Optics, Perceptual, Sharpness

DOF and Diffraction: Object Side

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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.

A Simple Two-Lens Model

The maestro, Alan Robinson, suggested a two-thin-lens model for this problem so that’s what we are doing, as shown in Figure 1.  The two lenses touch.  The first object-facing lens is focused on an on-axis object a distance u from the front element. The second image-facing lens projects the image onto the observation plane a distance z from the back element.   A second on-axis object at variable distance u_2 from the front element will let us characterize out-of-focus effects.

To make things simpler let’s initially consider the two thin lenses separately: the object facing lens will be treated geometrically in this article; the image facing lens will instead incorporate all diffraction effects and will be discussed in the next.

We can do this because, to a good approximation and paraphrasing Goodman[1], the effect of diffraction is to convolve the ideal image with the Fraunhofer diffraction pattern of the lens pupil.  It is in fact possible to apply all diffraction effects during passage of light from the object to the front element of a lens or, alternatively and equivalently, from the back element to the image.  We chose the latter because of the path we have traveled.

Our objective will be to calculate the point spread function (PSF) and consequent modulation transfer function (MTF) on the sensing plane of on axis-point sources in object space:  MTF is the basis of many quantitative indicators of the ‘sharpness’ of an imaging system and as such a good aid to determine what should be acceptably in and out of focus, thereby yielding a meaningful depth of field (DOF) for the given setup.

The Geometric Object Side

At its simplest, the job of a lens is to convert incoming spherical wavefronts into plane wavefronts and vice-versa.

Figure 2. Spherical wave to plane wave transformation of thin lens when the object is positioned at its focal length.  Under licence, rotated, labeled.

For instance an on-axis point source located in front of a lens at its focal length f_u, will produce a spherical field U that, upon reaching the lens, will be transformed into plane waves U' as shown above.  The image-facing lens discussed in the next article will then complete the job by performing the opposite function, receiving plane waves and transforming them into spherical waves converging onto the observation plane at its own separate focal length.

In-Focus Object

A sphere can be easily characterized by high-school geometry.  However, for reasons that will become apparent later, it is useful in our case to express the diverging spherical field of the point source U in complex notation.  We will therefore describe the field generated by the on-axis point source at u as a function of location along the extension of the optical axis z and the radial axis of the lens perpendicular to it \rho, with quadratic approximation (Goodman 4.2.1 p. 69)

(1)   \begin{equation*} U(z,\rho) = e^{i\frac{k}{2z}\rho^2} \end{equation*}

with k = \frac{2\pi}{\lambda} the wave number and \lambda the mean wavelength of light.

Figure 3. Diverging spherical wave and opposite phase transformation provided by the lens for objects positioned at its focal length.

The spherical field arrives at the lens and it interacts with it.  The phase transformation provided by a thin lens with refractive index n on the arriving wavefront is controlled by the curvature of its surfaces characterized by radii R_1 and R_2 as follows (Goodman 5.1.3, p. 100)

(2)   \begin{equation*} t(\rho) = e^{-i\frac{k}{2f}\rho^2} \end{equation*}

also with quadratic approximation. Constant f collects the physical properties of the lens.  In air it is defined as \frac{1}{f} \equiv (n-1)(\frac{1}{R_1}-\frac{1}{R_2}) and referred to as its focal length.  This is the well known lens maker equation for thin lenses.

The field after the lens U' is equal to input field U times lens transformation t.  If the object is positioned on-axis at the focal length of a lens, z is equal to f so the exponential terms of Equations (1) and (2) add to zero, Ut is equal to 1 and the resulting field is the phaseless plane wavefront shown above.

Out of Focus Objects

What happens if the object is not at in-focus position u but at out-of-focus position u_2?  In such a case the diverging spheres described by Equation (1) will no longer match the lens transformation of Equation (2) upon arrival, the exponentials will not cancel and the field out of the lens will not be a plane wavefront but will carry a phase error.

Figure 4. An object that is not on the focal plane of the lens is out of focus.  The out of focus object a distance u2 from the lens results in p-v optical path difference W2.

The phase error introduced by the out of focus position can be calculated geometrically by working in two dimensions, exploiting a sphere’s radial symmetry.  With a few simplifying assumptions the Optical Path Difference (W) of a spherical wave centered at on-axis location u_2 compared to the in-focus one at u is clearly a function of radial distance \rho from the center of the lens intersected by the optical axis

(3)   \begin{equation*} W(\rho) \approx -[\sqrt{ u^2 - \rho^2 } - \sqrt{ u_2^2 - \rho^2 } - (u - u_2)] \end{equation*}

The minus sign in front is there as a reminder that the optical axis z is positive to the right of the lens (see the Appendix for a slightly different solution which gives the same result).  At the lens’ widest aperture, with diameter D, we therefore have a peak-to-valley wavefront error compared to an object at f_u

(4)   \begin{equation*} W_2 \approx -[\sqrt{ f_u^2 - \frac{D^2}{4} } - \sqrt{ u_2^2 - \frac{D^2}{4} } - (f_u - u_2)] \end{equation*}

Expressed in phasor notation and subject to paraxial limitations, the field U_2 from an on axis out-of-focus object at distance u_2 from the lens is therefore no longer a plane wave once it has gone through it but

(5)   \begin{equation*} U_{2}' =e^{i k W_2 \rho^2} \end{equation*}

Plus Simple Aberrations

Aberrations are deviations from the ideal lens that result in wavefront errors compared to the reference Gaussian sphere.  They can be imagined to be occurring in a fictitious phase plate existing in the aperture, whose effect is multiplied into the field from the front element U_{2}'.

For this exercise, let’s just assume third order spherical aberrations that vary with \rho^4 and are typically present in the center of the field of view of most current lenses.  After the fictitious aberration face plate, the field from an on-axis out-of-focus point source a distance u_2 from the lens will then carry the following total wavefront error to the exit pupil, assuming SA3 peak-to-valley coefficient W_{040}:

(6)   \begin{equation*} U_{2}'' = e^{i k (W_2 \rho^2 + W_{040} \rho^4)} \end{equation*}

We have gotten this far by adopting a touching-two-thin-lens model and making only three macro assumptions: that our system behaves according to scalar theory; the paraxial approximation; and that  diffraction effects can be fully accounted for either in the object or image side of light travel – and we chose the latter.  We’ll take it from here in the next article.

Appendix: Validating W2

An alternative solution for W_2, the path error of out-of-focus point u_2, is as shown

In this case, with paraxial approximation Equation (3) becomes

(7)   \begin{equation*} W(\rho) \approx \sqrt{ \rho^2 + u^2} - \sqrt{ \rho^2 + u_2^2} - (u - u_2) \end{equation*}

The result in practice is the same as for Equation (3).  For instance with u = 5m, u_2 = 4.5m, f_z = 50mm and f-number N = 2.8, W_2 = 0.8858um in both cases.

Recalling from the defocus article that in image space d_z=-8W_2N^2, we calculate a shift along the optical axis of 55.6um with respect to the published focal plane.  We can compare this result to what would be obtained by applying the lens equation twice

(8)   \begin{equation*} \frac{1}{f_z} = \frac{1}{u} + \frac{1}{z} \end{equation*}

yielding 56.7um.  The slight discrepancy is due to  different approximations.

Notes and References

1. Introduction to Fourier Optics, 3rd Edition. Joseph W Goodman.
2. Thanks to Alan Robinson for his excellent mentorship skills.
3. Basic Wavefront Aberration Theory for Optical Metrology, James C. Wyant and Katherine Creath.

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