How to Draw and Read Ray Diagrams Draw the image forming device(s), i.e., lens(es), mirror(s), on the optical axis A. Draw a principal plane (line) H for each device perpendicular to optical axis. For a (thin) lens, its principal plane passes through the middle of the lens; for a mirror, its principal plane passes through the point where the reflecting mirror surface intersects the optical axis: If you know where the original object is located, trace some schematic ray through each device to mark the “incoming”, “not incoming”, “outgoing”, “not outgoing” side for each device, e.g., as shown above; see also the “Microscope” example on how to do this. Remember: “incoming” is where the ray enters/strikes the device; “outgoing” is where the ray leaves/emerges from the device. Hence, for a lens, “incoming” and “outgoing” are always on opposite sides of the lens; but for a mirror “incoming” and “outgoing” are always on the same side of the mirror. Lens Principal Plane H(Lens) Mirror Principal Plane H(Mirror) Optical Axis A In, Not Out Out, Not In In, Out Not In, Not Out Orig. Object

If you know the focal length, f, of the device mark the incoming focal point, F, and the outgoing focal point, F’, on the optical axis A, following these sign rules: If f>0: F on “In”, F’ on “Out” If f<0: F on “Not In”, F’ on “Not Out” Alternatively, if you know, for example, on which side(s) of the device F and/or F’ are located, i.e., whether F is on “In” or “Not In” and/or whether F’ is on “Out” or “Not Out”, you can Infer from the foregoing rules whether f>0 or f<0. Also, if you know that F and F’ are on opposite sides of the device, you can infer that the device is a lens; whereas, if F and F’ coincide on the same sides of the device, you can infer that the device is a mirror. In the two examples below, the placement of F, relative to “In”, and of F’, relative to “Out”, is shown for a convergent (convex) lens, i.e., a lens with f>0; and for a convex (divergent) mirror mirror, i.e., a mirror with f<0. By the sign rule above, F is placed on “In” and F’ on “Out” for the lens; but F is placed on “Not In” and F’ on “Not Out” for the mirror. Mirror with f<0 H Optical Axis A H In, Not Out Out, Not In In, Out Not In, Not Out F F’ F=F’ |f| Lens with f>0

Once you have properly identified “In” and “Out” and correctly placed F and F’ on the optical axis A, you can start to draw a ray diagram to find the image for a given object; or vice versa to find the object for a given image. Here are the rules, to find the image, Q’, for a given object point, Q, which must not be on the optical axis A: 1) Draw the P-ray: a straight line through Q, parallel to A. 2) Extend the P-ray to intersect H; label that intersection point Q P 3) Draw the P’-ray: a straight line through Q P and F’. 4) Draw the F-ray: a straight line through Q and F. 5) Extend the F-ray to intersect H; label that intersection point Q F 6) Draw the F’-ray: a straight line through Q F, parallel to A. 7) Extend P’-ray and F’-ray, in either direction, until they intersect: that’s the image point Q’. Here are two examples, using the same lens (f>0) and mirror (f<0) shown on previous slide: H In, Not Out Out, Not In Lens with f>0, d>f, gives: real, Inverted Image, d’>0, m<0 Object A F F’ F’-ray P’-ray QFQF Q ’ P-ray F-ray QPQP Q Image Mirror with f 0, gives: virtual, erect image, d’ 0 H A In, Out Not In, Not Out P-ray F-ray QPQP Q F=F’ Q ’ F’-ray P’-ray QFQF Object Image

Use the same rules “in reverse”, to find the object point, Q, for a given image point, Q’, which, again, must not be on the optical axis A: 1) Draw the P’-ray: a straight line through Q’ and F’. 2) Extend the P’-ray to intersect H; label that intersection point Q P 3) Draw the P-ray: a straight line through Q P, parallel to A. 4) Draw the F’-ray: a straight line through Q’, parallel to A. 5) Extend the F’-ray to intersect H; label that intersection point Q F 6) Draw the F-ray: a straight line through Q F and F. 7) Extend P-ray and F-ray, in either direction, until they intersect: that’s the object point Q. The same two examples, using the same lens (f>0) and mirror (f<0) shown on previous slide, also illustrate these rules in reverse: H In, Not Out Out, Not In Lens with f>0, d>f, gives: real, Inverted Image, d’>0, m<0 Object A F F’ F’-ray P’-ray QFQF Q ’ P-ray F-ray QPQP Q Image Mirror with f 0, gives: virtual, erect image, d’ 0 H A In, Out Not In, Not Out P-ray F-ray QPQP Q F=F’ Q ’ F’-ray P’-ray QFQF Object Image

Here are some other examples: Q ’ In, Not Out Out, Not In Lens with f>0, f>d>0, gives: virtual, erect Image, d’ 0 Object: Q A F F’ P-ray F-ray QPQP Q QFQF H F’-ray P’-ray Image: Q ‘ Mirror with f>0, f>d>0, gives: virtual, erect image, d’ 0 H Not In, Not Out In, Out P-ray F-ray QFQF F=F’ Q Q ’ F’-ray QPQP A P’-ray Object: Q Image: Q ‘

In the Galilean telescope, a virtual object is presented to a divergent eyepiece lens: Img.1, formed by Lens 1 (objective lens) at its focal point F 1 ’, serves as virtual object, Obj.2, for Lens 2 (eyepiece). The ray diagram for Lens 2 forming the virtual Img.2 from virtual Obj.2, drawn according to the same 7 ray diagram rules, is shown below. The required focal points of Lens 2, F 2 and F 2 ’, are placed on the “Not In” and “Not Out” side of Lens 2, resp., since f 2 <0. From virtual object point Q 2, virtual image point Q 2 ‘ is constructed, with both d 2 <0 and d 2 ’<0. F2’F2’F2F2 F1’F1’ F-ray (Lens 2) P-ray (Lens 2) Obj.1 Img.2 Img.1 = Obj.2 d1d1 d1’d1’ d2d2 d2’d2’ In, Not Out Out, Not In Q 1 ’ = Q 2 Divergent Lens 2: f 2 <0Convergent Lens 1: f 1 >0 Q2’Q2’ H2H2 H1H1 A QPQP QFQF F ’-ray P ’-ray