Gravitational Lensing With Adobe Photoshop

Extract from the zip archive the plug-in itself, the file grlens13.8bf, and copy it into your plug-in directory (usually C:\PHOTOSHP\PLUGINS). Next time you run Adobe Photoshop, it will recognize a new category in the Filter menu, Astronomy, with Gravitational Lenser 1.3 in it. Note that the filter is available only if you have a RGB (16.7-million-color) image opened, since it doesn't work with other types (in particular grayscale images or indexed color GIFs).

This is the menu of Gravitational Lenser 1.3:

Position of the lens within the image

The actual range is [0,X] and [Y,0], where X is the width and Y is the height of the image in pixels. Note the reversed limits of the vertical shift slider – the origin of the coordinate system (x=0, y=0) is in the top left-hand corner of the image. The default value for the first two sliders is 128, understood by the filter as X/2 and Y/2, respectively, that is the image's center.

Einstein radius

The basic characteristic of a gravitational lens with circular symmetry (see below). The slider changes it from 0 to the minimum of X/2 and Y/2. For example, when the image you are lensing have width (X) 1000 and height (Y) 1200 pixels, and the slider shows 51, the Einstein radius of the applied gravitational lens is 100 pixels, because (51/255)*(1000/2) equals 100.

RGB Color

Since Gravitational Lenser 1.3 is intended first of all as an educational tool, it adds a circle of the Einstein radius to the lensed image. Its RGB color can be changed by the last three sliders. The default color is R=255, G=255, B=255, nothing else than pure white. The same color applies also to a strangely shaped area around the lens. This is an undefined area where the filter tries to map pixels from outside the image. Curiously enough, these sliders shows the actual values!

Underlying Geometry and Physics

An attempt to model with Filter Factory the appearance of the starry skies during a fall into a black hole would give you a pretty hard time, but a typical gravitational lens is fortunately much easier to code. Under most circumstances, we can take advantage of several considerable simplifications (not independent of each other):

Thin screen (lens) approximation

Makes use of the fact that most of the light deflection occurs within a distance from the lens which is comparable to the impact parameter and much smaller than the total length of the light path from the source to the observer. The lens is no longer considered a three-dimensional object. Instead, it is approximated by a thin mass sheet perpendicular to the optical axis, referred to as the lens plane (yellow in the Figure 1 below). The effect of the lens depends only on projection of its mass to this plane. The light ray consists of two straight lines which meet at the lens plane.

Small angles approximation

Assumes that the angles b, h, and the deflection angle a are very small and their tangents can be replaced by the angles themselves (measured in radians). Note this condition is intimately related to the thin screen approximation – given the distances between the observer and the lens, and the lens and the source are required to be much greater than the impact parameter, the light ray can reach the observer only if the deflection angle is small enough.

Weak field approximation

Requires that the light passes through a weak field with the absolute value of the Newtonian gravitational potential much smaller than the square of the speed of light c. In addition, the relative (peculiar) velocities of the observer, the lens and the source must be much smaller than c.

Again, this is related to the small angles approximation because the weak field implies the small deflection angle.

Source Code

The lens equation is surprisingly easy to code with the Filter Factory programming language. If the lens characterized by the Einstein radius of 100 pixels is situated in the middle of the image, the entire code reads as follows:

R: rad(d,m–100*100/m,z)

G: rad(d,m–100*100/m,z)

B: rad(d,m–100*100/m,z)

(Note that since gravitational lenses are achromatic, the code is identical for all three colors R, G, and B.)

Mathematically speaking, the Filter Factory statement

rad(function₁(d),function₂(m),z)

phi_old = function₁(phi_new)

r_old = function₂(r_new)

In particular, the statement

rad(d,m,z)

leaves the image as it is, while the statement

rad(d,m/2,z)

enlarges it twice.

For those familiar with Filter Factory programming, here is the complete source code of Gravitational Lenser 1.3:

R:

put(x-val(0,0,X),0),put(y-val(1,Y,0),1),
put(get(0)*get(0)+get(1)*get(1),2),
put(val(2,0,min(X/2,Y/2)),6),
put(get(0)*get(6)*get(6)/get(2),3),
put(get(1)*get(6)*get(6)/get(2),4),
put(sqr(get(2))==get(6) || get(2)==0 || (x-get(3) > X)
|| (x-get(3) < 0) || (y-get(4) > Y) || (y-get(4) < 0),5),
get(5) ? ctl(3+z) : src(x-get(3),y-get(4),z)

G:

get(5) ? ctl(3+z) : src(x-get(3),y-get(4),z)

B:

get(5) ? ctl(3+z) : src(x-get(3),y-get(4),z)