超快光學 第03章 脈沖(1)

1、Ultrashort Laser Pulses I Description of pulses Intensity and phase The instantaneous frequency and group delay Zeroth and first-order phase The linearly chirped Gaussian pulse Prof. Rick Trebino Georgia Tech An ultrashort laser pulse has an intensity and phase vs. time. 1 02 ( )

2、exp ().().titcctI tE Neglecting the spatial dependence for now, the pulse electric field is given by: Intensity Phase Carrier frequency A sharply peaked function for the intensity yields an ultrashort pulse. The phase tells us the color evolution of the pulse in time. ( )I t Electric field E (t) Tim

3、e fs The real and complex pulse amplitudes ( )exp( )( )EI ttit Removing the 1/2, the c.c., and the exponential factor with the carrier frequency yields the complex amplitude, E(t), of the pulse: This removes the rapidly varying part of the pulse electric field and yields a complex quantity, which is

4、 actually easier to calculate with. ( )I tis often called the real amplitude, A(t), of the pulse. ( )I t Electric field E (t) Time fs The Gaussian pulse where tHW1/e is the field half-width-half-maximum, and tFWHM is the intensity full-width-half-maximum. The intensity is: 2 01/ 2 0 2 0 ( )exp( /) e

5、xp2ln2( /) exp1.38( /) HWe FWHM FWHM E tEt Et Et t t t For almost all calculations, a good first approximation for any ultrashort pulse is the Gaussian pulse (with zero phase). 2 2 0 2 2 0 ( )exp4ln2( /) exp2.76( /) FWHM FWHM I tEt Et t t Intensity vs. amplitude The intensity of a Gaussian pulse is

6、2 shorter than its real amplitude. This factor varies from pulse shape to pulse shape. The phase of this pulse is constant, (t) = 0, and is not plotted. Its easy to go back and forth between the electric field and the intensity and phase: The intensity: Calculating the intensity and the phase Im ( )

7、 arcta( )n Re ( ) E t t E t (t) = ImlnE(t) The phase: Equivalently, (ti) Re Im E(ti) I(ti) I(t) = |E(t)|2 Also, well stop writing “proportional to” in these expressions and take E, E, I, and S to be the field, intensity, and spectrum dimensionless shapes vs. time. The Fourier Transform To think abou

8、t ultrashort laser pulses, the Fourier Transform is essential. ( )( ) exp()tit dt EE 1 ( )( ) exp() 2 tit d EE We always perform Fourier transforms on the real or complex pulse electric field, and not the intensity, unless otherwise specified. The frequency-domain electric field The frequency-domain

9、 equivalents of the intensity and phase are the spectrum and spectral phase. Fourier-transforming the pulse electric field: 1 02 ( )exp ().().titcctI tE yields: 1 2 1 0 002 0 ( )exp () exp () () )i Si S E The frequency-domain electric field has positive- and negative-frequency components. Note that

10、and are different! Note that these two terms are not complex conjugates of each other because the FT integral is the same for each! The complex frequency-domain pulse field Since the negative-frequency component contains the same infor- mation as the positive-frequency component, we usually neglect

11、it. We also center the pulse on its actual frequency, not zero. So the most commonly used complex frequency-domain pulse field is: Thus, the frequency-domain electric field also has an intensity and phase. S is the spectrum, and is the spectral phase. ( )exp( )( )Si E The spectrum with and without t

12、he carrier frequency Fourier transforming E (t) and E(t) yields different functions. We usually use just this component. ( )E ( ) E The spectrum and spectral phase The spectrum and spectral phase are obtained from the frequency-domain field the same way the intensity and phase are from the time-doma

13、in electric field. Im( ) arctan Re ) ( ) ( E E Im ln( )( ) E or 2 ()S E Intensity and phase of a Gaussian The Gaussian is real, so its phase is zero. Time domain: Frequency domain: So the spectral phase is zero, too. A Gaussian transforms to a Gaussian Intensity and Phase Spectrum and Spectral Phase

14、 The spectral phase of a time-shifted pulse ()exp() ( )f taia FFRecall the Shift Theorem: So a time-shift simply adds some linear spectral phase to the pulse! Time-shifted Gaussian pulse (with a flat phase): Intensity and Phase Spectrum and Spectral Phase What is the spectral phase? The spectral pha

15、se is the phase of each frequency in the wave-form. t 0 All of these frequencies have zero phase. So this pulse has: ( ) = 0 Note that this wave-form sees constructive interference, and hence peaks, at t = 0. And it has cancellation everywhere else. 1 2 3 4 5 6 Now try a linear spectral phase: ( ) =

16、 a . By the Shift Theorem, a linear spectral phase is just a delay in time. And this is what occurs! t (1) = 0 (2) = 0.2 (3) = 0.4 (4) = 0.6 (5) = 0.8 (6) = To transform the spectrum, note that the energy is the same, whether we integrate the spectrum over frequency or wavelength: Transforming betwe

17、en wavelength and frequency The spectrum and spectral phase vs. frequency differ from the spectrum and spectral phase vs. wavelength. () (2c/) ( )( )SdSd Changing variables: S(2c/) 2c 2 d S() S(2c/) 2c 2 The spectral phase is easily transformed: 2 2 (2/ ) c Scd 2 2dc d 2 c The spectrum and spectral

18、phase vs. wavelength and frequency Example: A Gaussian spectrum with a linear spectral phase vs. frequency vs. Frequencyvs. Wavelength Note the different shapes of the spectrum and spectral phase when plotted vs. wavelength and frequency. Bandwidth in various units (1/)c In frequency, by the Uncerta

19、inty Principle, a 1-ps pulse has bandwidth: = 1/2 THz c (1/)/c So (1/) = (0.5 1012 /s) / (3 1010 cm/s) or: (1/) = 17 cm-1 In wavelength: 41 (800nm)(.8 10 cm)(17 cm ) Assuming an 800-nm wavelength: using t 2 1 (1/) 2 (1/) or: = 1 nm In wave numbers (cm-1), we can write: The temporal phase, (t), conta

20、ins frequency-vs.-time information. The pulse instantaneous angular frequency, inst(t), is defined as: The Instantaneous frequency 0 ( ) inst d t dt This is easy to see. At some time, t, consider the total phase of the wave. Call this quantity 0: Exactly one period, T, later, the total phase will (b

21、y definition) increase to 0 + 2: where (t+T) is the slowly varying phase at the time, t+T. Subtracting these two equations: 00 ( )tt 00 2()tTtT 0 2 ()( )TtTt Dividing by T and recognizing that 2/T is a frequency, call it inst(t): inst(t) = 2/T = 0 (t+T) (t) / T But T is small, so (t+T) (t) /T is the

22、 derivative, d /dt. So were done! Usually, however, well think in terms of the instantaneous frequency, inst(t), so well need to divide by 2: inst(t) = 0 (d /dt) / 2 While the instantaneous frequency isnt always a rigorous quantity, its fine for ultrashort pulses, which have broad bandwidths. Instan

23、taneous frequency (contd) Group delay 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, tgr( ( ) ), given by: tgr() = d / d A similar derivation to that for the instantaneous f

24、requency can show that this definition is reasonable. Also, well typically use this result, which is a real time (the rads cancel out), and never d/d, which isnt. Always remember that tgr() is not the inverse of inst(t). Phase wrapping and unwrapping Technically, the phase ranges from to . But it of

25、ten helps to “unwrap” it. This involves adding or subtracting 2 whenever theres a 2 phase jump. Example: a pulse with quadratic phase Wrapped phaseUnwrapped phase The main reason for unwrapping the phase is aesthetics. Note the scale! Phase-blanking When the intensity is zero, the phase is meaningle

26、ss. 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. (i) Re Im E(i)

27、S(i) Without phase blanking Time or Frequency With phase blanking Time or Frequency Phase Taylor Series expansions We can write a Taylor series for the phase, (t), about the time t = 0: where where only the first few terms are typically required to describe well- behaved pulses. Of course, well cons

28、ider badly behaved pulses, which have higher-order terms in (t). Expanding the phase in time is not common because its hard to measure the intensity vs. time, so wed have to expand it, too. 2 012 ( ). 1!2! tt t 1 0t d dt is related to the instantaneous frequency. Frequency-domain phase expansion ()

29、2 0 0 012 ( ). 1!2! Its more common to write a Taylor series for (): As in the time domain, only the first few terms are typically required to describe well-behaved pulses. Of course, well consider badly behaved pulses, which have higher-order terms in (). 0 1 d d where is the group delay! 0 2 2 2 d

30、 d is called the group-delay dispersion. Zeroth-order phase: the absolute phase The absolute phase is the same in both the time and frequency domains. An absolute phase of /2 will turn a cosine carrier wave into a sine. Its usually irrelevant, unless the pulse is only a cycle or so long. Different a

31、bsolute phases for a single-cycle pulse Notice that the two four-cycle pulses look alike, but the three single- cycle pulses are all quite different. f(t)exp( i0) F()exp( i0) Different absolute phases for a four-cycle pulse First-order phase in frequency: a shift in time By the Fourier-transform Shi

32、ft Theorem, f(t 1) F()exp(i1) Time domainFrequency domain 1 0 1 20 fs Note that 1 does not affect the instantaneous frequency, but the group delay = 1. First-order phase in time: a frequency shift By the Inverse-Fourier-transform Shift Theorem, 11 ()( )exp()Ff tit Time domainFrequency domain 1 0/fs 1 .07 /fs Note that 1 does not affect the group delay, but it does affect the instantaneous frequency = 1. Second-order phase: the linearly chirped pulse A pulse can have a frequency that