Determining Wave-Optics Mesh Parameters for Modeling Complex Systems of Simple Optics

Transcription

1 Determining Wave-Optics Mesh Parameters for Modeling Complex Systems of Simple Optics Dr. Justin D. Mansell, Steve Coy, Liyang Xu, Anthony Seward, and Robert Praus MZA Associates Corporation

2 Outline Introduction & Motivation Aperture Imaging Into Input Space Finding the Field Stop & Aperture Stop Mesh Determination for Complex Systems Conclusions

3 Introduction to WaveTrain WaveTrain Wave-Optics Modeling Tool Based on tempus FREE for government work WaveTrain is becoming the industry standard for waveoptics modeling An investment in WaveTrain or tempus is not lost because for government use they are: open-source & nonproprietary tempus can work with existing modeling software. no duplication of effort or need to learn too much new software 3

4 Huygens Principle In 678 Christian Huygens expressed an intuitive conviction that if each point on the wavefront of a light disturbance were considered to be a new source of a secondary spherical disturbance, then the wavefront at any later instant could be found by construction the envelope of the secondary wavelets. -J. Goodman, Introduction to Fourier Optics (McGraw Hill, 968), p. 3. 4

5 Simple Fourier Propagator U = P U & Notation Simplification h dx dy k, h = exp j y [( ) ( ) x x + y ] Q = exp jk r F( U ) = U exp exp( jk) P = jk j k Quadratic Phase Factor (QPF): Equivalent to the effect a lens has on the wavefront of a field. ( x x + y y ) dx dy Fourier Transform Multiplicative Phase Factor: Takes into account the overall phase shift due to propagation U ' = P Q 5 F ( U Q )

6 Convolution Propagator Two FTs Steps: Fourier transform multiplication by the Fourier transformed kernel an inverse Fourier transform Advantage: Allows control of the mesh spacing Remaining Question for FFT Implementation: What mesh sie and spacing should be used? U h U = P = exp F( h) = = = H j P F P F U k = h dx [( ) ( ) ] x x + y y exp dy [ ( )] jπλ f + f ( F( h) F( U )) ( H F( U )) x y 6

7 Prior Work: Rules of Thumb Siegman gives guidance for single propagations as follows : Number of samples between N and 8N where N is the Fresnel number and Guard band of. to.5 times the halfwidth of the aperture itself. (Lasers, 8.3) Another General Procedure: Double the mesh sie and reduce the spacing by and see if the answer matches the lower resolution one. 7

8 Picking Mesh Parameters for Simple Systems 8

9 Adequate Phase Sampling In most situations, the most rapidly varying part of the field is the QPF. In a complex field, the phase is reset every wavelength or π radians. To achieve proper sampling, sampling theory dictates that we need two samples per wave. Phase (radians) Samples Inadequate Sampling Adequate Sampling 9

10 0 Mesh Sampling: Angular Bandwidth D D D θ max = D D λ δ δ λ D θ max = D D λ δ δ λ D D λ δ δ + Fully General Result Simplified Result

11 Virtual Adjacent Apertures Wrap Around Now that we know the mesh sampling intervals (δ and δ ), we need to know how big a mesh we need to use to accurately model the diffraction. The Fourier transform assumes a repeating function at the input. This means that there are effective virtual apertures on all sides of the input aperture. We need a mesh large enough that these virtual adjacent apertures do not illuminate our area of interest. This allows us to avoid wraparound by using a guard band.

12 Mesh Sie: Avoid Wrap-Around D θ = λ δ D D +θ D δ D θ = λ NOTE: Drawn for δ /δ =D /D and maximum mesh spacing. Fully General Result D D λ δ δ δ δ N + +

13 Mesh Determination Rules of Thumb Mesh Sample Spacing Mesh Sie δ λ λ and δ D D r r N 6N f _ eff = 6 λ for maximumδ and δ Approximation: Mesh spacing should be bigger than half the diffraction limited radius from the other end. Approximation: Mesh sie should be bigger than 6 times the effective Fresnel number. S. Coy, "Choosing Mesh Spacings and Mesh Dimensions for Wave Optics Simulation" SPIE (005). 3

14 Determining Fourier Propagation Mesh Parameters for Complex Optical Systems of Simple Optics 4

15 Introduction Wave-optic mesh parameters can be uniquely determined by a pair of limiting apertures separated by a finite distance and a wavelength. D D λ An optical system comprised of a set of ideal optics can be analyed to determine the two limiting apertures that most restrict rays propagating through the system using field and aperture stop techniques. 5

16 Definitions of Field & Aperture Stop Aperture Stop = the aperture in a system that limits the cone of energy from a point on the optical axis. Field Stop = the aperture that limits the angular extent of the light going through the system. NOTE: All this analysis takes place with ray optics. 6

17 Optical System D= Example System D=5 D=5 f=00 f=00 D=5 A L L A Input Input Space Plane Plane 3 Plane A 50 A L L 7

18 Procedure for Finding Stops /3 Find the location and sie of each aperture in input space.. Find the ABCD matrix from the input of the system to each optic in the system.. Solve for the distance ( image ) required to drive the B term to ero by inverting the input-space to aperture ray matrix. This matrix is the mapping from the aperture back to input space. 3. The A term is the magnification (M image )of the image of that aperture. M 0 i = M image input space _ to _ aperture M Ai Ci image image Bi M D = i C = B D = C image Ai = Ci + A B i D 0 i / M 8

19 Procedure for Finding Stops /3. Find the angle formed by the edges of each of the apertures and a point in the middle of the object/input plane. The aperture which creates the smallest angle is the image of the aperture stop or the entrance pupil A 50 A L 9 L

20 Procedure for Finding Stops 3/3. Find the aperture which most limits the angle from a point in the center of the image of the aperture stop in input space. This aperture is the field stop A 50 A L 0 L

21 Example: Fourier Propagation Input Space A 50 L A L D = mm, D = 5 mm, λ = μm, = 0.5 m Minimal Mesh = 400 x μm = 3.75 mm

22 Example System Modeled Optical System D= D=5 D=5 f=00 f=00 D=5 A L L A Input Plane Plane 3 Plane 4

23 N=04, δ=6.6 μm Over-Sampled Input Plane Plane 3 Plane 4 3

24 N=5, δ=9.4 μm Minimal Sampling Input Plane Plane 3 Plane 4 4

25 N=56, δ=3.3 μm Under Sampled Input Plane Plane 3 Plane 4 5

26 Conclusions We have devised a procedure to reduce a complex system comprised of simple optics into a pair of the most restricting apertures using the concepts of field stop and aperture stop. With these two apertures, a wavelength, and a distance, we can determine the mesh parameters for this system. Limitation: Does not include possibility of softedged apertures or aberrations, but they can be added. 6

27 Questions? (505) x 7