Figure1. To construct a light pulse, the electric component of the plane wave should be multiplied with a bell shaped function.

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1 Introduction The Electric field of a monochromatic plane wave is given by is the angular frequency of the plane wave. The plot of this function is given by a cosine function as shown in the following graph. Figure1. To construct a light pulse, the electric component of the plane wave should be multiplied with a bell shaped function. E y = Re{E o exp(-τt 2 +iω o t)} Figure2. Normally, this bell shaped function is chosen to be a Gaussian function described by E y = Re{E o exp(-τt 2 +iω o t)}. To analyze the spectral content of the light pulse, consider the plain wave with an angular frequency ω o. The Fourier transform of the plain wave is a Dirac distribution function of δ(ω o ).

2 Figure3. Width of the cosine function grows larger (T o ), the Fourier transform tends toward a direct distribution of zero width. The Fourier transform of a Gaussian pulse is also a Gaussian function. Light pulse with larger times which are in the range of few hundreds of fs or ps can be measured with mostly by direct methods that are common. But after the use of ultra short pulses such as pulses shorter than hundred fs, reliable method is needed for the characterization. These methods need a model to extract experimental data, ie, the correlation function described in math books. In laser pulses this test function is the pulse signal coming from laser because the test function cannot be synthesized in short time scale. Hence, for this case, the correlation function is called an auto correlation function that is constructed by an interferometric setup. The generation of ultrafast optical pulses involves shaping and compression using special techniques based on linear and non-linear dispersive optical components in addition to the various switching techniques. Pulse Characteristics The optical pulse of an ultrafast optical field with narrow bandwidth can be characterized by three properties. Understanding these basic properties is essential to study the characteristics. They are categorized into temporal & spectral, Guassian & chirped pulses and Spatial characteristics.

3 Temporal & Spectral Characteristics In optics, a narrow bandwidth pulse is considered as an optical field of finite time duration. Figure4. The wave function of an optical pulse is given by U(r,t) where U(t)= A(t)exp(jω o t) for a particular position. ω o is the central angular frequency. The optical intensity of the pulse is given by I(t) where I(t)= U*(t).U(t)=ІU(t) І 2 =І A(t) І 2 the intensity of these pulses is proportional to the time constant τ which describes by Gaussian function, Lorenzian or Hyperbolic functions. The phase of the optical pulse is given by arg{a(t)}. The spectral density S() which is defined as ІV() І 2 where V() is the Fourier transform of the pulse ( (U(t)exp(-j2π t)dt). The Fourier Transform of the complex envelope A()= A() exp(-j2π t)dt=v(- o ) is centered at =0. Temporal & Spectral Width:- The Temporal width of a pulse is the width of intensity defined as I(t)=U(t) which is denoted by τ FWHM. And the Spectral width of a pulse is the width of a spectral intensity defined as S()= ІV() І 2 which is denoted by Δ. The Spectral width is inversely proportional to the temporal width for narrow pulses and shows a linear dependence on each other. The Spectral Intensity S() is related to wavelength λ by the equation, If Δ «o then

4 For ultra narrow pulses with large, the above relationship gives as For an example, for 2fs optical pulse, spectral width of Δ= 220THz corresponds to Δλ= 847nm at λo = 1μm, the spectrum gives a broad shape. Chirped Pulses A Chirped pulse is said to be chirped or Frequency modulated if its instantaneous frequency is time varying which the instantaneous frequency is given by If Also if φ(t)= at 2 /τ 2 where a is the chirp parameter, then the instantaneous frequency is a linear function of time. This type of pulse is called linearly chirped pulses. Gaussian & Chirped-Gaussian Pulses The Gaussian function included optical pulse given in the form of intensity I(t) α exp(-2t 2 /τ 2 ). A transform-limited Gaussian pulse or a simple Gaussian pulse has a complex envelope with constant pulse of Gaussian magnitude A(t)= A o exp(-t 2 /τ 2 ) and intensity I=I o exp(-2t 2 /τ 2 ), where I o (= ) is the peak intensity. The temporal width of the Gaussian function Is τ FWHM = 2ln(2τ) and the spectral width of the Gaussian function is Δ= 0.375/τ. Chirped-Gaussian Pulse The Chirped-Gaussian pulse has a magnitude of A(t)= A o exp(-t 2 /τ 2 ).exp(jat 2 /τ 2 ) where at 2 /τ 2 is the phase of the pulse. Summarized table of parameters of chirped-gaussian pulse is shown below.

5 Table1. Spatial Characteristics The characteristics of the optical pulse waves obey the fundamental wave equation where U(r,t) is the wave function. This tells that the wave form is a pulsed plane wave or in the form of spherical wave. The plane wave has a solution of U(r,t)= A(tz/c).exp(jω o (t-z/c)) and intensity of I(t-z/c). At any time, the travelling pulse has a width of cτ as shown in the following figure. Figure5. The other solution of the wave equation is the spherical pulse wave of U(r,t)=(1/r).g(tz/c).exp(jω o (t-z/c)) where g(t) is an arbitrary function. At any time, spherical shell has a radial width of cτ.

6 Pulse Detection The detection of a pulse of an ultra fast and ultra narrow has been a challenge due to slow photo detectors. The gate or shutter associated with a photo detector measure the optical intensity and pulse phase at different time delays and these detected pulses are measured using interferometric techniques and optical spectrum analyzers. Another major issue is that the detecting equipments alter the pulse before it has been detected. But this can be minimized by clever design and accurate signal processing techniques. The different detecting parameters such as intensity, spectral intensity, phase detection and spectrogram measurements have been reported in the following section. Measurement of Intensity The intensity of an ultra narrow optical pulse can be measured as a photocurrent by a photo detector. The photocurrent measure is varied according to the pulse duration which may be greater or smaller than the response time of the photo detector. As an example, for a detector with shorter response time compared to pulse duration, the measured photocurrent is given by i(t) = RAI(t) where R is the responsivity (A/W) and A is the active area of the detector. On the other hand when the detector response time is much larger than the pulse duration the photocurrent is given by where τ c is the time constant of the receiver circuit. Measurement of spectral intensity Spectral; intensity of an optical pulse can be measured using a system of spectral filters tuned to receive a set of frequencies/wave lengths which is called an optical spectrum analyzer. The received signal is in the form of energy then the resulted measurement is the spectral intensity S(). Michealson interferometer is an example for spectrum analyzer which consists of beam splitter to produce two beams from one with a delay time. The resulted is a fringe pattern similar to the output of conventional spectrum analyzer. Measurement of Phase To study an optical pulse generated from a source, the detector must keep an account of phase in addition to the intensity. Spectral interferometry is an applicable technique used to extract the phase information with respect to a reference pulse.

7 Figure6. The optical pulse delayed by a fixed time τ is added to a known reference pulse U r (t) of same frequency. The interferogram is generated by the summation of Fourier transform of those two pulse. The resulted interferogram is a fringe pattern as a function of frequency with a phase difference of φ() - φ r (). The disadvantage of this method is the necessity of a known reference pulse. Measurement of Spectrogram Spectrogram consists of a gate controlled by a gating function which is delayed by time τ (This Gating function is a time delayed optical pulse which behaves as a function of time). Then the product of this gating function and optical pulse (U(t).W(t-τ)) is measured by a spectrum analyzer at each time delay τ. This time delay is introduced using a moving mirror as shown in the schematic below. Figure7.

8 FROG Technique FROG (Frequency Resolved Optical Gating) method is one of the common and most reliable methods improved to determine the ultrafast light pulse shape. In addition to the amplitude of the pulse, phase can be revealed by using this method. The pulse which to be determined is splits into 2 parts that crossed in a non-linear medium which creates a signal pulse and this signal pulse is spectrally resolved as a function of delay time. The PG-FROG (Polarization Gating FROG technique is one of the common geometries of the FROG technique that a linearly polarized pulse is sent through a pair of crossed polarizer as shown in the figure below. A 45 degree angle oriented polarizer rejected the pulse which is going through it in the absence of the gating pulse. The spectrometer measure the spectrogram as a function of delay time between probe pulse and the gate pulse. Figure8. Applications of Ultrafast Optics The ultrafast optics has opened-up a broad area in optics due to its unique properties. Dr. Ahmed Zewail from Caltech, won the nobel prize in Chemistry in 1999 because of the tremendous studies he carried out using femtosecond pulse to determine the molecular interactions in a chemical reaction. Ultrafat optics has following unique properties or characteristics that lead to a broaden area yet to be unveiled. a. High time resolution- for excitation and measurement of ultrafast waves. b. High spatial resolution- due to very short pulse, this leads to a microscopy and imaging applications. c. High bandwidth.

9 In Laser controlled chemistry, ultrafast optics is used to identify the nuclear motions of chemical reactions which is in the range of 100fs. In a addition to these applications, in electronics, ultrafast laser generate subpicosecond electrical pulse which produce faster electrical devices. References:- Fundamentals of Photonics by B.E.A. Saleh & M.C. Tiech, 2nd Edition, Femtosecond Laser Pulses: Principles & Experiments by C.Rulliere, Ultrafasts Optics by Andrew M. Weiner, Nonlinear Fiber Optics by Govind P. Agrawal, 4th Edition, 2007.