When Is the Quadratic Lens Phase Good Enough?

Key Points

• The familiar quadratic lens phase is directly connected to the the Fourier transform property of a thin lens.
• Diffraction-limited focusing requires a hyperboloidal phase profile, which eliminates spherical aberration at normal incidence. The quadratic form is its paraxial approximation, with the discrepancy growing rapidly toward the edge of a fast lens.
• At sufficiently low f-number, the difference becomes visible in the propagated field as focal shift, longitudinal broadening, and residual aberration (largely primary spherical) that can remain even after refocusing.

Overview

There are a few closely related questions about lens phase profiles that come up fairly often in Fourier optics and metasurface design. Why is a lens commonly represented using a quadratic phase? Why does an exact focusing phase have a hyperbolic form? Are these simply two different ways of describing the same lens, and when does the distinction actually matter?

Some papers and textbooks touch on these points individually, but we have seen enough questions and occasional misconceptions around them that it is worth treating the distinction briefly. The subtlety is that the quadratic and hyperbolic phase profiles arise naturally from somewhat different viewpoints. In the paraxial regime they agree closely, but for sufficiently fast optics the distinction becomes important.

The Quadratic Phase in Fourier Optics

In paraxial Fourier optics, a thin lens is represented by the familiar quadratic phase

$$\phi_q(r)=-\frac{k r^2}{2f},$$

where $r$ is the radial coordinate, $f$ is the focal length, and $k=2\pi/\lambda$. This phase has an important role beyond simply being a convenient approximation to a focusing lens. As discussed in standard Fourier optics texts such as Goodman, the quadratic phase introduced by a thin lens combines with the quadratic phase associated with Fresnel propagation. At the back focal plane, under the usual paraxial assumptions, these terms cancel in such a way that the field is proportional to a scaled Fourier transform of the electric field one focal length in front of the lens. This is quite a remarkable property and one that is heavily exploited in optical information processing and Fourier domain filtering (e.g., using 4f systems).

This is the origin of the familiar statement that a lens performs a Fourier transform. The important qualification is that this result is derived within paraxial Fourier optics. Both the Fresnel propagation kernel and the quadratic lens phase are part of the same approximation, so the Fourier transform property should be understood within that framework rather than as an exact result for arbitrarily fast lenses.

Exact Focusing Gives a Different Phase

If instead we ask what phase profile causes a normally incident plane wave to arrive at a focal point with equal optical path length across the aperture, the result is

$$\phi_h(r) = -k\left(\sqrt{f^2+r^2}-f\right).$$

This is the familiar hyperbolic phase commonly used for metalenses and other fast diffractive optics. The connection between the two expressions becomes clear by expanding the square root:

$$\sqrt{f^2+r^2}-f = \frac{r^2}{2f} -\frac{r^4}{8f^3} +\frac{r^6}{16f^5} -\cdots.$$

The quadratic phase is based on the leading term. When $r/f$ is small, the higher-order terms are negligible and the two prescriptions are essentially equivalent. As the aperture becomes larger relative to the focal length, however, the neglected terms grow rapidly, with the leading error varying as $r^4$.

Figure 1: Optical path difference in waves between the quadratic and hyperbolic lens phases as a function of normalized pupil radius for several f-numbers. The focal length is held fixed while the aperture diameter is varied. The discrepancy is small for slower lenses but increases rapidly toward the edge of the pupil as the f-number decreases.

What Happens to the Focus?

The phase difference itself is straightforward to calculate, but the more useful engineering question is what it does to the actual optical field. To examine this, we propagated both phase profiles at a wavelength of 632 nm while holding the focal length fixed at 1 mm and varying the aperture diameter.

Figure 2 shows the on-axis intensity through focus. For the slower lens cases, the quadratic and hyperbolic profiles produce very similar axial responses. As the lens becomes faster, however, the quadratic profile increasingly departs from the exact result. The peak shifts away from the nominal focal plane and the longitudinal response broadens.

This behavior is closely related to spherical aberration. The quadratic approximation does not maintain equal optical path from all radial positions across the pupil to the intended focus, and the leading neglected term has the same fourth-order radial dependence commonly associated with primary spherical aberration. The result is therefore more than a simple numerical difference between two phase equations.

Figure 2: On-axis intensity as a function of propagation distance for quadratic and hyperbolic phase profiles at several f-numbers. Each comparison uses the same wavelength and focal length. As the f-number decreases, the quadratic phase produces increasing focal shift and longitudinal broadening relative to the hyperbolic phase.

The axial intensity curves are useful quantitatively, but a two-dimensional cross section makes the change in focal structure easier to see. Figure 3 shows x-z intensity maps for the same cases, with the hyperbolic results on the top row and quadratic results below. Each panel is normalized independently to its own peak intensity, so these plots are intended to show the shape and location of the focal region rather than relative peak efficiency.

For the slower lenses, the focal regions remain quite similar. At lower f-number, the quadratic case develops a visibly longer and shifted focal structure, making the breakdown of the paraxial approximation much more apparent. This degradation of the focus wherein it is longitudinally extended is characteristic of spherical aberration, and this becomes more significant as the f-number decreases for the quadratic lens case. At normal incidence, the hyperbolic lens phase does not exhibit spherical aberration.

Figure 3: x-z intensity cross sections for the hyperbolic phase (top row) and quadratic phase (bottom row) for each f-number. Each panel is normalized independently to its own peak intensity to emphasize changes in focal position and spatial structure rather than absolute peak intensity.

Is It Just Defocus?

One important point is that comparing the two lenses only at $z=f$ can make the quadratic result appear worse than it really is. Part of the difference is simply that its maximum intensity occurs at a different axial position. A more useful comparison is therefore to evaluate the quadratic phase both at the nominal focal plane and at its own best-focus plane.

Figure 4 makes this distinction explicit. Refocusing can recover some of the loss associated with focal shift, particularly for more moderate f-numbers. At sufficiently low f-number, however, the best-focus quadratic profile still does not reproduce the transverse distribution of the exact hyperbolic lens. Residual phase error remains after defocus has been removed, which leads to a reduction in peak intensity for the quadratic case.

Figure 4: Transverse intensity profiles comparing the hyperboloidal lens at the nominal focus, the quadratic lens at the nominal focus, and the quadratic lens at its own best-focus plane. Refocusing recovers part of the degradation associated with focal shift but residual broadening remains for sufficiently fast lenses.

Which Phase Should You Use?

For ordinary paraxial Fourier optics calculations, the quadratic phase remains extremely useful. It is internally consistent with Fresnel propagation and is fundamental to the familiar Fourier-transform property of lenses. The issue arises when that same phase profile is carried into a regime where the paraxial approximation is no longer adequate and is then interpreted as an exact focusing lens.

For fast diffractive optics and metalenses, the distinction can become significant. The hyperbolic phase follows directly from the equal optical path length requirement for focusing, while the quadratic phase follows from the paraxial approximation underlying Fresnel propagation. Neither expression is arbitrary as they answer closely related questions under different assumptions. The practical takeaway is therefore not that the quadratic lens phase is incorrect. It is to recognize the approximation being made and to check whether the aperture and focal length place the system in a regime where the neglected higher-order terms in the phase expansion materially affect the focusing behavior.

References

Goodman, Joseph W. Introduction to Fourier optics. Roberts and Company publishers, 2005.

Smith, Warren J. Modern optical engineering: the design of optical systems. 2008.

Aieta, Francesco, et al. “Aberrations of flat lenses and aplanatic metasurfaces.” Optics express 21.25 (2013): 31530-31539.

Matsushima, Kyoji, and Tomoyoshi Shimobaba. “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields.” Optics express 17.22 (2009): 19662-19673.