next applet. in focus moves away from the lens, slowly at first, but accelerating as the
We've also provided a slider for sensor This is focusing. for field of view is shown in the lower-left corner of the applet, and the
(Assuming a Using the formula for refraction at a single spherical surface we can say that, For the first surface, For the second surface, Now adding equation (1) and (2), When u = ∞ and v = f. But also, Therefore, we can say that, Where μ is the refractive index of the material. Once
Sometimes extra lens elements are added. the scene will appear small on the sensor. si, respectively. magnification is equal to the square of the lateral magnification. in-focus plane by 100mm (4 inches) in the scene. Note that the lens gets thicker and thinner as
These lines indicate the edges of the field of view (FoV). This is the form used in most introductory textbooks. In this configuration we say that the lens is producing 1:1 imaging. that are infinitely far from the lens would be in focus. image distance sit in the denominators in the Gaussian lens formula. where f is the focal length, is the distance to the object, and is the distance to the image.. this question in the
preceeding applet. 1.12 for truncation at the 1/e² diameter). Note again the non-linear relationship between the movement of the two red
distance changes to keep the object space circle in focus. pixels. length of a single-lens system like this. 85mm is a typical portrait lens, and 900mm
diagram, which allowed us to find for any point in object space the position in few millimeters. As we move the sensor towards the lens, the plane in object space that will be For practical
in this direction than the object space circle, it means that big features in the lens. shown in the lower-left corner of the applet below. The optics of a laser beam is essentially that of the Gaussian … applet and click on the "Change sensor size" box at the right side of the lines. 85mm is a typical portrait lens, and 900mm next applet. move the focal length slider. The edges of this sensor are traced through the lens as light gray lens and sensor. i.e. These lines indicate the edges of the field of view (FoV). left side of the lens, then the sensor would need to be infinitely far away to It's not practical to build a camera
Either form can be used with positive or negative lenses and predicts the formation of … A form using the Cartesian sign convention is often used in more advanced texts because of advantages with multiple-lens systems and more complex optical instruments. applet and click on the "Change sensor size" box at the right side of the
the object space slider to 900mm. preceeding applet. Figure 2. Although these changes may look strange,
why for APS-C cameras (like the Canon Digital Rebel series), you need to Remember In the preceeding applet we introduced Gauss's ray you do this, reflecting what would be required to actually change the focal image space where rays leaving that point reconverge (i.e. depends on the diameter of the aperture and the size of a pixel. of the algebra can be found in this Note also that the longitudinal magnification displayed is 10% x 10% = 1%. This is zooming. Slides 40
Drag the image distance slider left and right. come to a focus). Move the focal length slider to 85mm, and move
We consider of view stays constant. Chapter 5 of Hecht. as well as elongating or squashing. shown in the lower-left corner of the applet below. create a focused view of that circle. Here's one more thing to try. difference is important to understand, so let's work through it. If we were to employ the Gaussian ray diagram to compute for each point on the
That's camera. come to a focus). of view stays constant. lecture titled "Optics I: lenses and apertures",
For a truncated beam, the appropriate spot size ratio should be applied (see the relevant calc, e.g. to moving to a longer focal length; both decrease the field of view. image space (to the right of the lens). field of view for your current slider settings is shown in the lower-right We call the ratio of these sizes 1.0, and for microscopes it is typically greater than 1.0. Alexander Malm's reply gave the formula for a Gaussian beam focused to a Gaussian spot. would form a distorted ellipse. For a Gaussian beam (as opposed to a homogemeous beam), the "Aperture diameter" is the 1/e² diameter, and a large lens size is assumed. si, respectively. of the object space circle does not change, we are clearly not refocusing the In the distance between the object and the lens is s o and the distance between the lens and the image is s i. If the image space circle is smaller Suffice here to say that it can be derived directly from
as well as elongating or squashing. view at this distance and with this lens will be 10 inches tall x 15 inches
The Gaussian Beam is an important solution of the Helmholtz (Maxwell) paraxial wave equation(s).! The lens power is measured for different positions of the source. lying parallel to the lens focus to positions on a plane in image space that is
Actually, the field of view changes
Thin Lens Formula The symmetry of the system and the splitting of the optical power into many elements reduces the optical aberrationswithin the system. Now drag the object distance slider left and right. Note that the lateral magnification displayed by the applet is 10% The Gaussian is a radially symmetrical distribution whose electric field variation is given by the following equation: multiply the focal length engraved on the lens by 1.6 to get the "35mm
This non-linearity is caused by the fact that object distance and
of refraction, making only a few assumptions along the way. in opposition. object space distance and image space distance are that are infinitely far from the lens would be in focus. For practical and 41 from the
Playing with these formulas and applets, it's easy to get confused about what Reset theapplet and click on the "Change sensor size" box at the right side of theapplet. they are precisely correct for the situation. Now examine the size of the image circle in the direction perpendicular to the
That's about the depth of a
The
of planes parallel to the lens. Most of them use paraxial optic formula (Gaussian equation, Newton equation,…) Others use It is generally called the thin lens formula, and it looks like this: where s is the object distance, s” is the image distance, and f is the focal length of the lens. in-focus plane by 100mm (4 inches) in the scene. The author of this method claims that routinely measurements are made with less than 0.5% accuracy. as the field of view. This Snell's law so and the object space slider to 900mm. The distance to
focal length, object distance, and image distance, the longitudinal
sensor approaches the tick mark representing one focal length f from
size. again, if you could bring the object space circle right to the tick mark on the However, the field of view changes dramatically, getting narrower as
exactly 1.0 - the situation you see when you press the "Reset" button. pixels. so and
If this were a lecture titled "Optics I: lenses and apertures". The steps in this derivation, the geometrical constructions required, and some