Gaussian lens formula

Applet: Katie Dektar

Technical assistance: Andrew Adams

Text: Marc Levoy

In the preceeding applet we introduced Gauss's ray diagram, which allowed us to find for any point in object space the position in image space where rays leaving that point reconverge (i.e. come to a focus). One implication of Gauss's diagram is that points on a plane in object space lying parallel to the lens focus to positions on a plane in image space that is also parallel to the lens. Thus, we can talk about how distances in object space (to the left of the lens in the diagram below) relate to distances in image space (to the right of the lens). This relationship, which depends on the focal length f of the lens, is given by the Gaussian Lens Formula, shown in the lower-left corner of the applet below. The standard symbols for object space distance and image space distance are so and si, respectively.

It is beyond the scope of this applet to derive the Gaussian lens formula. Suffice here to say that it can be derived directly from Snell's law of refraction, making only a few assumptions along the way. The steps in this derivation, the geometrical constructions required, and some of the algebra can be found in slides 15-23 of this lecture, or in simplified form in slides 24-26 of the same lecture. A complete derivation can be found in any undergraduate textbook on optics, such as Chapter 5 of Hecht's Optics.

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Focusing in photography

Drag the image distance slider left and right. Note how the other slider moves in opposition. Note also how the two red circles in the applet move. Remember from our earlier discussion that these circles can be treated as the location of planes parallel to the lens. Think of the right circle as a digital sensor. As we move the sensor towards the lens, the plane in object space that will be in focus moves away from the lens, slowly at first, but accelerating as the sensor approaches the tick mark representing one focal length f from the lens. If we could bring the sensor right to the tick mark in the applet, i.e. make image distance = f, then only points in object space that are infinitely far from the lens would be in focus. If this were a photographic camera, we would say it is "focused on infinity".

Now drag the object distance slider left and right. For features in object space at this distance from the lens, the sensor would have to be placed at the distance computed by the applet in order for these features to be in focus. Note again the non-linear relationship between the movement of the two red circles. This non-linearity is caused by the fact that object distance and image distance sit in the denominators in the Gaussian lens formula. Once again, if you could bring the object space circle right to the tick mark on the left side of the lens, then the sensor would need to be infinitely far away to create a focused view of that circle. It's not practical to build a camera with a sensor that can move too far from the lens; how far it can move determines the closest distance on which you can focus. For practical photographic lenses, this minimum focus distance ranges from a few feet to a few millimeters.

The relationship between focusing and zooming

Playing with these formulas and applets, it's easy to get confused about what it means to change the focus on a camera versus zooming a camera. The difference is important to understand, so let's work through it. Reset the applet and click on the "Change sensor size" box at the right side of the applet. A cartoony-looking sensor appears, composed of a column of square pixels. The edges of this sensor are traced through the lens as light gray lines. These lines indicate the edges of the field of view (FoV). The formula for field of view is shown in the lower-left corner of the applet, and the field of view for your current slider settings is shown in the lower-right corner. Now move the image distance slider by small amounts. The distance to the in-focus plane in object space changes, as you would expect, but the field of view stays constant. This is focusing. Actually, the field of view changes slightly. This is due to the perspective transform discussed in a preceeding applet.

Reset the applet again, click once more on the "Change sensor size" box, and move the focal length slider. Note that the lens gets thicker and thinner as you do this, reflecting what would be required to actually change the focal length of a single-lens system like this. The applet has been designed to hold object distance constant when you change focal length, so instead the image distance changes to keep the object space circle in focus. Since the location of the object space circle does not change, we are clearly not refocusing the camera. However, the field of view changes dramatically, getting narrower as the focal length increases, as one would expect when moving from a wide-angle to a telephoto lens. This is zooming. We've also provided a slider for sensor size. Perhaps you already know that moving to a smaller sensor is equivalent to moving to a longer focal length; both decrease the field of view. That's why for APS-C cameras (like the Canon Digital Rebel series), you need to multiply the focal length engraved on the lens by 1.6 to get the "35mm equivalent focal length".

Lateral and longitudinal magnification

As you moved the focal length slider left and right in the previous paragraph, you might have noticed that the circle in image space gets bigger or smaller, as well as elongating or squashing. Although these changes may look strange, they are precisely correct for the situation. If we were to employ the Gaussian ray diagram to compute for each point on the object space circle the corresponding point in image space (sometimes called the conjugate point), the locus of all such points in image space would form a distorted ellipse.

How should we interpret these changes in the shape of the image space circle? Think first of the circle's size in the direction parallel to the lens. This direction is also parallel to the sensor. If the image space circle is smaller in this direction than the object space circle, it means that big features in the scene will appear small on the sensor. We call the ratio of these sizes the lateral magnification produced by the lens for these object and image distances. For photographic cameras this ratio is typically less than 1.0, and for microscopes it is typically greater than 1.0. For macro lenses, it may be exactly 1.0 - the situation you see when you press the "Reset" button. In this configuration we say that the lens is producing 1:1 imaging.

Now examine the size of the image circle in the direction perpendicular to the lens and sensor. if the image space circle is smaller in this direction than the object space circle, it means that if a feature in the scene moves axially (towards or away from the lens) by a given distance, the sensor doesn't need to move as far to keep that feature in focus. We call the ratio of these sizes the longitudinal magnification produced by the lens. It rises and falls with the lateral magnification, but not in proportion; otherwise the image space circle wouldn't become elliptical.

Although it's beyond the scope of this applet, it can be shown that for any focal length, object distance, and image distance, the longitudinal magnification is equal to the square of the lateral magnification. Slides 40 and 41 from the lecture titled "Optics I: lenses and apertures", reproduced at left, sketch the proof. A complete derivation can be found in Chapter 5 of Hecht. It is this non-linear relationship between longitudinal and lateral magnification that gives the image space replica of an object space cube its characteristic frustum shape in the preceeding applet.

Relationship to depth of field

Here's one more thing to try. Move the focal length slider to 85mm, and move the object space slider to 900mm. 85mm is a typical portrait lens, and 900mm is a typical subject distance for a closeup shot of a human face. (Assuming a horizontally held full-frame camera having a 24mm x 36mm sensor, the field of view at this distance and with this lens will be 10 inches tall x 15 inches wide.) Note that the lateral magnification displayed by the applet is 10% (meaning 0.1x). This makes sense, because the sensor is 1/10 as high and wide as the field of view.

Note also that the longitudinal magnification displayed is 10% x 10% = 1%. This means that for every 1mm you move the lens or sensor, you'll shift the in-focus plane by 100mm (4 inches) in the scene. That's about the depth of a human face, and it shows how delicate accurate focusing is! How much of the face will be "in focus" at once in this situation is another question, which depends on the diameter of the aperture and the size of a pixel. We consider this question in the next applet.

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