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Abstract—Angularly selective mirrors (ASMs) are proposed as a means to expand the mode area and modal discrimination of microchip lasers. ASMs used as ...
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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 44, NO. 7, JULY 2008

Enhancement of the Mode Area and Modal Discrimination of Microchip Lasers Using Angularly Selective Mirrors Jean-Francois Bisson, Nikolay Lyndin, Ken-ichi Ueda, and Olivier Parriaux

Abstract—Angularly selective mirrors (ASMs) are proposed as a means to expand the mode area and modal discrimination of microchip lasers. ASMs used as output couplers selectively vectors over a narrow angular range, while reflect incoming they transmit more inclined components. The eigenvalue problem of a microchip resonator equipped with a Gaussian ASM is solved analytically in the paraxial optics approximation using the ABCD matrix formalism. The narrow angular distribution of the reflected beam produces, through the laws of diffraction, a significant increase of the mode size and improved transverse mode discrimination, at the expense of higher oscillation threshold due to larger output coupling losses. Simulations performed using the parameters of Yb3+ -doped YAG material show that one order of magnitude increase of the mode area can reasonably be achieved without causing overheating and thermal fracture. ASMs can be directly deposited on the active material in the form of a resonant grating mirror. This technology involves only planar batch processes that retain the mass production advantage of microchip lasers. The significant increase of brightness of microchips expected from this innovation will give rise to more effective and more compact devices and new applications. Index Terms—Diffraction gratings, microchip lasers, power scaling, solid-state lasers.

I. INTRODUCTION

M

ICROCHIP lasers are laser-diode-pumped platelets of active material dielectrically coated on both faces to form a monolithic resonator. Their short length makes it possible to operate in single-longitudinal mode, deliver the shortest available self- -switched pulses, the highest repetition and modulation rates among solid-state lasers, and they are also mass producible since thousands of identical microchip lasers can be cut from a single wafer [1]. Because of their short length, microchips have a small mode waist, generally in the order of 50 m, that entails a small output power per transverse mode and makes them prone to multimode operation

Manuscript received August 22, 2007; revised January 24, 2008. This work was supported by the 21st Century COE program of the Ministry of Education, Science and Culture of Japan. J. F. Bisson and K. Ueda are with the Institute for Laser Science, University of Electro-Communications, Tokyo 182-8585, Japan (e-mail: bisson@ils. uec.ac.jp; [email protected] ). N. Lyndin is with the Institute of General Physics of the Russian Academy of Science, Moscow 119991, Russia (e-mail: [email protected]). O. Parriaux is with the Hubert Curien Laboratory, St-Etienne 42000, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JQE.2008.921381

at high power [2]. Moreover, compact microchip -switched lasers delivering several millijoules (mJ) pulses in a single transverse mode are desired for many applications where high pulse energy and high brightness are needed from a compact geometry. Examples include laser spark plug for automotive industry [3], laser range finding [4], and on-site laser-induced breakdown spectroscopy [5]. However, the pulse fluence is limited to a fraction of the saturation parameter of the active material, typically in the order of a few joules per centimeter square (J/cm ), which limits the pulse energy in one mode to less than the millijoule level [6], [7]. One key to solving this issue is to increase the mode area and modal discrimination. Large transverse mode delivery from short resonators has been actively researched (see [8], [9], and the references therein), because it allows the extraction of a larger amount of power per mode from a compact geometry. However, this has been very difficult to achieve with stable resonator. A short and wide mode can more easily be achieved by using unstable resonators [10]. While unstable resonators are widely used in applications involving high gain materials, such as CO or copper vapor, recently unstable designs have also been shown to work well with rare-earth-doped materials [11], [12]. The most successful design of unstable resonator is probably the combination of a geometrically unstable resonator with a graded reflectivity mirror (GRM) with a Gaussian [13] or super-Gaussian [14], [15] reflectivity distribution. The unstable resonator geometry is used to increase the mode volume and the transverse mode discrimination, while the GRM provides smooth, Gaussian-like mode profile. The fabrication of a GRM involves the use of tapered multilayers [16] or complex, bulky elements [17], [18]. GRMs are difficult and relatively expensive to fabricate and, in several aspects, are incompatible with microchip configurations. Because of their inherent center of symmetry and need for alignment, they disqualify the microchip as a mass-producible, low-cost device if thousands of microchips are to be cut from a single wafer. Next, since most microchips are generally geometrically stable, the stability being ensured either by thermal lensing [19], gain guiding [20], or loss aperture [21], adding a GRM to a microchip would merely contribute to further reduction of the waist size, which contradicts the purpose of a GRM. In this paper, we propose the angularly selective mirror (ASM) to achieve large-mode-area microchip lasers. ASMs reflect a narrow range of wavevectors near the optical axis, while more inclined vectors are almost totally transmitted. Since the angular distribution and the transverse extent of the mode are connected through a Fourier transform, a narrower

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BISSON et al.: ENHANCEMENT OF THE MODE AREA AND MODAL DISCRIMINATION

distribution of perpendicular components entails a larger mode size. A mode size of about 200 m is predicted for a 0.1 angular spread, which is significantly larger than the typical mode size of microchips. ASM and GRM can be viewed as a Fourier pair. Indeed, the former can in principle be obtained by placing a GRM at the focal plane of a lens, but this is not an adequate solution for microchip lasers. Achieving a narrow angular reflectivity from a planar geometry is a technological challenge, but it can be realized by using abnormal reflection from resonant grating mirrors [2], [22], [23]. These are all-dielectric structures composed of two submirrors: a standard dielectric multilayer reflector and a resonant grating submirror composed of a high-index layer or a set of layers, which act as a slab waveguide and in which a corrugation grating is fabricated by using photolithographic and dry etching techniques. Unlike tapered reflectivity mirrors, ASMs are fully compatible with thin, flat-flat resonator structures and preserve the translation invariance of the microchip wafer, which in turn ensures that the microchip lasers remain mass-producible. This paper aims at evaluating the potential of associating ASMs with microchip lasers to increase both the mode area and modal discrimination. In Section II, we provide an analytical description of the eigenmode of a microchip laser equipped with a Gaussian ASM output coupler. In Section III, we show simulation results using the parameters of Yb -doped Y Al O garnet sample. We show that an ASM-microchip can produce a significant increase of the mode area and an efficient suppression of higher order transverse modes. In Section IV, we discuss the thermal limits of ASM-equipped solid-state microchip lasers. We show that a microchip with mode area enlarged by one order of magnitude can realistically operate without suffering excessive heating or thermal fracture. In Section V, we discuss one attractive practical realization of ASMs using abnormal reflection from resonant grating mirrors. Finally, we summarize our results in Section VI. II. ANALYTICAL DESCRIPTION OF A MICROCHIP LASER EQUIPPED WITH A GAUSSIAN ANGULAR SELECTIVE MIRROR In this section, we first give a short overview of the ABCD matrix formalism. Then, we derive the ABCD matrix of a Gaussian ASM mirror. We then find the value of the complex radius , that makes the equation self-consistent in a round-trip inside the resonator; this gives rise to a quadratic equation with two solutions. We identify the physically realistic confined solution. Then, the stability of the confined solution is determined by the eigenvalues of the round-trip ABCD matrix. Then, we derive the formulas for the effective reflectivity, the modal discrimination, i.e., round-trip loss ratio of the higher mode to the fundamental mode, and the far-field output beam pattern distribution. A. ABCD Matrix Formalism Resonators which support Gaussian beams as their eigenfunctions can conveniently be described in the paraxial approximation using the ABCD matrix formalism [24]. We will show that this is the case for a standard microchip equipped with an ASM with a Gaussian angular reflectivity distribution with zero

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pedestal reflectivity. Depending on the technology used to realize the ASM, the actual angular response of the ASM may slightly differ from the Gaussian distribution, cf. Section V, but we expect the Gaussian model to capture the main features of the ASM-equipped microchip. The calculation of the mode structure at some plane is done by tracing the evolution of the complex parameter, defined as (1) where is the radius of curvature of the wave front. A di. is the verging wave is represented by a positive value of waist size and is the wavelength inside the medium is the wavelength in vacuum. We of refractive index and use for convenience the reduced parameter . The complex amplitude of a Gaussian beam is, in one dimension (2) The transformation of the reduced parameter by an optical element is obtained in the paraxial approximation by using the Huygens integral [25]

(3) are elements of the complex ABCD ray where , , and matrix of the optical element. Note that the element is defined by the fact that the ABCD matrix is unitary. Inserting (2) into (3) and integrating, we obtain [25, Sec. 20.2] (4) where the new parameter

is given by (5)

which gives the transformation law of a Gaussian beam by an optical element defined by the two-by-two matrix . B. ABCD Matrix of the ASM We suppose that the output coupler is a mirror, of which the angular complex reflection distribution is (6) The mirror has an on axis amplitude reflectivity of and reflects -vectors well only within an incident angle in the order of .1 We consider an incoming axi-symmetrical Gaussian 1The equations derived are also valid for complex values of  as long as the real part of  is positive. The imaginary part corresponds to a lensing effect, which is often encountered when using abnormal reflection from resonant grating mirrors, see Section V.

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beam, with reduced parameter, incident on the ASM. The corresponding amplitude distribution is given by

(7) where is the radial coordinate. We then take the Fourier trans, form of (8) Fig. 1. Scheme of the microchip laser with ASM resonant grating mirror. Thermal lensing is modeled by a thin lens of focal length f . The equivalent periodic structure corresponds to an unfolded resonator twice as thick, with ASMs placed on both sides and with a thin lens placed in the center of the f = . resonator with effective f

After reflection by the ASM, the angular spectrum becomes

=

(9)

where

2

is the translation matrix through the distance

We identify

(13) (10)

where and are the complex parameters of the beam before and after incidence on the ASM, respectively. Equation (10) can also be expressed in terms of ABCD matrix by comparison with (5). The matrix of an ASM, , is

is the refraction matrix due to the thermal lens with effective focal length (14) Performing the matrix product, one finds

(11) (15a) (15b)

C. Calculation of the Parameter of the Eigenmodes of a ASM-Equipped Microchip and Stability Analysis We now consider the whole microchip composed of a flat-flat active medium of thickness , index . The rear face is assumed to be HR-coated, while the front face is coated with a Gaussian ASM. Thermal lensing is assumed to take place as a result of heat load and lateral heat flow. We assume axial symmetry, i.e., we neglect any astigmatism that may take place due to optical or thermo-mechanical anisotropy. The value of the focal distance, associated with the thermal lens is in general much longer, e.g., more than several centimeters, than the resonator thickness, e.g., millimeter size. Hence, we can reasonably approximate the distributed thermal lensing by a thin lens placed at the center of the microchip. The round-trip propagation is equivalent to a propagation over thickness 2 , while the effective focal length in the unfolded resonator, , is reduced by half, Fig. 1. We neglect aperture loss, which frequently takes place in quasi-four level active materials. The latter could be easily taken into account, but these complications are unnecessary for the understanding of the basic characteristics of ASM. We also neglect the effect of the gain distribution on the mode structure. We consider the round-trip in the resonator starting from the incidence on the ASM. The round-trip matrix is written as (12)

(15c) (15d) For an eigenmode of the resonator, the parameter must reproduce itself after a round-trip inside the resonator (16) The two solutions of this equation,

, are [25]

(17) Only the solution with a real positive waist parameter, cf. (1), is retained. In all the cases we simulated, satisfied this condition. The physically realistic solution must also be perturbation stable, i.e., a small deviation from the solution , , must decay over a round-trip inside the resonator. A small deviation becomes, after round-trip [25], [26] (18)

BISSON et al.: ENHANCEMENT OF THE MODE AREA AND MODAL DISCRIMINATION

where (19) are the eigenvalues of the matrix and is the half trace of .2 Hence, the stability with respect to a parameter requires the modulus small perturbation of the of the perturbation eigenvalue, to be smaller than 1. In all the simulated cases, the self-consistent solution with positive real waist, , was also found to be perturbation stable, i.e., . Hence, the solution was found to meet the three criteria for the physically meaningful Gaussian eigenmode of the resonator: 1) self-consistency after a round-trip; 2) real positive waist; 3) stability of the solution against perturbations. D. Effective Reflectivity of the Output Coupler and Modal Discrimination The effective reflectivity of the ASM for the fundamental , is obtained by calculating the ratio of integrated mode, power spectrum in the Fourier space after and before incidence on the ASM

(20)

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distribution of a higher order Hermite–Gauss modes of indices and , , into the Huygens integral (3) and then perform the integration, we find that the effective reflectivity and power loss per round-trip are [25, Sec. 20.5]

(24) For example, if the on axis reflectivity, , is unity and the effective reflectivity of the ASM for the fundamental mode is 20%, then the reflectivity of the next Hermite–Gauss mode is only 4%. The gain required to achieve lasing becomes significantly higher for the higher order modes, so that effective suppression of higher order modes is expected. This result contrasts with standard stable resonators, for which are unity and modal discrimination in the latter eigenvalues is considerably weaker. In general, the output coupling losses provided by the ASM, of Hermite–Gauss mode of order , normalized to that of the fundamental mode, , is given by (25)

E. Output Pattern of the Far-Field Emission We assume that the ASM has no scattering or absorption loss; the total transmission factor is thus given by where is given by (20) (26) The output beam profile in vacuum is

(21) (27) where . The integration range was extended to the whole real axis to get analytical estimates. The effective reflectivity of the ASM is also equal to the change of modulus square of the wave in a round-trip inside the cold resonator because the ASM is the only lossy element involved in the modeling. From (4), the amplitude of the wave in a roundtrip changes by a factor: where the square exponent comes from the 2-D geometry. Hence, the fraction of the power remaining after a round-trip inside the cold resonator is (22) which reduces to [25] (23) Equations (21) and (23) are equivalent. The effective reflectivity of the ASM strongly decreases as the mode order increases: if we insert the appropriate



2

should not be confused with the wavelength 

=

=nn.

The angular intensity density distribution is reduced by a factor at the refraction and is the refracted angle. The far-field output profile is depressed on the axis. Numerical calculation of near-field and far-field pattern for a 1-D structure will be shown in Section V. III. SIMULATION RESULTS Simulations using the ABCD formalism were carried out for a microchip laser with a thickness of mm with the refractive index of Y Al O , . For these examples, the parameter is assumed to be real. The influence of the angular width of the ASM on the waist parameter, measured before incidence on the ASM, is shown in Fig. 2 for three values of focal length of thermal lensing: m, 10 cm, and 1 cm. For example, a waist size ranging between 100 and 300 m is obtained for depending on the focusing power of the thermal lens. The latter depends on the heat load density, the cooling architecture, in particular, how the lateral heat flow compares to longitudinal heat flow and whether quasi-continuous-wave (CW) pumping or CW pumping is used (cf. Section IV for realistic estimates). The waist for large

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Fig. 2. Beam waist as a function of the modulus of the angular width of the ASM,  . The thickness of the microchip is d 1 mm, the refractive index is that of YAG n = 1:82. The calculation was perform for a thermal lens of f = 100, 10, or 1 cm.  = 10 (0.057 ) enables more than a ten-fold increase in the mode area. The dotted line is the ASM-limited waist value from (29).

=

Fig. 3. Effective reflectivity of the output coupler as a function of  for various = 1. values of the thermal lensing, 1=ff , for R

values of is constant and equal to that of the standard microchip without ASM [19, Ch. 5]

(28) At the other end of the graph, for small values of , the waist is inversely proportional to the angular width of the ASM angular reflectivity. The waist parameter of a Gaussian beam with angular width is given by (29) This formula, plotted as a straight dotted line in Fig. 2, approximates the mode waist of the microchip laser when the angular width of the ASM is about one order of magnitude smaller than the natural divergence of the mode propagating in the ASM-free microchip. The effective reflectivity of the ASM is shown in Fig. 3 as a function of the angular width of the ASM for a few values of thermal lensing and for the case . We see that a small value of produces larger output coupling losses, because a smaller fraction of the angular power spectrum of the beam is reflected. We also note that the output coupling losses increase with the focusing power of the thermal lens. This is due to the fact that, after reflection on the ASM, thermal lensing converts, during the round-trip propagation, a collimated beam into converging rays at the next incidence on the ASM. The ratio of round-trip losses for the first higher order mode to that of the fundamental mode , calculated from (25) is shown in Fig. 4. The modal selectivity increases as decreases and as the thermal lensing increases. IV. ORDERS OF MAGNITUDE AND THERMAL LIMITS The ASM enables an increase of the mode size and effective suppression of higher order transverse modes, but this is at the price of higher oscillation threshold due to higher output coupling losses. Hence, the question arises as whether the

Fig. 4. Ratio of round-trip losses for the first higher order mode to that of the fundamental mode as a function of the angular selectivity of the ASM.

higher pumping power needed to exceed the threshold of laser oscillation and to get good conversion efficiency will not cause overheating or thermal fracture, especially in low-gain materials such as rare-earth-doped materials. Consider a 1-mm-thick, 10% atomic concentration, Yb-doped YAG microchip equipped with an ASM with and . We suppose the focal length of thermal lensing is cm. We will see a posteriori that this value for is a reasonable estimate. According to Fig. 3, (0.057 ) an ASM with angular bandwidth of produces an effective reflectivity of about 20%. According to Fig. 2, the waist size will be about 200 m, i.e., about three times the size of the microchip without ASM. The required gain at threshold is estimated to be about cm , which is roughly one third of the small signal gain at strong pump of 20 cm [10]. The pumping power density required to achieve is estimated at about 75% of the absorption saturation parameter of 20 kW/cm at nm [10]. Assuming good matching of pump beam distribution with the fundamental mode of the microchip, the required pump power at threshold is about 19 W. A microchip Yb:YAG microchip laser with similar geometry is already demonstrated at this level of pumping

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power in the quasi-CW pumping [2]. If we assume that about 10% of the pump power absorbed by Yb is converted to heat [27], then about 2 W of heating power is generated. Now, even if we use aggressive water cooling at the back surface, the heat transfer coefficient will not exceed 10 W/cm K [28], [29], so that the Biot number [30] (30) will remain well below unity. This means that the lateral heat transfer dominates the longitudinal heat transfer. This result contrasts with thin-disk laser architectures, wherein an external mirror is used to increase and recirculation of the pump beam is used to enable the use of thinner layer of active layer, thereby generating higher values of and favoring longitudinal heat flow. In the case of microchip lasers, the lateral heat flow is dominant because the thickness-waist aspect ratio of the microchip is generally much larger than one, e.g., millimeter thickness versus tens of micrometers wide. The dominant lateral heat transfer implies that we can estimate the thermal lensing with the same formula as for side-cooled rod geometries [19, Ch. 7] (31) where is the heating power, m is the radius of the pumping spot, taken equal to the mode radius, the thermal conductivity W/m K and K is the thermooptic coefficient for Yb-doped YAG. The estimated focal lens is about cm, which is close to the initially assumed value of thermal lensing. Similarly, the dominant lateral heat flow implies that the maximum heating power per unit thickness, not the power density, mainly determines the maximum stress inside the microchip. The maximum heating power per unit thickness can be estimated by [19, Ch. 7] (32) where is the thermal shock parameter. For YAG ceramic, is in the order of 10 W/cm, so that a maximum of of heating power for 1-mm-thick YAG microchip can be wasted without causing thermal fracture. Hence, even for an ASM with a 20% effective reflectivity, the pumping power required to exceed the fracture limit is about one order of magnitude larger than the estimated power at threshold. Note that, for largely dominant lateral heat transfer, the larger mode area does not contribute to an increase of the fracture limit. However, the brightness is significantly improved due to the larger mode area, which enables a better overlap between the pump and the fundamental mode distributions, and due to the improved modal discrimination. The situation is different in the case of pulses emitted from a passively -switched microchip laser: larger mode area means larger pulse energy because the saturation fluence of the gain medium generally limits the output fluence. Simulations [6], [7] indicate that pulse fluence in the order of 5%–10% of the saturation fluence of the gain material can at best be achieved with a passively -switched laser. For Yb:YAG, the saturation fluence

Fig. 5. Scheme of the active medium equipped with an ASM. The ASM is made from an AR bilayer, a corrugated buffer layer, and four thin layers forming the waveguide.

is in the order of 10 J/cm [10]. Hence, if the mode radius is about 200 m, then single-mode pulses with pulse energy close to 1 mJ should be achievable. Nonwithstanding the resistance to optical damage of this structure, cf. Section V-A, ASMs provide a solution to expand the mode area, and hence the maximum pulse energy in one mode, which is needed for several applications. V. REALIZATION OF ASMS USING ABNORMAL REFLECTION FROM A RESONANT GRATING MIRROR A. Practical Considerations Abnormal reflection from resonant grating mirror structures can be used to achieve narrow enough angular reflectivity spectrum to increase the mode area of a microchip [23], [31], [32]. Resonant grating mirrors are all-dielectric structures composed of two submirrors: a standard dielectric multilayer and a coupled corrugated slab waveguide structure. A schematic drawing of the ASM is shown in Fig. 5. The ASM is made of an antireflection (AR) coating and a buffer layer on top of which a corrugation grating is etched. The buffer role is to separate the ASM from the active medium to prevent modes leakage losses. On top of the grating, a set of four layers acts as a waveguide. An ASM with only one high refractive index layer would also support waveguide mode, but to provide AR properties of ASM outside the resonance the special four-layers ASM was needed. The grating structure couples the incoming light into counter-propagating modes of the waveguide under normal incidence when the grating spatial frequency is equal to the propagation constant of the waveguide mode. The grating and the waveguide structure are designed so that this condition is satisfied at the lasing wavelength. The light coupled into the waveguide propagates inside the waveguide structure and is radiated back inside the microchip over a transverse distance inversely proportional to the radiation loss coefficient. The grating acts as a beam expander, Fig. 5. When carrying out the expansion in terms of vectors, it is well known that an expanded beam is obtained from a narrower angular distribution. In general, a shallower grating is needed to achieve a narrower angular bandwidth [33].

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13-fold enhancement of electric field in the structure described in this section. This will probably cause a reduction of the resistance to optical damage of the ASM. This is especially problematic for passively -switched microchip applications. Measurement of optical damage threshold of similar resonant grating mirror structures with nanosecond -switched pulses, and comparison with the calculated electric field distribution inside the structure, showed that the drop of damage threshold at resonance was roughly commensurate with the enhancement of the electric field [34]. The resistance to optical damage of the ASM in the -switched mode remains an important issue for the applicability of the ASM to passively -switched microchips.

Fig. 6. Reflectivity versus wavelength (nm) and angle (degrees) for the structure shown in Fig. 5, with the layer thicknesses and indices described in the text. The multilayer acts as an AR coating or a high-reflection mirror out of or in resonance, respectively.

A grating depth in the order of a few tens of nanometers is required to achieve a mode size of a few hundred micrometers. Actual design parameters for the realization of the ASM for the 1-D grating are a stack alternating lowand high index starting with the low-index on the YAG substrate. Layer thicknesses are from the down to top, cf. Fig. 5: 278, 222, 1000, 37, 59, 129, and 168 nm. The grating period is 612 nm and grating depth is 30 nm. The corresponding reflectivity calculated as a function of the incident angle from the microchip and wavelength is shown in Fig. 6. At nm, the reflectivity displays a full-width at half-maximum of about 0.1 deg. However, it is seen in Fig. 6 that the angular reflectivity distribution depends on the wavelength and the emission linewidth. Thin resonators with homogeneously broadened active medium, such as Yb:YAG, will emit single longitudinal mode if the pumping power does not exceed a few times the threshold pumping power. However, the oscillation of several longitudinal longitudinal modes is expected at pump powers several times larger than threshold because of spatial hole burning. An effective linewidth in the order of less than 1 nm appears reasonable even at the highest pump power. If we consider a 1-nm linewidth window in Fig. 6 centered at around 1030 nm, we see that the effective angular response of the microchip is only slightly broadened by the spectral emission bandwidth. The ability to achieve uniform grating depth and period and accurate knowledge of the physical parameters relevant to the modeling of this structure are essential to produce narrow ASM with proper spectral characteristics. The waveguide mode excitation at resonance is generally accompanied by an enhancement of the electric field in the slab waveguide. The field enhancement was calculated for an incident plane at normal incidence at the resonance wavelength using the Chandezon method. The field in the resonant structure was found to be about 30 times larger than for conditions outside the resonance. However, if we consider the fact that the incident beam contains a distribution of angular wavevectors exceeding the resonance bandwidth, then the effective field enhancement is reduced by roughly the square root of the effective reflectivity. Hence, we expect roughly a

B. Analytical Description of the Resonant Grating Mirrors and Numerical Results in 1-D In general, the field reflection coefficient from a waveguide grating with two resonances can be approximated as follows [35]:

(33) where and are parameters of the waveguide mode resonance poles in the reciprocal space. These parameters strongly depend on the incident wavelength; are coefficients in the expression of the Fresnel reflection coefficient out of resonance; they determine the reflection pedestal on which the resonance peaks of the first two terms are based

(34) where , , and are the incident wavenumber, incidence angle, and resonance angle, respectively. Under normal incidence, the two resonances correspond to the “left” and “right” waveguide modes coupling. At the exact resonance wavelength, the normal incidence symmetry condition imposes

(35) and for odd indices. In the present application of a resonant grating mirror to an ASM, it is preferable to prevent any interference between the resonant reflection and the pedestal reflection. Therefore, the resonant mirror will be designed so as to set the reflection pedestal to zero, i.e., . This condition gives rise to an almost symmetrical resonance reflection with respect to the wavelength. In this case, we can also neglect coefficients of even indices and (33) becomes simply (36)

where . Popov, Mashev, and Maystre analysis [32] of symmetrical grating and single waveguide resonance in lossless structures

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shows that abnormal reflection always reaches 100%. This is true if only one mode is excited at a time. Under normal incidence, however, if one mode is excited, the mode propagating backwards is also excited and the second order of the coupling grating couples the two waveguide modes to each other. This renders the analysis of resonant reflection more complex and there has not been so far a phenomenological analysis demonstrating theoretically that 100% resonant reflection is always possible. However, the long numerical modeling practice of the authors tends to confirm that the statement of [37] is also likely to be valid in the case of double resonance under normal inci. In addition, numerical simdence, so we may set ulation shows that under the conditions formulated above, the following relation is essentially fulfilled: (37) Neglecting the real part of and taking into account that , (36) takes the form (38) where and are the incident angle and the resonance curve width. The sign in the denominator depends on the polarization and on the type of corrugation, i.e., double- or single-sided corrugated waveguide. This function can reasonably be approximated by a Gaussian profile in the form of (6) where is a complex parameter which can be estimated by minimizing the integral of the square modulus of the difference between (6) and (38)

where

is the complex conjugate of

. We find that the value (39)

minimizes this integral. The modulus and phase angular profile of function (38) with positive sign corresponding to the structure described in Section V-A, and its Gaussian approximation (6) are shown in Fig. 7(a) and (b), respectively. The complex value of with negative imaginary part produces a converging beam after the reflection. The modulus of the theoretical and the Gaussian approximation are quite close but a larger discrepancy exists in the phase distribution. However, the lensing effect of the ASM is very small, less than a wavelength over the beam aperture, and the equivalent focusing power of the ASM, in the order of inverse meter, is generally much smaller than that of the thermal lens in a microchip under normal operating conditions; therefore it can be considered to play a negligible role. The distribution of the fundamental mode was numerically calculated in one transverse dimension using the angular response shown by the solid line in Fig. 7. The intensity distribution of the fundamental mode outside the microchip is shown in Fig. 8. The diameter of the fundamental mode, in the order of 400 m diameter, is consistent with 2-D analytic results found

Fig. 7. Theoretical angular reflection coefficient (solid line), (a) in modulus and (b) in phase , and comparison with the Gaussian angular profile (dashed line) used for the 2-D simulations.

for the Gaussian ASM mirror of similar angular width response. The curved phase response profile of the ASM causes the beam to be focused away from the microchip at a distance of about 2.9 cm from the ASM output coupler. Although the field distribution at the output coupler displays oscillations, Fig. 9 left, the intensity distribution at the focus displays more favorable properties, since the central lobe contains about 80% of the total power, Fig. 10 left. The far-field pattern, Fig. 9 right, displays a depressed profile resulting from the higher reflectivity of the ASM at small angles. This could be undesirable for some applications. However, if a 100- m-wide slit is placed at the waist in order to remove the satellite lobes, then a more adequate farfield profile with a central lobe is obtained, Fig. 10 right. In this case, the product of the beam width by the angular spread in the far field indicates a good beam quality, approaching the Fouriertransform limit obtained for Gaussian beams. Hence, adequate beam quality can reasonably be expected from an ASM-equiped microchip. VI. CONCLUDING REMARKS We have shown that ASMs make it possible to achieve a significant increase of the mode area by controlling the far-field beam size. ASMs also act as a mode filter, thereby enhancing the modal discrimination and suppressing higher order transverse modes. ASMs can be realized with abnormal reflection from resonant grating mirrors. The use of ASMs is not limited to the

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TABLE I COMPARISON OF THE GRADED REFLECTIVITY MIRROR WITH THE ANGULAR-SELECTIVE MIRROR

Fig. 8. Numerical calculation of the field propagated outside the microchip.

(a)

(b)

Fig. 9. (a) Numerical calculations of the transverse square field amplitude distribution at the output coupler. (b) Far-field angular distribution.

ASMs are fully compatible with thin, flat-flat resonator structures and, because of the absence of center of symmetry, preserve the translation invariance of the microchip wafer. The use of planar batch processing, involving only multilayer deposition, photolithography and etching of grating, ensures that microchip lasers remain mass-producible. The fabrication of a ASM-equipped microchip is under way and experimental results will be shown soon. REFERENCES

(a)

(b)

Fig. 10. (a) Numerical calculations of the square field amplitude distribution at the waist, 2.9 cm from the output coupler. (b) Far-field angular distribution when a 100-m-slit filter is placed at the waist in order to cut out the secondary lobes.

microchip lasers and could be used with other resonator geometries and other gain media. When comparing ASM with the traditional graded reflectivity mirror, the duality between the two elements is quite evident, since one is just the Fourier transform of the other. The graded reflectivity mirror with Gaussian reflectivity profile placed inside an unstable resonator contributes to stabilize the geometrically unstable cavity and to constrain the mode dimensions like a soft aperture. On the other hand, the ASM plays a key role in expanding the mode area of a geometrically stable resonator. The properties of ASMs and GRMs are compared and summarized in Table I.

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by combining diffractive optical elements to achieve polarization-stabilized microchip ceramic laser or radially-polarized beams. His interests also concern ceramic laser materials, especially the influence of grain boundaries on thermal transport, optical damage and optical scattering losses, etc. Since 2007, he also works as a Research Scientist at Amada, where he develops very-high-averagepower lasers processes for metal cutting and welding.

Nikolay Lyndin was born in Vitebsk, Belorus, in 1953. He received the Ph.D. degree in physics from the Moscow Physics and Technology Institute, Moscow, Russia, in 1979. Between 1979 and 1985, he was Senior Research Scientist at the Radio-Communication Research Institute, Moscow. From 1985 to 1991, he was Senior Lecturer of General Physics course at the Instrument Manufacturing Institute. He was Technical Director of Fiber Optics Technology Ltd, Moscow. In 1992, he joined the Moscow General Physics Institute as Senior Research Scientist in the field of integrated optics. He was appointed for two years as Senior Physicist at Thermo Labsystems Affinity Sensors, Cambridge, U.K., for supervising the development of grating coupled biosensors and related readout instrumentation. Since 2003, he has been with the Moscow General Physics Institute as Senior Research Scientist. His current interests are in multilayer gratings theory and applications. During the past decade he made several stays at the Université Jean Monnet as Invited Professor. He is coauthor of more than 100 publications and of a number of patents. He has developed and offers grating modelling computer codes based on the true-mode, the “C” and the corrected RCWA methods.

Ken-ichi Ueda was born in Osaka, Japan, in 1946. He received the Ph.D. in Physical Chemistry from the University of Tokyo, Tokyo, Japan, in 1977. Between 1971 and 1976, he was a Senior Researcher of JEOL R&D Center. Between 1976 and 1981, he was a Research Associate of Physics Department of Sophia University, Tokyo. In 1981, he joined to the Institute for Laser Science (ILS), e University of Electro-Communications (UEC), Tokyo, as an Associate Professor. He has been a Professor of ILS/UEC since 1999. He is a Director of ILS since 1995 and appointed to a Foreign Professor of Harbin Institute of Technology, Shenzhen Graduate School, China, in 2003. He is interested in a wide area of laser science during his carrier, for example, inorganic liquid lasers, glass lasers, CO2 lasers, excimer lasers, ultra-stabilized lasers, ultrahigh quality optics, ceramic lasers, and high power fiber lasers. He applied laser technique to IFE drivers, gravitational wave detection, and industrial applications. Dr. Ueda is a Fellow of OSA and JSAP. He is the recipient of the Sakurai Award, Quantum Electronics and Optics Award, Nikkei BP Award, and others for the development of ceramic lasers and high power fiber lasers.

Olivier Parriaux was born in Combrement-le-Petit, Switzerland, in 1943. He received the Ph.D. degree in physics in 1976 from the Federal Institute of Technology,Lausanne, Switzerland, for studies in the field of microwaves. During 1976–1979, he was a Postdoctoral Member of University College of London, U.K., working in the field of fiber and integrated optics. During 1979–1980, he held a half-year postdoctoral position at Lebedev Physical Institute, Moscow, Russia. From 1980 to 1983, he was Maitre de Conferences at the Institut National Polytechnique, Grenoble, France, where he was also teaching guided-wave optics as an external Lecturer until 1995. In 1983, he joined the CSEM Swiss Center for Electronics and Microtechnology, where he led the development of early industrial applications of fiber and integrated optics for sensors and microsystems. He has been the European manager of two large European projects, the most recent being EU-922 FOTA which focuses on optical gratings and diffractive optical elements which he finalized from FriedrichSchiller-Universität-Jena, Jena, Germany, as an Invited Professor. After one year as Directeur de Recherche Associé at the Centre National de la Recherche Scientifique (CNRS), he was appointed Professor at Jean Monnet University, Saint-Etienne, France, in 1997. He also received a Docentship from Twente University, The Netherlands, in 1996. His current interests are in the achievement of new functionalities in optical gratings and waveguide couplers for their industrial exploitation in sensors, microsystems and optical communications. He is the author of more than 130 publications and holds a number of patents. Dr. Parriaux was Chairman of the 6th European Conference on Integrated Optics (ECIO) Conference in 1993 and of the 7th International Workshop on Waveguide Theory in 1999.