Tag Archives: f-ratio

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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DOF, Optics, Sensors, Sharpening, Sharpness, Spatial Resolution

Diffracted DOF Aperture Guides: 24-35mm

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As a landscape shooter I often wonder whether old rules for DOF still apply to current small pixels and sharp lenses. I therefore roughly measured  the spatial resolution performance of my Z7 with 24-70mm/4 S in the center to see whether ‘f/8 and be there’ still made sense today.  The journey and the diffraction-simple-aberration aware model were described in the last few posts.  The results are summarized in the Landscape Aperture-Distance charts presented here for the 24, 28 and 35mm focal lengths.

I also present the data in the form of a simplified plot to aid making the right compromises when the focusing distance is flexible.  This information is valid for the Z7 and kit in the center only.  It probably just as easily applies to cameras with similarly spec’d pixels and lenses. Continue reading Diffracted DOF Aperture Guides: 24-35mm →

DOF, IQ, Perceptual, Sharpness, Spatial Resolution

DOF and Diffraction: 24mm Guidelines

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After an exhausting two and a half hour hike you are finally resting, sitting on a rock at the foot of your destination, a tiny alpine lake, breathing in the thin air and absorbing the majestic scenery.  A cool light breeze suddenly rips the surface of the water, morphing what has until now been a perfect reflection into an impressionistic interpretation of the impervious mountains in the distance.

The beautiful flowers in the foreground are so close you can touch them, the reflection in the water 10-20m away, the imposing mountains in the background a few hundred meters further out.  You realize you are hungry.  As you search the backpack for the two panini you prepared this morning you begin to ponder how best to capture the scene: subject,  composition, Exposure, Depth of Field.

Figure 1. A typical landscape situation: a foreground a few meters away, a mid-ground a few tens and a background a few hundred meters further out.  Three orders of magnitude.  The focus point was on the running dog, f/16, 1/100s.  Was this a good choice?

Depth of Field.  Where to focus and at what f/stop?  You tip your hat and just as you look up at the bluest of blue skies the number 16 starts enveloping your mind, like rays from the warm noon sun. You dial it in and as you squeeze the trigger that familiar nagging question bubbles up, as it always does in such conditions.  If this were a one shot deal, was that really the best choice?

In this article we attempt to provide information to make explicit some of the trade-offs necessary in the choice of Aperture for 24mm landscapes.  The result of the process is a set of guidelines.  The answers are based on the previously introduced diffraction-aware model for sharpness in the center along the depth of the field – and a tripod-mounted Nikon Z7 + Nikkor 24-70mm/4 S kit lens at 24mm.
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Optics, Quantization, Sensors, Spatial Resolution

Wavefront to PSF to MTF: Physical Units

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In the last article we saw that the intensity Point Spread Function and the Modulation Transfer Function of a lens could be easily approximated numerically by applying Discrete Fourier Transforms to its generalized exit pupil function \mathcal{P} twice in sequence.[1]

Numerical Fourier Optics: amplitude Point Spread Function, intensity PSF and MTF

Obtaining the 2D DFTs is easy: simply feed MxN numbers representing the two dimensional complex image of the Exit Pupil function in its uv space to a Fast Fourier Transform routine and, presto, it produces MxN numbers representing the amplitude of the PSF on the xy sensing plane.  Figure 1a shows a simple case where pupil function \mathcal{P} is a uniform disk representing the circular aperture of a perfect lens with MxN = 1024×1024.  Figure 1b is the resulting intensity PSF.

Figure 1a, left: A circular array of ones appearing as a white disk on a black background, representing a circular aperture. Figure 1b, right: Array of numbers representing the PSF of image 1a in the classic shape of an Airy Pattern.
Figure 1. 1a Left: Array of numbers representing a circular aperture (zeros for black and ones for white).  1b Right: Array of numbers representing the PSF of image 1a (contrast slightly boosted).

Simple and fast.  Wonderful.  Below is a slice through the center, the 513th row, zoomed in.  Hmm….  What are the physical units on the axes of displayed data produced by the DFT? Continue reading Wavefront to PSF to MTF: Physical Units →

Optics, Photometry, Quantization, Radiometry, Sensors, Spatial Resolution

Aberrated Wave to Image Intensity to MTF

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Goodman, in his excellent Introduction to Fourier Optics[1], describes how an image is formed on a camera sensing plane starting from first principles, that is electromagnetic propagation according to Maxwell’s wave equation.  If you want the play by play account I highly recommend his math intensive book.  But for the budding photographer it is sufficient to know what happens at the Exit Pupil of the lens because after that the transformations to Point Spread and Modulation Transfer Functions are straightforward, as we will show in this article.

The following diagram exemplifies the last few millimeters of the journey that light from the scene has to travel in order to be absorbed by a camera’s sensing medium.  Light from the scene in the form of  field  U arrives at the front of the lens.  It goes through the lens being partly blocked and distorted by it as it arrives at its virtual back end, the Exit Pupil, we’ll call this blocking/distorting function P.   Other than in very simple cases, the Exit Pupil does not necessarily coincide with a specific physical element or Principal surface.[iv]  It is a convenient mathematical construct which condenses all of the light transforming properties of a lens into a single plane.

The complex light field at the Exit Pupil’s two dimensional uv plane is then  U\cdot P as shown below (not to scale, the product of the two arrays is element-by-element):

Figure 1. Simplified schematic diagram of the space between the exit pupil of a camera lens and its sensing plane. The space is assumed to be filled with air.

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Equivalence, Formats, Sensors

Equivalence in Pictures: Focal Length, f-number, diffraction, DOF

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Equivalence – as we’ve discussed one of the fairest ways to compare the performance of two cameras of different physical formats, characteristics and specifications – essentially boils down to two simple realizations for digital photographers:

  1. metrics need to be expressed in units of picture height (or diagonal where the aspect ratio is significantly different) in order to easily compare performance with images displayed at the same size; and
  2. focal length changes proportionally to sensor size in order to capture identical scene content on a given sensor, all other things being equal.

The first realization should be intuitive (see next post).  The second one is the subject of this post: I will deal with it through a couple of geometrical diagrams.

Continue reading Equivalence in Pictures: Focal Length, f-number, diffraction, DOF →