Paraxial wave propagation and lens phase

The engineering question

A collimated field reaches a converging lens. Immediately behind the lens, has its intensity already concentrated into a focus? The lens first changes the field’s phase across its aperture. Propagation then combines those phase-shifted contributions into a different spatial field. Keeping these two operations separate explains how focusing works and which approximations support the familiar lens equation.

Section 2.1 supplied the complex amplitude and the rule for adding coherent fields. We now apply them between planes, while retaining Chapter 1’s signed propagation coordinates.

Wavefront normals connect phase to ray direction

A wavefront is a surface of equal phase. Where a smooth local phase and a local propagation direction are meaningful, that direction is parallel to the wavefront normal, given by the phase gradient:

In the meridional x–z plane, the signed angle to the positive z axis follows from the transverse-to-axial ratio of these components. For forward propagation with \partial\Phi/\partial z>0,

For a small angle, the arctangent is approximately its argument. In the local plane-wave interpretation, \partial\Phi/\partial z=k_z\approx k for a forward paraxial field, so

For example, a plane wave traveling in the meridional plane has spatial phase \Phi(x,z)=k_xx+k_zz+\phi_0. Its phase derivatives are \partial\Phi/\partial x=k_x and \partial\Phi/\partial z=k_z, giving

A phase that increases toward positive x tilts the local direction upward. A phase that decreases with x tilts it downward. A constant phase offset changes neither direction nor curvature. The same local relationship applies to the curved wavefronts considered next; it is not a claim that every interfering field, including its zeros, has one unambiguous local ray.

Spherical waves give a quadratic transverse phase

In a homogeneous medium, the scalar wave equation admits spherical-wave solutions as well as the plane-wave solution introduced in Section 2.1. The ideal scalar spherical wave below satisfies the source-free wave equation away from its point source. Its quadratic approximation describes a small region near the axis; the entire spherical wave is not globally paraxial.

Consider an ideal point source on the z axis at (0,0,z_s). Its outgoing spherical wave has the exact spatial form

Here s=\sqrt{x^2+y^2+(z-z_s)^2} is the distance from the source. C\ge0 is a real, constant source coefficient with units of field amplitude times length, so C/s is in V/m; \phi_0 is a constant source phase offset. The positive spatial phase matches our e^{-i\omega t} convention.

Choose an observation plane perpendicular to the z axis at coordinate z>z_s. Its axial distance from the source is R=z-z_s>0. If the source is at the origin, R=z. The distance R is constant across this fixed plane and changes when the plane moves along the axis. On the plane, define the transverse radius as r=\sqrt{x^2+y^2}. Then

For r/R\ll1, approximate the slowly changing amplitude factor by 1/s\approx1/R, while retaining the quadratic change in phase and the full on-axis phase:

The real amplitude and full spatial phase are therefore

The amplitude is approximately uniform across a small transverse region of a fixed plane. Moving the plane changes R, so its on-axis amplitude falls as 1/R and its on-axis intensity falls as 1/R^2. The quadratic term alone is the transverse phase difference relative to the axis, not the full spatial phase:

A converging spherical field approaches a point ahead of the plane. Its local form uses e^{-iks}/s, where distance s to that point decreases along propagation. Introduce a signed wavefront radius: R>0 for divergence, R<0 for convergence, and 1/R=0 for a plane wave. Both spherical cases then share the paraxial form

Using the phase-gradient relation introduced earlier gives the local ray angle:

Above the axis, x>0: positive R gives an outward angle, while negative R gives an inward angle toward the axis. See Figure 2.4(c,d) for examples of these local directions.

The radius is signed from the local wavefront geometry; it is not an unsigned object distance. The amplitude A(x,y;R) retains electric-field units. A spherical wave’s amplitude is not exactly uniform over a plane, even when its quadratic phase is a good approximation.

An ideal thin lens changes phase locally

Write U^- and U^+ for the fields immediately before and after the lens at z=0. An ideal lens of signed focal length f applies

We choose the lens transmission phase at its center as the reference, omitting a spatially uniform lens phase delay. This convention removes only that uniform delay, not the input field’s on-axis phase. Inside the clear aperture, this multiplier has unit magnitude, so |U^+|^2=|U^-|^2. The intensity does not suddenly concentrate at the lens plane. An aperture that clips the field is an additional amplitude operation and can remove power.

For the converging lens illustrated in Figure 2.6, an incident field with planar wavefronts acquires the negative quadratic phase -kr^2/(2f). This matches the paraxial phase of spherical wavefronts converging toward (0,0,f). The field magnitude remains unchanged immediately across the lens; subsequent propagation brings the contributions together into a finite focal spot. The focal region in the figure is schematic.

From lens phase to the thin-lens equation

Now consider an incident field with spherical wavefronts of signed radius R_{\mathrm{in}} at the lens plane. Its paraxial form is

Here A^-=|U^-|\ge0 is the real field amplitude and \Phi^-(0,0) is its on-axis phase. Spherical wavefronts specify the phase geometry; A^-(x,y) allows a general illumination profile, including a uniform one. For the near-axis point-source approximation introduced above, the amplitude is approximately uniform across a small illuminated region of the lens plane. Its profile does not enter the following phase-curvature calculation because the ideal phase-only lens preserves it. Adding the incident transverse phase difference to the lens’s quadratic phase gives:

The output phase has the same quadratic form as the paraxial phase of a spherical wave. We can therefore describe its wavefronts within this approximation by a new signed radius R_{\mathrm{out}}, with a plane wavefront as the zero-curvature limit. Matching the phase curvature does not imply that the output amplitude has the form of an exact spherical-wave solution.

Matching this quadratic coefficient to the output spherical-wave phase gives the curvature rule:

A plane input has 1/R_{\mathrm{in}}=0, so a converging lens produces R_{\mathrm{out}}=-f. The phase-gradient relation also gives \theta_{x,\mathrm{out}}=\theta_{x,\mathrm{in}}-x/f, matching the thin-lens ray-angle change from Chapter 1.

For a spherical field from an axial object, R_{\mathrm{in}}=-z_{\mathrm{o}}. A paraxial real image at z_{\mathrm{i}}>0 corresponds to R_{\mathrm{out}}=-z_{\mathrm{i}}. Thus the curvature rule recovers the signed imaging relation:

This predicts a geometric conjugate or collimation condition. A finite illuminated field propagates to a finite wave pattern rather than an infinite-intensity point. Sections 2.3–2.4 develop the Fourier and diffraction-PSF descriptions; Gaussian-beam parameters are introduced in Section 2.6.

Paraxial propagation combines fields from a plane

How does a known complex field at one transverse plane determine the field downstream in a homogeneous medium? Consider an input plane at z and an observation plane at z+L, separated by L>0.

The distance from a small input patch at (x',y',z) to the observation point (x,y,z+L) is

In the scalar diffraction model, each patch acts as a mathematical secondary source. Its local complex field supplies amplitude and phase; a complex propagation weight, the kernel K_L, gives its contribution to the observation point:

In the small-angle scalar diffraction model, the propagation weight is approximately

Here s is the geometric path length. Scalar diffraction theory supplies the normalization; the prefactor already uses the near-axis approximation for the propagation weighting. We next approximate the path length in the phase.

For small transverse separations compared with L, use the same paraxial path expansion introduced for the spherical wave:

The propagation phase ks then separates into an axial part and an additional part from transverse separation:

Substituting this phase approximation into the propagation weight gives the Fresnel kernel:

Here \lambda=\lambda_0/n is the wavelength in the propagation medium, and k=2\pi/\lambda. The kernel has units of inverse area, so multiplying by the patch area leaves field units. Summing the contributions from the entire input plane gives the Fresnel integral:

The integral adds complex field contributions: matching phases reinforce, while differing phases can partly or fully cancel. Calculate intensity only after the sum. This is not Chapter 1’s incoherent intensity-PSF convolution. At L=0, use the input field directly rather than the integral’s singular-looking prefactor.

The paraxial path approximation must give adequate accumulated phase accuracy over the contributing region, propagation distance, and aperture required by the task. Small angles alone do not guarantee that phase accuracy. Section 2.3 describes the same propagation through spatial-frequency components.

For the lens example, take U(x,y,0)=U^+(x,y), the field immediately after the lens. Propagate downstream to U(x,y,L) to determine how its changed phase produces the intensity distribution.

Explore: lens phase and downstream intensity

Choose plane-to-focus, diverging-to-collimated, or finite-image illumination, then adjust input vergence V_{\mathrm{in}}=1/R_{\mathrm{in}}, focal length, and observation distance. Positive input vergence here means a diverging wavefront because R_{\mathrm{in}}>0. Compare the meridional wavefront schematic, lens-plane phase, and calculated central intensity profile. Each setting describes a stationary monochromatic field; there are no animation or pulse-propagation controls.

Worked examples

Further reading

MIT OpenCourseWare, 2.71 Optics: wave-optics lecture notes develop propagation from an aperture and the quadratic phase of a thin lens. The notation here retains our medium wavelength and signed axial convention; Fourier-plane interpretations are developed in Section 2.3.