AY 105 Lab Experiment #3: Fundamentals of spectroscopyThis week you will set up a spectrograph on the optical table and use it to vi-sually resolve the D-lines of neutral sodium (NaI D1= 5895.93˚A,2P1/2−2S1/2; andD2= 5889.96˚A,2P3/2−2S1/2). You will also iinvestigate how spectral resolution de-pends upon such spectrograph design parameters as grating (or prism) dispersion, inputslit width, camera EFL, camera-collimator angle, etc.1 Diffraction grating spec t r ographConfiguration. You need to set up on your optical table a Na arc lamp light source(powered by a high voltage supply), opal glass diffuser, entrance slit, and Newtonianreflecting collimator (45◦flat mirror and paraboloid mirror). This setup should producea collimated beam parallel to the long axis o f the optical table, with the front portio nof the table left empty to accommodat e the camera axis of the spectrograph. Oppo sitethe collimator needs to be a diffraction grating assembly. Sketch the configuration inyour notebook, and record the collimating mirror focal length, grating grooves/mmand blaze angle θB. Mark on your sketch the grating blaze direction; is this the correctorientation to use if the camera is to be located on the front of the ta ble? (See theinstructor or T.A. if the grating blaze direction isn’t what yo u expect). What blazewavelength, λB, do you derive for your grating from the grating equation,mλa= (sin α + sin β) cos γ, (1)evaluated in first-order Littrow configuration (m = 1; α = β = θB; and γ = 0)? Is t hisreasonable? In which order would you expect the NaD lines to be brightest with thisgrating? When the gro up working at the other table has derived λBfor their grating,compare your results with theirs and comment on any differences in your lab notebook.Turn o n the spectral line lamp power supply and let the Na arc lamp warmup. Meanwhile, you will need to position your camera lens on the front portion ofthe optical table and align its optical axis with the center of the grating. Keep thecollimator axis to camera axis angle at the grating small, but do not obstruct any partof the collimator beam to the grating with the camera lens. Turn the micrometer to areading of zero to reveal the two bolt holes in the bottom (stationary) portion of thetranslation stage. Since the tapped holes in the table comprise a rectangular grid, youwill only be able to use one bolt hole in the tra nslation stage. You may use the slottedpost-holder “bases” as stops to prevent the translation stage from rotating around thesingle bolt attaching it to the optical table.Determine the angle (which in lecture we called 2θ) between the collimator andcamera axes as viewed from t he grating. Next, turn the knob which rotates the gr ating1Ay 105 Spring 2009 Experiment 3 2so that the grating is normal to the collimator axis (if you are using the “cylindrical”grating mount; α = 0) or to the camera axis (if you are using the “rectangular” gratingmount, pre-loaded by an external spring; β = 0). Record the count er reading at thisposition. With this counter reading as the zero-point, you can use the counter tomeasure the grating angle α (or β). As a working hypothesis, assume the counteron the cylindrical mount reads α in units of arcminutes; check this by rotating thegrating until the grating is normal to the camera axis (βc= 0), and compare α at thissetting to the trigonometrically derived 2θ f r om above. Is the agreement consistentwith the hypothesis that the counter reads in arcminutes? Note that the two leastsignificant digits on the cylindrical grating rotat or range from 0 0 - 59 (and so the twomost significant digits on this unit are then in degrees if indeed the two least significantdigits correspond to arcminutes). In the case of the rectangular grating mount, adoptas a working hypothesis a conversion factor of −280 counts per degree for angle β.What relationship between α and β does Equation 1 predict for the zeroth order(m = 0) from the diffraction grating? Given tha t the geometry of the spectrographfixes α − βc= 2θ, what values of α and βcwill produce a zeroth order image in yourspectrograph? Calculate the grating counter setting corresponding to this value of α(cylindrical) or βc(rectangular) and if possible rotate the grating accordingly. Positionthe micrometer eyepiece on- axis behind t he camera lens and find a location to boltit in place (make sure the optical axis of the eyepiece is parallel to the optical table,and coaxial with the camera lens axis). With the lens barrel focus set at ∞ translatethe camera lens along its axis using the micrometer until a focus is achieved (the frontsection of the eyepiece should penetrate the rear opening of the lens at the ∞ focussetting). Fine tune first the eyepiece alignment and possibly also the grating rotation tocenter the zeroth order image horizontally in the eyepiece. Record the grating rotationsetting in your notebook, and use any offset from your prediction to revise your valueof θ.Substitute α − 2θ = βc, (the value of β at the center of the camera field of view)for β in Equation 1 a nd solve for α in terms of the other variables. The t r ig onometricidentitysin A + sin B = 2 sinA + B2cosA − B2(2)may help you simplify your solution analytically. Evaluate your solution to find α forthe NaD lines in first, second, and third orders, and tabulate these angles (both indegrees and in grating counter units) in your lab notebook for f uture reference. Ro t atethe grating to the first order α setting and confirm that the m = 1 Na spectrum appearsin the eyepiece.Method. Return to Equation 1 and solve it for sin β, the sine of the anglebetween the gr ating normal and the outgoing diffracted rays as a function of λ, m,a cos γ, and the angle of incoming rays α. Take the par tial derivative of sin β withrespect to λ, a nd show that substitution from the grating equation and elimination ofα leads to the following expression for the dispersive power (i.e., the angular changeAy 105 Spring 2009 Experiment 3 3in the direction of outgoing rays with a change in wavelength ∆λ/λ) of a diffractiongrating:λ∂β∂λ= 2 tan βccos2θ + sin 2θ. (3)The trigonometric identitiessin(A + B) = sin A cos B + cos A sin B (4)sin A cos B =12sin(A + B) +12sin(A − B) (5)may be helpful in deriving this expression (you can work o n the derivation at homelater if it doesn’t wor k out for you right away, so don’t