Relating axial motion of optical elements to focal shift

Transcription

1 Relating aial motion o optical elements to ocal shit Katie Schwertz and J. H. Burge College o Optical Sciences, University o Arizona, Tucson AZ 857, USA katie.schwertz@gmail.com ABSTRACT In this paper, simple relationships are presented to determine the amount o ocal shit that will result rom the aial motion o a single element or group o elements in a system. These equations can simpliy irst-order optomechanical analysis o a system. Eamples o how these equations are applied are shown or lenses, mirrors, and groups o optical elements. imitations o these relationships are discussed and the accuracy is shown in relation to modeled systems. Keywords: ocal shit, optomechanics, aial motion, deocus, alignment. INTRODUCTION When designing and tolerancing an optical system, typically a ull computer model o the system is created and a complete tolerance analysis is done. It is good practice to understand what the epected outcome o a simulation is beore dedicating time to the endeavor. First order calculations, estimations, rules o thumb, and eperience can all provide means with which to determine an epected outcome -. The equations presented in this paper provide a simple way to estimate the amount o ocal shit that will occur rom the aial motion o an optical element, or group o elements, in a system based on the cone o light entering and eiting the element(s). Related equations eist or determining the amount o image motion due to the decenter and tilt o an optical element 3. These equations can be used to estimate the tolerances needed on the aial position o an optical element or the epected amount o ocal shit rom a given aial motion. They can also be used to determine which element in a system will aect ocus the most.. Sign Convention All calculations and ormulas in this paper ollow the sign convention as presented in Field Guide to Geometrical Optics 4. All angles are deined relative to a reerence line, where counterclockwise angles are positive and clockwise angles are negative. The reerence line in this case is the optical ais (by convention, the z-ais). Figure : Sign convention or deining angles All distances are deined relative to a reerence point or plane, where distances (arrows) to the right and above the reerence point are positive and distances (arrows) to the let and below the reerence point are negative. Optical System Alignment, Tolerancing, and Veriication IV, edited by José Sasián, Richard N. Youngworth, Proc. o SPIE Vol. 7793, SPIE CCC code: X/0/$8 doi: 0.7/ Proc. o SPIE Vol Downloaded From: on 08/6/03 Terms o Use:

2 Figure : Sign convention or deining distances. IAGE OTION EQUATIONS. Image motion or reractive suraces in terms o magniication Consider a biconve lens at inite conjugates that is shited in the + ẑ direction by an amount, Δ z. As a result, the ocus will shit a given amount, Δ. z Figure 3: Shiting an optical element by an amount, Δ z, will cause the ocus to shit by Δ z. Using the geometry o the system above, the ollowing relationships can be deined: o' = o + Δ Equation z i = i + ' Equation Rearranging these two equations yields: Δ o = o o' = Δ Equation 3 z Proc. o SPIE Vol Downloaded From: on 08/6/03 Terms o Use:

3 Δ i = i i' = Equation 4 From the imaging equation, + = Equation 5 i o where is the ocal length o the lens, we can dierentiate to obtain, i o + i o = 0 Equation 6 When i and o are small, we can make the estimation that i Δi and o Δo. Rearranging Equation 6 we ind, Δi Δo i = o = m Equation 7 where m is the magniication o the lens. Inserting Equations 3 and 4 rom above and rearranging we ind the epression: Δ z = m Equation 8 ( ) This derivation can then be etended to a system with two lenses by knowing the shit in the image o the irst lens Δ ) serves as the shit in the object or the second lens ( Δ ). The resulting relationship is ound to be: ( z z ( m ) m Δ z = Equation 9 where m is the magniication o the irst lens and m is the magniication o the second lens. Again, this epression can be etended to a system with N lenses where element undergoes an aial shit. The epression is simply multiplied by the magniication o the individual lenses ollowing the shited element. enses beore the shited element will not be aected. ( m )( m )( m ) K( m ) + + N Δ z = Equation 0. Image motion or reractive suraces in terms o numerical aperture The magniication o an element can be deined in terms o the paraial marginal ray angle, u, and the inde o reraction, n, surrounding the lens. The angles are deined as they occur at the element in a system, not at ininite conjugates. Figure 4: The marginal ray angles at a speciic element Proc. o SPIE Vol Downloaded From: on 08/6/03 Terms o Use:

4 nu m = n u Equation The numerical aperture () deines the cone o light entering or eiting an optic in terms o the real marginal ray angle, U, and can be approimated using the marginal ray angle. This approimation is valid or slow systems (small ray angles) and systems that have little or no pupil aberrations. = nsin U nu Equation The magniication can then be written in terms o : m = Equation 3 We can substitute this epression or magniication into Equation 0 to ind: + + N = K Equation 4 ' ( )' ( )' + + im where im is the numerical aperture at the image plane. The light eiting each element will enter the ollowing element at the same angle, meaning that, )', etc. The equation then reduces to: ' = + ( + = + ' z = Δ z Equation 5 im Δ.3 Image motion or reractive suraces in terms o lens ocal length and beam diameter The angle o light eiting a lens can be epressed in terms o the angle o the incoming light and the lens ocal length and beam diameter incident on the optic by the paraial raytrace equation: y n' u' = nu Equation 6 = ocal length o lens y = height o light incident on the optic (marginal ray height) Again using the approimation or in Equation and inserting this epression into Equation 5, we can derive another equation or ocal shit based on the properties o the lens: y y = im Equation 7 However, ater comparing the results o the epression above to the actual results rom a ray trace model, it was ound that this equation has very large error in comparison to the prior two equations presented. It is included here to provide insight as to the sensitivity o a given optic based on the ocal length, height o the beam on the optic, and angle o light entering the optic. This equation behaves dierently or three dierent regions where y >>, Proc. o SPIE Vol Downloaded From: on 08/6/03 Terms o Use:

5 y <<, or y. The plots below show the ratio o ocal shit to element shit as the ratio o y increases (i.e. increasing /#) or the dierent regions. When y, the ratio is approimately zero. Figure 5: Amount o ocal shit due to element shit based on the /# o the element These graphs can be used to get a general idea o the amount o ocal shit that will occur or a given element motion based on the element ocal length, beam diameter, and entering. The most sensitive elements o a system can then easily be determined. As evident by the graph, elements with a large /# will be very insensitive to element aial motion..4 Image motion or relective suraces The equations derived above can also be applied to relective suraces, but with a sign correction. Due to the act that the numerical aperture terms are squared, the change in direction o the light ater relection (and thereore the change in sign) does not aect the equation. For relective suraces, the sign in Equation 0 and Equation 5 must be changed rom subtraction to addition: ( + m )( m )( m ) K( m ) Δ z = Equation N ' + z = Δ z Equation 9 im Δ where Δ z is the aial motion o the mirror. Proc. o SPIE Vol Downloaded From: on 08/6/03 Terms o Use:

6 3. ERROR AYSIS For small ocal lengths and large incident beam diameters (i.e. ast /# lenses), the relationships presented here have greater error. As seen rom Equation 5, and knowing that #, Equation 0 the error in these equations will be proportional to the (/#) o the lens. The waveront error due to deocus can also be ound. Deocus is given by the epression 4 : Δ W0 = Equation 8 # im ( ) Substituting in Equations 5 and 0, we ind: ( ) Δ W0 = ' Equation This allows or direct calculation o the amount o deocus in the image based on the aial motion and the entering and eiting values or the given element. 4. BASIC EXAPES A series o eamples are shown below or simple systems in which the relationship between the optical element motion and the ocal shit is already known, or can be easily determined. For each eample shown, the element motion is in the positive direction, so the sign o Δ z is taken as positive. I the optical element motion is in the ẑ direction, the sign o Δ z is taken as negative. 4. Positive ens with Object at Ininity For a reractive surace with an object at ininity, the light will ocus at the ocal length o the lens and the magniication is zero. I we move the lens aially an amount ±, the image will move the same distance, in the same direction. Figure 6: Positive lens with object at ininity (m = 0) Using Equation 8 we ind: ( m ) = ( 0 = Δ z = ) Using Equation 5 and knowing that light is entering the lens parallel to the optical ais ( = 0) image space is the same as ', we ind: and the in Proc. o SPIE Vol Downloaded From: on 08/6/03 Terms o Use:

7 ' ' 0 = = im ' = 4. Powered irror with Object at Ininity Similar to the lens above, the magniication o a powered mirror with an object at ininity is zero. Since light is entering parallel to the optical ais, is zero and the in image space is the same as. ' Figure 7: Concave mirror with object at ininity (m = 0) Using Equation 8 we ind: Using Equation 9 we ind: ( + m ) = ( + 0 = Δ z = ) + + ' ' 0 = = im ' = 4.3 Plane Parallel Plate Inserting a plane parallel plate into a beam will displace the beam aially, but once inserted, moving it aially will not cause additional image motion. The angle o light eiting the plate is the same as the angle o light entering it, so the magniication o the plate is. Figure 8: Plane parallel plate in a converging beam (m = ) Using Equation 8 we ind: Using Equation 5 we ind: = = ( m ) = ( ) = 0 ' im = im = 0 Proc. o SPIE Vol Downloaded From: on 08/6/03 Terms o Use:

8 4.4 Flat irror Inserting a lat mirror into a beam will old the optical path, but will not add power. The magniication o a lat mirror is and the ocus will shit twice the amount that the mirror is shited aially. From the law o relection we know the = angle o the light beore and ater relection will be the same ( ) '. Figure 9: Flat mirror in a converging beam (m = ) Using Equation 8 we ind: Using Equation 9 we ind: ( + m ) = ( + ) = = + ' = = = im 5. VERIFICATION OF EQUATIONS 5. Veriication o individual lens motion in a Cooke triplet To veriy the validity o the relationships presented above, individual elements were perturbed in a Cooke triplet. The sample Cooke triplet design provided with the Zema sotware was used 5. The distance to the object was set to 00mm (instead o ininity) to demonstrate the validity o the equations or a inite conjugate system. A summary o the relevant system properties is given below. Table : Cooke Triplet System Data Parameter Value Eective Focal ength 50.4 mm Working F/# 6.5 Entrance Pupil Diameter 0 mm Image Space ens ocal length mm ens ocal length mm ens 3 ocal length 4.8 mm Proc. o SPIE Vol Downloaded From: on 08/6/03 Terms o Use:

9 Figure 0: Cooke Triplet design used to compare the actual ocal shit to the predicted values or individual lenses Each individual lens was shited aially in the + ẑ direction while the other two lenses were held in a ied position. The shit o the marginal ray ocus was recorded or each lens shit. The graph below compares the predicted ocal shit rom Equation 5 to the actual ocal shit recorded rom the Zema simulation. Figure : Graph comparing predicted vs actual ocal shit or an individual lens Proc. o SPIE Vol Downloaded From: on 08/6/03 Terms o Use:

10 Because these equations are derived rom geometrical optics principals, they are valid or small shits and are more accurate or lower powered suraces (i.e. larger incidence angles make the paraial assumptions less valid). This can be seen in the graph, where ens has less power (longer ocal length) than ens or ens Veriication o group lens motion in a Double Gauss System The relationships above can also be applied to a group o elements in a system that shit together. The sample Double Gauss system provided with the Zema sotware was used to demonstrate the validity o the equations or the shit o a group o elements. A summary o the relevant system properties is given below. Table : Double Gauss System Data Parameter Value Eective Focal ength 99.5 mm Working F/# 3 Entrance Pupil Diameter 33.3 mm Image Space 0.65 Focal length o shited group 78.6 mm Figure : Double Gauss design used to compare the actual ocal shit to the predicted values or a group o elements The angle o the light entering the irst optic and the angle o the light eiting the inal optic are used or the calculations. Proc. o SPIE Vol Downloaded From: on 08/6/03 Terms o Use:

11 Figure 3: Numerical apertures used or the ollowing calculations The group o lenses was shited aially in the + ẑ direction and the shit o the marginal ray ocus was recorded. The predicted ocal shit rom Equation 5 was then compared to the shit o the marginal ray ocus in the ray trace simulation. Figure 4: Graph comparing predicted vs actual ocal shit or a group o lenses Since the eective ocal length o the shited group o elements is much larger than each o the individual elements in the Cooke triplet, the predicted values are accurate or larger shits (>mm compared to >0.5mm or the Cooke triplet) Proc. o SPIE Vol Downloaded From: on 08/6/03 Terms o Use:

12 5.3 Veriication o individual mirror motion in a Ritchey-Chretien telescope To veriy the validity o the relationships given or relective elements, the primary and secondary mirrors in a Ritchey- Chretien telescope were individually shited aially while the other element remained ied. The sample Cassegraintype Ritchey-Chretien telescope provided with the Zema sotware was used. A summary o the relevant system properties is given below. Table 3: Ritchey-Chretien system data Parameter Value Eective Focal ength 75. mm Working F/#.7 Entrance Pupil Diameter 50 mm Image Space Primary mirror ocal length 37.4 mm Secondary mirror ocal length -45. mm Figure 5: Ritchey-Chretien design used to compare the actual ocal shit to the predicted values or individual mirrors The graph below shows the ocal shit predicted by Equation 5 compared to the actual ocal shit rom Zema. Proc. o SPIE Vol Downloaded From: on 08/6/03 Terms o Use:

13 Figure 6: Graph comparing predicted vs actual ocal shit or a mirror As discussed previously, the amount o error in the estimation is proportional to the (/#) o the shited elements. The /# o the mirrors are larger than the previous two eamples, so the error in the estimation is greater here or the same amount o aial motion. 6. CONCUSION Presented in this paper are simple equations or estimating the ocal shit rom the motion o an optical element or group o elements. In summary: ' ± = Δ z E im Δ z = ( ± m )( m )( m ) K( m ) E + + N Δ z = ocal shit rom nominal or reractive suraces + or relective suraces Δ z E = aial shit o the element(s) = entering numerical aperture (estimated by angle o light entering element, relative to the optical ais) ' = eiting numerical aperture (estimated by angle o light eiting element, relative to the optical ais) Proc. o SPIE Vol Downloaded From: on 08/6/03 Terms o Use:

14 im = numerical aperture in the image plane m = magniication o the element These equations are valid or small motions and the amount o error rom actual ocal shit decreases proportional to /(/#) o an element. Determining the amount o waveront error due to deocus was also presented. These equations can be applied to determine which elements in a system aect ocal shit the greatest amount or or simpliied irst order tolerance analysis o how the aial motion o a lens, mirror, or group o lenses is coupled to ocal shit. REFERENCES [] Friedman, Edward and John ester iller., [Photonics Rules o Thumb: Optics, Electro-optics, Fiber Optics, and asers], cgraw-hill, New York, (004). [] Schwertz, Katie, [Useul Estimations and Rules o Thumb or Optomechanics]. Unpublished aster s Thesis, retrieved rom (00). [3] Burge, J. H., An easy way to relate optical element motion to system pointing stability, Proc. SPIE 688 (006). [4] Greivenkamp, J. E., [Field Guide to Geometrical Optics], SPIE, Bellingham, (004). [5] Zema Development Corporation, Bellevue WA. Proc. o SPIE Vol Downloaded From: on 08/6/03 Terms o Use: