2.3 Additional Relations

Transcription

1 3 2.3 Additional Relations Figure 2.3 identiies additional relations, indicating te locations o te object and image, and te ratio o teir eigts (magniication) and orientations. Ray enters te lens parallel to te optical axis. Tereore, te point at wic it crosses te axis ater exiting te lens is te second ocal point F 2. Ray 2 passes troug te ront ocus F and leaves te lens parallel to te optical axis. Te two nodal points are special cardinal points because tey lie on te optical axis. A ray aiming at N leaves te lens rom N 2 in te same direction as it entered te lens, indicated wit ray 3. empty space (iatus) e ray d ray 2 d 2 ray 3 P P 2 optical axis F F 2 N N 2 P 3 P 4 l bl l l ront principal plane rear principal plane Figure 2.3 Positive lens and its cardinal points. Focal points, F and F 2. Nodal points, N and N 2. Principal points, P,P 2,P 3, and P 4.

2 4 Capter 2 Te principal planes are actually sperical suraces, but tey can be treated as planes in te paraxial region, wic is te region close to te optical axis, were te sine and tangent o te ray angles are close to eac oter, and to te angle expressed in radians. As a point o reerence, sin 5 deg ¼ , and tan 5 deg ¼ Five degrees also equals te 72t part o te 2p ull 360-deg circle. Tereore, 5 deg represents ð2p 360Þ 5¼ p 36 ¼ rad. It is a limit o acceptable accuracy in wat is termed te paraxial region. Tis paraxial treatment is useul even wen rays are ar rom te optical axis because, wit te assumption o te trigonometric unctions equality, te equations become linear wereby a simple scaling eect is acieved. Te caracteristic eature o te principal planes is tat te magniication between tem is unity, wic means tat te rays are transerred at te same eigt rom te ront principal plane to te rear principal plane. In general, a positive lens is used to orm an image o an object wit a certain magniication. Sign Convention Distances to te let o te ront principal plane and eigts below te optical axis are negative; distances to te rigt o te rear principal plane and eigts above te optical axis are positive. Tis indicates tat te ligt is assumed to travel rom te let to te rigt. Te ocal lengts o a positive lens, including te ront and back ocal lengts, are positive. It is extremely important to very careully apply te agreed-upon sign convention.

3 5 Te ollowing equations reer to te call-outs in Fig. 2.3: Magniication m ¼ l 0 Back ocal lengt Front ocal lengt b l ¼ l ¼ l ¼ 0 : (2.3) ðn Þt nr : (2.4) ðn Þt nr 2 : (2.5) Te distance rom te vertex o te ront surace to te ront principal plane is ðn Þt d ¼, (2.6) nr 2 and te distance rom te vertex o te rear surace to te rear principal plane is ðn Þt d 2 ¼ : (2.7) nr To demonstrate wat is meant by careully observing te sign convention, we derive te so-called Gaussian expression* or te location o te image. In Fig. 2.3, it can be seen tat te ollowing relations exist: ¼ ð 0 Þ l 0 0 ¼ ð 0 Þ l *Joann Carl Friedric Gauss, a German matematician, lived rom 777 until 855 and is well-known or is many contributions in te ields o matematics and pysics.

4 6 Capter 2 Rearranging leads to l 0 ¼ ð 0 Þ, (2.8) l ¼ 0 ð 0 Þ : (2.9) Subtracting Eq. (2.9) rom Eq. (2.8) yields l 0 l ¼ ð 0 Þ 0 ð 0 Þ ¼ ð 0 Þ ð 0 ¼ Þ : Te inal Gaussian orm is usually presented as l 0 ¼ l þ : (2.0) as To ind te image location, one rewrites Eq. (2.0) to read l 0 ¼ l l þ : (2.) Exercise 2 Find te image location and eigt o an object 5 mm ig, located 50 mm to te let o te vertex o te lens discussed in exercise. Approac and Solution Given are l d ¼ 50 mm; ¼ 5 mm. From exercise we know tat te ocal lengt ¼ 00 mm, te lens tickness t ¼ 5 mm, te index o reraction n ¼.5, and te rear radius o te lens R 2 ¼ 200 mm.

5 7 Using Eq. (2.6), we ind te location o te ront principal plane ðn Þt ð.5 Þ5 00 d ¼ ¼ ¼ 2.5 mm: nr 2.5 ð 200Þ l 0 ¼ Wit tat, l ¼ 50 d ¼ ¼ 52.5 mm. Equation (2.) yields l l þ ¼ 52.5 þ 00 ¼ 52,500 ¼ 2, mm: 52.5 Te magniication is ound wit Eq. (2.3), i.e., m ¼ l 0 l ¼ 2, ð 52.5Þ 9. Te image eigt, also using Eq. (2.3), is 0 ¼ m ¼ ð 9Þ5 ¼ 95 mm. To ind te distance o te image rom te vertex o te rear surace o te lens, we must subtract d 2 rom l 0. Using Eqs. (2.7) and (2.), we obtain l 0 d 2 ¼ l 0 ðn Þt ¼ 2, nr ð.5 Þ5 00 ¼ 2, mm: Negative Lens, Focal Lengt, and Back Focal Lengt A negative lens as a sape as sown in Fig Since its ocal lengt is negative, te locations o te ocal points are in reverse order compared to te positive element, as indicated in Fig By coosing te ront radius R ¼ 60 mm, rear radius R 2 ¼ mm, tickness t ¼ 5 mm, and again