Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Electric field E Magnetic field H The light wave is comprised of an electric field and a magnetic field. The magnetic field, H is always perpendicular to the electric field. Phase these two waves are in phase Phase 1/2 l difference = 180 deg these two waves are out of phase A2 A1 A1+A2 Superposition Add amplitudes for waves that are in phase A2 A1-A2 A1 Superposition Subtract amplitudes for waves that are out of phase by 180 deg Superposition A2 A1 A1-A2=0 Total destructive interference A1 = A2 but the waves are out of phase by 180 deg. Mutual Coherence Two waves are said to be mutually coherent when the phase difference between the two waves does not change over time. (i.e. the crest of the first wave is always a fixed distance from the crest of the second wave) When the phase difference between two waves varies over time, the waves are said to be mutually incoherent. Mutual Coherence • Coherent sources are generally derived from the same source. That way, both waves have the same wavelength and the same random fluctuations in phase*. *The wavetrain from any source (including a laser) is not constant but undergoes random changes in phase Coherence Length The distance over which a wave can interfere with itself * * or …. The average length of a wavetrain coherence length for.. •laser: •low-coherent laser: •sun: many meters 10 nm 2 mm Examples 1.What is the intensity of two mutually coherent waves, one with amplitude 5 and another with amplitude 13 and a phase difference between the two of a) 90 degrees? b) 180 degrees? 2. What is the intensity of two mutually incoherent waves, one with amplitude 5 and another with amplitude 13? Consider this example… If two mutually coherent waves of amplitude 5 and 10 respectively have a combined intensity of 135, what is the phase difference between them? I coherent = 52 + 102 + 2 5 10 cos ??? = 135 125 + 100 cos(???) = 135 100 cos(???) = 10 cos(???) = 0.1 ??? = 84.26 deg Interference Young’s Double Slit single light source Young’s Double Slit I screen single light source Young’s Double Slit Calculation d Slit separation = a d sin = a y tan = s d y = a s d l 2 = y s d = ay this is the distance s 2 ay this is the distance converted to phase ls Young’s Double Slit Calculation I coherent = E1 + E2 2 = A1 + A2 + 2 A1 A2 cos1 - 2 2 2 d = 2 A + 2 A cos 2 l 2ay 2 2 = 2 A + 2 A cos ls 2 2 Substitute in the expression for phase difference Young’s double slit • Maxima occur whenever ml s y= , m = 0, 1, 2... a y – position on screen m – counter l – wavelength s – distance from aperture to screen a – slit separation y Young’s double slit interference pattern for monochromatic light ml s y= , m = 0, 1, 2... a m= 3 m= 2 m= 1 m=0 m=-1 m=-2 m=-3 Young’s doubleslit interference pattern for white light Example • Given an aperture with a 0.1 mm slit spacing, a wavelength of 500 nm, and a screen held at a distance of 2 m. What is the separation between maxima? • What is the separation for 400 nm light? Lloyd’s mirror interference S mirror S’ Fresnel’s double prism S’1 interference S S’2 two thin prisms Michelson Interferometer Deformable Mirrors Michelson Interferometer to Characterize Actuator Deflection of a MEMS DM. Applications of Interference Retinal Interference Patterns Potential Acuity Meter cataract The laser beams bypass the cataract and generate scatter-free, high resolution interference fringes on the retina to test retinal function prior to cataract removal. Thin Film Interference What happens to a reflected wave when n2 > n1? n2 n1 incident wave reflected wave Reflected wave is shifted in phase by 180º (1/2 cycle) Thin Film Interference What happens to a reflected wave when n2 < n1? n2 n1 incident wave reflected wave n2 < n1 Reflected wave continues with no change in phase Reflectance of an AR Coating reflectance (%) no ARC with ARC 4 3 2 1 400 550 l 700 Why do ARCs Appear Purplish? • green reflection is eliminated • some reddish and bluish reflectance remains (see graph) • mixture of red and blue has purplish hue • reflected color will change with angle since effective thickness of coating changes Thin Film Problem • What is the reflectance of a glass (n=1.5) surface with a MgFl2 coating (n=1.38) optimized for 550 nm light for 1. 550 nm light? 2. 400 nm light? Step 1 • What is the thickness of the coating? tdest l 1 550 1 = = = 99.64 nm 4n 4 1.38 c Step 2 • What is the amplitude of reflectance at the surfaces? nc - nair 1.38 - 1 r1 = = = 0.16 nc + nair 1.38 + 1 ng - nc 1.5 - 1.38 r2 = = = 0.0417 ng + nc 1.5 + 1.38 Step 3 • For 550 nm light…. I coherent = E1 + E2 = A + A2 + 2 A1 A2 cos p1 - p2 2 2 1 2 p1 - p2 = 180 since they are out of phase I coherent = A + A2 - 2 A1 A2 = 2 1 2 Step 4 • For 400 nm light, what is the phase difference? 2 99.64 waves = = 0.687 waves 400 1.38 phase = 0.687 2 = 4.32 radians Step 5 • For 400 nm light I coherent = E1 + E2 = A + A2 + 2 A1 A2 cos p1 - p2 2 2 1 2 I coherent = A + A2 - 2 A1 A2 cos 4.32 = 2 1 2 Newton’s Rings Summary • If the phase changes are common to both surfaces (eg ARC), then tdest m+ 1 l 2 = , tconst 2 n2 m = 0,1, 2... m l = , m = 0,1, 2... 2 n2 Summary • If the phase changes are not common to both surfaces (eg soap bubble, or oil), then m l tdest = tconst 2 n2 , m = 0,1, 2... m+ 1 l 2 = , 2 n2 m = 0,1, 2... Fringes of Equal Thickness Problem • Two flat microscope slides, 10 cm long, are touching at one end and are separated by three microns on the other. How many dark interference bands will appear on the slide if you look at the reflection for 450 nm light? Diffraction and Resolution Diffraction “Any deviation of light rays from a rectilinear path which cannot be interpreted as reflection or refraction” Sommerfeld, ~ 1894 Huygen’s Principle Huygens' principle applied to both plane and spherical waves. Each point on the wave front AA can be thought of as a radiator of a spherical wave that expands out with velocity c, traveling a distance ct after time t. A secondary wave front BB is formed from the addition of all the wave amplitudes from the wave front AA. Fresnel Diffraction Fraunhofer Diffraction • Also called far-field diffraction • Occurs when the screen is held far from the aperture. • Occurs at the focal point of a lens Diffraction and Interference • diffraction causes light to bend perpendicular to the direction of the diffracting edge • interference due to the size of the aperture causes the diffracted light to have peaks and valleys rectangular aperture square aperture ??? circular aperture Airy Disc Airy Disk 1.22 l = a angle subtended at the nodal point l wavelength of the light a pupil diameter (minutes of arc 500 nm light) distance from peak to 1st minimum 1.22 l = a angle subtended at the nodal point 2.5 l wavelength of the light 2 a pupil diameter 1.5 1 0.5 0 1 2 3 4 5 pupil diameter (mm) 6 7 8 Point Spread Function vs. Pupil Size 1 mm 2 mm 3 mm 4 mm Perfect Eye 5 mm 6 mm 7 mm Unresolved point sources Rayleigh resolution limit Resolved Rayleigh Resolution Limit At the Rayleigh resolution limit, the two points are separated by the angle… 1.22 l min = a min angle subtended at the nodal point l wavelength of the light a pupil diameter This is the same as the distance between the max and the first minimum for one Airy disk!!! (minutes of arc 500 nm light) minimum angle of resolution min 1.22 l = a min angle subtended at the nodal point 2.5 l wavelength of the light 2 a pupil diameter 1.5 1 0.5 0 1 2 3 4 5 pupil diameter (mm) 6 7 8 Minutes of arc 20/10 5 arcmin 2.5 arcmin 20/20 1 arcmin convolution 6 mm 20/20 E 3 mm 1 mm DH 20/20 E uncorrected corrected First light AO image of binary star k-Peg on the 3.5-m telescope at the Starfire Optical Range September, 1997. min 1.22 l 1.22 900 10-9 = = = 0.064 seconds of arc a 3.5 About 1000 times better than the eye! Keck telescope: 10 m reflector: about 4500 times better than the eye Point Spread Function vs. Pupil Size 1 mm 2 mm 3 mm 4 mm Perfect Eye Typical Eye 5 mm 6 mm 7 mm 2.5.7: Image quality as a function of pupil size optical quality (arb. units) Best overall quality ~ 2 - 3 mm 0 2 4 pupil size (mm) 6 8 Polarization Direction of Polarization vertical horizontal diagonal Any Polarization can be Expressed as a Sum of a Vertical and a Horizontal Component A Asin (vertical component) y diagonal polarization x Acos (horizontal component) I = A , I x = A cos , I y = A sin 2 2 2 2 2 Unpolarized Light Most light is unpolarized. •sun •incandescent lamp •candlelight Circular and Elliptical Polarization =? E E linear polarization random polarization E circular polarization E unpolarized light E elliptical polarization Generating Polarized Light Polarizing Filters polarized light out unpolarized light in Example • Unpolarized light is incident on a polaroid filter whose orientation is vertical (90 degrees). It is followed by a filter whose orientation is 180 degrees. If 100 units of intensity are incident on the pair of filters, how many units of light will emerge? Example • If you add a 3rd filter oriented 45 degrees from the horizontal in between the two original filters, how much light emerges? Polarization by Reflection Ep Es B Es is the component of the polarization that is parallel to the reflecting surface. Ep is the component of polarization that is perpendicular to Es. Es Es Ep Polarization by Reflection Rs is the reflectance of the Es component. Rp is the reflectance of the Ep component. 20 reflectance (%) Rs n=1.5 Rp 14.8 % At 90º, both Rs and Rp are 100 % 15 10 5 56.3 0 30 60 angle (deg) 90 Brewster’s angle n B = arctan n Polarization by Scattering Applications of Polarization • Haidinger’s brushes • Polarizing sunglasses – reflections from flat surfaces (roads, water, snow, carhoods) are horizontally polarized. – These are suppressed by having glasses that transmit only the vertically polarized component of light • Reducing specular reflections LCD screen Calcite Haidinger’s Brush Disc Hyperpigmentation Glaucoma Suspect Depolarized Parallel Polarized Courtesy of Steve Burns and Ann Elsner, Schepens Eye Research Institute, Boston, MA Randomly Polarized GDX Laser Diagnostic Technologies linear polarization linear polarization strong elliptical polarization weak elliptical polarization Thick NFL Thin NFL GDX Image: AR left eye