It might be expected that lenses appropriate for a range of sensor sizes could be produced by simply scaling the same designs in proportion to the crop factor.[9] Such an exercise would in theory produce a lens with the same F-number and angle of view, with a size proportional to the sensor crop factor. In practice, simple scaling of lens designs is not always achievable, due to factors such as the non-scalability of manufacturing tolerance, structural integrity of glass lenses of different sizes and available manufacturing techniques and costs. Moreover, to maintain the same absolute amount of information in an image (which can be measured as the space bandwidth product[10]) the lens for a smaller sensor requires a greater resolving power. The development of the 'Tessar' lens is discussed by Nasse,[11] and shows its transformation from an f/6.3 lens for plate cameras using the original three-group configuration through to an f/2.8 5.2 mm four-element optic with eight extremely aspheric surfaces, economically manufacturable because of its small size. Its performance is 'better than the best 35 mm lenses – but only for a very small image'.

In summary, as sensor size reduces, the accompanying lens designs will change, often quite radically, to take advantage of manufacturing techniques made available due to the reduced size. The functionality of such lenses can also take advantage of these, with extreme zoom ranges becoming possible. These lenses are often very large in relation to sensor size, but with a small sensor can be fitted into a compact package.

Small body means small lens and means small sensor, so to keep smartphones slim and light, the smartphone manufacturers use a tiny sensor usually less than the 1/2.3" used in most Bridge cameras. At one time only Nokia 808 PureView used a 1/1.2" sensor, almost three times the size of a 1/2.3" sensor. Bigger sensors have the advantage of better image quality, but with improvements in sensor technology, smaller sensors can achieve the feats of earlier larger sensors. These improvements in sensor technology allow smartphone manufacturers to use image sensors as small as 1/4" without sacrificing too much image quality compared to budget point & shoot cameras.[12]

For calculating camera angle of view one should use the size of active area of the sensor. Active area of the sensor implies an area of the sensor on which image is formed in a given mode of the camera. The active area may be smaller than the image sensor, and active area can differ in different modes of operation of the same camera. Active area size depends on the aspect ratio of the sensor and aspect ratio of the output image of the camera. The active area size can depend on number of pixels in given mode of the camera. The active area size and lens focal length determines angles of view.[13]

Semiconductor image sensors can suffer from shading effects at large apertures and at the periphery of the image field, due to the geometry of the light cone projected from the exit pupil of the lens to a point, or pixel, on the sensor surface. The effects are discussed in detail by Catrysse and Wandell .[14] In the context of this discussion the most important result from the above is that to ensure a full transfer of light energy between two coupled optical systems such as the lens' exit pupil to a pixel's photoreceptor the geometrical extent (also known as etendue or light throughput) of the objective lens / pixel system must be smaller than or equal to the geometrical extent of the microlens / photoreceptor system. The geometrical extent of the objective lens / pixel system is given by

${\displaystyle G_{\mathrm {objective} }\simeq {\frac {w_{\mathrm {pixel} }}{2{(f/\#)}_{\mathrm {objective} }}}}$,

where w_pixel is the width of the pixel and (f/#)_objective is the f-number of the objective lens. The geometrical extent of the microlens / photoreceptor system is given by

${\displaystyle G_{\mathrm {pixel} }\simeq {\frac {w_{\mathrm {photoreceptor} }}{2{(f/\#)}_{\mathrm {microlens} }}}}$,

where w_{photoreceptor} is the width of the photoreceptor and (f/#)_microlens is the f-number of the microlens.

So to avoid shading,

${\displaystyle G_{\mathrm {pixel} }\geq G_{\mathrm {objective} }}$, therefore ${\displaystyle {\frac {w_{\mathrm {photoreceptor} }}{{(f/\#)}_{\mathrm {microlens} }}}\geq {\frac {w_{\mathrm {pixel} }}{{(f/\#)}_{\mathrm {objective} }}}}$

If w_{photoreceptor} / w_pixel = ff, the linear fill factor of the lens, then the condition becomes

${\displaystyle {(f/\#)}_{\mathrm {microlens} }\leq {(f/\#)}_{\mathrm {objective} }\times {\mathit {ff}}}$

Thus if shading is to be avoided the f-number of the microlens must be smaller than the f-number of the taking lens by at least a factor equal to the linear fill factor of the pixel. The f-number of the microlens is determined ultimately by the width of the pixel and its height above the silicon, which determines its focal length. In turn, this is determined by the height of the metallisation layers, also known as the 'stack height'. For a given stack height, the f-number of the microlenses will increase as pixel size reduces, and thus the objective lens f-number at which shading occurs will tend to increase. This effect has been observed in practice, as recorded in the DxOmark article 'F-stop blues'[15]

In order to maintain pixel counts smaller sensors will tend to have smaller pixels, while at the same time smaller objective lens f-numbers are required to maximise the amount of light projected on the sensor. To combat the effect discussed above, smaller format pixels include engineering design features to allow the reduction in f-number of their microlenses. These may include simplified pixel designs which require less metallisation, 'light pipes' built within the pixel to bring its apparent surface closer to the microlens and 'back side illumination' in which the wafer is thinned to expose the rear of the photodetectors and the microlens layer is placed directly on that surface, rather than the front side with its wiring layers. The relative effectiveness of these stratagems is discussed by Aptina in some detail.[16]

• Full-frame digital SLR
• Sensor size and angle of view
• 35 mm equivalent focal length
• Film format
• Digital versus film photography
• List of large sensor interchangeable-lens video cameras
• List of sensors used in digital cameras
• Angle of view
• Crop factor
• Field of view

• Eric Fossum: Photons to Bits and Beyond: The Science & Technology of Digital, Oct. 13, 2011 (YouTube Video of lecture)
• Joseph James: Equivalence at Joseph James Photography
• Simon Tindemans: Alternative photographic parameters: a format-independent approach at 21stcenturyshoebox
• Compact Camera High ISO modes: Separating the facts from the hype at dpreview.com, May 2007
• The best compromise for a compact camera is a sensor with 6 million pixels or better a sensor with a pixel size of >3μm at 6mpixel.org