Lecture 2: Optics / C2: Quantum Information and Laser Science

Transcription

1 Lecure : Opics / C: Quanum Informaion and Laser Science Ocober 9, 8 1 Fourier analysis This branch of analysis is exremely useful in dealing wih linear sysems (e.g. Maxwell s equaions for he mos par), when we wan o go beyond plane wave wih monochromaic frequency. The basic idea is ha any periodic signal, V () (say a scalar a presen), wih period τ p, can be represened as a sum of sines and consines wih discree frequencies. Le Period ec. V () = a n cos(nω p + α n ) (1) n= where ω p = π/τ p, a n = ampliude of componen a frequency nω p, α n = phase of he componen a frequency nω p. Then he se of real numbers {α n, a n compleely specify he signal, once τ p is specified. They can be found by using he orhogonaliy properies of he series: V n = 1 τp d V ()e inωp 1 τp = d a n cos(n ω + α n )e inωp τ p where V n = τ p n = A n,n () n = A n,n = 1 τp d a n τ p ei(n+n )ω p+iα n + a n [ ] = a τp n eiαn e i(n +n)ω p τ p i(n + a n e iαn + n)ω p τ p where he Kronecker-dela symbol is defined by: e i(n n )ω p iα n [ e i(n n)ω p i(n n)ω p = a n eiαn δ n τ,n + a n e iαn δ n p τ,n (3) p δ n,m = 1 n = m ] τp = oherwise. (4) 1

2 Then he coefficiens are; n > V n = α ne iαn n < V n = α ne iαn n = V = a (5) and in general V n = V n. The funcion V () can herefore by expressed as V () = V n e inωp V n = 1 τp d V ()e inωp (6) τ p Fourier Transforms If a funcion V () is non-periodic, i can no be expanded in a Fourier series. This is because τ p, and he sum becomes an inegral. Recall ha for a periodic funcion: V () = V n e inωp (7) Define a coninuous funcion Ṽ (ω) such ha he coefficiens V n are samples of his funcion Then V () = Now ake he limi δω, wih nδω ω V () = A n = Ṽ (nδω)δω π. (8) δω π Ṽ (nδω)einδω (9) π Ṽ (ω)eiω (1) This defines he funcion V () as he Fourier Transform of a conjugae fucion Ṽ (ω). Since V () is real, V () = V (), so Ṽ (ω) = (V ) ( ω), which is analogous o he propery of he Fourier coefficiens V n = V n in he series expansion. As we have seen, i is ofen convenien o work wih a complex signal field, raher han a real one; and i is useful o define he analyic signal, which is he Fourier Transform of he half specrum of Ṽ (ω) associaed wih posiive frequencies. By definiion V () = = π Ṽ (ω)eiω π Ṽ (ω)eiω + π Ṽ (ω)eiω = 1 U () + 1 U() (11)

3 The funcion U() is he analyic signal associaed wih V (). Is real and imaginary pars are he cosine and sine ransforms of he specrum Ṽ (ω) = a(ω)eiα(ω). I is useful, no only because i makes he mahemaics more compac, bu also because i is closely relaed o he envelope of he real signal. The analyic signal is a convenien ool for calculaing properies of non-monochromaic fields such as Poyning vecor modulus or he inensiy: I = 1 T d (E()H()) = n 1 T d E () (1) T T Z T T Now ake V () o be an aperiodic signal signal, wih non-zero value only in some small range of, and le T ; in he inegral (so long as T > range of suppor of V ()) I = n 1 Z T I = n 1 Z T d V () π Ṽ (ω) = n 1 Z T π Ṽ (ω). (13) I makes sense from he poin of view of conservaion of energy ha he inensiy can be expressed in he same form for boh ime-domain and frequency-domain signals. Thus he power densiy per uni frequency inerval P (ω) is defined by so P (ω) = I (14) P (ω) = n Z 1 π Ṽ (ω) (15) 3 Some imporan properies of Fourier ransforms (F Transformaion operaion) 1. Lineariy: F {αg + βh = αf{g + βf{h (16) The F.T. of he sum of wo funcions is he sum fo he ransforms of each.. Similariy: If F{g() = G(ω), hen: F{g(a) = 1 ( ω ) a G a (17) 3

4 A scale change in... leads o he inverse change in he conjugae domain, e.g. compressing he ime domain (a < 1) expands he specrum. 3. Shif Theorem: If F{g() = G(ω) hen: 4. Parseval s Theorem: F{g( a) = G(ω)e iωa (18) d g() = The oal power in he signal is he same in boh domains. π G(ω) (19) 5. Convoluion Theorem: If F{g() = G(ω) and F{h() = H(ω), hen { F d g( )h( ) = G(ω) H(ω) () Compac noaion: F {g() h() = G(ω) H(ω) (1) 6. Auocorrelaion Theorem: (special case of Convoluion Theorem) { F d g( )g( ) = G(ω) () This is closely relaed o he measuremen of specra of opical fields. 7. Fourier Inegral Theorem FF 1 {g() = F 1 F{g() = g() (3) The Fourier ransformaion operaion has an inverse, excep a poins of disconinuiy. 4 Generalized Funcions: he Dirac δ funcion The familiar idea of he Kronecker-dela symbol can be exended o coninuous indices: Kronecker-dela: δ nn = 1 if n = n = oherwise (4) This funcion arose naurally in he developmen of Fourier series. For example, i is obvious ha he Fourier coefficiens for he sinusoidal funcion: are V () = cos(mω p ) (5) V n = 1 n = m, n = m = oherwise (6) or V n = 1 δ nm + 1 δ n, m. (7) Now define a densiy funcion δ(ω), such ha: + δ(ω) = 1 (8) 4

5 wih he propery The dela funcion has a sifing propery δ(ω) = ω = δ(ω) = oherwise (9) f(ω)δ(ω ω ) = f(ω ) (3) Tha allows i o pick ou a cerain value form he funcion wih which i is convolved. The Dirac dela funcion can be seen as he limi of a series of progressively more spiky es funcions. or alernaively: N δ(ω) = lim N N d d π e iω = π e iω (31) δ() = lim N δ N(); lim N Ne N π ; lim Nrec(N); N lim N sin(n) N N (3) For example: he funcion δ N () = sin(n) (33) has he desired sifing and normalizaion properies, as does he funcion δ N () = N rec(n) (34) The Fourier ransform of he rec(...) funcion is he sinc(...) funcion; so hese wo sequences are 5

6 -1/ 1/ Fourier Transform pairs. This can be seen formally from he following calculaion; 5 Sampling Theorem δ N (ω) = = N d δ N ()e iω 1/N = N eiω iω 1/N d e iω 1/N 1/N = N eiω/n e iω/n iω = eiω/n e iω/n iω/n = sin(ω/n) ω/n = sinc(ω/n). Ofen in he laboraory we can no measure a coninuous variable; we sample i using a discree deecor a a se of specified posiions, for example. Thus if he field is represened by he analyic signal U( r, ), we usually end up wih a se of numbers represening he signal, say {U n, where U n = U( r n, n ) δ r {{ δ sampling volume (35) Under wha condiions can we say ha he sample se is a faihful represenaion of he signal iself? We can use he Fourier series o answer his quesion. Firs we need an inermediae resul. Le f(x) be an aperiodic inegrable funcion. Then g(x) = f(x + n) is a periodic funcion (since g(x + m) = g(x), wih m ineger). Therefore g(x) mus have a Fourier series represenaion. g(x) = f(x + n) = f m e imx (36) m= where f m = 1 dx g(x)e imx = 1 dx f(x + n)e imx (37) 6

7 The sum and inegral can be combined; 1 dx dx Leing x = x + n So ha f m = g(x) = f m = dx f(x + n)e imx (38) dx f(x )e im(x n) = e imn f(m) = f(m) (39) f(x + n) = m= f m e imx (4) Now consider he special case f(x) = δ(x). Subsiuing his in Eqns.(39) and (4), we find he Poisson Sum Formula: δ(x + n) = e imx. (41) m= This is very useful for represening sampled funcions, because we can hink of a sampled funcion as being a produc of he coninuous funcion wih a comb of Dirac dela funcions. Un =Us() U() comb() = n n+1 n n+1 Then if U s () represens he sampled funcion, so U n = U s ( n ) ; U s () = U() δ( nτ s ) (4) where τ s is called he sampling rae. The specrum of his funcion is he convoluion { Ũ s (ω) = Ũ(ω) F δ( nτ s ) (43) and he Fourier ransform of he second erm is { F δ( nτ s ) = d δ( nτ s )e iω = where in he las sep we used he Poisson Sum Formula. Therefore Ũ s (ω) = Ũ(ω) d e in/τs iω (44) δ(ω n/τ s ) (45) is a periodic funcion, wih he specrum of he signal occuring in every period, as shown in he figure. I is clear ha we can ge back he orginal funcion U() by simply filering one of hese replica specra and aking is inverse ransform. Bu his will only work if he replicas do no overlap. The condiion for non-overlapping is ha 1/τ s is broader in frequency han he specrum of he pulse iself. This leads o he Nyquis sampling heorem: To faihfully reconsruc a pulsed signal of bandwidh (FWHM) ω, he sampling rae in he ime domain mus be greaer han τ s = π/ ω. 7

8 8 ω