PHYS 450 Fall semester Lecture 05: Dispersion and the Prism Spectrometer. Ron Reifenberger Birck Nanotechnology Center Purdue University

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1 PHYS 450 Fall semester 06 Lecture 05: Dispersion and the Prism Spectrometer Ron Reifenberger Birck Nanotechnology Center Purdue University Lecture 05 Prisms Dispersion of Light n n As early as the 3th century, accounts of six-sided crystals of natural quartz used to generate rainbows. Exploited by Newton between 666 and 67 n n sin in ( i s ) r The refractive index of a glass prism is greater for violet light, smaller for red light.

2 Prism Fundamentals A prism is a transparent optical element with flat, polished surfaces that refract light. At least two of the flat surfaces must have an angle between them. Apex Angle α Prism Function: Dispersion Beam steering Beam manipulation Polarization altering Beamsplitting Base 3 i) Prisms bend light without focusing it Important Properties ii) The angle of deflection does not depend where the light hits the surface of a prism (IMPORTANT) light/latest/bending light_en.html 4

3 Important Properties (cont.) iv) Beam splitting application v) Change Angle of light beam 44 o 40 o Light ray thru prism becomes parallel to base 47 o 5 Internal reflections can be an issue! 6 3

4 Dispersion of EM Wave: n varies with f d c AmaterialpropertyofglassesistheAbbe number, which quantifies the amount of dispersion that a glass exhibits. It is a function of the refractive index of a material at three well-defined wavelengths: f (486.nm), d (587.6nm), and c (656.3nm). The Abbe number is defined as Values of Abbe number range from Crown glasses have v d > 50 (less dispersive) Flint glasses with v d < 50 (more dispersive) can be measured 7 The Glass Map Designations that end in K refer to a crown glass. Those that end in F refer to flint glass. 8 4

5 The Basic Idea Atomic%0Structure/Spectra/ModernAtom5.html 9 Gas Discharge Tube - Discrete Wavelengths (Ångström 868: 000 spectral lines, Sweden) H Spectral Lines for Common Gases Atomic%0Structure/Spectra/ModernAtom5.html He Hg Glass tube, evacuated and backfilled to low pressure with a gas like hydrogen, helium, neon, etc. U 0 5

6 Prism Spectrometer Caution: high voltage =angle of deviation See Appendix for discussion of Vernier scale Two Important Quantities A. Angle of deviation δ B. Resolving Power, two closely spaced wavelengths Two wavelengths: = -Δ & 6

7 A. Calculating the angle of deviation Apex a sin i nsin t Interface b nsin i sin t i t t i t i (i) (ii) (iii) (iv) θ i δ=δ +δ δ δ P θ t Q θ i R θ t Interface i t t i t i (v) n Base prism air 3 from b t i nsin t : sin n sin sin sin sin cos cos sin t t t t sin sin cossin t n sin sin t n cos sin t n n t n int now use a n i i t sin sin sin sin cos s sin sin sin cos sin from v : i t Working it out trig identity n i i sin sin sin cos sin i now move n inside sqrt Main Result A 4 7

8 n i i Very complicated! Simplify? small angle approximation for δ isin sin sin cos sin exact, but ugly for i, sini i for, sin, cos isin n i i 0.0% -.0% i -4.0% i sin n i n -6.0% -8.0% -0.0% i i sin n... i n i sin n... i i n... i Light with highest index n... n... approximate of refraction bends most [(sin(φ)-φ)/sin(φ)] x 00% Percent Error in Small Angle Approximation Angle φ (in degrees) 5 Plot δ vs. θ i n sin sin sin cos sin i i i δ min Angle of minimum deviation 6 8

9 Angle of minimum deviation - condition of high symmetry - air incident ray θ i θ t θ i δ min θ t refracted ray n prism B 7 Implications t i iv itt i v it t when condition for minimum deviation i min i AND t i since t i vi t i t sin n sin Snell ' s Law,st face solving for n min sin sin sin t sin i n Main Result B, valid when δ=δ min Forms the basis of precise measurements of the index of refraction n as a function of. Measure δ min () with a carefully constructed prism withapreciselyknownvalue of gives n() with high precision. 8 9

10 Calculus Review. Calculate small change f in some function f. Chain Rule: If some function g depends on some variable y and y depends on another variable x, then g becomes a function of x as well and we can write f(x) f df dx dg dx dg dy dy dx x x df f~ dx x 9 B. Resolving Power ability to resolve two closely spaced wavelengths Assume condition for minimum deviation Incident Light Two wavelengths: = -Δ & air n prism δ δ min s L L D δ L Refracted Light How does the deviation angle δ change with wavelength? from calculus : d d from geometry : L L D D min tan θ i B What is resolving power? 0

11 What is smallest measureable value for the angle Δ? Smallest value for Δ is set when ΔL assumes the smallest possible value that can be resolved. This condition is typically specified by setting ΔL smallest =. From geometry From calculus L D L smallest smallest D D d see Appendix min smallest d dmin dmin dn B dn D D D d dn d D d dn B Main Result C d Typical results (for a prism with 4 cm base at =589 nm) Material Schott Technical Designation (nm) n dn/d (in nm) - dδ/d (in mrad/nm) minimum Δ in nm Crown glass BK E Flint glass SF E

12 Up Next Diffraction Gratings 3 Appendix: Evaluating dδ min /dn θ i π-(/+θi ) θ i D θ i s θ i n prism B air s D θ t =θ i First, some geometry: D scos i B B sin s s sin B B cosi D cosi sin sin D cosi B sin

13 Estimating dδ min /dn min sin sin i n Result B sin t sin min sin nsin min sin nsin min d dn min sin nsin d dn sin nsin n sin i n t t i cosi d sin A sin xsin A dx x sin A sin d min dn n sin but at minimum deviation t from Eq.( vi) sin d min dn sin sin Snell ' s Law sin sin d min dn sin B D 5 Appendix: Reading the angular Vernier Scale 0 index Vernier scale Main scale 3 Scales line up, say here 0 o 0.5 o = =30 =60 0 Read from the 0 index on Vernier scale Reading: between o and o Upon inspection: reading is between 0 and 30 Vernier scale tick mark reading: + 40 Reading: o = o 40 =.38 o This image is somewhat distorted. In the actual case, only one tick mark on the Vernier scale should match up with one line on the Main scale. Most hand-held calculators do not understand angles in degrees and minutes. Must convert! 6 3

14 Reading the angular Vernier Scale (continued) Vernier scale Main scale Read from the 0 index on Vernier scale Reading: between 53 o and 54 o Upon inspection: reading is greater than 53 o 30 Vernier scale tick mark reading: 8 The image is somewhat distorted. In the actual case, only one tick mark on the Vernier scale should match up with one line on the Main scale. Most hand-held calculators do not understand angles in degrees and minutes. Must convert! Reading: 53 o =53 o 38 = o 7 Which diagram is correct? 8 4