Newton's Prism Experiment and Goethe's Objections

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2. Snell’s law and Sellmeier’s equation The refraction of a ray which enters from a medium with refraction index n 1 into a medium with refraction index n 2 is defined by Snell’s law. The angles α and β are shown on the next page.The refraction index for air is practically 1.0. sin( b) n1 1 = = sin( a ) n2n Goethe said: Willebrord Snellius (1591-1626) knew the fundamentals but he did not know the definition by sine functions [2, Erster Teil, Fünfte Abteilung ]. The refraction index as a function of the wavelength can be modelled by Sellmeier’s equation: 2 B1 n C 2 l ( l) = 1+ 2 l - 1.80 n 1.75 1.70 1.65 1.60 1.55 1 2 B2 l + 2 l - C 2 SF15 BK7 1.50 0.3 0.4 0.5 0.6 0.7 λ / µ m 0.8 2 B3 l + 2 l - C 3 The structure of this function is essentially based on Maxwell’s theory of continua [4]. The function has three poles which are outside the spectrum of visible light. The poles indicate resonance phenomena without damping. In reality the peaks are finite because of damping. The actual parameters are found by measurements, here for special types of flint and crown glass. The refraction of flint glass is stronger. Because of the steeper slope the dispersion is stronger as well. Combinations of flint glass and crown glass can be used for achromatic lens or prism systems which show no dispersion for two distinct wavelengths and nearly no dispersions for the others. Goethe knew this already. Tables are supplied by Schott AG [15], here used for flint glass SF15. Thanks to Danny Rich. The Sellmeier equation and the data for crown glass BK7 were found at Wikipedia [16]. 6 Glass SF15 B1 1.539259 B2 0.247621 B3 1.038164 C1 0.011931 C2 0.055608 C3 116.416755 Glass BK7 B1 1.039612 B2 0.231792 B3 1.010469 C1 0.006001 C2 0.020018 C3 103.560653
3. Calculations for the ray path The path of the ray is calculated by intersections of line segments, using Snell’s law and some simple geometry: lines in parametric representation; solutions by Cramer’s rule. All direction vectors are normalized for unit length. The origin of u 1 is at C 1 . � 1 R1 C1 u1 n1 d1 g1 = È - sin( g) ˘ ÎÍ + cos( g) ˚˙ u1 = È - sin( r1) ˘ ÎÍ - cos( r1) ˚˙ d1 = È+ cos( r 1) ˘ ÎÍ - sin( r1) ˚˙ r1 = c1+ uu1 x1 = r1+ l d1 x2 = m g1 x1 = x2 mg1 - ld1 = r1 m g1x - l d1x = r1x mg - ld = r 1y 1y 1y � 1 Solution by Cramer’s rule dA = d1xg1y - d g dM = d1 r1 -d1 r1 m = dM/ dA p1 = m g1 a = g + r 1 1 1y 1x x y y x Refraction by Snell’s law sin( b ) = ( 1/ n)sin( a ) 1 1 P 1 cos( b1) = 1-sin ( b1) b1 = a tan2 [sin( b1),cos( b1)] d = g -b 1 1 2 d 0 � 1 g 1 � y g 2 7 � 2 P 2 d2 �2 � 2 C 2 u 2 n 2 g2 = È + sin( g) ˘ ÎÍ + cos( g) ˚˙ d0 = È+ cos( d1) ˘ ÎÍ + sin( d1) ˚˙ x1 = p1+ l d0 x2 = m g2 dA = d g -d g dM = d p -d p p2 = ( dM/ dA) g2 b2= g + d sin( a ) = n sin( b ) 2 2 n 1.5500 rho1 45.0000 rho2 45.0000 alf1 75.0000 bet1 38.5486 alf2 34.5311 bet2 21.4514 defl 49.5311 x R 2 0x 2y 0y 2x 0x 1y 0y 1x cos( a2) = 1-sin ( a2) a = atan 2 [ sin( a ),cos( a )] 2 2 2 u2 = È + sin( r2) ˘ ÎÍ - cos( r2) ˚˙ d2 = a2 -g d2 = È+ cos( d2 ) ˘ ÎÍ + sin( d2 ) ˚˙ x1 = p2 + l d2 x2 = c2 + mu2 s2 = p2 -c2 dA = d u -d u dM = d s -d s 2 2x 2y 2y 2x 2x 2y 2y 2x r = c +( dM/ dA) u 2 2 2
Page 1 and 2: Gernot Hoffmann Newton’s Prism Ex
Page 3 and 4: 1.2 Introduction / About Newton Isa
Page 5: 1.4 Introduction / Goethe says... S
Page 9 and 10: 5. Ray visualization and a photo We
Page 11 and 12: 7. Gamut compression 7.1 Gamut comp
Page 13 and 14: 9. Single ray / Illuminant Equal En
Page 15 and 16: 11. Single Ray / Linear Spectrum /
Page 17 and 18: 13. Multiple ray / Illuminants A, D
Page 19 and 20: 15. Conclusions The Conclusions ref
Page 21 and 22: 17. Reverse problem and inverse app
Page 23: 18.2 References [15] Schott AG Opti

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