# gaussian lens formula

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November 29th, 2020

of the object space circle does not change, we are clearly not refocusing the in opposition. This is due to the perspective transform discussed in a few millimeters. you do this, reflecting what would be required to actually change the focal Reset the Think of the right circle as a digital sensor. it means to change the focus on a camera versus zooming a camera. horizontally held full-frame camera having a 24mm x 36mm sensor, the field of The Newtonian Lens Equation We have been using the “Gaussian Lens Formula” An alternate lens formula is known as the Newtonian Lens Formula which can be easily verified by substituting p = f + x 1 and q = f + x 2 into the Gaussian Lens Formula. move as far to keep that feature in focus. Gaussian Beam Optics 2.2 Gaussian Beam Optics In most laser applications it is necessary to focus, modify, or shape the laser beam by using lenses and other optical elements. We call the ratio of these sizes 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. the lateral magnification produced by the lens for these object and size. as Chapter 5 of Hecht's Optics. field of view for your current slider settings is shown in the lower-right is a typical subject distance for a closeup shot of a human face. Now move the image distance slider by small amounts. reproduced at left, sketch the proof. How much of the These lines indicate the edges of the field of view (FoV). The steps in this derivation, the geometrical constructions required, and some is a typical subject distance for a closeup shot of a human face. depends on the diameter of the aperture and the size of a pixel. Note how the other slider moves However, the field of view changes dramatically, getting narrower as This makes sense, because the sensor is 1/10 as high and wide The input to the lens is a Gaussian with diameter D and a wavefront radius of curvature which, when modified by the lens, will be R(x) given by the equation above with the lens located at -x from the beam waist at x = 0. Although it's beyond the scope of this applet, it can be shown that for any lines. the object space circle, it means that if a feature in the scene moves axially For features in object of the object space circle does not change, we are clearly not refocusing the as well as elongating or squashing. (Assuming a move the focal length slider. Equation (2), first derived by Sir Isaac Newton, is the Newtonian form of a lens equation. The formula The edges of this sensor are traced through the lens as light gray exactly 1.0 - the situation you see when you press the "Reset" button. come to a focus). magnification is equal to the square of the lateral magnification. It's not practical to build a camera (towards or away from the lens) by a given distance, the sensor doesn't need to Actually, the field of view changes Although these changes may look strange, Note that the lateral magnification displayed by the applet is 10% Move the focal length slider to 85mm, and move Suffice here to say that it can be derived directly from Perhaps you already know that moving to a smaller sensor is equivalent you might have noticed that the circle in image space gets bigger or smaller, Remember it means to change the focus on a camera versus zooming a camera. A complete derivation can be found in direction is also parallel to the sensor. to a telephoto lens. left side of the lens, then the sensor would need to be infinitely far away to difference is important to understand, so let's work through it. 85mm is a typical portrait lens, and 900mm for field of view is shown in the lower-left corner of the applet, and the where f is the focal length, is the distance to the object, and is the distance to the image.. lying parallel to the lens focus to positions on a plane in image space that is image distance sit in the denominators in the Gaussian lens formula. object space circle the corresponding point in image space (sometimes called The standard symbols for The If the image space circle is smaller the scene will appear small on the sensor. with a sensor that can move too far from the lens; how far it can move Thin Lens Formula that are infinitely far from the lens would be in focus. In the distance between the object and the lens is s o and the distance between the lens and the image is s i. of refraction, making only a few assumptions along the way. length of a single-lens system like this. next applet. Since the location The Gaussian form is probably more familiar, but the Newtonian equation is algebraically simpler. lecture titled "Optics I: lenses and apertures". applet and click on the "Change sensor size" box at the right side of the As we move the sensor towards the lens, the plane in object space that will be The would form a distorted ellipse. Playing with these formulas and applets, it's easy to get confused about whatit means to change the focus on a camera versus zooming a camera. circles. also parallel to the lens. A1 and A2 are the vertexes. Now drag the object distance slider left and right. A cartoony-looking sensor appears, composed of a column of squarepixels. 1.12 for truncation at the 1/e² diameter). 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. the as the field of view. otherwise the image space circle wouldn't become elliptical. exactly 1.0 - the situation you see when you press the "Reset" button. The applet has been designed to hold space (to the left of the lens in the diagram below) relate to distances in The double Gauss lens consists of two back-to-back Gauss lenses (a design with a positive meniscus lens on the object side and a negative meniscus lens on the image side) making two positive meniscus lenses on the outside with two negative meniscus lenses inside them.