Handy shop formulas Ray Williamson, Ray Williamson Consulting 4984 Wellbrook Drive, New Port Rihey, FL 34653 www.ray-optis.om ABSTRACT A olletion of formulas is presented that are useful for those working in optis. INTRODUCTION Optial engineers and researhers have need of equations and formulas sattered throughout their textbooks. Shop personnel may not have textbooks and instead rely on little slips of paper of unknown provenane, dubious reliability, and often missing important qualifiers. My intention is to offer an aessible and reliable alternative. A version of this olletion will also appear in my upoming SPIE Field Guide to Optial Fabriation. 1. Quantities and yields, plane surfaes 1.1. Blok quantity vs. diameters, parallel blok: blok # parts 0.763 eff Where eff is the diameter plus one-sided spaing (or the enter-to-enter distane between properly spaed parts) 1.. Quantity per rail vs. diameters, wedge bar blok: b= a=3 a=0 blok eff a b a = # olumn spaings aross a entered zone of interest b = # part spaings vertially 1.3. Maximum oring yield, hex pak Area blank blank # parts 1.153 0.906 eff eff Where eff is the outer diameter of the ore drill 1.4. Maximum oring yield, square pak 1
. Spherial urves Area blank blank # parts 0.785 eff eff.1. Lensmaker s formula, thin lens 1 1 1/ EFL n 1 R1 R.. Lensmaker s formula, thik lens 1 1 1 n t 1/ EFL n 1 R1 R nr1 R.3. Diopter power (ophthalmi) per surfae 1 meter D EFL.4. Bevel up-tool radius for 45 bevel R b Where b is bevel leg length measured radial to part diameter.5. Bevel up-tool radius for bevel at to surfae normal R b sin Where b is bevel leg length measured radial to part diameter.6. Element tilt as a result of using deentered bevel as a mounting surfae b max b R min Where is tilt, b max and b min are greatest and least extents of leg lengths measured radial to part diameter, R is (onave) radius of side with deentered bevel used as mounting surfae.7. Sagitta, exellent approximation for R> s 8R.8. Sagitta, exat y y s 1 1 R R
s s R y B R-B y.9. Sagitta reported on ball-irle spherometer y y s 1 1 R B R B Where R is positive for onave, B is half-diameter of balls, and y is half-span of ball-irle as it ontats a plane..10. Fringe power differene vs. radius hange # fringes R 4R.11. Approximate spherial blok apaity, edge angle os N 6R 1 eff Where eff is the diameter plus one-sided spaing.1. Generator head angle R surfae wheel sin.13. Wedge in lenses whose enters are at radial distane r from the tool axis aused by grinding to non-ideal thikness rz R 1 sin.14. Change in thikness required to remove wedge (as above) z R ( ETV ) r.15. Slope angle at edge of spherial bloks ontaining 3 or 4 piees 1 eff 1 eff 3 sin sin 3R R
1 eff 1 eff 4 sin sin R R Where eff is part diameter plus one-sided spaing.16. Radius of pith pikup tool, onvex radii and onvex sags positive, all thiknesses positive, R 1 being bloked and R being piked up if R1 R t 0 eff pikup 1 least _ pith then R R t s t Where t is enter thikness, s is sagitta of side being piked up, and eff is part diameter plus one-sided spaing if R1 R t 0 then R R1 t t _ 3. Lens volume pikup least pith 3.1. Lens volume, ylindrial edge, with or without flat bevels s1 s V s1 R1 s R 3 3 4 Where s 1 and s are sagittas, R 1 and R are radii, is diameter, t e is edge thikness 3.. Lens volume, onial edge t e s1 s t e 1 1 V s1 R1 s R 3 3 1 Where s 1 and s are sagittas, R 1 and R are radii, 1 and are diameters, t e is edge thikness 4. Aspheri urves 4.1. Rotationally symmetri asphere equation Cr 4 6 8 s A 1/ 1r A r A3r A4r... 1 1 1 Cr Where C is urvature (reiproal radius), is oni onstant, r is radial distane from axis of symmetry, s is loal sagitta as referred to axis of symmetry, and A n s are oeffiients. NOTE: Confirm that oeffiients are intended for same units as used in evaluating sagitta, using sample sag table. 5. Thermal issues 5.1. Conversion, Celsius vs. Fahrenheit 5 9 9 3 5 C F 3 F C 5.. Conversion, Celsius vs. Kelvin C K 73.15 NOTE: Kelvins are fundamental units, not degrees.
5.3. Temperature intervals 1K 1C 1.8 F 5.4. Radius hange due to axial thermal gradient R t T R R T 5.5. Sagitta hange due to axial thermal gradient s T 8t 5.6. OPD hange due to temperature hange NOTE: not linear for large exursions dn OPD tt n tt dt dndt dt Last term above is small ompared with first term and an be dropped in most ases for onveniene: dn OPD tt n dt 5.7. Settling time for gradients to deay to 1/e Ct p k Where is time, is speifi gravity, C p is speifi heat, t is thikness, is thermal diffusivity 5.8. Settling time for gradients to deay to 10% Ct p.3 k 6. Mehanial issues 6.1. Bend due to uniform pressure for plane parallel irular, simply supported and lamped s ss 5P y 4Et 3 4 s 0.176Py 3 Et 4 Where s is sagitta at the point of interest, E is Young s modulus, P is pressure, y is radial distane from enter of part to point of interest, and t is thikness. 6.. Bend under self-weight for plane parallel irular, simply supported on edge s ss 0.176 y Et 4 Where s is sagitta at the point of interest, E is Young s modulus, P is pressure, y is radial distane from enter of part to point of interest, and t is thikness. 6.3. Minimum thikness, pressure-bearing window, simply supported and lamped
t 0.55 P SF t 0.433 ss eff f P SF eff f Where t is minimum thikness, eff is unsupported diameter, P is pressure, SF is Safety Fator, and f is rupture strength NOT YOUNG S MODULUS. NOTE: Safety Fator is in quotes beause it does not guarantee safe operation. It is a margin fator applied to the predited pressure at whih rupture would our for stati loads, no internal strains, and no surfae srathes. Typial Safety Fator for non-ritial appliations is 4, but higher fators and additional preautions are neessary for situations where injury, great expense, toxiity, or loss of life or mission are possible due to failure. 7. Angles NOTE: most autoollimators are mirror-reading, meaning that they report the tilt angle of a mirror whih is half the beam deviation of its refletion. These equations indiate the angles between beams, not the value reported in the autoollimator. 7.1. Snell s law nsin nsin 7.. Critial angle (internal), Brewster s angle (external) 1 n C sin n n n 1 B tan 7.3. Beam deviation vs. wedge, small angle, normal inidene n 1 Where is beam deviation, is wedge 7.4. Beam deviation vs. wedge, 1 st surfae normal, exat sin 1 nsin Where is beam deviation, is wedge 7.5. External 90 bank shot refletions 1 4 Where 1 and are returns from opposing sides, is error in 90 7.6. Internal bank shot refletions, 90, 60, 45 4n 6n 8n 1 90 60 45 Where 1 and are returns from opposing sides, is error in physial angle 7.7. First surfae vs. seond surfae refletions, small angles 1 n Where 1 and are returns from opposing sides, is error in physial angle 8. Centering 8.1. Beam deviation angle vs. wedge n 1
Where is single-pass transmitted beam deviation, is physial wedge 8.. Single radius of urvature deenter r Rtan Where r is deenter, R is radius, and is physial wedge (at physial enter of part, or equivalently aross diameter) 8.3. Image deenter vs. wedge angle r n 1 EFL tan EFL tan Where r is deenter, EFL is effetive foal length, is physial wedge in lens, and is transmitted beam deviation 8.4. Edge thikness variation or axial runout ETV tan Where ETV is edge thikness variation, is part diameter, and is physial wedge angle 8.5. Lateral displaement of lens huked on one side needed to restore entration R1 R tan R1 R tan r R R t n R R t 1 1 1 Where r is lateral displaement, R s are radii, is physial wedge, t is enter thikness, and is transmitted beam deviation 8.6. Element tilt as a result of using deentered bevel as a mounting surfae b b R max min Where is tilt, b max and b min are greatest and least extents of leg lengths measured radial to part diameter, R is (onave) radius of side with deentered bevel used as mounting surfae 9. Fringe sale fators, normal inidene OPD n h a-) 1st surfae refletion, 1 1 d-g) DPTWF, e-b) nd surfae refletion, f-h) internal fringes, OPD n h h OPD n n1 h1 n n3 h OPD n n h n h 1 9.1. Ratio of fringes to wavelengths of OPD a b d e g f n 1 t h 1 h h n 1 1 n 3
1:1 at the plane of interferene. Always. 9.. Test plate fringes OPD h h # fringes Where h is surfae form error 9.3. Internal fringes vs. thikness variation OPD nt n h h 1 9.4. 1 st surfae refleted wavefront per refletion, AOI = 0, embedded in index n 1 OPD RWF n h h # fringes 1 9.5. nd surfae refleted wavefront per refletion, AOI = 0, embedded in index n1 OPD n n1 h1 nh 9.6. Single pass transmitted wavefront, AOI = 0, embedded in indies n 1 and n 3 OPD n n h n n h 1 1 3 10. Fringe saling fators, oblique inidene i j k l j-i) oblique 1 st surfae refletion k-l) oblique DPTWF 10.1. Refleted wavefront per pass, oblique inidene, embedded in index n1 OPD n hos 1 Where n 1 is index outside of item under test, h is surfae height variation, and is angle of inidene 10.. Single pass transmitted wavefront, oblique inidene, in air OPD h n sin os Where h is surfae height variation, n is index inside of item under test, h is surfae height variation, and is angle of inidene.