10 INTRODUCTION AND REVIEW In optics one is usually interested in the time-average power flow.This is calculated in complex notation as follows.First consider scalar functions f(t)and g(t),where f(t)=Refeio and 8(t)=Regeiou (1.31) The time average of f(t)g(t)is given by ®=2Re{fg} (1.32) Here (..)denotes the time average and indicates a complex conjugate.Similarly,if f(t),g(t),f,and g now denote vectors,the time average of f x g is given by f×g)=Re{t×g} (1.33) Using these relations,the time-average Poynting vector for the plane waves of eqs.(1.25) and (1.30)becomes E×由=/层货 (1.34) where we have assumed that s and p are real.Power is carried along the direction of k. In the case of a nonmagnetic material,we can write the magnitude of the time-average Poynting vector,commonly called the intensity I,in the following useful form: I=lE×H1=i0cn|ao (1.35) 1.3 REVIEW OF LASER ESSENTIALS We will shortly discuss in some detail methods by which lasers can be made to produce ultrashort light pulses.First,however,we give a brief and simple review of lasers in general. More detail can be found in texts on lasers,such as [7,8]. 1.3.1 Steady-State Laser Operation Schematic drawings of two simple laser geometries are shown in Fig.1.1.Both lasers consist of a set of mirrors and a gain medium.The gain medium is an optical amplifier which coherently amplifies light passing through it.The mirrors may be curved or planar and together make up the laser cavity or resonator.The cavity is aligned so that light reflects back and forth again and again,passing along the same path every time.If we imagine even a very weak light intensity in the cavity (due to spontaneous emission from the gain medium),then for sufficiently high gain,the intensity increases from one round trip through the laser to the next,eventually resulting in an intense beam.In steady state the gain per round trip must equal the loss.Part of the light passes through the partially transmissive output coupler,and this forms the output laser beam,which can be used for experiments
10 INTRODUCTION AND REVIEW In optics one is usually interested in the time-average power flow. This is calculated in complex notation as follows. First consider scalar functions f (t) and g(t), where f (t) = Re f e˜ jωt and g(t) = Re ge˜ jωt (1.31) The time average of f (t)g(t) is given by fg = 1 2 Re f˜g˜ ∗ (1.32) Here · · · denotes the time average and ∗ indicates a complex conjugate. Similarly, if f(t), g(t), ˜f, and g˜ now denote vectors, the time average of f × g is given by f × g = 1 2 Re ˜f × g˜ ∗ (1.33) Using these relations, the time-average Poynting vector for the plane waves of eqs. (1.25) and (1.30) becomes E × H = 1 2 ε µ E˜ 0 2 k k (1.34) where we have assumed that ε and are real. Power is carried along the direction of k. In the case of a nonmagnetic material, we can write the magnitude of the time-average Poynting vector, commonly called the intensity I, in the following useful form: I = |E × H| = 1 2 ε0cn E˜ 0 2 (1.35) 1.3 REVIEW OF LASER ESSENTIALS We will shortly discuss in some detail methods by which lasers can be made to produce ultrashort light pulses. First, however, we give a brief and simple review of lasers in general. More detail can be found in texts on lasers, such as [7,8]. 1.3.1 Steady-State Laser Operation Schematic drawings of two simple laser geometries are shown in Fig. 1.1. Both lasers consist of a set of mirrors and a gain medium. The gain medium is an optical amplifier which coherently amplifies light passing through it. The mirrors may be curved or planar and together make up the laser cavity or resonator. The cavity is aligned so that light reflects back and forth again and again, passing along the same path every time. If we imagine even a very weak light intensity in the cavity (due to spontaneous emission from the gain medium), then for sufficiently high gain, the intensity increases from one round trip through the laser to the next, eventually resulting in an intense beam. In steady state the gain per round trip must equal the loss. Part of the light passes through the partially transmissive output coupler, and this forms the output laser beam, which can be used for experiments.
REVIEW OF LASER ESSENTIALS 11 Laser Output Output beam beam Gain Gain HR OC HR (r1≈100%) (r2) →Z HR (a) (b) Figure 1.1 Schematic of(a)a linear cavity laser and(b)a ring laser.HR.high reflector:OC.output coupler. In the linear or Fabry-Perot laser cavity shown in Fig.1.1a,light passes along the same path,traveling from left to right and right to left.The light makes two passes through the gain medium per round trip.In the ring cavity shown in Fig.1.1b,light passes through the gain medium only once per round trip.In the diagram we have assumed unidirectional operation (i.e.,light is traveling around the ring in only one direction).In real ring lasers either unidirectional or bidirectional operation is possible.Both linear and ring geometries have been used in femtosecond laser design. For our purposes we usually consider the gain mediumas a black box.The physics of laser gain media is usually analyzed using quantum mechanics in courses on laser fundamentals. Here we usually stick to a classical description,although we will use the indispensable energy-level concept,which does come from quantum mechanics.Common laser media used for ultrafast lasers include impurity-doped crystals or glasses such as Ti:sapphire, Nd:YAG or Nd:glass,organic dyes,doped optical fibers,and semiconductor heterojunction diodes.Note that in thermodynamic equilibrium,materials can absorb light but cannot amplify it.To achieve gain,power must be supplied to the medium to promote electrons into excited-state energy levels.When the electrons are"pumped"to the excited state at a sufficiently high rate (i.e.,when enough power is supplied),a population inversion,in which the population of electrons in an excited energy level exceeds that in a lower level,can be achieved.A population inversion is a necessary condition to achieve optical gain.Power can be supplied to the laser medium in many different ways.Optical pumping,in which absorption of pump photons from a flashlamp or an external laser promotes the electrons to the excited state,is often used in ultrashort-pulse lasers.Other pumping methods include current injection in semiconductor diode lasers or electric discharges in gas lasers.The need for a pump source to obtain optical gain can be likened to the need to plug in an electronic amplifier. To achieve steady-state laser operation,the electric field must repeat itself after a round trip through the laser cavity.Therefore,let us now consider a single round trip of a monochro- matic field through a cavity.To be specific,we consider the linear cavity of Fig.1.la.The optical path length between mirrors is denoted /the total optical path length is therefore 21. The electric field amplitude of a monochromatic plane wave propagating the +z direction can be written E(z,t)=Re Eoei(o-k2) (1.36)
REVIEW OF LASER ESSENTIALS 11 HR z (r1≈100%) (r2 ) OC Gain Laser Output beam HR HR (a) (b) OC Gain Output beam Figure 1.1 Schematic of (a) a linear cavity laser and (b) a ring laser. HR, high reflector; OC, output coupler. In the linear or Fabry–Perot laser cavity shown in Fig. 1.1a, light passes along the same path, traveling from left to right and right to left. The light makes two passes through the gain medium per round trip. In the ring cavity shown in Fig. 1.1b, light passes through the gain medium only once per round trip. In the diagram we have assumed unidirectional operation (i.e., light is traveling around the ring in only one direction). In real ring lasers either unidirectional or bidirectional operation is possible. Both linear and ring geometries have been used in femtosecond laser design. For our purposes we usually consider the gain medium as a black box. The physics of laser gain media is usually analyzed using quantum mechanics in courses on laser fundamentals. Here we usually stick to a classical description, although we will use the indispensable energy-level concept, which does come from quantum mechanics. Common laser media used for ultrafast lasers include impurity-doped crystals or glasses such as Ti:sapphire, Nd:YAG or Nd:glass, organic dyes, doped optical fibers, and semiconductor heterojunction diodes. Note that in thermodynamic equilibrium, materials can absorb light but cannot amplify it. To achieve gain, power must be supplied to the medium to promote electrons into excited-state energy levels. When the electrons are “pumped” to the excited state at a sufficiently high rate (i.e., when enough power is supplied), a population inversion, in which the population of electrons in an excited energy level exceeds that in a lower level, can be achieved. A population inversion is a necessary condition to achieve optical gain. Power can be supplied to the laser medium in many different ways. Optical pumping, in which absorption of pump photons from a flashlamp or an external laser promotes the electrons to the excited state, is often used in ultrashort-pulse lasers. Other pumping methods include current injection in semiconductor diode lasers or electric discharges in gas lasers. The need for a pump source to obtain optical gain can be likened to the need to plug in an electronic amplifier. To achieve steady-state laser operation, the electric field must repeat itself after a round trip through the laser cavity. Therefore, let us now consider a single round trip of a monochromatic field through a cavity. To be specific, we consider the linear cavity of Fig. 1.1a. The optical path length between mirrors is denoted l; the total optical path length is therefore 2l. The electric field amplitude of a monochromatic plane wave propagating the +z direction can be written E(z, t) = Re E0e j(ωt−kz) (1.36)
12 INTRODUCTION AND REVIEW where the field is now taken as a scalar for convenience.Eo is the field amplitude just to the right of the high reflector at=0,@is the angular frequency,k =on/c is the propagation constant,andn =n'+in"is the complex refractive index(n'and n"are both real numbers). The field just before the output coupler is written E=Re Eoe(/e)n"Isejote-j(wleXn'1s+la) (1.37) In eq.(1.37)n'and n"refer to the gain medium;the regions outside the gain medium are taken as air,withn=1.lg and la are the physical lengths of the gain medium and air region, respectively.We can now identify I=n'lg+la as the optical path length.We also see that for n">0,the field has been amplified. The field at z=0,corresponding to a round-trip path through the resonator,is obtained by reflecting off the output coupler with amplitude r2,passing back through the cavity,and then reflecting off the high reflector with amplitude r1.Although ideally we would have r=100%,we keep the variable ri in our expressions to account for any imperfections of the high reflector as well as any other losses in the laser cavity besides those arising from the output coupler.The resulting expression for the field is as follows: E=Rerre2(len"ls Eoe kote-j2ole (1.38) To satisfy the steady-state requirement,eq.(1.38)must be equal to the initial field, eq.(1.36),evaluated at 2=0:E=Re[Eoej].This leads to two conditions:the gain condition and the phase condition.The gain condition. rire2olenn"ls =1 (1.39) states that the round-trip gain exactly balances the round-trip loss.The phase condition, 2o1 =2mz (1.40) requires that thethe round-trip phase shift is equal to an integer m times 2.This means that the laser is only allowed to oscillate at certain discrete angular frequencies,given by mπC Om (1.41) These frequencies are known as the longitudinal modes of the cavity.In terms of the fre- quency f =@/2,the mode frequencies are mc fm=21 (1.42) and the mode spacing(taking n'as frequency independent)is △f=fm-fm-l= (1.43)
12 INTRODUCTION AND REVIEW where the field is now taken as a scalar for convenience. E0 is the field amplitude just to the right of the high reflector at z = 0, ω is the angular frequency, k = ωn/c is the propagation constant, and n = n ′ + jn′′ is the complex refractive index (n ′ and n ′′ are both real numbers). The field just before the output coupler is written E = Re E0e (ω/c)n ′′lg e jωte −j(ω/c)(n ′ lg+la) (1.37) In eq. (1.37) n ′ and n ′′ refer to the gain medium; the regions outside the gain medium are taken as air, with n = 1. lg and la are the physical lengths of the gain medium and air region, respectively. We can now identify l = n ′ lg + la as the optical path length. We also see that for n ′′ > 0, the field has been amplified. The field at z = 0, corresponding to a round-trip path through the resonator, is obtained by reflecting off the output coupler with amplitude r2, passing back through the cavity, and then reflecting off the high reflector with amplitude r1. Although ideally we would have r1 = 100%, we keep the variable r1 in our expressions to account for any imperfections of the high reflector as well as any other losses in the laser cavity besides those arising from the output coupler. The resulting expression for the field is as follows: E = Re r1r2e 2(ω/c)n ′′lgE0e jωte −j(2ω/c)l (1.38) To satisfy the steady-state requirement, eq. (1.38) must be equal to the initial field, eq. (1.36), evaluated at z = 0: E = Re E0e jωt . This leads to two conditions: the gain condition and the phase condition. The gain condition, r1r2e (2ω/c)n ′′lg = 1 (1.39) states that the round-trip gain exactly balances the round-trip loss. The phase condition, 2ωl c = 2mπ (1.40) requires that the the round-trip phase shift is equal to an integer m times 2π. This means that the laser is only allowed to oscillate at certain discrete angular frequencies, given by ωm = mπc l (1.41) These frequencies are known as the longitudinal modes of the cavity. In terms of the frequency f = ω/2π, the mode frequencies are fm = mc 2l (1.42) and the mode spacing (taking n ′ as frequency independent) is f = fm − fm−1 = c 2l (1.43)
REVIEW OF LASER ESSENTIALS 13 Laser gain Longitudinal modes △f=c/2l Laser output Frequency Figure 1.2 Laser gain and cavity loss spectra,longitudinal mode locations,and laser output for multimode laser operation. A frequency-domain view of basic laser operation is pictured schematically in Fig.1.2. A comparison of the gain vs.loss is shown at the top,with the locations of the longitudinal modes shown below.The resonator loss is taken as frequency independent,while the gain is assumed to have a bandpass spectral response.Laser oscillation occurs only for those modes where the gain lies above the loss line.In the situation shown,gain exceeds loss for several longitudinal modes,and multiple output frequencies appear simultaneously.This is called multimode operation. Cavity loss Small-signal gain Saturated gain II11111111111 Longitudinal modes △f=c/21 Laser output Frequency Figure 1.3 Gain and loss spectra,longitudinal mode locations,and output of a homogeneously broadened laser under single-mode operation
REVIEW OF LASER ESSENTIALS 13 f c Figure 1.2 Laser gain and cavity loss spectra, longitudinal mode locations, and laser output for multimode laser operation. A frequency-domain view of basic laser operation is pictured schematically in Fig. 1.2. A comparison of the gain vs. loss is shown at the top, with the locations of the longitudinal modes shown below. The resonator loss is taken as frequency independent, while the gain is assumed to have a bandpass spectral response. Laser oscillation occurs only for those modes where the gain lies above the loss line. In the situation shown, gain exceeds loss for several longitudinal modes, and multiple output frequencies appear simultaneously. This is called multimode operation. Small-signal gain f c Figure 1.3 Gain and loss spectra, longitudinal mode locations, and output of a homogeneously broadened laser under single-mode operation
14 INTRODUCTION AND REVIEW To obtain monochromatic or single-mode laser radiation,it is usually necessary to insert a frequency-dependent loss element (a filter)to ensure that gain exceeds loss for only a single longitudinal mode.This situation is sketched in Fig.1.3.Note,however,that the steady-state laser gain condition [eq.(1.39)]requires that the gain exactly equal rather than exceed the loss.For this reason we now distinguish between small-signal gain and saturated gain.Small-signal gain is gain available under conditions of zero or at least very weak light intensity and depends only on the properties of the laser medium and the pump level.For weak spontaneous emission to build up to produce a significant laser intensity, the small-signal gain must indeed exceed the loss.However,as the intracavity intensity increases,the small-signal condition is violated.As laser photons are amplified in the laser medium through stimulated emission,at the same time electrons in the excited energy state are stimulated back down to the lower-energy state.The intracavity field extracts energy that was stored in the gain medium by the pump and reduces the number of excited-state electrons available for amplification.As a result,the gain is reduced.The actual gain that results,known as the saturated gain,depends on the properties of the laser medium,the pump level,and the intracavity laser intensity.Note that saturation phenomena are quite familiar in the context of electronic amplifiers,where the full amplifier gain is available only for input signals below a certain voltage level;for higher voltage levels the output may appear to be clipped. To clarify the role of gain saturation in lasers,let us imagine that the pump intensity is increased slowly from zero in a single-mode laser cavity.As long as the small-signal gain remains below the loss,the laser intensity is zero.When the pump intensity is sufficient to raise the small-signal gain to equal the loss exactly,the laser reaches threshold.The laser power is still zero at this point.When the pump is increased above threshold,the laser power increases,and this saturates the gain.For a given pump level,the laser intensity builds up just enough to maintain the saturated gain at exactly the loss level,as pictured in Fig.1.3. Thus,pump power above the threshold value is converted into stimulated emission.As a result,the laser intensity increases linearly with pump power above threshold. In our discussion we have implicitly assumed that the gain medium is broadened ho- mogeneously.This means that all the excited-state electrons have identical gain spectra,so that the overall gain is simply the gain spectrum per electron times the population differ- ence between upper and lower laser states.Gain saturation in a homogeneously broadened medium results from a decrease in this population difference,and therefore the saturated gain has the same spectral shape as the small-signal gain. Inhomogeneously broadened media and inhomogeneously broadened lasers.in which different excited-state electrons have different spectra,are also possible.This may result. for example,due to the fact that different impurity ions in a glass laser experience different local environments.As shown in Fig.1.4a,the overall absorption or gain spectrum is then determined by both the individual homogeneous lines(associated with electrons with identi- cal resonance frequencies)and the inhomogeneous distribution function G(-So).When the inhomogeneous distribution is wide compared to the homogeneous linewidth,the net absorption or gain spectrum is determined mainly by G(@-o)and peaks at the center of the inhomogeneous distribution 9o.As shown in Fig.1.4b,the saturation behavior of an in- homogeneously broadened medium interacting with a narrow-bandwidth laser field is quite different than saturation behavior for homogeneous broadening.In particular,saturation induces a"spectral hole"in the vicinity of the laser field,while the spectrum is essentially unchanged for frequencies much more than a homogeneous linewidth away from the laser frequency.Because inhomogeneously broadened lasers are not in common use for ultrashort
14 INTRODUCTION AND REVIEW To obtain monochromatic or single-mode laser radiation, it is usually necessary to insert a frequency-dependent loss element (a filter) to ensure that gain exceeds loss for only a single longitudinal mode. This situation is sketched in Fig. 1.3. Note, however, that the steady-state laser gain condition [eq. (1.39)] requires that the gain exactly equal rather than exceed the loss. For this reason we now distinguish between small-signal gain and saturated gain. Small-signal gain is gain available under conditions of zero or at least very weak light intensity and depends only on the properties of the laser medium and the pump level. For weak spontaneous emission to build up to produce a significant laser intensity, the small-signal gain must indeed exceed the loss. However, as the intracavity intensity increases, the small-signal condition is violated. As laser photons are amplified in the laser medium through stimulated emission, at the same time electrons in the excited energy state are stimulated back down to the lower-energy state. The intracavity field extracts energy that was stored in the gain medium by the pump and reduces the number of excited-state electrons available for amplification. As a result, the gain is reduced. The actual gain that results, known as the saturated gain, depends on the properties of the laser medium, the pump level, and the intracavity laser intensity. Note that saturation phenomena are quite familiar in the context of electronic amplifiers, where the full amplifier gain is available only for input signals below a certain voltage level; for higher voltage levels the output may appear to be clipped. To clarify the role of gain saturation in lasers, let us imagine that the pump intensity is increased slowly from zero in a single-mode laser cavity. As long as the small-signal gain remains below the loss, the laser intensity is zero. When the pump intensity is sufficient to raise the small-signal gain to equal the loss exactly, the laser reaches threshold. The laser power is still zero at this point. When the pump is increased above threshold, the laser power increases, and this saturates the gain. For a given pump level, the laser intensity builds up just enough to maintain the saturated gain at exactly the loss level, as pictured in Fig. 1.3. Thus, pump power above the threshold value is converted into stimulated emission. As a result, the laser intensity increases linearly with pump power above threshold. In our discussion we have implicitly assumed that the gain medium is broadened homogeneously. This means that all the excited-state electrons have identical gain spectra, so that the overall gain is simply the gain spectrum per electron times the population difference between upper and lower laser states. Gain saturation in a homogeneously broadened medium results from a decrease in this population difference, and therefore the saturated gain has the same spectral shape as the small-signal gain. Inhomogeneously broadened media and inhomogeneously broadened lasers, in which different excited-state electrons have different spectra, are also possible. This may result, for example, due to the fact that different impurity ions in a glass laser experience different local environments. As shown in Fig. 1.4a, the overall absorption or gain spectrum is then determined by both the individual homogeneous lines (associated with electrons with identical resonance frequencies) and the inhomogeneous distribution function G(ω − 0). When the inhomogeneous distribution is wide compared to the homogeneous linewidth, the net absorption or gain spectrum is determined mainly by G(ω − 0) and peaks at the center of the inhomogeneous distribution 0. As shown in Fig. 1.4b, the saturation behavior of an inhomogeneously broadened medium interacting with a narrow-bandwidth laser field is quite different than saturation behavior for homogeneous broadening. In particular, saturation induces a “spectral hole” in the vicinity of the laser field, while the spectrum is essentially unchanged for frequencies much more than a homogeneous linewidth away from the laser frequency. Because inhomogeneously broadened lasers are not in common use for ultrashort