Tag Archives: macro

Fourier Optics, Geometrical Optics, MTF, PSF, wavefront

Fourier Optics and the Complex Pupil Function

Leave a comment

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.

Continue reading Fourier Optics and the Complex Pupil Function →

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

Diffracted DOF Aperture Guides: 24-35mm

12 Comments

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

2 Comments

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.
Continue reading DOF and Diffraction: 24mm Guidelines →

IQ, Optics, Perceptual, Sharpness, Spatial Resolution

DOF and Diffraction: Setup

1 Comment

The two-thin-lens model for precision Depth Of Field estimates described in the last two articles is almost ready to be deployed.  In this one we will describe the setup that will be used to develop the scenarios that will be outlined in the next one.

The beauty of the hybrid geometrical-Fourier optics approach is that, with an estimate of the field produced at the exit pupil by an on-axis point source, we can generate the image of the resulting Point Spread Function and related Modulation Transfer Function.

Pretend that you are a photon from such a source in front of a f/2.8 lens focused at 10m with about 0.60 microns of third order spherical aberration – and you are about to smash yourself onto the ‘best focus’ observation plane of your camera.  Depending on whether you leave exactly from the in-focus distance of 10 meters or slightly before/after that, the impression you would leave on the sensing plane would look as follows:

Figure 1. PSF of a lens with about 0.6um of third order spherical aberration focused on 10m.

The width of the square above is 30 microns (um), which corresponds to the diameter of the Circle of Confusion used for old-fashioned geometrical DOF calculations with full frame cameras.  The first ring of the in-focus PSF at 10.0m has a diameter of about 2.44\lambda \frac{f}{D} = 3.65 microns.   That’s about the size of the estimated effective square pixel aperture of the Nikon Z7 camera that we are using in these tests.
Continue reading DOF and Diffraction: Setup →

Optics, Perceptual, Sharpness, Spatial Resolution

DOF and Diffraction: Image Side

Leave a comment

This investigation of the effect of diffraction on Depth of Field is based on a two-thin-lens model, as suggested by Alan Robinson[1].  We chose this model because it allows us to associate geometrical optics with one lens and Fourier optics with the other, thus simplifying the underlying math and our understanding.

In the last article we discussed how the front element of the model could present at the rear element the wavefront resulting from an on-axis source as a function of distance from the lens.  We accomplished this by using simple geometry in complex notation.  In this one we will take the relative wavefront present at the exit pupil and project it onto the sensing plane, taking diffraction into account numerically.  We already know how to do it since we dealt with this subject in the recent past.

Figure 1. Where is the plane with the Circle of Least Confusion?  Through Focus Line Spread Function Image of a lens at f/2.8 with the indicated third order spherical aberration coefficient, and relative measures of ‘sharpness’ MTF50 and Acutance curves.  Acutance is scaled to the same peak as MTF50 for ease of comparison and refers to my typical pixel peeping conditions: 100% zoom, 16″ away from my 24″ monitor.

Continue reading DOF and Diffraction: Image Side →

Optics, Perceptual, Sharpness

DOF and Diffraction: Object Side

1 Comment

In this and the following articles we shall explore the effects of diffraction on Depth of Field through a two-lens model that separates geometrical and Fourier optics in a way that keeps the math simple, though via complex notation.  In the process we will gain a better understanding of how lenses work.

The results of the model are consistent with what can be obtained via classic DOF calculators online but should be more precise in critical situations, like macro photography.  I am not a macro photographer so I would be interested in validation of the results of the explained method by someone who is.

Figure 1. Simple two-thin-lens model for DOF calculations in complex notation.  Adapted under licence.

Continue reading DOF and Diffraction: Object Side →

Optics, Quantization, Sensors, Spatial Resolution

Wavefront to PSF to MTF: Physical Units

2 Comments

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 →