Pulsed Solid State Laser Systems Using ABCD Matrix Method: a Review

Nonlinear Optics, Quantum Optics, Vol. 51, pp. 265–316 ©2019 Old City Publishing, Inc. Reprints available directly from the publisher Published by license under the OCP Science imprint, Photocopying permitted by license only a member of the Old City Publishing Group.

Pulsed Solid State Laser Systems Using ABCD Matrix Method: A Review

,∗ Y. S. NADA1,J.M.EL-AZAB2,S.M.MAIZE1 AND Y. H. E LBASHAR3

1Physics Department, Faculty of Science, Menouﬁa University, Shebeen-Elkoom, Menouﬁa, Egypt 2National Institute of laser Enhanced Sciences, Cairo University, Giza, Egypt 3Egypt Nanotechnology Center (EGNC), Cairo University, Giza, Egypt

Received: March 12, 2019. Accepted: April 14, 2019.

Although the laser diode and fiber laser are better than CW microchip lasers as they have the smallest size, compact design and wide range of emission laser wavelengths, the Q-switched solid state microchip laser is better than laser diode and fiber laser, because the energy storage capacity and laser induced threshold damage of laser diode are less than those of Q-switched microchip laser and long cavity of the fiber laser prevents achieving very short laser pulses. Yb:YAG was chosen as active medium due to its simple energy level scheme, long upper laser level lifetime (951 μs) which is suitable for storing energy, low quantum defects, large emission cross section, and large absorption cross-section for InGaAs laser emission. The combination of Cr4+: YAG and a doped- YAG gain medium, such as Yb:YAG, is particularly attractive from the point of view of an extremely robust device. Since both materials use the same host crystal, YAG, they can be diffusion-bonded to each other in a way that blurs the distinction between a monolithic and a composite- cavity device. Both materials have the same thermal and mechanical properties and the same refractive index, and the bond between them can be sufficiently strong that the composite device acts in all ways as if it were a single crystal. Due to better mechanical and thermal properties compared to the glass and single crystal, transparent ceramics became high power end-pumping lasers candidate in numerous fields. Ceramic active medium can be heavily and homogeneously doped with laser- active ions.

Keywords: ABCD Matrix, Pulsed lasers, laser systems, ultra short pulse lasers

∗ Corresponding author: e-mail: [email protected]

265 266 Y. S. NADA et al.

1. INTRODUCTION 1.1. Solid state lasers According to the case of active medium, lasers can be classified into solid, liquid or gas. The first invented laser was solid state laser (Ruby laser) [1]. Laser science has many branches like laser safety, material processing, and measurements, and all these applications depend on the active medium wavelength and switching technique [2–22]. The achieving gain with a given solid state laser system performance, the host material with its unique macroscopic and microscopic properties, the activator ions, and optical pump source should be selected self-consistently [23]. Solid state laser materials must have sharp fluorescent lines, strong absorption band and high quantum efficiency. Solid state laser materials are mainly crystal or glass doped by transitions or rare earth ions, in which optical transitions can occur between states of inner incomplete electron shells, (4f-4f) transition for rare earth and (3d-3d) transition for transition ions doped materials. The laser radiation of solid state laser emits in the range of visible and near –IR spectrum (0.4-3μm) [23, 24]. Solid state laser material can be expressed as a physical system of active ions doped in host materials (crystal or glass). For best understanding of the solid state laser system, it can be approached by interaction among active ions with certain electronic states, that be immersed in the local electrostatic crystal field of the host, a phonon field, and a photon field (see Figure 1) [25].

1.1.1. Active ions With an efﬁcient absorption of the pump radiation and an efﬁcient emission, the ions will be useful for giving the optical dynamics of laser material at

FIGURE 1 Physical system of a solid state laser. PULSED SOLID STATE LASER SYSTEMS 267

FIGURE 2 (a) Weak electron–phonon coupling of RE ions; (b) relatively strong electron–phonon coupling of TM ions described by the configurational coordinate model. certain wavelengths. In the case of ions with excellent absorption and weak emission, or vice versa, the solution is using two ions in the same host material, the first called sensitizer, which absorbs the pump energy, and the other called activator which emits the laser radiation. This operation takes place by having efficient nonradiative energy transfer from the sensitizer to the activator through the strong overlap of the emission spectrum of the sensitizer and absorption spectrum of the activator [25]. Due to the electronic transitions of the active ions in the local ligand field environment of the host material, the optical spectral properties of a laser material are determined, and the active ions categorized into two main types. The first type is the transition metal ions (TM ions) which have the nonshielded 3d-electrons that can couple eas- ily with the phonons of the surrounding oxygen ligands, resulting in broad (3d-3d) transition bands. The other type are the rare earth ions (RE ions) with 4f-electrons which are shielded by the electrons of 5s and 5p orbitals leading to narrow (4f-4f) transitions, due to the weak interaction with the crystal field produced by the ligands as seen in Figure 2 [25-26].

1.1.2. Host material For achieving a highly efficient solid state laser material the active medium with a high optical quality should be fabricated with the absence of absorption and scattering centers. Therefore, choosing the host material depends on its good optical properties, as inhomogeneous propagation of light through the crystal, which leads to poor beam quality, comes from the variation in the refractive index of the host material. Also, choosing the host material 268 Y. S. NADA et al. depends on its good mechanical and thermal properties such as, hardness, thermal conductivity and fracture strength. On the other side, the possibility of scaling the growth of impurity-doped crystal with maintaining high production should be obtained [23, 25-26]. The active ion properties such as: the size, disparity, valence, and the spectroscopic properties limit the number of useful materials for the solid state lasers. So, the crystal must have lat- tice sites that have local crystal fields of symmetry and strength required for achieving the desired spectroscopic properties and can accept ions with long radiative lifetimes with emission cross sections near 10−20 cm2 [23]. The two main groups of host materials are crystalline solids and glasses. Compared to glasses, the crystalline materials have the advantages of higher thermal conductivity, narrower fluorescence linewidths, and larger hardness. However, the crystalline materials show problems of poor optical quality and doping homogeneity as well as narrower absorption lines and limited dimension of laser medium. Rods of glasses up of to 1m in length and over 10 cm in diameter and disks of up to 90 cm in diameter and several centimeters thick, which are useful in high energy applications, have been produced with higher optical quality [23].

1.2. Ceramic laser “Ceramics" is gotten from the Greek keramos, meaning porcelain and pottery. Ceramics are composed of randomly oriented microcrystallites, as shown in Figure 3, so the opaque and translucent cement and clay, used often in tableware, cannot be used as a laser medium. It is due to the existence of many scattering sources, such as grain boundary phases, residual pores and

FIGURE 3 Schematic presentation of the microstructure of conventional transparent ceramics, light scattering and the attenuation of input power through the ceramic body. Strong scattering owing to (28) a grain boundary, (30) residual pores, (32) secondary phase, (33) double refraction, (5) inclusions and (6) surface roughness in ceramics prohibits applications in optics [6]. PULSED SOLID STATE LASER SYSTEMS 269

FIGURE 4 Brief overview of the fabrication process of highly transparent Nd: YAG ceramics. secondary phases, which give rise to considerable scattering losses that block laser oscillation in the translucent ceramic laser gain medium [27]. In 1995 the first, highly efficient Nd: YAG ceramic laser medium with average grain size of 50 μm, was fabricated by a solid state reaction method using a high purity powder (> 99.99 wt% purity) [28]. Also Nd: YAG material with average grain size of 10 μm [29] and 3-4 μm [30] was made by using Nanocrystalline Technology and Vacuum Sintering method (NTVS) as illustrated in Figure 4. The first demonstration of polycrystalline Nd-doped YAG ceramics with high conversion efficiency of pore-free polycrystalline Nd:YAG ceramics and high optical quality comparable to the commercial one as well as single crys- tals with new structures were fabricated from ceramic with Solid–State Crys- tal Growth (SSCG) method, as illustrated in Figure 5 [31]; exhibiting a high degree of transparency. Dense Nd:YAG ceramic samples were prepared by conventional sintering and post- Hot Isostatic Pressing (HIP) treatment [32- 33]. As a result of these different techniques, transparent ceramics with very high optical quality were obtained, because. On the macroscopic scale these materials show no refractive index fluctuation or double refraction. On the microscopic one no secondary phases, residual pores or optically inhomogeneous parts are observed [27]. Comparing ceramic composite and composite crystalline Nd:YAG rod used for high power diode end–pumping laser system 270 Y. S. NADA et al.

FIGURE 5 Fabrication process of a polycrystalline and a single crystal of Nd:YAG material.

[34-35] a maximum CW laser output power of 113 W, and a maximum optical to optical transformation efficiency of 47% were achieved for ceramic rod. For crystalline rods these numbers were: 121 W fowerpower and 48% for efficiency. Also, most of the radiative spectral properties such as, cross sections of the intermanifold transitions, emission cross sections of the inter-Stark transitions as well as the fluorescence branching ratios of ceramic and single crystal Nd:YAG, are in good agreement [25].Therefore, ceramic technology can be considered as a good approach for overcoming the technological and economic problems of using single crystal. It has emerged as a promising candidate because it can be produced in large volumes, what is very suitable for high-power laser generation. It can also provide a gain medium for fiber lasers with high beam quality and can also be heavily and homogeneously doped with laser-active ions [27]. The Yb:YAG laser characteristics indicate a great potential for highly efficient and high power Diode Pumped Solid State (DPSS) lasers [36-40]. In 2003, the first cw Yb:YAG laser beam with 345 mw output power and 26% slope efficienc, was demonstrated. It was fabricated by nanocrystalline technology and a vacuum sintering method [37]. An output power 7.7 W with a slope efficiency of 60% in continuous wave (cw) operation and an output power of 9 W with a slope efficiency of 73% and a threshold of 1 W at 1030 nm in the quasi-continuous wave (qcw) operation were obtained [36]. Output power of up to 5.1W with slope efficiency up to 44.3% (cw) has been achieved. 4.4 W power at 48 kHz, with 7-ns pulses were obtained by using passive Q-switching with Cr:YAG saturable absorber [38]. An average output power of 3.80W with a pulse duration of 433 fs at a repetition rate of 90.9 MHz and a peak power of 96.5kW was achieved at the wavelength of 1050 nm using a 2% output coupler in mode locked PULSED SOLID STATE LASER SYSTEMS 271

FIGURE 6 Energy levels in Yb:YAG.

Yb-YAG ceramic laser [39]. A maximum average output power of 2.55 W, with pulse duration of 11.5 ns, repetition rate of 5.71 kHz, pulse energy of 446.25 μJ and peak power of 38.8 kW were obtained by using composite Yb:YAG/Cr4+:YAG/YAG crystal with passive Q-switching at 1030 nm [40]. These results are mainly due to the distinctive physical properties of ytter- bium ion as a long life time emission, a broad emission band and a wide range of tunability, a simple energy level scheme (just two manifolds), a low quantum defect; this in addition to the accessibility of suitable ceramic hosts with good thermo-mechanical properties [36,37].

1.3. Microchip laser Solid state laser systems are composed of discreet optical components that must be carefully assembled and critically aligned. Thus, the laser assembly must be carried out by trained technicians what results in a large cost and consuming time. Also the laser is of large size and become unreliable. The solution to these problems of high cost, time consuming, large size and unre- liability is the Microchip laser. The term ‘microchip laser’ was coined at MIT Lincoln Laboratory in the early 1980s to draw an analogy between this new class of device and semiconductor electronic microchips with their inherent small size, reliability, and low-cost mass production [26,41].

1.3.1. Microchip laser construction Microchip lasers are miniature diode-pumped solid-state devices formed by dielectrically coating thin platelets of gain media [42]. As illustrated in Figure 7 a microchip laser consists of a small piece of solid-state gain medium polished ﬂat and parallel on two sides. Cavity mirrors are dielectrically deposited directly onto the polished surfaces and the 272 Y. S. NADA et al.

FIGURE 7 Monolithic microchip laser. laser is longitudinally pumped with a semiconductor diode laser. Microchip lasers that consist of a single dielectrically coated material are referred to as monolithic. There are several aspects of the microchip laser that make it interesting. First, the cavity length can be made sufficiently short that only a single longitudinal mode falls under the gain profile, thereby ensuring single- frequency output. By using a flat-flat cavity and longitudinally pumping the microchip laser, the TEM00 mode is strongly favorized and only one transverse mode may oscillate. When only one spatial mode oscillates (both longitudinal and transverse), the two orthogonal polarization modes of the laser compete for exactly the same gain and the device operates in a single polarization. Another extremely important aspect of microchip lasers is that all of their surfaces are flat. This allows large wafers of microchip lasers to be polished and coated before the wafers are cut into 1-mm-square pieces, with each piece being a complete device. As a result of their small size and simple fabrication, microchip lasers are potentially mass producible at low cost. Other interesting aspects, for dynamic applications of microchip lasers, are their short cavity lifetimes and small mode volumes. It allows for rapid changes of the operating parameters of the lasers, leading to high rates of frequency modulation and a short-pulse operation. Yb:YAG material was successfully used in microchip lasers operating at 1.05 mm [43], which is becoming more popular, particularly in passively Q-switched devices because of its longer upper-state lifetime [26].

1.3.2. Pumping microchip lasers Microchip lasers are pumped by semiconductor diode lasers. In the simplest conﬁguration one places the microchip laser in close proximity to the output PULSED SOLID STATE LASER SYSTEMS 273

FIGURE 8 Photograph of passively Q-switched microchip laser bonded to the ferrule of the fiber used to pump it. facet of the diode with no intervening optics [41]. An alternative configuration for pumping microchip lasers uses a fiber coupled diode laser. This configuration decouples the diode-laser system from the microchip cavity and offers several practical advantages. It facilitates an extremely compact laser head which can fit into small places, coupled to the rest of the system by a single, flexible optical fiber, as shown in Figure 8 [26].

1.3.3. Passively Q-switched laser A passive Q-switching provides an attractive alternative to the active one for many applications. A passively Q-switched laser contains a gain medium and a saturable absorber. As the gain medium is pumped it accumulates energy and emits photons. Over many round trips in the resonator the photon flux sees gain, fixed loss, and a saturable loss. If the gain medium saturates before the saturable absorber andthe laser will tend to oscillate in cw mode. On the other hand, if the photon flux builds up to a level that saturates, or bleaches, the saturable absorber first, the resonator will see a dramatic reduction in intracavity loss and the laser will Q-switch, generating a short, intense pulse of light, without the need for any high-voltage or high-speed switching electronics. In addition to simplicity of implementation, the advantages of a passively Q-switched laser include the generation of pulses with a well-defined energy and duration that are insensitive to pumping conditions. In contrast to actively Q-switched lasers, the population inversion at the time the cavity Q-switches is fixed by material parameters and the design of the laser. In the absence of pulse bifurcation it is identical from pulse to pulse. The most com- monly used saturable absorber for passive Q-switching of microchip lasers is Cr4+: YAG. It has been used to Q-switch Nd:YAG microchip lasers operating at 1.064 mm [44,45], 946 nm [45], and 1.074 mm [46]; Nd:YVO4 microchip lasers operating at 1.064 mm [47]; Nd:GdVO4 microchip lasers operating at 1.062 mm [48]; and Yb:YAG microchip lasers operating at 1.03 mm [49, 50]. The combination of Cr4+: YAG and a doped-YAG gain medium, such as Nd:YAG, is particularly attractive from the point of view of an extremely robust device. Since both materials use the same host crystal, YAG, they can 274 Y. S. NADA et al.

FIGURE 9 Schematic presentation of a composite-cavity Yb:YAG/Cr4+:YAG passively Q-switched microchip laser. be diffusion-bonded to each other in a way that blurs the distinction between a monolithic and a composite-cavity device. Both materials have the same thermal and mechanical properties and the same refractive index, and the bond between them can be sufﬁciently strong that the composite device acts in all ways as if it were a single crystal.

2. ATOMIC PROCESSES 2.1. Light absorption and emission Absorption and spontaneous emission processes are the two main atomic processes required to obtain an incoherent light emission. In case of producing laser, stimulated emission process takes place. This process is similar to spontaneous emission except that emission is stimulated to occur by a “seed” photon, which is “cloned” to produce two identical photons which are coherent photon as they have the same wavelength and are in phase [51]. As illustrated in Figure 10, both the absorption and the stimulated emission are stimulated processes and are opposite of each other. Stimulated emission requires an initial photon: an atom already at the upper energy level is stimulated to produce a photon by proximity to an existing photon. In the course of the process the original photon is not destroyed but rather passes through the medium with an identical clone produced in the process [51]. To introduce probabilities of these absorption and emission processes, let us consider two energy levels system as illustrated in Figure 11, with N1 as PULSED SOLID STATE LASER SYSTEMS 275

FIGURE 10 Atomic processes in laser.

FIGURE 11 Schematic presentation of (a)Spontaneous emission (b) Stimulated emission (c) Absorption processes.

the population of the ground state and N2 as the population of the excited state. For the spontaneous emission (see Figure 11(a)) the decay rate of the excited state (dN/dt)sp must be proportional to the population N2. dN2 =−AN2 (1) dt sp where A is a positive constant named the rate of spontaneous emission or Einstein A coefﬁcient. The quantity (τsp = 1/A) is the spontaneous emission (radiative) life time. For the stimulated emission (Figure 11(b)) the decay rate can be written as dN2 =−W21 N2 (2) dt st

dN2 where ( dt )st is the rate at which transition from upper energy level to lower energy level (2-1) occurs due to the stimulated emission and (W21) is the rate of stimulated emission which has the dimension of (time−1) and depends on 276 Y. S. NADA et al. both the 2-1 transition and the ﬂux of incident photons. For a plane wave it can be written that, W21 = σ21θ, where θ is the photon ﬂux of the incoming wave and σ21 is the stimulated emission cross section, which depends on the transition characteristics. Similarly the stimulated absorption can be expressed as follows dN1 =−W12 N1 (3) dt ab dN1 where the derivative dt ab is the transition rate from lower to the upper energy level (1-2) due to absorption (Figure (c)) and (W12) is the rate of absorption process which can be written as,

W12 = σ12 (4) where σ12 is the absorption cross section which depends on the 1-2 transition characteristics. Rate equations that describe the population in the laser levels and number of photons in the laser cavity can be derived by using the probabilities of the absorption and emission processes.

2.2. Population rate equation In this chapter we will derive the population rate equation for two -, three -, four - and quasi-three level lasers which are considered in the current study.

2.2.1. Population rate equations for two level lasers In the two level lasers, the rate of change of N2 due to both absorption and stimulated emission is given by

dN 2 = (absorption rate)N − (stimulated emission rate)N (5) dt 1 2

The stimulated processes are characterized by the stimulated emission and absorption cross-sections σ21 and σ12, respectively. Einstein showed at the beginning of the twentieth century that, if the two levels are nondegenerate, one has W21 =W12 and thus σ21 = σ12 [52]. Thus we can write,

dN 2 = absorption rate(N − N ) (6) dt 1 2 and

dN 2 = σ(N − N ) (7) dt 1 2 PULSED SOLID STATE LASER SYSTEMS 277

Similarly the rate of change of N1 due to both absorption and stimulated emission can be expressed as,

dN 1 = σ(N − N ) (8) dt 2 1

Also the spontaneous emission has an effect on the population of these two levels. Thus the complete rate equations will be obtained by adding the term of spontaneous emission from Equation (28) to Equations (7 & 8).

dN 2 = σ(N − N ) − A N (9) dt 1 2 21 2

dN 1 = σ(N − N ) + A N (10) dt 2 1 12 2

We must account also for the pumping process that produces the (positive) population inversion (N2 − N1). To do this the most simply we add a term Wp to the population equation and call it the pumping rate into the upper level. Therefore Equation (36) will read as,

dN 2 = σ(N − N ) − A N + W (11) dt 1 2 21 2 p

2.2.2. Population rate equations for three level laser In fact, in the two level system the probabilities of the excitation and de- excitation are the same, i. e., by irradiating the medium, the radiation will induce both upward transitions 1– 2 (absorption) and downward transitions 2 –1 (stimulated emission). Therefore there no population inversion takes place. The solution of this problem is by using of a third energy level, as in the three-level laser inversion scheme, which is shown in Figure 12 [53].

FIGURE 12 Three-level laser. Level 1 is the ground stat. Laser oscillation occurs on the 2 –1 transition. 278 Y. S. NADA et al.

In this three level system, such as the ruby laser, level 3 is depopulated rapidly via fast decay to the upper laser level 2 which does not give the pumping process much chance to act in reverse and repopulate the ground level 1. Also, it can be considered that the pumping process occurs directly from lower laser level (level 1) to the upper laser level (level 2), because any atom in level 3 converts quickly to level 2. The change of population of dN1 =− level (28) due to pumping is; dt PN1 , where (P) is the pumping rate which is the number of atoms per cubic centimeter per second that are excited from level 1 to level 3. According to the assumption that, level 3 decays very rapidly to level 2 we can write for the rate of change of population of level 2 due to pumping the following expression:

dN dN dN 2 ≈ 3 =− 1 = PN (12) dt dt dt 1

Atoms in level 2 are decaying form level 2 to level 1 via the spontaneous and stimulated emissions, respectively. Thus the changes in the populations of these two levels can be written as:

dN 1 = σ(N − N ) + A N − PN (13) dt 2 1 21 1 1 and

dN 2 = PN − σ(N − N ) − A N (14) dt 1 2 1 21 1

2.2.3. Population rate equation for Four level laser As illustrated in Figure 13 another useful system for achieving population inversion is possible. The lower laser level (level 1) is not the ground level but, an excited level that can itself decay into the ground level what enhances the population inversion [53]. In four level lasers such as Nd:YAG the pumping proceeds from the ground level 0 to level 3, which, as in the three-level laser, decays rapidly into the upper laser level 2. Also level 1 is decaying rapidly into ground level, thus we can write N3 = N1 ≈ 0 and, the population inversion (N2 − N1) ≈ N2 = N. The change in level 2 population due to the spontaneous emission and stimulated emission with the effect of pumping process can be expressed in the following equation.

dN 2 = PN − σN − A N (15) dt 21 PULSED SOLID STATE LASER SYSTEMS 279

FIGURE 13 A four-level laser. The lower laser level 1 decays into the ground level 0.

2.2.4. Population rate equation for a Quasi-three level system In a quasi-three level lasers such as Yb:YAG which is the active medium used in this study; the lower laser level is sublevel of the ground level states which are strongly coupled and in thermal equilibrium therefore non-negligible fraction of the ground state population is present in the lower laser level because the energy separation between sublevels 0 and 1 is comparable to KT. This results in laser photon absorption. See Figure 14 [52]. As illustrated in Figure 14 pumping state decays very rapidly into the upper laser level, so that we can deal with N1 and N2 as the total population in the laser. The rate equations for upper state laser sublevels can be

FIGURE 14 Quasi – three- level laser. 280 Y. S. NADA et al. written in terms of the total populations N1 and N2 as follows;

dN N 2 = PN − (σ N − σ N ) − 2 (16) dt 1 e 2 a 1 τ where σe and σa are the emission and absorption cross sections of laser photons, respectively.

2.3. Photons rate equation Laser rate equations are derived on the basis of a simple notion that there should be a balance between the total number of atoms undergoing a transition and the total number of photons being created or annihilated [52]. The change in intensity of the laser beam I(ν) due to the stimulated emission and absorption is described in plane wave approximation by the following equation [53]: 1 ∂ ∂ g2 − ) I (ν) = σ(ν) N2 − I (ν) (17) c ∂t ∂z g1

As we postulate that these two levels of the same degeneracy, i. e. g1 = g2 and by neglecting the population of the ground state, Equation (44) will take + the form in the forward direction of the propagation, where I(ν) is the intensity of the laser beam in the forward direction of propagation.

∂ I +(ν) 1 ∂ I +(ν) + = σ(ν)N I +(ν) (18) ∂z c ∂t 2

− In the backward direction of propagation case, where (I(ν)) is the corresponding intensity of the laser beam, the equation can be written as,

∂ I −(ν) 1 ∂ I −(ν) − + = σ(ν)N I −(ν) (19) ∂z c ∂t 2

Addition of the two Equations (18 & 19) gives

∂(I +(ν) − I −(ν)) 1 ∂(I +(ν) + I −(ν)) + = σ(ν)N (I +(ν) + I −(ν)) (20) ∂z c ∂t 2

There is a very little variation of I +(ν)&I −(ν) with z in many lasers [30], therefore Equation (46) can be approximated by:

d (I +(ν) + I −(ν)) = cσ(ν)N (I +(ν) + I −(ν)) (21) dt 2 PULSED SOLID STATE LASER SYSTEMS 281

FIGURE 15 Energy level diagram of (a) Yb:YAG laser medium and (b) Cr4+:YAG saturable absorber.

If the gain medium does not completely ﬁll the resonator i.e. l < L, where L is the cavity length and l is the active medium length, the left side of Equation 20,which is independent of z, is simply multiplied by L. However, the right side is multiplied by l, which is less than L, Thus,

d (I +(ν) + I −(ν))L = cσ(ν)N (I +(ν) + I −(ν))l (22) dt 2

Since the number of photons inside the cavity is proportional to the total intensity I +(ν) + I −(ν) we may write,

d = σ ν l (2 ( )N2) (23) dt tr

= 2L where is the photon ﬂux and (tr c ) is the cavity round trip time.

2.4. Steady state laser rate equations model Two differential equations which control the process of stimulated emission in the period that after the switching, had been formulated with concerning only with the fast switching case in which no significant changes of population inversion takes place. Also, the processes of slow effects in comparison to the formation of the giant pulse, such as continued pumping and spontaneous emission have been neglected. The population inversion, total energy and peak power can be obtained by solving these two equations [54]. Gener- alized theory which includes the effect of the absorber in the cavity flux and population inversion rate have been formulated and results in three rate equations [55]. Solution of these rate equations for the key laser parameter such as the output energy and pulse width had been obtained without considering the saturable absorber excited state absorption [56]. The three rate equations 282 Y. S. NADA et al. considering the Cr:YAG saturable absorber excited state absorption had been solved to obtain pulse characteristics such as output energy, peak power and pulse width [57]. In this Chapter a laser rate equation model will be introduced to describe the behavior of Yb:YAG/Cr:YAG composite ceramic passively Q switched microchip laser. Including the excited state absorption of the saturable absorber, the pump term and the reabsorption loss of the quasi- three-level laser medium, the modified coupled rate equations of photons and the saturable absorber in the passively q-switched resonator are given in Refs [55–58]. d 1 = [ 2σ (ν)Nl − 2σg Ngls − 2σe Nels − ln − L − 2αLl ] (24) dt tr R

dN N =−γσcN − + W (25) dt τ p

dNg Nso − Ng =−σgcNg + (26) dt τs

Ng + Ne = Nso (27)

All the mentioned parameters are defined in Table 1. Equations (24-26) describes the behavior of the resonator photon flux with time. This equation includes all mechanisms contributing to the intracavity photon flux. The first term (2σ Nl) is the gain of the active medium. The second term (−2σg Ngls ) and the third term (−2σe Nels ) express the losses of the intracavity photon flux as a result of absorbing atomic transitions of the q-switch ground and excited states over the saturable absorber length, − 1 respectively. The fourth term ( ln R ) is the useful loss due to laser output extracted from the cavity. The fifth term (−L) is the remaining loss due to other cavity optical processes, such as material scattering, quantum defects, etc. the sixth term (−2αLl) is the loss due to the reabsorption of quasi-three- level Yb:YAG laser material. Equation (25) represents the time behavior of the population inversion density of the laser gain medium as a result of stimulated emission, spontaneous emission and the volumetric pump rate into the upper laser level which is proportional to the continuous-wave pump power, αl Wp = Pp[1 − e ]/(hvp A pl), respectively. where (Pp) is the incident pump power, (hvp) is the pump photon energy, (A p) is the pump beam area and (α) is the absorption coefficient of gain medium at the pump wavelength [58]. ?? is the inversion reduction factor which indicates the population inversion change in the laser active medium when the population inversion of the upper laser level is decreased by one. It is equal 2 for Yb3+ doped three-level PULSED SOLID STATE LASER SYSTEMS 283

TABLE 1 Rate equation model parameters. Symbol Meaning The photon density in the laser cavity

tr Cavity round trip time = 2lc/c

lc Optical length of the laser cavity c Speed of light in vacuum σ Stimulated emission cross section of the laser medium N Population inversion density of the gain medium l Length of the laser gain medium

σg Absorption cross section of the ground state of the saturable absorber

Ng Ground state population of the saturable absorber

ls Length of the saturable absorber

σe Absorption cross section of the excied state of the saturable absorber

Ne excited state population of the saturable absorber R Reﬂectivity of the output coupler L Nonsaturable intracavity round-trip dissipative optical loss

αL Absorption coefﬁcient at lasing wavelength of gain medium γ Inversion reduction factor τ Life time of the upper laser level of the gain medium

Wp Volumetric pump rate into the upper laser level

Nso Total population density of the saturable absorber

τs Excited state life time of the saturable absorber solid-state lasers [58–60]. Equation (26) shows the variation of the population density of the q-switch element ground state depending on stimulated absorption and the spontaneous emission, respectively. As the photon ﬂux increases, the ground state population density decreases until bleaching. The effect of pumping process and spontaneous decay of the laser population inversion density during pulse duration can be neglected, because they take a quite long time in comparison to the pulse build up time. Therefore the Equation (25) becomes:

dN =−γσcN (28) dt

Before saturation of the saturable absorber, the initial population inversion density (Ni ) can be approximated to zero. Because, the photon density is low 284 Y. S. NADA et al. the most of the population of the saturable absorber is in the ground state. Thus we can assume that

d = 0, N = N , N = 0, N = N . (29) dt g so e i

By substituting in Equation (24), the initial population inversion can be written as 1 2σg Nsols + ln + L + 2αLl N = R (30) i 2σl

After the bleaching of the saturable absorber with highest photon intensity, almost all the saturable absorber ground state population is excited. Thus we can assume that

d = 0, N = 0, N = N , N = N . (31) dt g e so th

By substituting in Equation (24), the threshold population inversion (Nth) can be written as: 1 2σe Nsols + ln + L + 2αLl N = R (32) th 2σl

Using Equation (32) we rewrite Equation (24) in the form:

d = (2σ Nl − 2σ N l) (33) dt t th

Dividing Equation (33) by Equation (28) in order to get;

d l N − N =− th (34) dN lc γ N

Equation (34) can be integrated from the time of Q-switch opening to an arbitrary time (t) by

(t) N(t) l N d =− 1 − th dN (35) lcγ N i Ni PULSED SOLID STATE LASER SYSTEMS 285 what gives ∼ l Ni (t) − i = Ni − N − Nth ln (36) γ lc N

The value of the initial photon density can be neglected, because it is very small compared with the photon density during the laser output pulse. There- fore Equation (36) can be rewritten as; ∼ l Ni (t) = Ni − N − Nth ln (37) γ lc N

According to Equation (33) the maximum photon density takes place when N = Nth. Therefore Equation (36) can be rewritten as: ∼ l Ni peak = Ni − N − Nth ln (38) γ lc Nth

At the end of the Q-switched pulse, the photon density is equal to zero and the laser population inversion density is reduced to a value below Nth. The ﬁnal population inversion density (N f ) can be derived from Equation (36) by setting N = N f and = 0. Thus we can write, Ni Ni − N f − Nth ln = 0 (39) N f

Equation (39) is transcendental, and can be solved with the aid of Lambert W- function which has the property of x = W(x) exp[W(x)] by changing Equation (39) into the form Ni Ni − N f = Nth ln (40) N f

We take the exponential of both sides of Equation (40) and change the for- mula into: N f N f Ni −Nth − exp − = Ni exp − (41) Nth Nth Nth 286 Y. S. NADA et al.

Equation (41) could be solved via Lambert function as by putting W(x) = −N f /Nth. Then it becomes Ni −NthW(x)exp[W(x)] = Ni exp − (42) Nth

Ni Ni Comparing Equation (42 with x = W(x) exp[W(x)], then x =− exp(− ), Nth Nth and with W(x) =−N f /Nth, we get (N f ) in the form Ni Ni N f =−NthW − exp − (43) Nth Nth

The instantaneous power extracted from the cavity by the output mirror can be calculated as shown in Ref. [38]: 1 hν Alc ln P(t) = R φ(t) (44) t where (hν) is the photon energy, and (Alc) is the resonator volume occupied by the photons. By using Equation (38) the peak power can be obtained in the form: Alchν Ni Ppeak = ln Ni − Nth − Nth ln (45) γ tr Nth

Laser output energy can be calculated by integrating Equation (44) over time from zero to inﬁnity (Q-switch transition) and changing the variable of integration from time to population inversion by using Equation (28) [59].

∞ ∞ 1 hν Alc ln E = P(t)dt = R (t)dt (46) tr 0 0

Or for a real system

N 1 i hν Alc ln dN E = R (47) γσctr N N f PULSED SOLID STATE LASER SYSTEMS 287

FIGURE 16 Experimental setup of Yb3+:YAG/Cr4+:YAG diode end pumped passively Q-switched microchip laser.

After integrating one gets

ν 1 Ni h A ln R ln N E = f (48) 2σγ

The full duration at half maximum of the laser pulse (tp) can be approximated by the ratio of the energy E to peak power. Assuming a triangle pulse shape, where (P) is its height, and E is its area, the laser pulse duration at a half- maximum (FWHM) is given by [59]:

E tp = (49) Ppeak

2.5. Laser System Configuration In this study we use the laser rate equation model that explained before to calculate the laser parameters such as output power, energy and pulse duration for laser diode end-pumped passively Q-switched Yb3+:YAG/Cr4+:YAG microchip laser as illustrated in Figure 16. In this Figure a monolithic laser coated Yb3+:YAG/Cr4+:YAG diode end pumped passively Q-switched microchip laser to form a resonator is shown. The length of the active medium is about 1.2 mm. this active medium left hand side is coated with high-reflection of 1 030 nm and anti-reflection of 940 nm pumping diode laser beam which is transferred to the active medium via an optical system composed of optical fiber and setup of two lenses of 8mm focal length to have a pump beam diameter of 66 μm. The other surface of Cr4+:YAG part is antireflection coated for 1030 nm lasing wavelength. A plane-parallel output coupling mirror was used as output coupler [61,62]. The parameters that had been used in our model are summarized in Table 2. 288 Y. S. NADA et al.

TABLE 2 Values of input parameters used in the rate equation model. Parameter Value −18 2 σg Ground state absorption cross section of Cr:YAG 4.6x10 cm −19 2 σe Excited state absorption cross section of Cr:YAG 8.2x10 cm σ Emission cross section of Yb:YAG ceramic 2.2x10−20 cm2 −19 hνp Pump photon energy 2.12x10 J hν Laser photon energy 1.39x10−19 J τ Life time of Yb:YAG ceramic 951 μs 4+ τs Life time of Cr :YAG ceramic 4.3 μs L Loss 0.05 3+ CYb Yb ions concentration 10 at% 4+ 17 −3 Nso Cr ions concentration 5x10 cm l Thickness of Yb:YAG ceramic 0.12 cm 4+ ls Thickness of Cr :YAG ceramic 0.15 cm λ Laser wavelength 1030 nm

λp Pump wavelength 940 nm Laser beam waist 60 μm

p Pump beam waist 66 μm

3. LIGHT PROPAGATION 3.1. Ray matrix In geometrical optics the light propagation is described in terms of rays [53]. Light rays propagate along straight lines (in free space) and experience a declination if they pass through an optical element like a lens [63]. As illustrated in Figure 17, r1 and r2 refer to the radial positions of the ray. Z refers to the optical axis which is deﬁned by the center of symmetry of the

FIGURE 17 Light propagation in geometrical optics by rays propagating along straight lines in free space. PULSED SOLID STATE LASER SYSTEMS 289

ﬁrst optical element and is perpendicular to this elements’ front surface. θ1 and θ2 refer to the inclination angles. The ray vector r1 at a given input plane IP of the optical element can be characterized by two parameters, namely its radial displacement r1 (z1) from the z axis and its angular displacement θ1. Likewise the ray vector r2 at a given output plane OP can be characterized by its radial r2 (z2) and angular displacements θ2. Thus we can write;

r2 = r1 + l tan θ1 (50)

According to the paraxial approximation, tan(θ1) = θ1 Equation (50) can be rewritten in the form:

r2 = r1 + lθ1 (51)

We will assume, as it is implicit in our deﬁnition of the ray displacement r and slope θ, that we have a cylindrical symmetry about the z axis. The slope of a ray is taken to be positive or negative depending on whether the displacement r is increasing or decreasing in the direction of propagation [53]. From the Figure we ﬁnd that,

θ2 = θ1 (52)

In the matrix notation we can write equations (51-52) as r 10 r 2 = 1 (53) θ2 θ2 1 θ1

Given the initial ray with displacement r1 and slope θ1, modiﬁcation in its propagation through a distance lcan be calculated via this Equation (53). Therefore the ray matrix or ABCD matrix of the propagation of the beam through this distance is. AB 1 l = (54) CD 01

3.1.1. Ray matrix of planar refractive surface

If a light beam passes from one medium with index of refraction n1 to another with index of refraction n2, the rays are refracted at the interface (Figure 18). 290 Y. S. NADA et al.

FIGURE 18 Refraction at the planar boundary surface between two media with different index of refraction according to Snell‘s law and in paraxial approximation we can write

θ sin θ n 1 ≈ 1 = 2 (55) θ2 sin θ2 n1

Therefore

n1 θ2 = θ1 (56) n2

As illustrated in Figure 18, both incident ray and refracted ray has the same position at the interface, thus we can write;

r2 = r1 (57)

In the matrix notation, Equations (56-57) can be written as: r2 = 10 r1 n1 (58) θ θ2 θ 2 n2 1

Thus, the ray matrix or ABCD matrix of the beam that refracted on a planner surface is 10 n1 (59) θ2 n2

3.1.2. Ray matrix of a spherical refractive surface Figure 19 shows the refraction at a spherical surface. The surface geometry is characterized by the radius of curvature R. For a convex surface which means that the center of curvature is to the right of the surface and the ray is arriving PULSED SOLID STATE LASER SYSTEMS 291

FIGURE 19 Refraction at a spherical surface. from the left (as shown), the radius of curvature is positive. A negative radius of curvature deﬁnes a concave surface [63]. Both the incident ray and the refracted ray have the same position at the refraction surface, thus we can write;

r2 = r1 (60)

According to the paraxial approximation and Snell‘s law which will be applied for angles α1 and α2 we can write; α sin α n 1 ≈ 1 = 2 (61) α2 sin α2 n1

From geometry presented in the Figure 19 it follows that

α1 = θ1 + β (62) and

α2 = θ2 + β (63) where r β ≈ tan β = 1 (64) R

Thus we can write

n1 − n2 n1 θ2 = r 1 − θ1 (65) n2 R n2 292 Y. S. NADA et al.

In matrix the notation we can rewrite Equations (Eqs 60 & 65) in the form: ⎡ ⎤ 10 r2 ⎣ ⎦ r1 = n1 − n2 n1 (66) θ2 θ1 n2 R n2

From Equation (66) the ray matrix or ABCD matrix of refractive spherical surface is given by ⎡ ⎤ 10 ⎣ ⎦ n1 − n2 n1 (67) n2 R n2

Putting R =∞, the ABCD matrix of a planner dielectric surface can be obtained.

3.1.3. ABCD matrix of thin lens and spherical mirror The knowledge of ABCD matrix of beam propagation in free space and ABCD matrix of refraction at a spherical dielectric interface is sufﬁcient to describe arbitrary optical systems since the ray transfer matrices for all optical elements can be derived by using these two fundamental ones [63]. For beam propagation through successive optical elements, the ray starts on the left plane having a ray vector V1. This ray vector is transformed into the ray vector V2 by the ﬁrst optical element. The second element generates the ray vector V3 on the third plane and so the fourth. In the case of n optical elements, we will have n equations:

V2 = M1V1

V3 = M2V2

V4 = M3V3 (68) ......

Vn+1 = Mn Vn where M1, M2, M3 ...... Mn are the ray matrices of the successive optical elements, and therefore;

Vn+1 = (Mn Mn−1 Mn−2 ...... M3 M2 M1)V1 = MV1 (69)

Thus, the resulting ray transfer matrix M is obtained by multiplying all individual matrices in the opposite order (i.e. right to left) of the ray propagation [63]. PULSED SOLID STATE LASER SYSTEMS 293

In the thin lens approximation, any change in ray position or angle inside the medium is neglected which means that we do not propagate the ray between the two surfaces. Thus the ray transfer matrix MTL of a thin lens is the product of two transfer matrices for refraction at a spherical interface [63]. Let us assume that the convex surface of the thin lens has radius of curvature ( R1 ) and the concave surface has a radius of curvature( –R2 ), thus the ray matrix of the thin lens can be obtained via multiplying the matrices of the two dielectric surfaces as follows; ⎡ ⎤ ⎡ ⎤ 1010 ⎣ − ⎦ ⎣ − ⎦ MTL = n1 n2 n2 n1 n2 n1 (70) −n2 R2 n1 n2 R1 n2 or ⎡ ⎤ ⎡ ⎤ 1010 ⎣ − ⎦ ⎣ − ⎦ MTL = n2 n1 n2 n1 n2 n1 (71) n2 R2 n1 n2 R1 n2 or = 10 MTL − 1 (72) f 1 where f is the focal length of the lens used and 1 n − n 1 1 = 2 1 − (73) f n1 R1 R2

A spherical mirror with radius of curvature R provides the same ray transformation properties as a lens [42]. Thus the ray matrix of spherical mirror can be written as = 10 Msm − 2 (74) R 1

3.2. Stability of the cavity A stable cavity can be defined as the cavity that can be designed from per- fectly aligned mirrors that keep ray near the optical axis [64]. If we follow the beam (Figure 20(a)), we find that the beam propagate through a distance d, then is reflected from curved mirror M2, then it prop- agates again through the distance d and is reflected from the curved mirror 294 Y. S. NADA et al.

FIGURE 20 (a) Optical cavity showing a ray bouncing back and forth between the mirrors; (b) lens- waveguide equivalent to the mirror system shown in (a).

M1. Then it re starts all over again. A setup of successive lenses can redirect the beam in the same manner as shown in Figure 20(b). It can be called the equivalent-lens wave guide. The ray matrix of a beam propagating through distance d followed by a thin lens of focal lens f can be expressed as;

M = Mlens + Mdistance (75)

= 10 1 d M − 1 (76) f 1 01 or = 1 d M − 1 − d (77) f 1 f

By using Equation (77) the ray matrix for the unit cell that is determined in Figure 20 can be expressed as: 1 d 1 d M = (78) − 1 1 − d − 1 1 − d f1 f1 f2 f2 PULSED SOLID STATE LASER SYSTEMS 295 or ⎡ ⎤ 1 − d d + d 1 − d ⎢ f2 f2 ⎥ M = ⎣ ⎦ (79) − 1 − 1 1 − d 1 − d 1 − d − d f1 f2 f1 f1 f2 f1

For successive round trips through the cavity, we will ﬁnd a second, different equation for the ray that passes the various planes of the succeeding unit cells.

rm+1 = Arm + Bθm (80) where m is an integer that refers to the different planes. Using equation (80) it can be written in the from:

1 θ = (r + − Ar ) (81) m B m 1 m

The slope at plane (m + 1) is equal to

θm+1 = Crm + Dθm) (82)

Substituting θm from Equation (81) into Equation (82) we have:

D θ + = Cr + (r + − Ar ) (83) m 1 m B m 1 m

Similar to Equation (81) the slope at plane (m + 1) can be written as:

1 θ + = (r + − Ar + ) (84) m 1 B m 2 m 1

Equations (83-84) are equal. Thus we can write:

1 D (r + − Ar + ) = Cr + (r + − Ar ) (85) B m 2 m 1 m B m 1 m

As AD − BC = 1, for any round trip [43], and by combining the terms we can write;

A + D r + − 2 ( ) r + + r = 0 (86) m 2 2 m 1 m

Here, if r has a maximum value, i. e., the ray makes the axis undulates between the two mirrors, the cavity is stable. If there is not any maximum 296 Y. S. NADA et al.

FIGURE 21 Example of the ray’s position at the various planes of the lens waveguide. value for r, the cavity is unstable, because the ray will has apposition more than the dimensions of the mirrors and escape from the cavity as illustrated in the following Figure 21: According to Figure 21, for a stable cavity we can assume the solution for r of the form:

iϕ m imϕ rm = r0 ( e ) = r0e (87)

By substituting it in Equation (86) one gets

A + D r ei(m+2)ϕ − 2 ( ) r ei(m+1)ϕ + r eimϕ = 0 (88) 0 2 0 0 or

A + D r eimϕ ei2ϕ − 2 ( ) eiϕ + 1 ] = 0 (89) 0 2

Here r is always real and r0 not equal zero for general case, also the exponential not equals zero, thus the second factor in Equation (89) equals zero.

A + D ei2ϕ − 2 ( ) eiϕ + 1 ] = 0 (90) 2

The solutions of Equation (90) are:

A + D A + D eiϕ = ± i [1 − ( )2 ] (91) 2 2

It is sufﬁcient for (A + D)/2 to be less than ±1 for ϕ to be real. This is the general condition for stability:

A + D 1 ≤ ≤ 1 (92) 2 PULSED SOLID STATE LASER SYSTEMS 297

By adding 1 to this Equation and then dividing by 2, we obtain

A + D + 2 0 ≤ ≤ 1 (93) 4

Returning to the speciﬁc case of the two-spherical-mirrors. Cavity, Equation (79) gives the elements of the ABCD matrix of the unit cell. Therefore A + D + 2 1 d d d d = 1 − − + 1 − 1 − + 2 (94) 4 4 f2 f1 f2 f1 or A + D + 2 d d d2 = 1 − − + (95) 4 2 f2 2 f1 4 f1 f2

Thus A + D + 2 d d = 1 − 1 − (96) 4 f1 f2

From Equation (96), if we put d g1 = 1 − (97) R1 and d g2 = 1 − (98) R2 where R1 and R2 are the radii of curvature of mirrors M1 and M2, respectively, we can write the condition of stability as;

0 ≤ g1g2 ≤ 0 (99)

Equation (99) can be graphed in the following Figure 22. That help in determination of stability of the cavity.

3.3. Gaussian beam It should be mentioned that, a ray is a path and is not a ﬁeld and does not have an amplitude, phase or spatial extent. Gaussian beam is a very important class 298 Y. S. NADA et al.

FIGURE 22 Stability curve of E-field solutions that can be achieved via an approximation analytical solution to the wave Equation yielding fields that are completely specified at all points in space, and consequently gives us a more complete wave description of the beams produced by the laser [52, 64]. For a monochromatic electromagnetic field with an uniform polarization (scaler approximation), the electric field can be described as:

E(x, y, z, t) = E(x, y, z) exp(iωt) (100) where the complex amplitude E must satisfy the wave Equation in scalar form. Thus it can be written as: ∇2 + k2 E(x, y, z) = 0 (101) with k = ω/c. The electric ﬁeld solution within the paraxial approximation, where the wave is assumed to be propagating at a small angle θ in z direction takes PULSED SOLID STATE LASER SYSTEMS 299 form:

E(x, y, z) == E0ψ(x, y, z) exp(−ikz) (102) where is a function varying little on a wavelength scale in z-coordinate, and tell us how the beam deviate from a uniform plane wave. By substituting of Equation (102) into Equation (101) we get;

∂ψ ∇2 ψ − 2ik = 0 (103) ⊥ ∂t where

∂2 ∂2 ∇2 = + (104) ⊥ ∂x2 ∂y2

Equation (103) is called the paraxial wave Equation. For a cylindrical system the Equation is 1 ∂ ∂ψ ∂ψ r − i2k = 0 (105) r ∂r ∂r ∂z

The solution of this partial differential Equation by using unknown functions p(z) and q(z) is; kr2 ψ = exp −i p(z) + (106) 0 2q(z) where 0 refers to the fundamental mode. Now our goal is the determination of 0 by converting the partial differential Equation (105) to ordinary differential Equation for the unknown functions p(z) and q(z). Thus;

∂ψ 2 2 2 0 −ip(z) − ikr −ip(z) − ikr 2ikr q (z) = e e q(z) (−ip (z) + e e q(z) (107) ∂z 4q2(z) or ∂ψ 2ikr2q (z) 0 = ψ −ip (z) + (108) ∂z 0 4q2(z)

∂ψ k2r 2q (z) −2ik 0 = ψ −2kp (z) + (109) ∂z 0 q2(z) 300 Y. S. NADA et al. and ∂ψ kr) 0 = ψ −i (110) ∂r 0 q(z) thus ∂ψ kr2) r 0 = ψ −i (111) ∂r 0 q(z) and ∂ ∂ψ ∂ kr2) ( r 0 ) = [ ψ −i ] (112) ∂r ∂r ∂r 0 q(z) Thus ∂ ∂ψ kr2) kr2) kr) ( r 0 ) = −2i ψ + −i −i ψ (113) ∂r ∂r q(z) 0 q(z) q(z) 0 and 1 ∂ ∂ψ k2r 2 2k) ( r 0 ) = − − i ψ (114) r ∂r ∂r q2(z) q(z) 0 By substituting Equations (109 & 114) into (105) one gets the following Equation for the ψ0 function: k2r 2 2k) k2r 2q (z) − − i ψ + −2kp (z) + ψ = 0 (115) q2(z) q(z) 0 q2(z) 0 Grouping ythe terms with equal power of r together one obtain: k2 i { (q (z) − 1) r 2 − 2k(p (z) + r } ψ = 0 (116) q2(z) q(z) 0

For the assumed form of 0 given by Equation (107), every factor of a power of r must be equal to zero. This yields two simple ordinary differential Equa- tions:

q (z) = 1 (117) i p (z) =− (118) q(z)

The solution of Equation (117) is

q(z) = q0 + z (119) PULSED SOLID STATE LASER SYSTEMS 301

Here q0 has the dimensions of z (length). We will discuss the case when q0 is 2 real. In that case the exponential term in Equation (107) exp[—ikr /2q(z)]is imaginary, i.e., its value is equal 1. This means that the phase changes rapidly with r while the amplitude is constant. This case does not describe the beam behavior. Therefore, the correct case of q0 is to be imaginary and q(z) is a complex number which can be written in the form:

q(z) = z + iz0 (120)

where q0 = iz0 and z0 is constant to be determined. Suppose the initial position at z = 0 therefore

q(z) = q0 + z (121)

q(z) = iz0 and Equation (107) takes the form;

kr2 ψ0 = exp(− ) exp(−ip(z = 0)) (122) 2z0

In Equation (122) the exponential term is real and thus the amplitude drops off quite rapidly with r, being down from its peak value of 1 at r = 0 to 0.368 1 = 2z0 2 and r k . The last above quantity is called the beam waist (the smallest spot size of the laser beam which is at the plane z = 0) and can be calculated from the following Equation:

2z λ z w2 = 0 = 0 0 (123) 0 k nπ

1 As illustrated in Equation (107) we are interested in q . Thus:

1 1 z z 1 λ = = − i 0 = − i 0 (124) + 2 + 2 2 + 2 π w2 q z iz0 z z0 z z0 R(z) n (z) where R(z) and W(z) are characteristic beam parameters to be discussed later. Substituting Equation (124) into Equation (107) gives;

kz r 2 ikzr2 ψ = exp [ − 0 ] exp [ − ] exp(−ipz) (125) 0 2 + 2 2 + 2 2(z z0) 2(z z0) 302 Y. S. NADA et al.

According to equation (125), when we move away from the axis, the phase changes faster and faster and the amplitude becomes insigniﬁcant with large r. The term multiplying r 2 in the ﬁrst exponential factor is a scale length, and we name it the spot size of the beam, which is now a function of z: 2 2z z 2 w2 = 2 + 2 = 0 + (z) (z z0) 1 (126) kz0 k z0

The beam waist varies with z as: λz 2 w2(z) = w2 1 + 0 (127) 0 π w2 n 0

From equation (124) one obtains for the R(z) function 2 1 z 2 πnw2 R(z) = (z2 + z2) = z 1 + 0 = z 1 + 0 (128) 0 λz z z 0

Now we must ﬁnd the function p(z) which is related to the q(z) as follows:

i i p (z) =− =− (129) q(z) z + iz0

dp(z) i =− (130) dz z + iz0

idp(z) i*i =− (131) dz z + iz0

z dz i dp(z) = (132) z + iz0 0

Giving after integration

p(z) = ln(z + iz0) − ln(izà) (133)

Thus

z + iz ip(z) = ln 0 (134) iz0 PULSED SOLID STATE LASER SYSTEMS 303 or

z ip(z) = ln( + 1) (135) iz0 or

z ip(z) = ln(1 − i ) (136) z0

Here we use the fact that; z z 1/2 z 1 − i = 1 + ( 0 )2 exp [ − i tan−1( ) ] (137) z0 z z0

Thus z 1/2 z ip(z) = ln 1 + ( 0 )2 − i tan−1( ) (138) z z0 and z −1/2 z exp(−ip(z)) = 1 + ( 0 )2 exp i tan−1( ) (139) z z0

Substituting Equations (125, 127, 128 & 138) into Equation (102) we get the complete expression for the fundamental or the lowest-order TEMo,o mode:

, , E(x y z) = E0 w 2 2 0 − r − − −1 z × − kr w exp w2 exp j kz tan ( ) exp j (z) (z) z0 2R(z) Amplitude factor Longitudinal phase Radial phase , , E(x y z) = E0 w 2 2 0 − r − − −1 z − kr w exp w 2 exp i kz tan ( ) x exp i (z) (z) z0 2R(z) (140) Amplitude factor Longitudinal phase Radial phase where λ z 2 w2(z) = w2 1 + 0 (141) 0 π w2 n 0 304 Y. S. NADA et al.

FIGURE 23 Gaussian beam parameters. and 2 z 2 πnw2 R(z) = z 1 + 0 = z 1 + 0 (142) z λ0z and

π w2 n 0 z0 = (143) λ0

3.4. ABCD matrix law Here, the transformation of the laser beam via thin lens will be discussed to have the generalization of this case which is the ABCD matrix law. As illustrated in Figure 24, an incoming spherical wave with a radius R1 exactly to the left side of thin lens with focal length f , is transformed into a spherical wave with the radius R2 exactly to the right side of the lens, where

1 1 1 = − (144) R2 R1 f

Phase fronts of laser beams will be transformed in the same way as those of spherical waves [65, 66]. In the case of laser beam diameter measured exactly at the surfaces of the thin lens, where they are the same, Equation (144) will be rewritten in the following form

1 1 1 = − (145) q2 q1 f PULSED SOLID STATE LASER SYSTEMS 305

FIGURE 24 Transformation of wave fronts via a thin lens. where q1 and q2 are the q parameters of the incoming and out coming laser beam, respectively, and

1 1 λ = − i (146) 2 q Rr πw where R is the wave front radius and w is the beam diameter. On the other hand in the case of measuring q1 and q2 at distances d1 and d2 from the thin lens, respectively, as shown in Figure 25. By using Equation (119) and Equation (146), q2 can be obtained from the relation.

(1 − d1/f )q1 + d1 + d2 − d1d2/f q2 = (147) −q1/f + (1 − d1/f )

In the case of more complicated laser cavities with different optical elements, it can be expressed as a sequence of multi-thin lenses [66]. Thus, with the elements of ABCD matrix of this sequence, q parameter at the ﬁnal plane

FIGURE 25 Distances and parameters of transformed beam. 306 Y. S. NADA et al. can be calculated via Equation

Aq1 + B q2 = (148) Cq1 + D

3.5. Temperature distribution There are sources of heat associated with the optical pumping process in solid state lasers such as quantum defect heating which is the difference in energy between the pump and laser photons, concentration quenching and nonradiative relaxation from the pump band to the ground state [23]. The time dependent temperature proﬁle of pulsed end-pumped lasers has been discussed in ref. [67-69]. It is determined by using heat Equation [67];

∂T C = A(r, t) + K ∇2T (149) ∂t where C is the heat capacity per unit volume, A accounts for heat deposition(source term), and K is the thermal conductivity. The heat source term in Equation (3.56) can be neglected as the heat distribution time is longer than that of thermal energy deposition via pump pulse. In this case Equation (3.56) can be written as; ∂T = K ∇2T (150) ∂t where κ = K/C is the thermal diffusivity. In the cylindrical coordinate, Equation (150) will be in the following form

∂T (r, z, t) ∂2 1 ∂ ∂2 = κ + + T (r, z, t) (151) ∂t ∂r 2 r ∂r ∂z2

The solution of Equation (151) could be obtained by separation of variables which can be accomplished by substituting;

T (r, z, t) = φ(r)ψ(z) f (t) (152)

The following ordinary differential equations will be obtained.

∂2ψ(z) + k2ψ(z) = 0 (153) ∂z2 ∂2φ(r) 1 ∂φ(r) + + ν2φ(r) = 0 (154) ∂r 2 r ∂r ∂(t) + κ(k2 + ν2) f (t) = 0 (155) ∂t PULSED SOLID STATE LASER SYSTEMS 307

The solutions of both Equation (153) and Equation (155) are, respectively, linear combination of sine and cosine

ψ(z) = A cos(kz) + B sin(kz) (156) and simple exponentials of the form f (t) = C exp −κ(k2 + ν2)t (157)

The solutions of Equation (154), which is the simplest Bessel Equation, are zero-order Bessel and Neumann functions. As the Neumann function diverges at r = 0, only the Bessel function contributes. With an initial temperature distribution, T1(r, z)att = 0 and the boundary condition T (r = a, z) = 0, corresponding to a ﬁxed surface temperature, which can be taken as zero since the heat transport Equation is linear, the general solution of Equation (157) can be obtained for a cylindrical rod of radius a and length l,

∞ ∞ α r mπz T (r, z, t) = A J n cos exp(−κλ t) (158) nm 0 a l nm n=1 m=0 where αn are the roots of the Bessel function J0 (a), and δnm is the mode constant, deﬁned as α 2 m 2 λ = n + π 2 (159) nm a l

The expansion coefﬁcients are

a l αnr dr dzT1(r, z)rJ0 a mπz A = 0 0 cos( ) (160) nm a2l 2 α l 4 J1 ( n)

Here the initial condition can be decomposed to a product of (r) and (z).

T1(r, z) = T0 exp(−γ z) for r ≤ r p (161)

T1(r, z) = T0 exp(−γ z) for r > r p (162) where γ is the absorption coefﬁcient of the gain medium at the pumping wavelength. The solution could be obtained as a product of radial and longitudinal parts, and can be written as the product (r, t)(z, t), where (r, t)isa 308 Y. S. NADA et al. solution to the problem of thermal conduction in an inﬁnitely long cylindrical rod of radius a and (z, t) describes thermal conduction in a region 0 ≤ z ≤ l, with l being the length of the crystal. The radial component of the solution is thus given by the following expansion

∞ α r φ(r, t) = B J n exp(−κα2/a2) (163) n 0 a n n=1 where the expansion coefﬁcients Bn are given by

r p αnr drT0rJ0 a B = 0 (164) n a2 2 α 2 J1 ( n)

The time-dependent radial temperature proﬁle thus becomes;

∞ α nr p α r p J1 a 1 nr p 2 2 φ(r, t) = 2T0 J0 exp(−κα t/a ) (165) a J 2 (α ) α a n n=1 1 n n

The longitudinal solution is a simple Fourier expansion of the initial temperature distribution and is written as

∞ 1 mπz κm2π 2t ψ(z, t) = C + C cos exp(− ) (166) 2 0 m l l2 m=1 where

l 2 mπz C = exp(−γ z) cos dz (167) m l l 0

With the initial temperature distribution given in Eqs. (161-162), this results in the longitudinal temperature evolution

ψ , = 1 − −γ (z t) γ l [1 exp( l) ∞ − π −γ π + 1 cos(m ) exp( l) m z −κ 2π 2 / 2 (168) 2 2 cos exp( m t l ) + mπ l m=1 1 lγ

Now the complete solution of the problem of heat conduction in the ﬁnite rod is given as the product of φ and . The starting temperature T0 is the PULSED SOLID STATE LASER SYSTEMS 309 temperature provided by the pump laser in an inﬁnitesimal slab at z=0.We now allow the solution

T (r, z, t) = φ(r, t)ψ(z, t) (169) to evolve until t = t, when a new laser shot deposits more energy. With the same pump-laser proﬁle used every time, the temperature proﬁle after the second pulse becomes

T2(r, z, t = 0) = φ(r,t)ψ(z,t)) + φ1(r)ψ1(z) (170) where, t is the time since the last laser shot. Because the Equation of heat conduction is linear, the propagation in time can be introduced for each term separately. If we introduce Xn(r) and m (z) for the time-independent functions of the expansions (165) and (169), αnr p 2r J1 1 α r X (r) = p a J n (171) n 2 α α 0 a J1 ( n) n a and 1 − cos(mπ) exp(−γ l) mπz = m (z) 2 cos (172) + mπ l 1 lγ

Thus, the temperature could be written as ∞ , , = T0 −κα2 + / 2 T2(r z t ) γ l exp n(t t) a Xn(r) n=1 ∞ 2 2 2 x 1 − exp(−γ l) + 2 exp −κm π (t + t)/l m (r) m=1 ∞ (173) + T0 { −κα2 / 2 γ l exp nt a Xn(r) m=1 ∞ 2 2 2 x 1 − exp(−γ l) + 2 exp −κm π t /l m (r) m=1

Through a rearrangement one gets: ∞ ! , , = T0 − −γ −κα2 / 2 { + −κ 2 / 2 T2(r z t ) γ l 1 exp( l) exp n t a Xn(r) 1 exp an t a n=1 ∞ ∞ + T0 −κ 2π 2 / 2 −κα2 / 2 2 γ l exp m t l m (r)exp n t a Xn(r) m=1 n=1 ! + −κ 2π 2 / 2 −κα2 / 2 x 1 exp m t) l exp n t a (174) 310 Y. S. NADA et al.

At the steady state, the temperature proﬁle at ﬁxed repetition rate can be expressed as [67]: " ∞ T T (r, z, t ) = 0 (1 − exp(−γ l) exp −κα2t /a2 X (r) } 2 γ l n n n=1 ∞ ∞ 1 T + 2 0 exp −κm2π 2t )/l2 1 − exp(−κα2t/a2 γ l n m= n=1 " # 1 (z)exp −κα2t /a2 X (z)x m n n − −κ 2π 2 / 2 −κα2 / 2 1 exp m t) l exp n t a (175)

3.6. Thermal focal length Due to the non-uniform temperature proﬁle within the crystal, the thermal induced optical path difference includes the change of refractive index with temperature, the end faces bulging out, and the stress induced birefringence. For YAG, the last two factors might be neglected as they are much smaller than the former part [70,71]. The refractive index near the center of the crystal (r = 0) is well described by a ﬁrst-order expansion around the temperature of the center, Tc [2]

∂n n(T ) =∼ n(T ) + | (T − T ) (176) c ∂T Tc c

Thus, the difference in the optical path length is given by the following expression [67] $ $ l ∂ $ n $ l(r) = $T T∞(r, z, t = 0) − T∞(r = 0, z, t = 0) dz (177) ∂T $ c 0

This shows that the thermal lensing is essentially determined by the integration of the temperature proﬁle along z. In Equation (175) all the z-dependent terms contain a factor cos (mμz/l) will vanish after integration. Only the ﬁrst term (constant in z) contributes, providing

$ ∞ ∂n $ T0 1 l(r) = $T 1 − exp(−γ l) x (Xn(r) − Xn(0)) ∂ c γ κα2t T = − − n n 1 1 exp a2 (178) PULSED SOLID STATE LASER SYSTEMS 311

By insertion the radial function Xn(r), the optical path difference can be written as: $ ∂ $ = n $ 2E0 − γ l(r) $Tc 1 exp( l) ∂T πr paC(Tc)

α r (179) n p ∞ J1 1 α r 1 a n − x 2 J0 1 2 = J (α ) αn a κα n 1 1 n 1 − exp − n να2 where C(Tc) is the heat capacity per unit volume of the laser crystal at the center temperature and the laser repetition rate ν = 1/t has been introduced. E0[1 − exp(2γ l)] is the total energy absorbed inside the crystal. Since the optical path length for rays through the center of the lens is longer than that for those going through the edge (marginal rays), the crystal acts like a focus- ing lens. It is tempting to expand the zero-order Bessel function in Equation (179) 2 around r = 0, J0(rαn/a) ≈ 1 − [rαn/(2a)] , in order to extract the spherical part of the optical path length difference.

l r ≈ ( ) κα2 α r n ∞ J n p exp − ∂n $ −2E r 2 1 α α να2 $ 0 1 − exp(−γ l) x n } Tc 3 2 2 ∂T πr pa C(Tc) J (α ) 4 κα n=1 1 n 1 − exp − n να2 (180)

As this optical path length difference is quadratic in r, it can be presented as spherical lens with

r 2 l(r) ≈− (181) 2R where R is the curvature of the lens. The focal length of the thermal lens can be expressed as given in [50,

R f = (182) n − 1 where n is the medium index of refraction. 312 Y. S. NADA et al.

Consequently the thermal focal length can be written as [67]:

α3πr C(T ) f = p c / n − 1 ⎧ ⎫ κα2 ⎪ α r n ⎪ ⎪ ∞ n p − ⎪ ⎨ $ J1 exp ⎬ ∂n $ α να2 T E0 1 − exp(γ l) x αn ⎪∂T c J 2 (α ) κα2 ⎪ ⎩ n=1 1 n 1 − exp − n ⎭ να2 (183)

4. CONCLUSIONS

In this review paper we give the present state of art of research on pulsed solid state lasers, which find increasing interest and practical applications. In first Chapter (Introduction) we describe different solid state lasers, lasing atoms and hosting them materials. A particular attention is devoted to ceramic and microchip lasers and their pumping as well as to passively Q-switched lasers. Second Chapter describes the basic atomic processes behind the population inversion and stimulated emission quantum processes. Population rate equations for two –, three – and – four level lasers are given with their solutions. The steady-state rate equations model as well as the laser system configuration are also presented and discussed. In the last case using combination of Cr4+: YAG and a doped, such as Yb:YAG as gain medium is presented and discussed. The last Chapter (3) addresses the problem of light propagation in laser cavity using the ABCD matrix approach. In particular the cavity stability, medium heating and thermal focal lens formation are addressed and discussed.

REFERENCES

[1] C. Webb and J. Jones. (2004). Handbook of laser technology and applications; Institute of Physics Bristol. [2] J. M. El-Azab, Y. S. Nada, and S. M. Maize. (2017). Thermal Focal Length Control in Passively Q-switched End Pulsed-Pumped Ceramics and Single Crystal Yb3+:YAG/Cr4+:YAG Microchip Laser, Nonlinear Optics Quantum Optics, Concepts in Modern Optics, 48(2):113–122. [3] M. Darwiesh, A.F. El-Sherif, H.S. Ayoub, Y.H. El-sharkawy, M.F. Hassan, Y.H. Elbashar. (2019). “Design and implementation of underwater laser imaging test aquarium”, Journal of Optics, 48(1):145–153, DOI 10.1007/s12596-018-0506-6(2019). PULSED SOLID STATE LASER SYSTEMS 313

[4] H.S. Ayoub, M. Nazeeh, W.M. Hussein, A. El-Sherif, Y.H. Elbashar, “A brief study of equiradial automotive visual vehicular using laser shadowgraphic system", Nonlinear Optics Quantum Optics, in print. [5] Y.H. Elbashar, R.A. Ibrahem, J.A. Khaliel, D.I. Moubarak, A.S. Abdel-Rahaman, H. H. Hassan, “Optical spectroscopic analysis of bandpass absorption glass filter based on vana- dium copper oxide for laser safety applications", Nonlinear Optics Quantum Optics,in print. [6] Amr Attia, H.S. Ayoub, Bassam Ghazolin, A.F. El-Sherief, M. El-Gohary, Ahmed Emara, H. Moneib, Y.H. Elbashar. (2018). “In-Situ Soot visualization using Low power 405nm laser shadowgraphy for premixed and non-premixed flames”, Annals of the University of Craiova, Physics, Physics AUC, 28, 12–16. [7] H. S. Ayoub, A.F. El-sherif, A. M. Mokhtar, H. H. Hassan, S. A. Khairy, Y.H. Elbashar. (2018). “Study of optical synchronous laser shadow vibrometry for measuring Young’s modulus of tungsten at elevated temperatures", Lasers in Engineering, X15.Ayoh-JL. [8] Y.H. Elbashar, S.M.Hussien, J.A. Khaliel, D.I. Moubarak, A.S. Abdel-Rahaman, H. H. Hassan. (2018). “Optical spectroscopic analysis of sodium zinc phosphate glass doped cadmium oxide used for laser window protection", Annals of the University of Craiova, Physics, Physics AUC, 28, 57–72. [9] D.I. Moubarak, H. H. Hassan, T.Y. El-Rasasi, H. S. Ayoub, A.S. Abdel-Rahaman, S. A. Khairy, Y. Elbashar. (2018). “Measuring the internal friction for some rubbers doped nano carbon by laser Shadowgraphic technique”, Nonlinear Optics Quantum optics, 49(3- 4):295–310. [10] H. S. Ayoub, A.F. El-Sherif, H. H. Hassan, S. A. Khairy, Y.H. Elbashar. (2018). “A laser shadowgraphy method for studying the vibrations of incandescent solids", Lasers in Engi- neering - VIIII16.Ayos-JL. [11] M. Darwiesh, A.F. El-Sherif, H.S. Ayoub, Y.H. El-sharkawy, M.F. Hassan, Y.H. Elbashar. (2018). “Hyper-spectral laser imaging of underwater targets”, Journal of Optics, 47(4):553–560; doi.org/10.1007/s12596-018-0493-7 [12] D. I. Moubarak, H. H. Hassan, H. S. Ayoub, T. Y. El-Rasasi, Sh. A. Khairy, Y.H. Elbashar, A. S. Abdel-Rahaman. (2018). “A Laser Shadowgraphy for Measuring the Internal Fric- tion of Butadiene Acrylonitrile Rubber doped nano carbon by Pulse Excitation Tech- nique", Lasers in Engineering, VII13.Moudi-JL [13] H. S. Ayoub, A.F. El-Sherif, H. H. Hassan, S. A. Khairy, Y.H. Elbashar. (2018). “Dilatom- etry of Refractory Metals and Alloys Using Multi- Wavelength Laser Shadowgraphy of Filament Samples", Nonlinear Optics Quantum Optics, 49(1-2):51–61. [14] H.S. Ayoub, A. Attia, A. Emara, A. El-Sherif, Y.H. Elbashar. (2018). “Ultraviolet laser shadowgraphic system of measuring ev burner flame", Nonlinear Optics Quantum Optics, 49(1-2):79–90. [15] Y.H. Elbashar. (2017). “Surface Treatment of Aluminum 6061-T6 by using multi-shot Laser Plasma Shock Processing (LPSP) without Confinement", Lasers in Engineering, XI30.Elbay-JL. [16] D.I. Moubarak, J. A. Khaliel, T.Y. El-Rasasi, H. S. Ayoub, A.S. Abdel-Rahaman, S. A. Khairy, H. H. Hassan, Y.H. Elbashar. (2018). “Mechanical and laser shadowgraphy studies of nanosized carbon black loaded natural rubber”, Lasers in Engineering, III08.Modi-JL. [17] Hussam Hassan, Deia Moubarak, Jamal Khaliel, Hany Ayoub, Ahmed Abdel-Rahaman, Sherif Khairy, Tarek El-Rasasi, Yahia Elbashar. (2018). “Design and construction of optical laser shadowgraphy system for measuring the internal friction of high elasticity solids”, Nonlinear Optics Quantum Optics, 48(4):313–320. 314 Y. S. NADA et al.

[18] Y.H. Elbashar, A.E. Omran, J. A. Khaliel, A.S. Abdel-Rahaman, H.H. Hassan. (2018). “Ultraviolet transmitting glass matrix for low power laser lens", Nonlinear Optics Quan- tum Optics, NLOQO, 49(3-4):247–265. [19] A. Attia, H.S Ayoub, A. Emara, A. El-Sherif, Y.H. Elbashar. (2018). “Design and implementation of laser ignition system for two slot and four slot ev burners", Nonlinear Optics Quantum Optics, 49(1-2):127–142. [20] Y.H. Elbashar, D.A. Rayan, M.M. Rashad. (2017). “Protection Glass Eyewear Against a YAG Laser Based on a Bandpass Absorption Filter ", Journal of Silicon, 9(1):111–116; DOI 10.1007/s12633-015-9389-1. [21] Y.H. Elbashar, “Design and fabrication of protection window against YAG laser". (2017). Journal of Silicon, DOI 10.1007/s12633-016-9429-5, 9(5):711–715. [22] H. Hassan, Y. Ismail, A.M. Nassar, H. Abd Elhamid, H. S. Ayoub, Y.H. Elbashar, “Charge Independent DC Model for Floating Gate MOSFET used for ﬂash memory and electro- optic switching applications”, Nonlinear Optics, Quantum Optics, 50(4):263–278. [23] W. Koechner. (1976). Solid state laser engineering. New York: Springer-Verlag. [24] R. Powell. (1998). Physics of solid-state laser materials. New York: AIP Press/Springer. [25] B. Denker and E. Shklovsky. (2013). Handbook of solid-state lasers. Cambridge, UK: Woodhead Publishing. [26] Ikesue and Y. Aung. (2008). “Ceramic laser materials", Nature Photonics, 2(12):721–727. [27] Ikesue, T. Kinoshita, K. Kamata and K. Yoshida. (1995). “Fabrication and Optical Prop- erties of High-Performance Polycrystalline Nd:YAG Ceramics for Solid-State Lasers", Journal of the American Ceramic Society, 78(4):1033-1040. [28] J. Lu, M. Prabhu, K. Ueda, H. Yagi, T. Yanagitani, A. Kudryashov, and A. A. Kaminskii. (2001). “Potential of ceramic YAG lasers." Laser Physics-Lawrence, 11(10):1053–1057. [29] J. Lu, H. Yagi, K. Takaichi, T. Uematsu, J. Bisson, Y. Feng, A. Shirakawa, K. Ueda, T. 3+ Yanagitani and A. Kaminskii. (2004). “110 W ceramic Nd :Y3 Al5O12 laser", Applied Physics B: Lasers and Optics, 79(1):25–28. [30] Ikesue, Y. Aung, T. Yoda, S. Nakayama and T. Kamimura. (2007). “Fabrication and laser performance of polycrystal and single crystal Nd:YAG by advanced ceramic processing", Optical Materials, 29(10):1289–1294. [31] S. Lee, E. Kupp, A. Stevenson, J. Anderson, G. Messing, X. Li, E. Dickey, J. Dumm, V. Simonaitis-Castillo and G. Quarles. (2009). “Hot Isostatic Pressing of Transparent Nd:YAG Ceramics", Journal of the American Ceramic Society, 92(7):1456–1463. [32] M. Suárez, A. Fernández, J. Menéndez, M. Nygren, R. Torrecillas and Z. Zhao. (2010). “Hot isostatic pressing of optically active Nd:YAG powders doped by a colloidal processing route", Journal of the European Ceramic Society, 30(6):1489–1494. [33] D. Kracht, M. Frede, R. Wilhelm and C. Fallnich. (2005). “Comparison of crystalline and ceramic composite Nd:YAG for high power diode end-pumping", Opt. Express, 13(16):6212. [34] M. Pokhrel, N. Ray, G. Kumar and D. Sardar. (2012). “Comparative studies of the spectroscopic properties of Nd3+: YAG nanocrystals, transparent ceramic and single crystal", Optical Materials Express, 2(3):235. [35] Pirri, D. Alderighi, G. Toci and M. Vannini. (2009). “High-efﬁciency, high-power and low threshold Yb3+:YAG ceramic laser", Opt. Express, 17(25):23344. [36] K. Takaichi, H. Yagi, J. Lu, A. Shirakawa, K. Ueda, T. Yanagitani and A. Kaminskii. 3+ (2003). “Yb -dopedY3Al5O12 ceramics– A new solid-state laser material", Phys. Stat. Sol., (a), 200(1):R5-R7. PULSED SOLID STATE LASER SYSTEMS 315

[37] Agnesi, L. Carrà, F. Pirzio, G. Reali, S. Veronesi, J. Tharayil Thomas, M. Tonelli, J. Li, Y. Pan and J. Guo. (2014). “Ceramic Yb:YAG for multiwatt compact passively Q-switched lasers", Optics Communications, 315, 208–212. [38] S. Nakamura, Y. Hikita, H. Sone, T. Ogawa and S. Wada. (2014). “High-power diode- pumped mode-locked Yb:YAG ceramic laser", Opt. Rev., 21(3):401–403. [39] W. Jiang, Y. Liu, W. Chen, S. Zhu, Z. Chen, G. Zhang, Y. Chen and Z. Chen. (2015). “Composite Yb:YAG/Cr 4+:YAG/YAG crystal passively Q-switched lasers at 1030 nm", Appl. Opt., 54(7):1834. [40] J. Zayhowski and A. Mooradian. (1989). “Single-frequency microchip Nd lasers", Optics Letters, 14(1):24. [41] J. Zayhowski. (1999). “Microchip lasers", Optical Materials, 11(2-3):255–267. [42] T. Fan. (1994). “Aperture guiding in quasi-three-level lasers", Optics Letters, 19(8):554. [43] J. Zayhowski and C. Dill. (1994). “Diode-pumped passively Q-switched picosecond microchip lasers", Optics Letters, 19(18):1427. [44] J. Zayhowski. (1996). “Ultraviolet generation with passively Q-switched microchip lasers", Optics Letters, 21(8):588. [45] J. Zayhowski. (2000). “Passively Q-switched Nd:YAG microchip lasers and applications", Journal of Alloys and Compounds, 303–304, 393–400. [46] M.A. Jaspan, D. Welford, G. Xiao, and M. Bass. (2000). “Atypical behavior of Cr: YAG passively Q-switched Nd: YVO 4 microlasers at high-pumping rates." In Conference on Lasers and Electro-Optics, Optical Society of America, 39, 454. [47] J. Liu, B. Ozygus, S. Yang, J. Erhard, U. Seelig, A. Ding, H. Weber, X. Meng, L. Zhu, L. Qin, C. Du, X. Xu and Z. Shao. (2003). “Efﬁcient passive Q-switching operation of a diode-pumped Nd:GdVO4 laser with a Cr4+ : YAG saturable absorber", Journal of the Optical Society of America, B, 204(4):652. [48] J. Dong, A. Shirakawa and K. Ueda. (2006). “Sub-nanosecond passively Q-switched Yb:YAG/Cr4+:YAG sandwiched microchip laser", Applied Physics, B, 85(4):513–518. [49] Y. Zhou, Q. Thai, Y. Chen and S. Zhou. (2003) “Monolithic Q-switched Cr,Yb:YAG laser", Optics Communications, 219(1-6):365–367. [50] M. Csele. (2014). Laser modeling. Taylor & Francis Group, LLC, Philadelphia. [51] O. Svelto. (1976). Principles of lasers. New York: Plenum Press. [52] P. Milonni and J. Eberly. (2010). Laser physics. Oxford: Wiley-Blackwell. [53] W. Wagner and B. Lengyel. (1963). “Evolution of the Giant Pulse in a Laser", J. Appl. Phys., 34(7):2040. [54] Szabo and R. Stein. (1965). ‘Theory of Laser Giant Pulsing by a Saturable Absorber’, J. Appl. Phys., 36(5):1562. [55] J. Degnan. (1995). “Optimization of passively Q-switched lasers", IEEE Journal of Quan- tum Electronics, 31(11):1890–1901. [56] X. Zhang, S. Zhao, Q. Wang, Q. Zhang, L. Sun and S. Zhang. (1997). “Optimization of Cr4+-doped saturable-absorber Q-switched lasers", IEEE Journal of Quantum Electron- ics, 33(12):2286–2294. [57] J. Ma, J. Dong, K. Ueda and A. Kaminskii. (2011). ‘Optimization of Yb:YAG/Cr4+:YAG composite ceramics passively Q-switched microchip lasers’, Applied Physics,B, 105(4):749–760. [58] J. Degnan. (1989). “Theory of the optimally coupled Q-switched laser", IEEE Journal of Quantum Electronics, 25(2):214–220. 316 Y. S. NADA et al.

[59] R. Corless, G. Gonnet, D. Hare, D. Jeffrey and D. Knuth. (1996). “On the LambertW function", Advances in Computational Mathematics, 5(1):329–359. [60] J. El-Azab, H. Kandel, M. Khedr. (2014). “Numerical Study of Auto-Control and Stabi- lization of Pulse Repetition Rate of Passively Q-Switched Laser Systems” ACSE Jour- nal, ISSN: Print 1687–4811, Online 1687–482X, CD-ROM 1687–4838, ICGST LLC, Delaware, USA. [61] J. Dong, K. Ueda, A. Shirakawa, H. Yagi, T. Yanagitani and A. Kaminskii. (2007). “Composite Yb:YAG/Cr4+:YAG ceramics picosecond microchip lasers", Opt. Express, 15(22):14516. [62] N. Hodgson and H. Weber. (1997). Optical resonators. London: Springer. [63] J. Verdeyen. (1981). Laser electronics. Englewood Cliffs, N.J.: Prentice-Hall. [64] H. Kogelnik and T. Li. (1966). “Laser Beams and Resonators", Appl. Opt., 5(10):1550. [65] H. Kogelnik. (1965). “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including those with a Loss or Gain Variation", Appl. Opt., 4(12):1562. [66] R. Lausten and P. Balling. (2003). ‘Thermal lensing in pulsed laser ampliﬁers: an analytical model’, Journal of the Optical Society of America, B, 20(7):1479. [67] S. Wang, X. Wang, T. Riesbeck and H. Eichler. (2009). “Thermal lensing effects in pulsed end pumped Nd lasers at 940 nm", Laser Resonators and Beam Control XI, 7194:71940J– 1. [68] X. Chen, J. Wu, C. Wu, H. Sun, Y. Yu and G. Jin. (2015). “Analysis of thermal effects in a pulsed laser diode end pumped single-ended composite Tm:YAG laser", Laser Physics, 25(4):045003. [69] Pﬁstner, R. Weber, H. Weber, S. Merazzi and R. Gruber. (1994). “Thermal beam distor- tions in end-pumped Nd:YAG, Nd:GSGG, and Nd:YLF rods", IEEE Journal of Quantum Electronics, 30(7):1605–1615. [70] M. Born and E. Wolf. (1999). Principles of optics. Cambridge: Cambridge University Press. [71] Haichao Yu, Hanyang Li. (2018). “Brief review on pulse laser propulsion", Optics & Laser Technology, 100(1):57–74. Copyright of Nonlinear Optics, Quantum Optics: Concepts in Modern Optics is the property of Old City Publishing, Inc. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.