Diffraction Gratings Lenses Laser beam diffraction A lens
Diffraction Gratings & Lenses Laser beam diffraction A lens transforms a Fresnel diffraction problem to a Fraunhofer diffraction problem The lens as a Fourier Transformer Babinet’s Principle Diffraction Gratings & Spectrometers Examples of Fraunhofer diffraction: Randomly placed identical holes X-ray crystallography Laser speckle Particle counting Prof. Rick Trebino, Georgia Tech www. physics. gatech. edu/frog/lectures
Recall the Fraunhofer Diffraction formula The far-field light field is the Fourier Transform of the apertured field. E(x, y) = const if a plane wave Aperture transmission function that is: and: kx = kx 1/z The k’s are off-axis k-vectors. and ky = ky 1/z
Fraunhofer diffraction of a laser beam A laser beam typically has a Gaussian radial profile: w 1 w 0 No aperture is involved. z What will its electric field be far away? In terms of x 1 and y 1: The Fourier transform of a Gaussian is a Gaussian. or where:
Angular divergence of a laser beam The beam diverges. What will its divergence angle be? Recall that: The half-angle will be: The divergence half-angle will be: w 0 q z w 1
Gaussian Beams The Gaussian beam is the solution to the wave equation, or equivalently, the Fresnel integral, for a wave in free space with a Gaussian profile at z = 0. x The beam has a waist at z = 0, where the spot size is w 0. It then expands to w = w(z) with distance z away from the waist. The beam radius of curvature, R(z), at first decreases but then also increases with distance far away from the waist.
Gaussian Beam Math The expression for a real laser beam's electric field is given by: w(z) w 0 w(z) is the spot size vs. distance from the waist, R(z) is the beam radius of curvature, and y(z) is a phase shift. z R(z) Recall the phase factor in front of the diffraction integrals. This is the solution to the wave equation or, equivalently, the Fresnel diffraction integral.
Gaussian Beam Spot, Radius, and Phase w 0 w(z) z R(z) The expressions for the spot size, radius of curvature, and phase shift: where z. R is the Rayleigh Range (the distance over which the beam remains about the same diameter), and it's given by:
Gaussian Beam Collimation Twice the Rayleigh range is the distance over which the beam remains about the same size, that is, remains collimated. w 0 Collimation Waist spot Distance size w 0 l = 10. 6 µm l = 0. 633 µm _______________________. 225 cm 0. 003 km 0. 045 km 2. 25 cm 0. 3 km 5 km 22. 5 cm 30 km 500 km ______________________ Longer wavelengths and smaller waists expand faster than shorter ones. Tightly focused laser beams expand quickly. Weakly focused beams expand less quickly, but still expand. As a result, it's very difficult to shoot down a missile with a laser.
The Guoy Phase Shift The phase factor yields a phase shift relative to the phase of a plane wave when a Gaussian beam goes through a focus. y(z) Phase relative to a plane wave: p/2 -z. R -p/2 Recall the i in front of the Fresnel integral, which is a result of the Guoy phase shift.
Laser Spatial Modes Some Transverse Electro-Magnetic (TEM) modes Electric field Laser beams can have any pattern, not just a Gaussian. And the phase shift will depend on the pattern. The beam shape can even change with distance. Some beam shapes do not change with distance. These laser beam shapes are referred to as Transverse Electro. Magnetic (TEM) modes. The actual field can be written as an infinite series of them. The 00 mode is the Gaussian beam. Higher-order modes involve multiplication of a Gaussian by a Hermite polynomial.
Laser Spatial Modes Some Transverse Electro. Magnetic (TEM) modes Irradiance
Laser Spatial Modes Some particularly pretty measured laser modes (with a little artistic license…)
Diffraction involving a lens A lens has unity transmission, but it introduces a phase delay proportional to its thickness at a given point (x, y): where L(x, y) is the thickness at (x, y). Compute L(x, y): t(x, y) d neglecting constant phase delays.
A lens brings the far field in to its focal length. A lens phase delay due to its thickness at the point (x 0, y 0): If we substitute this result into the Fresnel (not the Fraunhofer!) integral: The quadratic terms inside the exponential will cancel provided that: Recalling the Lens-maker’s formula, z is the lens focal length! For a lens that's curved on both faces,
A lens brings the far field in to its focal length. This yields: If we look in a plane one focal length behind a lens, we are in the Fraunhofer regime, even if it isn’t far away! So we see the Fourier Transform of any object immediately in front of the lens! E(x, y) F {t(x, y) E(x, y)} t(x, y) A lens in this configuration is said to be a Fourier-transforming lens.
Focusing a laser beam Lens A laser beam typically has a Gaussian radial profile: 2 w 0 f 2 w 1 f What will its electric field be one focal length after a lens? or where: or: Look familiar? This is the same result for a beam diffracting!
How tightly can we focus a laser beam? Recall that we showed earlier that a beam cannot focus to a spot smaller than l/2. But this result 2 w 0 f 2 w 1 f seems to say that, if w 0 is huge, we can focus to an arbitrarily small spot w 1. What’s going on? The discrepancy comes from our use of the paraxial approximation in diffraction, where we assumed small-angle propagation with respect to the z-axis. So don’t use this result when the focus is extremely tight! A beam cannot be focused to a spot smaller than l/2.
Babinet’s Principle The diffraction pattern of a hole is the same as that of its opposite! Holes Neglecting the center point: Anti. Holes
The Diffraction Grating A diffraction grating is a slab with a periodic modulation of any sort on one of its surfaces. The modulation can be in transmission, reflection, or the phase delay of a beam. Diffraction angle, qm(l) First order Zeroth order Minus first order The grating is then said to be a transmission grating, reflection grating, or phase grating, respectively. Diffraction gratings diffract different wavelengths into different directions, thus allowing us to measure spectra.
Diffraction Grating Mathematics Begin with a sinusoidal modulation of the transmission: We need the t 0 term because t(x, y) usually isn’t negative. where a is the grating spacing. The Fraunhofer diffracted field is: Ignoring the y 0 -integration, the x 0 -integral is just the Fourier transform:
Diffraction orders x 1 Because x 1 depends on l, different wavelengths are separated in the +1 (and -1) orders. No wavelength dependence in zero order. z The longer the wavelength, the larger its diffraction angle in nonzero orders.
Diffraction Grating Math: Higher Orders What if the periodic modulation of the transmission is not sinusoidal? Since it's periodic, we can use a Fourier Series for it: Keeping up to third order, the resulting Fourier Transform is: A square modulation is common. It has many orders.
The Grating Equation An order of a diffraction grating occurs if: Scatterer a D where m is an integer. This equation assumed normal incidence and a small diffraction angle, however. We can derive a more general result, the grating equation, if we use a tilted beam, E(x, y), or if we recall scattering ideas: qm C Incident qi wave-front qm a A qi Scatterer B Potential diffracted wave-front AB = a sin(qm) CD = a sin(qi)
Diffraction-grating spectrometer resolution How accurate is a diffraction-grating spectrometer (a grating followed by a lens)? Two similar colors illuminate the grating. d cos( q m) d f 2 w 1 dq f Two nearby wavelengths will be resolvable if they’re separated by at least one spot diameter, 2 w 1. The diffraction grating will separate them in angle by dq, which will become f dq at the focal plane of the lens.
Diffraction-grating spectrometer resolution Recall the grating angular dispersion: d f So two nearby spots will be separated by: Setting this distance equal to the focused-spot diameter: or where N = # grating lines illuminated = d / a
Diffraction-grating spectrometer resolution Let’s plug in some numbers: 2 w 1 2 w 0 f f l ≈ 600 nm m=1 N = (50 mm) x (2400 lines/mm) = 120, 000 lines For simple order-of-magnitude estimates, 4 / p Resolution: ≈ 1: And the resolution, dl/l, depends only on the order and how many lines are illuminated!
Blazed Diffraction Grating By tilting the facets of the grating so the desired diffraction order coincides with the specular reflection from the facets, the grating efficiency can be increased. Efficient diffraction Specular means angle of incidence equals angle of reflection. Input beam Inefficient diffraction Even though both diffracted beams satisfy the grating equation, one is vastly more intense than the other.
Fraunhofer Diffraction: interesting example Hole pattern Randomly placed identical holes yield a diffraction pattern whose gross features reveal the shape of the holes. Square holes Round holes Diffraction pattern
The Fourier Transform of a random array of identical tiny objects Define a random array of two-dimensional delta-functions: Shift Theorem Sum of rapidly varying sinusoids (looks like noise) If Hole(x, y) is the shape of an individual tiny hole, then a random array of identically shaped tiny holes is: The Fourier Transform of a random array of identically shaped tiny holes is then: Rapidly varying Slowly varying
X-ray Crystallography The tendency of diffraction to expand the smallest structure into the largest pattern is the key to the technique of x-ray crystallography, in which x-rays diffract off the nuclei of crystals, and the diffraction pattern reveals the crystal molecular structure. This works best with a single crystal, but, according to theorem we just proved, it also works with powder.
Laser speckle is a diffraction pattern. When a laser illuminates a rough surface, it yields a speckle pattern. It’s the diffraction pattern from the very complex surface. Don’t try to do this Fourier Transform at home.
Particle detection and measurement by diffraction
Moon coronas are due to diffraction. When the moon looks a bit hazy, you’re seeing a corona. It’s a diffraction effect.
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