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Fourier Optics Phys 322 Chapter 11 Lecture 33 Fourier Optics Optical applications: Linear optical system Fraunhofer diffraction Other applications Optical application: linear system Function f(y,z) passes through an optical system and is transformed into g(Y,Z): gY ,Z L f y, z The system is linear if: • multiplying f(y,z) with constant a produces output ag(Y,Z) L af y, z aL f y, z • when the input is a sum of two (or more) functions, f1 (y,z) +f2 (y,z), the output would be g1 (y,z) +g2 (y,z), where g1 and g2 are outputs of f1 and f2, respectively L f1y, z f2 y, z L f1y, zL f2 y, z Optical application: linear system gY ,Z L f y, z Linear system is space invariant if it possesses the property of stationarity. changing the position of input merely changes the position of output without altering its functional form Idea behind: at each point in image plane g(Y,Z) is a linear superposition of outputs arising from each of the individual points on the object f(y,z) Optical application: linear system gY ,Z L f y, z f y, z f y', z' y' y z'z dy'dz' gY ,Z L f y', z' y'y z'z dy'dz' gY ,Z f y', z' L y'y z'z dy'dz' Impulse response Optical application: linear system g Y ,Z f y', z' L y'y z'z dy'dz' yz Impulse response: gY ,Z L f y, z L y' y z'z image of a single object point at (y’,z’) If we knew the impulse response of the system, we could construct the image g of any object f. Optical application: linear system Example: lens system Image: upright, same size I0(y,z) - irradiance in object plane Image blur due to diffraction etc: S y, z,Y ,Z 2f point spread function 2f Image intensity due to one element radiant flux I0(y,z)dydz dIi Y ,Z S y, z,Y ,Z I0 y, zdydz If the object emits incoherent light, intensities add up I Y ,Z S y, z,Y ,Z I y, z dydz i 0 Optical application: linear system I Y ,Z S y, z,Y ,Z I y, z dydz i 0 Meaning of S Suppose object is just a single luminous point at y0, z0 I Y ,Z S y, z,Y ,Z A y y z z dydz i 0 0 A = 1 W/m2 Ii Y ,Z AS y0, z0,Y ,Z Ideal system will show far-field diffraction, Airy distribution Optical application: linear system dIi Y ,Z AS y0, z0,Y,Z Space invariance: Spread function is the same for any x0,y0 (reality: spread function varies slightly with position) for MT=1 S y, z,Y ,Z S Y y,Z z For source in center:S Y ,Z I Y ,Z I y, z S Y y,Z z dydz i o convolution integral Linear systems summary For simplicity, considering the case 1) incoherent light, and 2) MT = +1. The flux density arriving at the image point from dydz is dIi (Y, Z) S(y, z;Y, Z)I0 (y, z)dydz Point-spread function z Z I0 (y,z) Ii (Y,Z) y Y Ii (Y, Z) S(y, z;Y, Z)I0 (y, z)dydz Ii (Y, Z) S(y, z;Y, Z)I0 (y, z)dydz z Z Ii (Y,Z) Point-spread function I0 (y,z) Example: I0 (y, z) A (y y0 ) (z z0 ) y Y Ii (Y, Z) A (y y0 ) (z z0 )S(y, z;Y, Z)dydz AS(y0 , z0 ;Y, Z) The point-spread function is the irradiance produced by the system with an input point source. In the diffraction-limited case with no aberration, the point-spread function is the Airy distribution function. The image is the superposition of the point-spread function, weighted by the source radiant fluxes. Ii (Y, Z) S(y, z;Y, Z)I0 (y, z)dydz z I (y,z) Z 0 Ii (Y,Z) I0 (y, z) Ai (y yi ) (z zi ) i y Y Ii (Y, Z) Ai S(yi , zi ;Y, Z) i Transfer functions for characterization of optical systems: I (,Y Z ) S ( Y y , Z z ) I (,) y z dydz i 0 Iyz0 (,) Syz (,) YYY{(,IYZi )} {(,)} I0 yz {(,)} Syz Y {(,)}Syz Tk (YZ , k ) Optical transfer function T (OTF) Modulation transfer function M (MTF) M (,kkYZ )exp(, i kk YZ ) Phase transfer function (PTF) We use these functions to characterize the quality of an optical system and to reconstruct the original (unknown) object. Optical application: Fraunhofer diffraction y Y Light r P (screen) x Z z Each area dydz is a source of spherical waves: E y, zdydz i tkr Source strength per dE A e yz R unit area E y, zei y,zdydz dE A eik YyZz / R R aperture function dE A y, zeikYyZz/ Rdydz EY ,Z A y, z eikYyZz / Rdydz Optical application: Fraunhofer diffraction y Y Light r P (screen) x Substitute: Z kY kY / R z k kZ / R EY ,Z A y, z eikYyZz / Rdxdz Z For each point Y,Z E k ,k A y, z eikY ykZ z dydz Y Z there is corresponding spatial frequency The field distribution in the Fraunhofer diffraction is the Fourier transform of the field distribution across the aperture (aperture function) E kkYZ,, Y T yz 1 The inverse transform:T yz,, Y EkYZ k Optical application: single slit y Y E k ,k A y, z eikY ykZ z dydz Y Z Light EkY ,kZ F A y, z R k kY / R k kZ / R Z Y Z z A 0 when z b / 2 A y, z 0 when z b / 2 b/ 2 E k F A z A z eikZ zdz A eikZ zdz Z 0 b/ 2 k b Ek A bsinc z k kZ / R k sin Z 0 2 Z Got the same result Optical application: double slit Single A(x) A(x) Double slit slit A0 A0 E(k) E(k) Phys 322 Chapter 11 Lecture 33 Fourier Optics Other applications: Time and frequency domains FTIR and many more Time and frequency domain E(t)=f(t) - electric field depends on time (time domain) Energy flux ~|E(t)|2 = |f(t)|2 2 2 Total emitted energy: f t dt f t f t f *t F()=F {f(t)} Fourier transform (frequency domain) |F()|2 = measure of energy per unit frequency interval 2 1 * it 1 * it f t dt f t F e ddt F f t e dtd 2 2 2 2 1 2 |F()| - f t dt F d 2 power spectrum or Spectral energy Parseval’s formula distribution Emission spectrum of excited atoms/molecules Emission intensity is proportional to the number h h of molecules in excited state N dN N N N et N et / dt 0 0 t / rate of transition Emission intensity: I I0e for t0 (damping constant) t / 2 Et E0e cos0t = 1/ = lifetime F E et / 2 cos t eitdt 0 0 0 2 2 2 E0 / 4 F 2 2 2 0 / 4 Lorentzian profile full width at half maximum: fwhm = 0 natural linewidth Example problem: An excited state lifetime of a chlorophyll molecule is 5 ns. What is the natural linewidth of its emission spectrum, if transition maximum occurs at 670 nm. Express linewidth in nm fwhm = = 1/ = 1/(5 ns) = 2×1010 Hz c c 2 2 2 2.81015 Hz 0 0 0 0 0 0 c c c fwhm 2 2 2 2 0 / 2 0 / 2 0 -3 fwhm = 4.9×10 nm 0 Reality: fwhm ~ 10 nm - inhomogeneus broadening FTIR spectrometer (Fourier Transform InfraRed) IR detector What would be recorded if IR source Movable emits a single wavelength? mirror IR source 2 I A cos Mirror A cos c What if there are two wavelengths? N I Ai i cosi i1 c I A cos d c FTIR spectrometer: qualitative approach IR detector I A cos d Movable c mirror IR source Mirror Measure I() FT gives spectrum:AI Y FTIR spectrometer IR detector No sample Sample Movable mirror IR source Mirror With sample Transmittance spectrum of a sample (polystyrene film) FTIR formal math: Autocorrelation IR detector Electric field at detector: Movable mirror Ed t E t Et IR source / c Measure intensity averaged in time: Mirror I E t 2 E t E t 2 dt d I E 2 t dt 2 E t E t dt E 2 t dt I 2 E 2 t dt 2 E t E t dt Autocorrelation Constant background Autocorrelation c f t f t dt ff 2 Wiener-Khinchine theorem: Y cFff IR detector Movable I E t E t dt mirror 2 IR source Y IF Mirror c f t h t dt f h Cross-correlation: hf Extraction of signal from noise Sine function “hidden” in noise. Autocorrelation of white noise is 0. Spectral grating. E(t) - electric field depends on time (time domain) 1 Et F eixd 2 F E t eixdx 2 I() I() Isin I F m a sin a Grating performs a Fourier transform - intensity distribution on screen is the Fourier image of the incoming time-dependent EM field Pulse shaping. it i Et Ite ESe EoutthtEt in Altering E(t) - will cause change in pulse shape and in spectrum EHEout in Altering E() - will cause change in spectrum and in pulse shape www.physics.gatech.edu/gcuo/lectures/UFO13Pulseshaping.ppt Pulse shaping: optical Fourier transform S() S’() H( ) x ()x grating grating f f ff f f John Heritage, UC Davis Fourier Andrew Weiner, Purdue Transform Plane How it works: The grating disperses the light, mapping color onto angle.
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