Section 19. Dispersing Prisms

Transcription

1 19-1 Sectio 19 Dispersig Prisms

2 Dispersig Prism 19-2 The et ray deviatio is the sum of the deviatios at the two surfaces. The ray deviatio as a fuctio of the iput agle : si si si cossi

3 Prism Deviatio - Derivatio Positive: Negative:,, 1 1,,,, (180 ) Sell s Law: si 1 si1 si si si si1 1 1 si 2 si2 si si si si 2 sicos si si1 cossi1 cos si si si si cossi si si 1si cossi si si si cossi 1

4 Deviatio versus Iput Agle There is some iput agle for which the ray deviatio is miimized. The correspodig deviatio is the agle of miimum deviatio.

5 Miimum Deviatio At miimum deviatio, the ray path through a dispersig prism is symmetric. The ray is bet a equal amout at each surface. By sig covetio, the deviatio is egative for this prism orietatio. The agle of miimum deviatio is 1 2si si /2 The miimum deviatio coditio is for the measuremet of the idex of refractio: si / 2 si / 2 At miimum deviatio, 50% of the et deviatio occurs at each surface. / 2 For =

6 Miimum Deviatio Proof At miimum deviatio, assume path is ot symmetric: But both the forward ad reverse ray paths produce the same magitude of deviatio, assumed to be. Both ad produce miimum deviatio, but they are ot equal This is a cotradictio as there caot be two differet iput agles that produce the miimum. Therefore, at : The ray path is symmetric through the prism.

7 Miimum Deviatio Derivatio Refer back to derivatio of prism deviatio /2 1 si si / si / 2 si / 2 1 2si si /2 si / 2 si / 2

8 Idex Measuremet Prism Spectrometer Miimum deviatio provides a extremely accurate method of measurig the idex of refractio of a material. Slit Source Collimator - The telescope ad the collimator are mouted to a rotatio stage. - Use the telescope as a autocollimator to measure the prism apex agle α. - Measure the straight through agle (o prism) I 1. - Isert the prism ad observe the agle of the refracted beam. - Rotate the prism to obtai miimum deviatio ad measure this agle I 2. - Subtract I 1 from I 2 to determie the agle of miimum deviatio. I I mi 2 1 Reticle si / 2 I1 I2 mi 0 si / 2 Telescope 19-8 Accuracies of oe part i the sixth decimal place ca be obtaied. Limited by the slit width ad diffractio effects.

9 Idex Measuremet Critical Agle Techiques The critical agle for total iteral refractio is ofte used for idex measuremet. The sample is placed i cotact with a referece prism of higher idex. Diffuse Illumiatio Trasparet Sample Referece Prism Dark S R S 1 S C si R 19-9 C This istrumet is called a Abbe Refractometer. Some istrumets also measure the Abbe umber of the material. The typical accuracy for this method of idex measuremet is Critical agle techiques ca also be used i reflectio to measure the idex of opaque samples.

10 Les Orietatio for Miimum Spherical Aberratio Whe focusig light with a plao-covex les, it should be orieted with the curved side towards the log cojugate (the object side) the covex-plao orietatio. The miimum aberratio occurs whe the light is bet the same amout at each les surface. Object at Ifiity: 1.5 Covex-Plao Plao-Covex I the plao-covex cofiguratio, all of the ray bedig occurs at oe surface. I the covex-plao cofiguratio, the ray bedig is split betwee the surfaces. This miimizes the agles of icidece at the surfaces ad reduces the aberratios. Small agles make the situatio as close to the assumptios of paraxial optics as possible Note that i the miimum aberratio coditio, the edge of the les looks like a prism used at miimum deviatio equal ray bedig at both surfaces. The true miimum for = 1.5 is a slightly bicovex les. The plao surface has a log radius, but the differece is essetially isigificat.

11 Les Shape for Miimum Spherical Aberratio There is o bedig that completely elimiates spherical aberratio. It ca oly be miimized. Differet object/image cojugates require differet bedigs to miimize spherical aberratio. The optimum shape varies with idex. At high idex, as is ofte foud i the IR, more bedig occurs at the surface of the les. For the same focal legth as before, the frot surface becomes flatter ad the best shape is a meiscus. Object at Ifiity: Meiscus Equal bedig occurs at both surfaces of the les ad edge of the les looks like a prism used at miimum deviatio.

12 Dispersio of a Prism The details of the prism dispersio deped o the geometry used ad the idex dispersio curve. However, assumig the prism is used at or ear, the average prism dispersio over a wavelegth bad (F to C) ca be estimated. Dispersio of a prism d d d d d d d d d d d d C F d 1 2si si /2 d d 2si /2 cos a / d 0 d d 0 d d Glass Dispersio 0 d Blue light is deviated more tha red light. As icreases, icreases, but is egative. The magitude of the deviatio decreases.

13 Dispersio of a Prism Derivatio 1 2si si /2 d d 2si / si /2 si / 2 si / 2 si / 2 si / si /2 1si / si /2 cos /2 d d 2si /2 cos / 2

14 Glass Dispersio Example Sice the refractive idex is a complex fuctio of wavelegth, the glass dispersio ca be approximated by fiite differeces: d 2 1 d 2 1 Idex (BK7) Idex (F2) F.4861 m d.5876 m C.6563 m Glass dispersio for differet wavelegth pairs: / BK7 F2 F-C /m /m F-d /m /m d-c /m /m F-d F-C d-c F d C

15 Prism Dispersio Example d d d d d F C d d d d d d d 2si /2 cos a / 2 Example: 60 prism ( = 60 ) d light F-C glass dispersio BK7 0 F C F d d d d rad rad /m 0.073/m 4.18 /m /m /m 0.177/m 10.2 /m.0102 /m

16 Prism Spectrometer The dispersig power of a prism ca be used to measure the spectral cotet of a light beam. This icludes the spectrum of the source or the trasmissio/absorptio spectrum of a material placed i the beam. Slit Source Measuremet Sample Collimator Dispersig Prism Focusig Les Detector The source is focused o a slit, ad the light is the collimated. After dispersio by the prism, a secod les images the slit oto the detector plae. The slit images for each wavelegth are displaced producig a spectrum. Each wavelegth i the spectrum is spread over the width of the slit image. Each wavelegth is further blurred by diffractio due to the les diameter ad prism size.

17 Resolvig Power ad Littrow Prisms The Resolvig Power of the spectrometer is defied as R ca be icreased by usig a arrow slit ad large diameter beam. Large prisms are required for high-resolutio spectroscopy. Littrow Prism cofiguratios use a double pass through the prism for icreased dispersio: Slit Source R Slit Source Detector Mirror Detector Mirror o Prism Face