Tutorial Solutions. 10 Holographic Applications Holographic Zone-Plate

Transcription

1 10 Holographic Applications 10.1 Holographic Zone-Plate Tutorial Solutions Show that if the intensity pattern for on on-axis holographic lens is recorded in lithographic film, then a one-plate results. Compare the location of the foci of such a one plate with the foci for the comventional hologaphic lens. (A lithographic film is a film with a very high gamma designed to produce a binary output with T a 1 for high exposure and 0 for low exposure; all grey levels are lost). Solution The on-axis holographic lens is formed by the optical system P 0 So at P 0 the reference wave is r exp(ıφ 0 )=Constant, and the object wave is [taking the Fresnel Approximation], o 0 exp(ıκ)exp ıκ (x2 + y 2 ) 2 so that the intensity in P 0 is given by r 2 + jo 0 j 2 + 2ro 0 cos The phase term Φ 0 is an arbitrary parameter, so let κ + x2 + y 2 2 κ = Φ 0 2nπ so we get that the intensity in P 0 is given by κx r 2 + jo 0 j y 2 2ro 0 cos 2 This intensity pattern can be written as α + βcos π λ ρ 2 + Φ 0

2 where ρ 2 = x 2 +y 2, α = r 2 +jo 0 j 2 and β = 2ro 0.Wehavethatβ < α, so the intensity in positive as expected. Lithographic film is essentially binary, so it have a H-D curve of approximately D 4 0 Log(E ) c Log(E) so that if Exposure < E c ) T 1 Exposure > E c ) T 0 so if we then expose this film with then we get a pattern ατ = E c so we get a transmission function of the form, βcos() > 0 ) T 0 Black βcos() < 0 ) T 1 White ρ b with the cross over points, ρ b when so which occurs when cos ρ n = π λ ρ 2 = 0 s n 1 + λ 2 This pattern is called a one-plate which looks like: (see Hecht Optics for full details of the non-holographic analysis of this structure).

3 Even for the binary pattern, we can still take a polynomial expansion approximation for the amplitude transmittance T a, which we can write as where T a = T 0 + T 1 cos() + T 2 cos() 2 + T 3 cos() 3 +:::+ T n cos() n π cos() = cos λ ρ 2 and T i are constants. If we now illuminate this with a plane wave v = r exp(ıφ 0 ), then if we assume that φ 0 = 0the transmitted amplitude vt a = rt 0 + rt 1 cos() + rt 2 cos() 2 + rt 3 cos() 3 +:::+ rt n cos() n which, noting that cos() = 1 2 [exp() + exp(,)], we can write this as 1 2 n=0 T n exp nıκ (x2 + y 2 ) 2 Now if we note the expression for a phase of a lens of focal length is given by exp,ıκ (x2 + y 2 ) 2 then we see that the expression for the amplitude transmittance of the one plate is that of a series of lenses with focal lengths n so that the reconstruction will be /2 /3 / Holographic Diffraction Gratings Holographiclly made diffraction gratings are common is spectrometers, both as a replacement for simple ruled gratings and in more complex systems where curved, or imaging gratings are used.

4 1. Sketch a optical system to produce a simple parallel grating which can be used in a conventional laboratory spectrometer (these grating typically have 600 lines/mm). 2. Suggest a optical layout for a specrometer that uses a single imaging refective grating and a linear CCD array detector. 3. Suggest how the grating for the above spectrometer could be made Solution Part 1: To produce a simple parallel fringe grating we simple need to interfere two plane wave with angle θ between then where, for grating of spacing d,wehave sinθ = λ d so for a 600 line/mm grating, d = 1:667µm, and θ = 22:3 assuming that we are using He-Ne light with λ = 633:28nm. The required optical system is then relatively simple, being, From Laser Lens Beamsplitter Microscope Objective θ Lens Such a system is optically simple, and provided that good quality lenses and other optical components are used, then very high quality gratings can be produced relatively easily. Clearly there is the same stability problem as for normal holography, but is a lot easier than the mechanical ruling engine where the lines are cut mechanically! Aside: the grating is not normally recorder on silver halide material, it is much more usual to use photo-resist which is developed to give a surface relief pattern which is then coated with aluminum, or silver, to give a reflective grating. Additionally the grating is usually exposed with equal intensity in the two beams and with an exposure that gives a binary response, like the one plate above. In this case the, typically unwanted, higher orders are used for forming higher order spectra. Part 2: The simplest form of imaging spectrometer is shown below,

5 Light Source Holographic (Imaging) Grating Input Slit Long Wavelengths Linear CCD Detector Short Wavelengths where the slit is imaged onto a linear detector by a grating that incorporates a lens. This can either be done with a curved grating, or more commonly a holographic grating as discussed below. This is the basics of the imaging spectrometer used in Third Year laboratory, and vastly simplifies the design and cost of such instruments. Part 3: What you actually need in this system is a holographic lens that forms the image of the input slit onto the detector. Consider a holographic lens made in the following geometry. Microscope Objectives This will give an off-axis holographic lens as discussed is slide 7 of the Applications Lecture, it will however be a transmission lens with the real foci appearing on the opposite side to the reconstruction beam. If however the holographic lens is combined with a mirror, this will reflect the real foci back to the same side as the reconstruction as shown below: Reconstruction Beam Coating -1 order -2 order -3 order foci The + order diffraction will also be reflected back, as an expanding beam, but with careful consideration of the geometry this can be made to miss the wanted real foci. This mirror is

6 conveniently formed by aluminum or silver coating the holographic grating. When illuminated with polychromatic light the curved imaging grating diffracts the light giving a different effective focal length for each wavelength, so producing a spectrum about each foci. This is exactly what is wanted in the above spectrometer where only the,1 order is shown and used. This all look simple, but if you do the analysis fully you find that the spectrum about each foci is focused on a curve, as shown below: Reconstruction Beam Coating Spectrum which complicates the design to some extent. This can be compensated for by recording the hologram on a curved substrate, which is what is done in the expensive spectrometers Making a Hologon Sketch an optical beam arrangement that can be used to produce a single facet of a hologon. Solution Each section of the hologon is a diffraction grating with a variable grating spacing, so in its simplest form we to interfere two coherent beams with varying angles between then to give, Reference Beam Object Beam θ 0 θ1 where the fringe spacings will be given by d 0 = λ & d λ 1 = sinθ 0 sinθ 1 However the object beam is not expanding from a point, but from a line. This is best obtained with a negative cylindrical lens, as shown

7 so giving a total system of Beamsplitter M/S Objective Collimators Cylinderical Lens Note the use of mirrors to ensure the same optical pathlength in each arm of the system to ensure that the two beams are coherent.