Tag Archives: photons

Fourier Optics, Geometrical Optics, MTF, PSF, wavefront

Fourier Optics and the Complex Pupil Function

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In the last article we learned that a complex lens can be modeled as just an entrance pupil, an exit pupil and a geometrical optics black-box in between.  Goodman[1] suggests that all optical path errors for a given Gaussian point on the image plane can be thought of as being introduced by a custom phase plate at the pupil plane, delaying or advancing the light wavefront locally according to aberration function \Delta W(u,v) as earlier described.

The phase plate distorts the forming wavefront, introducing diffraction and aberrations, while otherwise allowing us to treat the rest of the lens as if it followed geometrical optics rules.  It can be associated with either the entrance or the exit pupil.  Photographers are usually concerned with the effects of the lens on the image plane so we will associate it with the adjacent Exit Pupil.

aberrations coded as phase plate in exit pupil generalized complex pupil function
Figure 1.  Aberrations can be fully described by distortions introduced by a fictitious phase plate inserted at the uv exit pupil plane.  The phase error distribution is the same as the path length error described by wavefront aberration function ΔW(u,v), introduced in the previous article.

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Fourier Optics, Geometrical Optics, intensity, MTF, Optics, PSF, wavefront

An Introduction to Pupil Aberrations

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As discussed in the previous article, so far we have assumed ideal optics, with spherical wavefronts propagating into and out of the lens’ Entrance and Exit pupils respectively.  That would only be true if there were no aberrations. In that case the photon distribution within the pupils would be uniform and such an optical system would be said to be diffraction limited.

Figure 1.   Optics as a black box, fully described for our purposes by its terminal properties at the Entrance and Exit pupils.  A horrible attempt at perspective by your correspondent: the Object, Pupils and Image planes should all be parallel and share the optical axis z.

On the other hand if lens imperfections, aka aberrations, were present the photon distribution in the Exit Pupil would be distorted, thus unable to form a perfectly  spherical wavefront out of it, with consequences for the intensity distribution of photons reaching the image.

Either pupil can be used to fully describe the light collection and concentration characteristics of a lens.  In imaging we are typically interested in what happens after the lens so we will choose to associate the performance of the optics with the Exit Pupil. Continue reading An Introduction to Pupil Aberrations →

DOF, intensity, Optics, wavefront

Pupils and Apertures

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We’ve seen in the last article that the job of an ideal photographic lens is simple: to receive photons from a set of directions bounded by a spherical cone with its apex at a point on the object; and to concentrate them in directions bounded  by a spherical cone with its apex at the corresponding point on the image.   In photography both cones are assumed to be in air.

In this article we will distill the photon collecting and distributing function of a complex lens down to its terminal properties, the Entrance and Exit Pupils, allowing us to deal with any lens simply and consistently. Continue reading Pupils and Apertures →

Equivalence, Exposure, IQ, Sensors

Equivalence and Equivalent Image Quality: Signal

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One of the fairest ways to compare the performance of two cameras of different physical characteristics and specifications is to ask a simple question: which photograph would look better if the cameras were set up side by side, captured identical scene content and their output were then displayed and viewed at the same size?

Achieving this set up and answering the question is anything but intuitive because many of the variables involved, like depth of field and sensor size, are not those we are used to dealing with when taking photographs.  In this post I would like to attack this problem by first estimating the output signal of different cameras when set up to capture Equivalent images.

It’s a bit long so I will give you the punch line first:  digital cameras of the same generation set up equivalently will typically generate more or less the same signal in e^- independently of format.  Ignoring noise, lenses and aspect ratio for a moment and assuming the same camera gain and number of pixels, they will produce identical raw files. Continue reading Equivalence and Equivalent Image Quality: Signal →

Exposure, Sensors

What is the Effective Quantum Efficiency of my Sensor?

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Now that we know how to determine how many photons impinge on a sensor we can estimate its Effective Quantum Efficiency, that is the efficiency with which it turns such a photon flux (n_{ph}) into photoelectrons (n_{e^-} ), which will then be converted to raw data to be stored in the capture’s raw file:

(1)   \begin{equation*} EQE = \frac{n_{e^-} \text{ produced by average pixel}}{n_{ph} \text{ incident on average pixel}} \end{equation*}

I call it ‘Effective’, as opposed to ‘Absolute’, because it represents the probability that a photon arriving on the sensing plane from the scene will be converted to a photoelectron by a given pixel in a digital camera sensor.  It therefore includes the effect of microlenses, fill factor, CFA and other filters on top of silicon in the pixel.  Whether Effective or Absolute, QE is usually expressed as a percentage, as seen below in the specification sheet of the KAF-8300 by On Semiconductor, without IR/UV filters:

For instance if  an average of 100 photons per pixel were incident on a uniformly lit spot on the sensor and on average each pixel produced a signal of 20 photoelectrons we would say that the Effective Quantum Efficiency of the sensor is 20%.  Clearly the higher the EQE the better for Image Quality parameters such as SNR. Continue reading What is the Effective Quantum Efficiency of my Sensor? →

Exposure, Sensors

How Many Photons on a Pixel

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How many visible photons hit a pixel on my sensor?  The answer depends on Exposure, Spectral power distribution of the arriving light and effective pixel area.  With a few simplifying assumptions it is not difficult to calculate that with a typical Daylight illuminant the number is roughly 11,760 photons per lx-s per \mu m^2.  Without the simplifying assumptions* it reduces to about 11,000. Continue reading How Many Photons on a Pixel →

Exposure

What Is Exposure

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When capturing a typical photograph, light from one or more sources is reflected from the scene, reaches the lens, goes through it and eventually hits the sensing plane.

In photography Exposure is the quantity of visible light per unit area incident on the image plane during the time that it is exposed to the scene.  Exposure is intuitively proportional to Luminance from the scene $L$ and exposure time $t$.  It is inversely proportional to lens f-number $N$ squared because it determines the relative size of the cone of light captured from the scene.  You can read more about the theory in the article on angles and the Camera Equation.

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