Chapter 3 Geometric Optics

Transcription

1 Chapter 3 Geometric Optics [Reading assignment: Goodman, Fourier Optics, Appendix B Ray Optics The full three dimensional wave equation is: (3.) One solution is E E o ûe i ωt± k r ( ). This is a plane wave. û is a unit transverse polarization vector, such that û k 0. r k The propagation vector is perpendicular to the phase fronts. These are planes of equal k r In geometric optics, we neglect the diffraction effects. We only consider the propagation of rays through optical systems. ray ray direction k r ray distance from axis optical axis A plane wave can be thought of as an infinite bundle of parallel rays. optical axis We can analyze the optical system by it s effect on individual rays. We describe a ray by 2 parameters: Geometric_optics_post.fm Chapter 3

2 r(z) : the displacement of the ray from the axis at distance z along the axis r ( z) dr - : the slope of the ray relative to the axis dz The propagation of the ray through space:, r, r L z z z z 2 L z 2 z z ( z 2 ) r ( z ) + Lr ( z ) Paraxial approximation: r small r tanθ sinθ ( z 2 ) r ( z ) θ For a thin lens, of focal length f r, r, z z o focal length f (3.2) ( z o ) - f r ( z o ) + r ( z o ) (3.3) Most optical elements can be described by a linear transformation previous examples. We can write this in matrix form: ( r, r ) (, ) as with the (3.4) (3.5) r 2 A B CD r r Mr (3.6) Geometric_optics_post.fm Chapter 3

3 Some ray matrices for common elements: Free space, length, L: L 0 Thin lens, focal length f: 0 - f Spherical mirror, with radius R: r R Notice that the determinant: AD BC. This holds generally for common elements. If the index is different on the input and output sides of the element, then cascaded ray matrices: AD BC n input n output. Consider the effect of several elements in a line M M 2 M 3 r o r r 3 The general result is: M tot M n M n M 2 M note: reverse order of ray encounters Since det( AB) det A det B, then AD BC n input n output. Conjugate, Focal, Principal Planes (These are defined in geometric optics.) Conjugate planes (points): object/image plane (point) pairs Focal planes (points): parallel rays incident from the left converge towards the rear (back) focal point. The focal plane is the plane to the optic axis through the focal Geometric_optics_post.fm Chapter 3

4 point. A point source at the front focal point causes parallel rays behind the lens. optical system optical system rear focal plane front focal plane Principal planes: These are important for thick lenses, or for optical systems. Geometric construction: P : the first principal plane is the plane where rays diverging from the front focal point intersect the corresponding collimated rays projected back. P P 2 : the second principal plane is the plane where the collinated rays from the left intersect the focused exit rays projected back If the ray matrix for propagation from P to P 2 is calculated, we get simple thin lens behavior P 2 (3.7) and the lens law is where z and are measured from P and P 2. f o z i z i z o Back to diffraction, which arises due to the finite lens aperture. In a complex lens system, how can we account for diffraction? We must find the most severely Geometric_optics_post.fm Chapter 3

5 limiting aperture. This is referred to as the aperture stop, P AS. Entrance : The image of the P A, viewed from object space Exit : The image of the P A, viewed from image space P AS exit P AS entrance exit entrance The entrance and exit planes do not, in general, coincide with the principal planes. P AS For a complicated system, in general the entrance and exit s are images of each other. exit entrance [Reading assignment: Goodman, Fourier Optics Ch. 6; 6.6 is optional The exit limits the spatial frequency components that can contribute to the image. [Equivalent view:] The entrance limits the frequency components that are intercepted from the object. For a diffraction limited system, a point source object causes an ideal spherical wave at the exit Geometric_optics_post.fm Chapter 3

6 coverging to the image point. Departures of the exit wavefront from a spherical wave are called aberrations. Recall the point spread function is a scaled Fourier Transform of the. huv (, ) Pxy (, ) j 2π exp ( ux+ vy) λ 2 dxdy z o z λz i i Defining the geometric optical ideal image (in some books this is called the Gaussian image ): (3.8) Where ζ, η U g ( ξ, η ) are scaled coordinates in object space U ξ M o -, M η - M (3.9) (3.0) The diffraction limited image amplitude is given by convolutions of the Gaussian image with the point spread function. U i ( uv, ) hu ( ξ, v η )U g ( ξ, η ) dξ dη (3.) Geometric_optics_post.fm Chapter 3