Ultrakurze Lichtimpulse und THz Physik. Einleitung 2. Darstellung ultrakurzer Lichtimpulse 2. Prinzip der Modenkopplung 2.2 Komplexe Darstellung ultrakurzer Lichtimpulse 2.2. Fourier Transformation 2.2.2 Zeitliche und spektrale Phase 2.2.3 Taylor Entwicklung der Phase 2.2.4 Nützliche Umrechnungen Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-
Elektrisches Feld im Frequenzbild The frequency-domain equivalents of the intensity and phase are the spectrum and spectral phase. Fourier-transforming the pulse electric field: X () t = I()exp{[ t i ω t φ()]} t + cc.. 2 yields: Note that φ and ϕ are different! X% ( ω) = S( ω ω ) exp{ i[ ϕ( ω ω )]} + 2 2 S( ω+ ω ) exp{ i[ ϕ( ω+ ω )]} The frequency-domain electric field has positive- and negative-frequency components. Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-2
Elektrisches Feld im Frequenzbild Since the negative-frequency component contains the same information as the positive-frequency component, we usually neglect it. We also center the pulse on its actual frequency, not zero. Thus, the most commonly used complex frequency-domain pulse field is: X% ( ω) S( ω) exp{ iϕ( ω)} Thus, the frequency-domain electric field also has an intensity and phase. S is the spectrum, and ϕ is the spectral phase. Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-3
Beispiel: Gauß-Puls Intensität und Phase eines Gauß- Pulses: The Gaussian is real, so its phase is zero. Time domain: Fouriertransformation eines Gauß-Pulses ist wieder ein Gauß-Puls Frequency domain: So the spectral phase is zero, too. Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-4
Die spektrale Phase Überlagerung von Wellen unterschiedlicher Frequenz Beispiel: spektrale Phase ϕ = ω ω 2 ω 3 ω 4 ω 5 ω 6 t All of these frequencies have zero phase. So this pulse has: ϕ(ω) = Note that this wave-form sees constructive interference, and hence peaks, at t =. And it has cancellation everywhere else. The spectral phase is the phase of each frequency in the wave-form. Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-5
Die spektrale Phase Lineare spektrale Phase: ϕ(ω) = aω. ϕ(ω ) = By the Shift Theorem, a linear spectral phase is just a delay in time. And this is what occurs! t ϕ(ω 2 ) =.2 π ϕ(ω 3 ) =.4 π ϕ(ω 4 ) =.6 π ϕ(ω 5 ) =.8 π ϕ(ω 6 ) = π Konstruktive Interferenz zu einem späteren Zeitpunkt Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-6
Die spektrale Phase Gruppenverzögerung: While the temporal phase contains frequency-vs.-time information, the spectral phase contains time-vs.- frequency information. So we can define the group delay vs. frequency, t gr (w), given by: τ gr (ω) = dϕ/ dω A similar derivation to that for the instantaneous frequency can show that this definition is reasonable. Also, we ll typically use this result, which is a real time (the rad s cancel out), and never dϕ/dν, which isn t. Instantane Frequenz und Gruppenverzögerung Always remember that τ gr (ω) is not the inverse of ω inst (t). Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-7
Die spektrale Phase Darstellung der Phase: Technically, the phase ranges from π to π. But it often helps to unwrap it. This involves adding or subtracting 2π whenever there s a 2π phase jump. Wrapped phase Unwrapped phase The main reason for unwrapping the phase is aesthetics. Note the scale! Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-8
Phase blanking: Die spektrale Phase Without phase blanking With phase blanking When the intensity is zero, the phase is meaningless. When the intensity is nearly zero, the phase is nearly meaningless. Phase-blanking involves simply not plotting the phase when the intensity is close to zero. The only problem with phase-blanking is that you have to decide the intensity level below which the phase is meaningless. Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-9
2.2.3 Die Taylorentwicklung der Phase Zeitliche Phase: We can write a Taylor series for the phase, φ(t), about the time t = : 2 t t φ() t = φ + φ + φ2 +...! 2! where dφ φ = is related to the instantaneous frequency. dt t= where only the first few terms are typically required to describe well-behaved pulses. Of course, we ll consider badly behaved pulses, which have higher-order terms in φ(t). Expanding the phase in time is not common because it s hard to measure the intensity vs. time, so we d have to expand it, too. Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-
Die Taylorentwicklung der spektralen Phase Spektrale Phase: It s more common to write a Taylor series for ϕ(ω): ( ω ω ) 2 ω ω ϕω ϕ ϕ ϕ! 2! ( ) = + + 2 +... where ϕ = dϕ dω ω = ω is the group delay! 2 d ϕ ϕ = is called the group-delay dispersion. 2 2 dω ω = ω As in the time domain, only the first few terms are typically required to describe well-behaved pulses. Of course, we ll consider badly behaved pulses, which have higher-order terms in ϕ(ω). Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-
Die Taylorentwicklung der spektralen Phase Wellenvektor, Gruppengeschwindigkeit und Gruppengeschwindigkeitsdispersion Die Phase auf Grund des Mediums ist: ϕ(ω) = k(ω) L d.h. Wirkung des Phasen-Operators im Frequenzraum Zur Berücksichtigung der Dispersion, entwickle die Phase (k-vector) in eine Taylor-Reihe: [ ] [ ] 2 k( ω) L = k( ω ) L + k ( ω ) ω ω L + k ( ω ) ω ω L+... 2 k( ω ) = ω v( ω ) φ k ( ω ) = v ( ω ) g d k ( ω) = dω v g Höhere Ordnungen sind typischerweise für Pulsdauern 5 fs zu berücksichtigen! Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-2
Die Taylorentwicklung der spektralen Phase Phasengeschwindigkeit Phasengeschwindigkeit: v φ ω = = k c n( λ) und Gruppengeschwindigkeit Gruppengeschwindigkeit: v g dk c = = dω dn n( λ) λ d λ Die Phase v φ unterscheidet sich von der Gruppe v g! dn n( λ) λ n( λ) d λ dn = λ = v c c c dλ φ v g Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-3
Phasen unterschiedlicher Ordnung ϕ( ω) ϕ ϕ. Ordnung: ω ω! = + + t φ() t = φ + φ + φ! Different absolute phases for a four-cycle pulse Different absolute phases for a single-cycle pulse The absolute phase is the same in both the time and frequency domains. f (t)exp(iφ ) F(ω )exp(iφ ) An absolute phase of π/2 will turn a cosine carrier wave into a sine. It s usually irrelevant, unless the pulse is only a cycle or so long. Notice that the two four-cycle pulses look alike, but the three single-cycle pulses are all quite different. Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-4
Frequenzbild. Ordnung: Phasen unterschiedlicher Ordnung Fourier-Transform Shift Theorem, f (t ϕ ) F (ω)exp(iωϕ ) ϕ( ω) ϕ ϕ ω ω! = + + Note that ϕ does not affect the instantaneous frequency, but the group delay = ϕ. Time domain Frequency domain ϕ = ϕ = 2 fs Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-5
Frequenzbild. Ordnung: Phasen unterschiedlicher Ordnung Zur Erinnerung: ϕ( ω) ϕ ϕ ω ω! = + + Lineare spektrale Phase führt zu einer Zeitverschiebung Time domain Frequency domain t ϕ = 2 fs Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-6
Zeitbild. Ordnung: t φ() t = φ + φ + φ! Phasen unterschiedlicher Ordnung By the Inverse-Fourier-Transform Shift Theorem, F( ω φ ) f( t)exp( iφ t) Note that φ does not affect the group delay, but it does affect the instantaneous frequency = φ. Time domain Frequency domain φ = / fs φ =.7 / fs Ultrakurze Lichtimpulse und THz Physik, WS7/8 4-7