MEFT / Quantum Optics and Lasers. Suggested problems from Fundamentals of Photonics Set 1 Gonçalo Figueira

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1 MEFT / Quantum Optics and Lasers Suggested problems from Fundamentals of Photonics Set Gonçalo Figueira. Ray Optics.-3) Aberration-Free Imaging Surface Determine the equation of a convex aspherical nonspherical) surface between media of refractive indexes n and n such that all rays not necessarily paraxial) from an axial point P at a distance z to the left of the surface are imaged onto an axial point P at a distance z to the right of the surface. Hint: In accordance with Fermats principle the optical pathlengths between the two points must be equal for all paths. Refraction at a convex spherical boundary The figure shows a spherical boundary of radius R between two media of refractive indexes n left) and n right). All paraxial rays originating from a point P =y,z ) in the z=z plane meet at a point P =y,z ) in the z=z plane. The z=z and z=z are said to be conjugate planes. Demonstrate that the following two relationships hold: y = n n z z y and n z + n z n n R.

2 .-) Fiber Coupling Spheres Tiny glass balls are often used as lenses to couple light into and out of optical fibers. The fiber end is located at a distance f from the sphere. For a sphere of radius a = mm and refractive index n =.8, determine f such that a ray parallel to the optical axis at a distance y =.7 mm is focused onto the fiber, as illustrated in the figure..3-3) Axially Graded Plate A plate of thickness d is oriented normal to the z axis. The refractive index nz) is graded in the z direction. Show that a ray entering the plate from air at an incidence angle θ in the y-z plane makes an angle θz) at position z in the medium given by nz)sinθz) = sinθ. Show that the ray emerges into air parallel to the original incident ray. Show that the ray position yz) inside the plate obeys the differential equation dy / dz) = n / sin θ )..4-) Special Forms of the Ray-Transfer Matrix ABCD) Consider the following situations in which one of the four elements of the ray-transfer matrix vanishes: a) Show that if A =, all rays that enter the system at the same angle leave at the same position, so that parallel rays in the input are focused to a single point at the output. b) What are the special features of each of the systems for which B =, C =, or D =? y θ = A B C D y θ.4-) A Set of Parallel Transparent Plates Consider a set of N parallel planar transparent plates of refractive indexes n, n,.., n N and thicknesses d, d,... d N, placed in air n = ) normal to the z axis. Using induction, show that the ray-transfer matrix is M = N d i i = n i Note that the order in which the plates are placed does not affect the overall ray-transfer matrix. What is the ray-transfer matrix of an inhomogeneous transparent plate of thickness do and refractive index nz)?

3 .4-4) Imaging with a Thin Lens Derive an expression for the ray-transfer matrix of a system comprised of free space/thin lens/free space, as shown in the figure. Show that if the imaging condition /d + /d = /f) is satisfied, all rays originating from a single point in the input plane reach the output plane at the single point y, regardless of their angles. Also show that if d = f, all parallel incident rays are focused by the lens onto a single point in the output plane..4-6) A Periodic Set of Pairs of Different Lenses Examine the trajectories of paraxial rays through a periodic system comprising a sequence of lens pairs with alternating focal lengths f and f, as shown in the figure. Show that the ray trajectory is bounded stable) if d f ) d f ).4-9) Ray-Transfer Matrix of a GRIN Plate Determine the ray-transfer matrix of a SELFOC plate i.e., a graded-index material with parabolic refractive index ny) n α y ) of thickness d.

4 . Wave Optics.-) The Paraboloidal Wave and the Gaussian Beam Verify that a paraboloidal wave with the complex envelope Ar) = A / z)exp[ ikx + y )/ z] satisfies the paraxial Helmholtz equation. Show that the wave with complex envelope Ar) = A / qz))exp[ ikx + y )/ qz) ], qz) = z + iz where z is a constant also satisfies the paraxial Helmholtz equation. This wave is the Gaussian beam...4) Intensity of a Spherical Wave Derive an expression for the intensity I of a spherical wave at a distance r from its center in terms of the optical power P. What is the intensity at r = m for P = W?.4-) Double-Convex Lens Show that the complex amplitude transmittance of the double-convex lens also called a spherical lens) shown in the figure is given by ) h exp ik x + y t x,y f, f = n ) ), +. * R - R You may prove this either by using the general formula.4-5) or by regarding the doubleconvex lens as a cascade of two plano convex lenses. Recall that, by convention, the radius of a convex/concave surface is positive/negative, so that R is positive and R is negative for the lens displayed in the figure. The parameter f is recognized as the focal length of the lens.

5 .4-5) Transmission Through a Diffraction Grating The thickness of a thin transparent plate varies sinusoidally in the x direction, [ ] dx,y) = d + cosπx / Λ) as illustrated in the figure. Show that the complex amplitude transmittance is tx,y ) = h exp[ i n )k d cosπx / Λ) ], h = exp[ i n + )k d ] b) Show that an incident plane wave traveling at a small angle θ i with respect to the z direction is transmitted in the form of a sum of plane waves traveling at angles θ q given by θ q = θ i + q λ Λ, q =,±,±,... Hint: expand the periodic function tx, y) in a Fourier series, and use the Jacobi-Anger expansion, exp[ iϕ coskx)] = J q ϕ)expiqkx), q = where J q is the Bessel function of the first kind of order q..5-) Interference of a Plane Wave and a Spherical Wave A plane wave traveling along the z direction with complex amplitude A exp-ikz), and a spherical wave centered at z = and approximated by the paraboloidal wave of complex amplitude, A / z)exp ikz)exp ikx + y )/ z [ ] interfere in the z = d plane. Derive an expression for the total intensity Ix,y,d). Assuming that the two waves have the same intensities at the z = d plane, verify that the locus of points of zero intensity is a set of concentric rings, as illustrated in the figure.

6 .5-6) Michelson Interferometer If one of the mirrors of the Michelson interferometer is misaligned by a small angle Δθ, describe the shape of the interference pattern in the detector plane. What happens to this pattern as the other mirror moves?

7 Selected Solutions.-3 n z + z) + y + n z z) + y = n z + n z Example for n =., n =.5, z =, z = 4: Use the following ray transfer matrices to represent the ball lens convex refraction, translation, concave refraction): M = n an n M 3 M M = f = y θ ) = a + n n) an n n M = a n + n a M 3 = n n a).3-3 For a given position z, divide the plate into N thin slabs of thickness dz such that z = N.dz. and each slab has a local refractive index ndz), ndz), n3dz) Apply Snell s law to each successive interface from the input surface, e.g.

8 sinθ = n dz )sinθ dz ) n dz)sinθ dz ) = n dz)sinθ dz ) n dz)sinθ dz ) = n 3dz)sinθ 3dz n[ N )dz]sinθ N )dz [ ] = n z ) )sinθ z Propagating until the output interface shows that the output ray has the same direction as the input one. Since the tangent of the ray angle corresponds to the local derivative of yz): tanθ z dy dz ) = dy z) dz = tan θ z ) n z)sinθ z) = sinθ ) = sin θ z) sin θ z) = n z ) sin θ.4- The ray transfer matrix for the first interface from n = into n = n ) and propagation d ) is: M = d = n n d n The ray transfer matrix for the nd interface from n = n into n = n ) and propagation d ) is: M = d n n = n n d n n So the total matrix after the first two slices becomes: M M = n n d n n After N plates, the total ray transfer matrix will be n d n M N M N M M = = d n + d n N d i i = n i Finally when the ray emerges into air from n = n N into n = n ): M plate = n N N d i i = n i For an infinitesimal thickness of each plate we would have n N n N * ) * n = N di i = n i

9 M plate = d dz n z ).4-4 d + d f ) M = d d f d f d f.4-6 y m θ m = A B C D m y θ A C B D = d f d f ) d f = d d f f + f d f f ) + d f f d f d f The condition of stability for the periodic trajectory is By defining A +D A +D = d d + d f f f f g = d f g = d f A + D = g ) g ) The conditions are: g ) g ) g ) g ) ) g ) g.4-9 M = cosαd α sinαd sinαd α cosαd

10 .5- I x,y,z ) = A + A + A A + A A z z I A = A I d = A I x,y,d ) = A + cos k x + y d The points of zero intensity are obtained when the cosine argument is an odd multiple of π: k x + y = πq + ), q =,, d x + y = q + d )λ