Chapter 2: Gaussian Optics - paraxial optics Mainly we will discuss – Analysis based on paraxial optics – Simple ray tracing – Aperture and stop – homeworks February 2001 optical system design (2-1) 1 Analysis based on paraxial optics • Paraxial approximation • Cardinal points • Paraxial ray tracing – YNU ray tracing – YUI ray tracing • Matrix Optics • Paraxial constants • Image equations • Specification of conjugate distances • Lens setup February 2001 optical system design (2-1) 2 Apertures and Pupils • Radiometric concepts • Optical System Layout and terminology – Thin lens – Radiance conservation – Photographic objective theorem – Irradiance by a flat circular –Magnifier source – Telescope – Cos4 law – Relay system – Vigentting – Telecentric lens – Computation of irradiance • Specifying lens • Stops and Pupils apertures – Specifying aperture and field of view – Special apertures February 2001 optical system design (2-1) 3 • Centered optical system – Optical axis – Basic law – Plane of incidence nsinθ = n'sinθ '⇒ nθ = n'θ ' • Formed by incident and refracted rays – Merdinal ray (子午線) n interface • On the plane of incidence • with optical axis n’ – Skew ray (歪斜線) • Optical axis is not on the plane of incidence February 2001 optical system design (2-1) 4 Approximation: a quick way for ray tracing • Now let us take a non- Tangent plane flat surface Index arc Index line R’ ρ ρ ρ ρ ρ ρ ρ n'e = ne + Gp G = n'( p • e) − n( p • e) = n'cos I'−ncos I February 2001 optical system design (2-1) 5 Cause of aberration • Aberration : the difference between real ray and paraxial ray February 2001 optical system design (2-1) 6 Cardinal point (基點) • Using cardinal points to characterize the performance of paraxial optics – Object space –Image space Focal points • Two points are enough Principal points to determine the Nodal points position of image February 2001 optical system design (2-1) 7 Paraxial optical system and aplanatic optical system second focal point: F’ First focal point: F P, P’: principal points (O and O’ off-axis where P and P’ on axis) February 2001 optical system design (2-1) 8 Paraxial optical system and aplanatic optical system principal planes changes to principal surface OSLO used aplanatic Ray aiming February 2001 optical system design (2-1) 9 Paraxial optical system and aplanatic optical system • For an aplantic lens, the effective refracting surface for ray coming from infinity is a sphere centered on the second focal point – Most real lens more closely resemble aplantic system than paraxial systems • Note: an aplantic lens obeys the Abbe sine condition. This condition was put into the calculation in common ray-tracing and thus to achieve an aplantic ray aiming (instead of paraxial ray aiming). February 2001 optical system design (2-1) 10 Nodal points: without angular deviation Unit positive angular magnification February 2001 optical system design (2-1) 11 (nodal point) Once the locations of the cardinal points of a system are known, it is straightforward to find how the system transforms rays from object to image space February 2001 optical system design (2-1) 12 Paraxial ray tracing y − y y u = − (y = y + tu) i = + u = yc + u ni = n'i' − r t nu − yφ y u'= I = U +θ; I'= U '+θ i'= + u'= yc + u' n' r power φ'= c(n'−n) February 2001 optical system design (2-1) 13 YNU ray tracing n'u'= nu − yφ t y = y + (nu) − n Idea of “SOLVE” n'u'−nu c = Angle solve y(n − n') y − y t = − height solve Two rays u February 2001 optical system design (2-1) 14 • Usually in ray tracing, we need to check two rays – (1) so called “a ray”: axial rays (marginal paraxial ray) – (2) so called “b ray”: principal ray (chief ray) – In last viewgraph, the subscripts of y (a, b) mean a ray and b ray correspondingly. February 2001 optical system design (2-1) 15 YUI ray tracing b ray a ray In OSLO use “pxt all” y = y− + tu to display this data i = u + yc n u'= u + ( −1)i n' February 2001 optical system design (2-1) 16 homework • Using visual C++ or the other computer language to implement the ynu ray tracing and yui ray tracing • And then, calculate a case of achromat lens • Using OSLO LT, and “pxt all” command to compare your result Radius thickness diameter Material 12.8018 0.434 3.41 BAK-1 -9.0623 0.321 3.41 SF-8 -37.6553 19.631 3.41 AIR Cemented achromat F-number: f/6 Focal length: 20 in Note: unit in inch From Laikin, lens design, p. 45 February 2001 optical system design (2-1) 17 Matrix Optics Translation matrix y j 1 t j / n j y j−1 = n ju j 0 1 n ju j T Refraction matrix y j 1 0 y j = n' j u' j −φ 1n ju j R February 2001 optical system design (2-1) 18 Transfer Matrix y A B y j = 1 n' u' j j C Dn1u1 Determinant =1 Determinant L Lab = n1(yb1ua1 − ya1ub1) (again, here subscripts a and b correspond a ray and b ray) Largange invariant But you do not have to use a ray and b ray to have a Lagrange invariant February 2001 optical system design (2-1) 19 Lagrange’s Law • How to calculate the magnification? –m=h’/h – Ray tracing (not good) • Using Lagrange invariant –Lab=n1yboua1=hn1ua1 (axial ray, zero height in object space) –Lab=nkybkuak=h’nkuak (axial ray, zero height on the image plabce) – Lagrange invariant hn1ua1=h’nkuak –M=h’/h=n1ua1/nkuak February 2001 optical system design (2-1) 20 Paraxial Constants • OSLO computes seven numbers (paraxial − eh constants) f '= u' e + u' h – Effective focal length (EFL) b a PIV = Lab = hnua nu – Larange (paraxial) invariant (PIV) TMAG = m = a n'u'a – Lateral mafnification (TMAG) nu GIH = h'= a h – Gaussian image height (GIH) n'u'a NAO – Numerical aperture (NA) NA = – F-number (FNB) TMAG nu NAO = nsinU = a – Petzval radius (PTZRAD) 2 1+ u a • The radius of curvature of the surface on which an image of an extended plan object would be formed (not a paraxial quantity, February 2001actually) optical system design (2-1) 21 • So, in designing a code for optical system design, at least – You need to know how to implement ynu and yui ray tracing (this is simple) – You need to know how to put Abbe sine conditions to have an aplanatic ray aiming (this is simple) – You need to know how to use Lagrange invariant to calculate fast (this is simple) – At least, seven paraxial constants should be implemented (this is simple) – You need to know how to implement “Solve” and “Pickup” to meet some design requirement and optimization needs (this is somehow difficult; you need to consult “numerical recipe”. But, it can be done.) February 2001 optical system design (2-1) 22 Specification of conjugate distance In OSLO, you can specify object and image distance referred to cardinal points, rather than physical lens surfaces February 2001 optical system design (2-1) 23 Lens Setup •In OSLO – Solve • Axial ray height solve (PY solve) – To specify the last thickness (the one before the image surface) • Chief ray height solve (PYC solve) – Final surface becomes the paraxial exit pupil • Angle solve – To constrain the f-number – Pickup • Force some surfaces to follow a fixing setting according to some specific surface (lens) February 2001 optical system design (2-1) 24.