20 INTRODUCTION AND REVIEW generation with solid-state lasers.Additionally,the Hermite-Gaussians form a complete orthogonal basis set,which means that an arbitrary spatial beam profile can be decomposed into a superposition of these functions. To calculate the propagation of Gaussian beams through linear optical systems,it is convenient to introduce a complex g parameter,such that 1 1 q(a)R(aπw2(2)n (1.63) The effect of the optical system can then be characterized by a bilinear transformation of q:namely, Jou Agin +B (1.64) Cqin+D The coefficients appearing in eq.(1.64)are usually written in matrix form.The matrices for a few common systems are as follows: A length d of a homogeneous medium: (8)=(0） (1.65a) A lens with focal length f(this matrix also applies to a spherical mirror of radius R if we set f=R/2): （&8）)=() (1.65b) A planar interface,perpendicular to the propagation direction,from a medium with index n to a medium with index n2: (&8)=(6) (1.65c) An important property of such matrices is that the effect of cascaded systems is computed via matrix multiplication.If M,M2,....MN represent the matrices for N cascaded systems, with the beam entering system 1 first and system N last,the effect of the cascaded system is given by MN...M2M1.For example,the matrix for a homogeneous slab of thickness d and index n and surrounded on either side by free space is (e8)=(6)0)(09a)=(（0) (1.66)
20 INTRODUCTION AND REVIEW generation with solid-state lasers. Additionally, the Hermite–Gaussians form a complete orthogonal basis set, which means that an arbitrary spatial beam profile can be decomposed into a superposition of these functions. To calculate the propagation of Gaussian beams through linear optical systems, it is convenient to introduce a complex q parameter, such that 1 q(z) = 1 R(z) − jλ πw2 (z)n (1.63) The effect of the optical system can then be characterized by a bilinear transformation of q: namely, qout = Aqin + B Cqin + D (1.64) The coefficients appearing in eq. (1.64) are usually written in matrix form. The matrices for a few common systems are as follows: A length d of a homogeneous medium: A B C D = 1 d 0 1 (1.65a) A lens with focal length f (this matrix also applies to a spherical mirror of radius R if we set f = R/2): A B C D = 1 0 −1/f 1 (1.65b) A planar interface, perpendicular to the propagation direction, from a medium with index n1 to a medium with index n2: A B C D = 1 0 0 n1/n2 (1.65c) An important property of such matrices is that the effect of cascaded systems is computed via matrix multiplication. If M1, M2, . . . , MN represent the matrices for N cascaded systems, with the beam entering system 1 first and system N last, the effect of the cascaded system is given by MN · · · M2M1. For example, the matrix for a homogeneous slab of thickness d and index n and surrounded on either side by free space is A B C D = 1 0 0 n 1 d 0 1 1 0 0 1/n = 1 d/n 0 1 (1.66)
REVIEW OF LASER ESSENTIALS 21 The matrices given above are also useful in the ray optics description of optical systems. Here one writes -(8)() (1.67) where xin and are the position and slope of the incoming ray,respectively,and similarly for the outgoing ray.For a given optical system,the matrixes for the ray optics description and the Gaussian beam (g parameter)description are identical.The ray optics description is sometimes helpful in identifying or interpreting the ABCD matrices.For example,the matrix in eq.(1.66)says that the slab of physical thickness d has an effective thickness of d/n.From the ray optics description,it is clear that this occurs due to bending of the rays toward the normal as they enter the medium. The g parameter and the ABCD matrices are useful in solving for the transverse mode of laser resonators.Assume that we have calculated the matrix for one complete round trip through the resonator,starting,for example,at a particular mirror M.Then one condition for an acceptable transverse mode is that the field pattern must exactly reproduce itself after one round trip,which can be formulated as 9= Aq+B (1.68) Cq+D A further condition is that for a stable mode,the beam radius must be finite,which means that q must have a nonzero imaginary part.From the ray optics perspective,this stability condition means that incoming rays remain confined after many passes through the system. Multiplying through by the denominator in eg.(1.68)yields a quadratic polynomial equation for g.The root of this equation with a negative imaginary part gives g at mirror M,and g in turn yields the mode size within the laser resonator.Not all resonators have stable solutions.A well-known and important example is the simple two-mirror cavity (Fig.1.7), consisting of mirrors with radii of curvature Ri and R2 which are separated by distance d. After calculating the round-trip ABCD matrix and requiring that the roots of eg.(1.68)have nonzero imaginary part,one obtains the stability relation 0<(-)(-)<1 (1.69a) or simply 0<8182<1 (1.69b) R Figure 1.7 Simple two-mirror optical resonator
REVIEW OF LASER ESSENTIALS 21 The matrices given above are also useful in the ray optics description of optical systems. Here one writes xout x ′ out = A B C D xin x ′ in (1.67) where xin and x ′ in are the position and slope of the incoming ray, respectively, and similarly for the outgoing ray. For a given optical system, the matrixes for the ray optics description and the Gaussian beam (q parameter) description are identical. The ray optics description is sometimes helpful in identifying or interpreting the ABCD matrices. For example, the matrix in eq. (1.66) says that the slab of physical thickness d has an effective thickness of d/n. From the ray optics description, it is clear that this occurs due to bending of the rays toward the normal as they enter the medium. The q parameter and the ABCD matrices are useful in solving for the transverse mode of laser resonators. Assume that we have calculated the matrix for one complete round trip through the resonator, starting, for example, at a particular mirror M. Then one condition for an acceptable transverse mode is that the field pattern must exactly reproduce itself after one round trip, which can be formulated as q = Aq + B Cq + D (1.68) A further condition is that for a stable mode, the beam radius must be finite, which means that q must have a nonzero imaginary part. From the ray optics perspective, this stability condition means that incoming rays remain confined after many passes through the system. Multiplying through by the denominator in eq. (1.68) yields a quadratic polynomial equation for q. The root of this equation with a negative imaginary part gives q at mirror M, and q in turn yields the mode size within the laser resonator. Not all resonators have stable solutions. A well-known and important example is the simple two-mirror cavity (Fig. 1.7), consisting of mirrors with radii of curvature R1 and R2 which are separated by distance d. After calculating the round-trip ABCD matrix and requiring that the roots of eq. (1.68) have nonzero imaginary part, one obtains the stability relation 0 < 1 − d R1 1 − d R2 < 1 (1.69a) or simply 0 < g1g2 < 1 (1.69b) d R R Figure 1.7 Simple two-mirror optical resonator
22 INTRODUCTION AND REVIEW where gi =1-d/RI and g2 =1-d/R2 are symbols often used to express the stability condition succinctly.We make use of this stability condition when we discuss Kerr lens mode-locking in Chapter 2. 1.4 INTRODUCTION TO ULTRASHORT PULSE GENERATION THROUGH MODE-LOCKING The single-mode laser discussed in Section 1.3 can have a very narrow optical frequency spectrum.This results in the well-known monochromaticity property of lasers.However, short pulse generation requires a broad optical bandwidth and hence multiple longitudinal mode operation.Even with the bandwidth-limiting filter removed,however,the saturated gain in a homogeneously broadened laser is below the loss level except for those modes near the peak of the gain spectrum,and this also limits the number of oscillating modes. Methods for forcing a great number of modes to oscillate to obtain broad bandwidths and ultrashort pulses are covered in detail later.In the following we analyze the time-domain characteristics of a laser which we assume already has multimode operation For a multimode laser we can write the electric field as follows: e(z,t)=Re Emej(@mI-km2+m) (1.70a) where 2mπC Dm=0+m△D=0+ (1.70b) L and km =Om (1.70c) Henceforth we use lowercase letters [e.g.,e(z,t)]for the time-domain representation of the field and capital letters (e.g.,Em)to refer to the frequency domain.Equation(1.70a)most accurately models a unidirectional ring laser,where the z coordinate refers to travel around the ring in the direction of laser oscillation,although the same essential results will also hold for a linear(Fabry-Perot)laser.The round-trip distance around the ring is denoted L(for a Fabry-Perot laser we would use L =21,where is the Fabry-Perot mirror separation).In this formulation we are assuming that the laser is oscillating in a single transverse mode, so that the spatial profile may be dropped. Equation (1.70a)can be rewritten as follows: (1.71a) -Refa(-)chu-ite) (1.71b)
22 INTRODUCTION AND REVIEW where g1 = 1 − d/R1 and g2 = 1 − d/R2 are symbols often used to express the stability condition succinctly. We make use of this stability condition when we discuss Kerr lens mode-locking in Chapter 2. 1.4 INTRODUCTION TO ULTRASHORT PULSE GENERATION THROUGH MODE-LOCKING The single-mode laser discussed in Section 1.3 can have a very narrow optical frequency spectrum. This results in the well-known monochromaticity property of lasers. However, short pulse generation requires a broad optical bandwidth and hence multiple longitudinal mode operation. Even with the bandwidth-limiting filter removed, however, the saturated gain in a homogeneously broadened laser is below the loss level except for those modes near the peak of the gain spectrum, and this also limits the number of oscillating modes. Methods for forcing a great number of modes to oscillate to obtain broad bandwidths and ultrashort pulses are covered in detail later. In the following we analyze the time-domain characteristics of a laser which we assume already has multimode operation. For a multimode laser we can write the electric field as follows: e(z, t) = Re m Eme j(ωmt−kmz+φm) (1.70a) where ωm = ω0 + m ω = ω0 + 2mπc L (1.70b) and km = ωm c (1.70c) Henceforth we use lowercase letters [e.g., e(z, t)] for the time-domain representation of the field and capital letters (e.g., Em) to refer to the frequency domain. Equation (1.70a) most accurately models a unidirectional ring laser, where the z coordinate refers to travel around the ring in the direction of laser oscillation, although the same essential results will also hold for a linear (Fabry–Perot) laser. The round-trip distance around the ring is denoted L (for a Fabry–Perot laser we would use L = 2l, where l is the Fabry–Perot mirror separation). In this formulation we are assuming that the laser is oscillating in a single transverse mode, so that the spatial profile may be dropped. Equation (1.70a) can be rewritten as follows: e(z, t) = Re e jω0(t−z/c) m Eme j[m ω(t−z/c)+φm] (1.71a) = Re a t − z c e jω0(t−z/c) (1.71b)
INTRODUCTION TO ULTRASHORT PULSE GENERATION THROUGH MODE-LOCKING 23 where al-3)=∑E.eraou-a] (1.71c) Thus,the electric field is the product of a complex envelope function a(t-/c)with the optical carrier ej(-z/e) This terminology is convenient because in many cases the carrier term,which oscillates with a period of just a few femtoseconds(for visible wavelengths),varies much more rapidly than the envelope function.(However,for the shortest pulses available today,comprising just a few optical cycles,this distinction is blurred.)Both carrier and envelope functions travel around the laser cavity at the speed of light.Furthermore,for a specific location in the cavity(z fixed),the envelope function is periodic with period T=2=21L (1.72) This corresponds to the time required for light to make one round trip around the resonator. If we are to say anything further about the shape of the envelope function,we need to specify the mode amplitudes Em and phasesm.If we assume that there are N oscillating modes all with equal amplitudes Eo and with the phases identically zero,eq.(1.71a)becomes easy to evaluate.The process(discussed later)by which the modes are held with fixed relative phases is known as mode-locking,and as we shall see,having all the phases equal is a particularly useful form of mode-locking.Equation(1.71a)now becomes (N-1)/2 e(z,t)= (-z/e) ejma-zle (1.73) -(N-1)/2 To evaluate this we substitute m'=m+(N-1)/2 and use the summation formula (1.74) m=0 Upon some further simplification,the result is e(z,t)=Re Eoejoo(-:/e)sin(N A/2)(t-z/e) (1.75) sin(△w/2)t-z/c)J The laser intensity (D),averaged over an optical cycle,is proportional to le(.t)2.At a specific cavity location,sayz=0,we have I~1Eol2sin2W△aw/2 (1.76) sin2(△ot/2)
INTRODUCTION TO ULTRASHORT PULSE GENERATION THROUGH MODE-LOCKING 23 where a t − z c = m Eme j[m ω(t−z/c)+φm] (1.71c) Thus, the electric field is the product of a complex envelope function a (t − z/c) with the optical carrier e jω0(t−z/c) . This terminology is convenient because in many cases the carrier term, which oscillates with a period of just a few femtoseconds (for visible wavelengths), varies much more rapidly than the envelope function. (However, for the shortest pulses available today, comprising just a few optical cycles, this distinction is blurred.) Both carrier and envelope functions travel around the laser cavity at the speed of light. Furthermore, for a specific location in the cavity (z fixed), the envelope function is periodic with period T = 2π ω = 2l c = L c (1.72) This corresponds to the time required for light to make one round trip around the resonator. If we are to say anything further about the shape of the envelope function, we need to specify the mode amplitudes Em and phases φm. If we assume that there are N oscillating modes all with equal amplitudes E0 and with the phases identically zero, eq. (1.71a) becomes easy to evaluate. The process (discussed later) by which the modes are held with fixed relative phases is known as mode-locking, and as we shall see, having all the phases equal is a particularly useful form of mode-locking. Equation (1.71a) now becomes e(z, t) = Re E0e jω0(t−z/c) (N −1)/2 −(N−1)/2 e j[m ω(t−z/c)] (1.73) To evaluate this we substitute m′ = m + (N − 1)/2 and use the summation formula q−1 m=0 a m = 1 − a q 1 − a (1.74) Upon some further simplification, the result is e(z, t) = Re E0e jω0(t−z/c) sin (N ω/2)(t − z/c) sin ( ω/2)(t − z/c) (1.75) The laser intensity (I), averaged over an optical cycle, is proportional to |e(z, t)| 2 . At a specific cavity location, say z = 0, we have I(t) ∼ |E0| 2 sin2 (N ωt/2) sin2 ( ωt/2) . (1.76)
24 INTRODUCTION AND REVIEW Intensity Intensity ~N2 Time Time (a) (b) Figure 1.8 (a)Mode-locked laser output with constant mode phase;(b)laser output with randomly phased modes. The resulting intensity profile is sketched in Fig.1.8a.Some of the key points are as follows: The output consists of a periodic series of short pulses,with period T=1/Af=L/c. ·The pulse duration is approximately△t=2r/W△w=l/W△f.Thus,the pulse du- ration is equal to the periodicity divided by the number of modes.Equivalently.the pulse width is equal to the inverse of the total laser bandwidth. The peak intensity is proportional to N2 Eol2.In comparison,the average intensity (averaged over the pulse period),given by the number of modes times the power per mode,is proportional to NEo-.Thus,the peak intensity of a mode-locked pulse is enhanced by a factor N. A very different situation arises if we assume that the mode phases are random,as in a garden-variety multimode laser.In this case the intensity profile takes on a random appearance,as shown in Fig.1.8b.Key points are as follows: The intensity fluctuates randomly about the average intensity value.The average in- tensity is~N|Eol2,which is the same as in the mode-locked case,although the peak intensity is a factor of N lower. .The time scale over which the fluctuations vary (also known as the correlation time) is still△t≈1/W△f. The intensity fluctuations are still periodic with period T=1/Af. In addition,in the garden-variety multimode laser,the random phases are likely to vary slowly with time as well.This introduces yet an additional degree of randomness to the multimode laser output.To avoid such fluctuations and obtain a well-defined output,one can either utilize single-mode lasers with narrow spectra,as in Section 1.3,or mode-locked lasers with narrow pulses and broad spectra. It is also worth noting that the carrier frequency @o is left arbitrary in this very simple treatment.Indeed,in many mode-locked lasers,the carrier frequency is allowed to drift.On the other hand,for some purposes it is important to stabilize and know the carrier frequency precisely.These issues are discussed in Section 7.5
24 INTRODUCTION AND REVIEW Intensity ~N2 ~N Time Time (a) (b) Intensity Figure 1.8 (a) Mode-locked laser output with constant mode phase; (b) laser output with randomly phased modes. The resulting intensity profile is sketched in Fig. 1.8a. Some of the key points are as follows: The output consists of a periodic series of short pulses, with period T = 1/ f = L/c. The pulse duration is approximately t = 2π/N ω = 1/N f. Thus, the pulse duration is equal to the periodicity divided by the number of modes. Equivalently, the pulse width is equal to the inverse of the total laser bandwidth. The peak intensity is proportional to N2 |E0| 2 . In comparison, the average intensity (averaged over the pulse period), given by the number of modes times the power per mode, is proportional to N |E0| 2 . Thus, the peak intensity of a mode-locked pulse is enhanced by a factor N. A very different situation arises if we assume that the mode phases are random, as in a garden-variety multimode laser. In this case the intensity profile takes on a random appearance, as shown in Fig. 1.8b. Key points are as follows: The intensity fluctuates randomly about the average intensity value. The average intensity is ∼N |E0| 2 , which is the same as in the mode-locked case, although the peak intensity is a factor of N lower. The time scale over which the fluctuations vary (also known as the correlation time) is still t ≈ 1/N f. The intensity fluctuations are still periodic with period T = 1/ f. In addition, in the garden-variety multimode laser, the random phases are likely to vary slowly with time as well. This introduces yet an additional degree of randomness to the multimode laser output. To avoid such fluctuations and obtain a well-defined output, one can either utilize single-mode lasers with narrow spectra, as in Section 1.3, or mode-locked lasers with narrow pulses and broad spectra. It is also worth noting that the carrier frequency ω0 is left arbitrary in this very simple treatment. Indeed, in many mode-locked lasers, the carrier frequency is allowed to drift. On the other hand, for some purposes it is important to stabilize and know the carrier frequency precisely. These issues are discussed in Section 7.5.