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Nonlinear optical holograms for spatial and spectral shaping of light waves
Sci. Bull. (2015) 60(16):1403–1415 DOI 10.1007/s11434-015-0855-3
www.scibull.com www.springer.com/scp
Review
Physics & Astronomy
Nonlinear optical holograms for spatial and spectral shaping of light waves Asia Shapira • Liran Naor • Ady Arie
Received: 8 June 2015 / Accepted: 8 July 2015 / Published online: 30 July 2015 Ó Science China Press and Springer-Verlag Berlin Heidelberg 2015
Abstract Shaping either the spatial or the spectral output of a nonlinear interaction is accomplished by introducing basic concepts of computer-generated holography into the nonlinear optics regime. The possibilities of arbitrarily spatially shaping the result of a nonlinear interaction are presented for different phase-matching schemes allowing for both one- and two-dimensional shaping. Shaping the spectrum of a beam in nonlinear interaction is also possible by utilizing similar holographic techniques. The novel and complete control of the output of a nonlinear interaction opens exciting options in the fields of particle manipulation, optical communications, spectroscopy and quantum information. Keywords Nonlinear optics Beam shaping Spectral shaping Computer-generated holograms
1 Introduction The scheme for nonlinear wave mixing is based on interaction between laser beams having a Gaussian profile. The generated beam is then shaped by various additional elements such as lenses, filters and holograms. However, as we will describe in this paper, it is now possible to realize these shaping tasks within the nonlinear converter, by suitable modulation of its nonlinear coefficient. The motivation for beam shaping comes from the fact it can save both cost and space compared with the alternative approach
A. Shapira (&) L. Naor A. Arie Department of Physical Electronics, School of Electrical Engineering, Tel Aviv University, 69978 Tel Aviv, Israel e-mail: [email protected]
of first frequency converting the beam and then manipulating it. In addition, such shaping techniques open new possibilities for all-optical control of beam parameters that cannot be achieved in linear optics [1]. Nonlinear optical shaping can be done not only in the spatial domain but also in the spectral domain, and in some cases, both spatial shaping and spectral shaping are possible, thereby enabling the realization of interesting wave functions such as light bullets [2]. One-dimensional shaping of the generated beam in a nonlinear process was first suggested by Imeshev et al. [3]. The generation of a flat top beam at the output of the crystal, i.e., near field, was achieved by changing the interaction length along the profile of the incoming Gaussian beam. This technique can be used for various onedimensional manipulations on the amplitude of the generated beam. Ellenbogen et al. [4] proposed a two-dimensional structure satisfying non-collinear phase matching, implementing an all-optical deflector. At the output of the crystal, two Gaussian beams were generated and their relative location could be controlled by different parameters in the experiment. Manipulating the phase of the output was suggested by Kurtz et al. [5] and demonstrated for linear phase arrays and lenses. In another study, Ellenbogen et al. [6] proposed using the transverse axis to the propagation direction to impose a cubic phase on the generated beam, when such a beam undergoes an optical Fourier transform and the resulting beam is a self-accelerating Airy beam [7]. This is a unique feature of this specific beam since other beams Fourier transform cannot necessarily be expressed in terms of a Gaussian beam multiplied by some defined phase function. A nonlinear structure that generated multiple focal points at the converted frequency was presented by Qin et al. [8].
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Two common characteristics exist in all of the abovementioned devices. First, they all allow manipulating the output beam of a nonlinear interaction only in a single dimension. A theoretical device for two-dimensional manipulation—in order to nonlinearly generate vortex beams—was previously proposed by Bahabad and Arie [9], but due to the fact that the main fabrication process for modulating the nonlinear coefficient, electric field poling of ferroelectric crystals [10], is a planar technique, it cannot yet be realized in practice. The second, and very important, common feature is the fact that none of the mentioned studies proposed a general approach for arbitrary shaping of the output beam; i.e., each study suggested a specific ad hoc type of solution for a specific type of desired beam. In recent years, arbitrary shaping of beams in a nonlinear process was demonstrated, by bringing the concept of computer-generated holography into nonlinear optics, first for one-dimensional beam shaping [11] and later for two-dimensional beam shaping [12, 13]. This concept was further extended for spectral shaping of the generated signal in a nonlinear three-wave mixing process [14, 15]. In this review, we present an overview of new research in the field of spatial and spectral shaping based on exploiting holographic techniques in nonlinear crystals. First, a theoretical background of nonlinear interactions is presented, followed by an introduction to basic concepts of holography. Next, different shaping schemes are described and evaluated.
2 Theoretical background 2.1 Three-wave mixing process and quasi-phase matching In a three-wave mixing process, two input beams of frequencies, x1 and x2, can generate a new wave of frequency x3. The generated wave may be equal to the sum or the difference of the two input waves, or the second harmonic of each one of them. In this review, we will mainly focus on the nonlinear process of second-harmonic generation SHG, where both incoming beams have the same fundamental frequency (FF), x1, and the generated second-harmonic (SH) beam has a frequency of x2 = x1 ? x1, although all the concepts we present here can also be utilized with other nonlinear optical interactions. The interaction taking place inside a nonlinear crystal between the two frequencies in SHG can be described in terms of two coupled-wave equations. These equations are derived from the wave equation using the slowly varying amplitude approximation [16]. The resulting equations are,
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dA1 2ix21 deff ¼ A2 A1 eiDkz ; dz k1 c2
ð1Þ
dA2 ix22 deff 2 iDkz ¼ A e ; dz k2 c2 1
ð2Þ
where A1 and A2 are the amplitudes of the two waves, k1 and k2 are the wave vectors, deff is the nonlinear susceptibility coefficient and Dk = 2k1 - k2. In the process of SHG, the annihilation of two photons with energy hx1 is followed by a generation of a single photon with energy hx2; energy is conserved. Achieving momentum conservation is more complicated due to dispersion. The parameter Dk is defined to asses this phenomena and is termed phase mismatch. As a result of phase mismatch, Dk = 0, different dipoles in the nonlinear crystal oscillate in different phases and this destructive interference results with a low conversion efficiency of the SHG. Under the assumption of an un-depleted pump beam, which occurs if A2 A1 throughout the entire process, an exact solution to the coupled equations, for a crystal with length l, is given by [16], ix1 deff 2 sin Dkl A2 ðz ¼ lÞ ¼ A l Dkl2 eiDkl=2 : ð3Þ n2 c 1 2 The effect of phase mismatch is clearly seen in the above expression. The two well established possible solutions to the phasemismatch problem are using the natural birefringence existing in many nonlinear crystals or quasi-phase matching (QPM) [17]. Birefringence is the dependence of the refractive index on the direction of polarization of the optical radiation. Finding the polarization combination allowing for phase matching is sometimes possible with angle tuning, i.e., setting the angle of beam propagation inside the crystal with regard to the different axes of the crystal. This method has several serious drawbacks: Since the method relies on given material properties of dispersion and birefringence, it is not always possible to find conditions in which phase matching is possible. Moreover, when the beams do not propagate along primary axes of the crystal, the effect of walk-off is observed [16]. In addition, this technique does not allow working with the more efficient diagonal components of the nonlinear susceptibility tensor, for example d33, which is accessed only when all beams are polarized along the Z-axis. In QPM [17], the basic idea is to overcome the phase mismatch by modulating the sign of the nonlinear coefficient. In ferroelectric crystals, this can be done by inverting the ferroelectric domain orientation. In the specific case of a periodic modulation in the sign of the nonlinear coupling coefficient, the nonlinear coefficient is dðzÞ ¼ dij Signðcosð2pz=KÞÞ;
ð4Þ
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where K is the period. For an infinitely long crystal, the nonlinear coefficient can also be described in terms of a Fourier series, 1 mp X 2 2pm dðzÞ ¼ dij sin z : ð5Þ exp i mp 2 K m¼1 When the above expression is substituted into Eqs. (1) and (2), it can be seen that choosing the period such that K = 2pm/Dk compensates for the phase mismatch. It should be noted that the above analysis is for plane waves, whereas tightly focused Gaussian beams are discussed in Ref. [18]. 2.2 Nonlinear diffraction If both beams, FF and SH, propagate in the same direction and the reciprocal wave vector G owing to the periodic alteration of the structure is also in this direction, the process is collinear. This is illustrated in Fig. 1a. However, the most general case is to consider the vectorial nature of ! ! ! the phase-matching condition, i.e., Dk ¼ 2k1 k2 ; where the vectors of the interacting waves and the crystal’s vector are at arbitrary directions. Phase matching can also be achieved if there is an angle between the FF and SH wave vectors. The vector corresponding to the periodic structure
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in Fourier space can have a component along two axes; this is schematically illustrated in Fig. 1b. The physical process by which spatial patterning is achieved in ferroelectric crystals is electric field poling [10], where a strong applied electric field is used to invert the internal polarization vector. This technique is planar, enabling to modulate the nonlinear coefficient only in the X and Y crystallographic coordinates of the crystal. The mask for defining the electrodes that induce the poling process and the subsequent poled structure are defined by standard lithography procedures. A specific case of interest is when the poled pattern in the crystal is one-dimensional and the FF propagates in perpendicular to it. This case is referred to as nonlinear diffraction, because the SH beam is diffracted due the FF beam passing through the crystal, in a similar manner to the linear diffraction of a light beam from a periodic amplitude or phase grating. The diffraction pattern, a symmetrical one in the case described here, is attained in the SH. A case where the FF propagates along the Z-axis of the crystal is illustrated in Fig. 1c. In this case, three different diffraction processes can be seen: Raman–Nath [19], Cerenkov [20] and Bragg [21]; insets (i), (ii) and (iii) show the wavevector diagram corresponding to the three processes. In Raman–Nath, the angles of diffraction are given by
Fig. 1 (Color online) Different schemes for phase-matching SHG. a A collinear scheme and b a general non-collinear scheme. c Nonlinear diffraction for three different cases: i - Raman–Nath, ii - Cerenkov and iii - Bragg
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sinðam Þ ¼
mG0 ; k2
ð6Þ
where m is the order of diffraction and G0 is the shortest reciprocal vector. In Cerenkov, the angle of diffraction is defined by cosðaÞ ¼
2k1 : k2
ð7Þ
In Bragg diffraction, both Eqs. (6) and (7) are satisfied. In both Raman–Nath and Cerenkov, only part of the vectorial condition for phase matching is satisfied. Satisfying the Bragg condition with small values of m, from Eq. (5), is very difficult in most materials and for wavelengths in the visible and near infrared because it requires submicron patterning of the crystal. A possible solution may be to work with high QPM orders, but this will sufficiently reduce the efficiency of the process, as suggested by Eq. (5) and demonstrated in Refs. [22, 23]. An additional limitation of an experimental setup based on light propagating along the Zaxis is a short interaction length, typically around 0.5–1 mm. Nonlinear diffraction can also be observed if the FF propagates along the Y-axis of the crystal. It has been shown that in this scheme Bragg diffraction can be observed for low QPM orders, with relatively long poling periods that can be easily fabricated [24]. 2.3 Computer-generated holography In conventional holography [25], information of an object is stored in a light-sensitive medium (e.g., a photographic plate) by recording the interference pattern between a reference beam and a beam reflected from the object. This information can later be restored and the object reconstructed by illuminating this recording with a reference beam. There are no limitations on the used object as long as its size is larger than the wavelength. The interference pattern can be described in terms of a two-dimensional intensity pattern, I ðx; yÞ ¼jR expði2px=T Þ þ Aðx; yÞ expði/ðx; yÞÞj2 ¼R2 þ Aðx; yÞ2 þ2RAðx; yÞ cosð2px=T þ /ðx; yÞÞ; ð8Þ where R expði2px=T Þ is the reference beam and Aðx; yÞ expði/ðx; yÞÞ is the object and the third term in the above expression is the key for reconstructing the recorded information about the object. Holograms can be used to reconstruct the object either in the near field or in the far field, depending on the relationship between the object and the complex wave front recorded in the hologram. In computer-generated holography [26], the first step of optically recording the hologram is replaced with a
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numerical computation. This is possible if the amplitude and phase of the object are known, and hence, the third term in Eq. (8) can be computed. The result of this computation is an amplitude transmittance function. In the case of a Fourier hologram, the object is formed in the Fourier space after the reference beam passes through the hologram and the encoded information in the hologram is the Fourier transform of the object. Intensive study was conducted for finding the optimized encoding technique for computergenerated holograms (CGH). Burch [27] found that instead of using the third term in Eq. (8), it is more beneficial to use the following expression for the amplitude transmittance of the hologram, tðx; yÞ ¼ 0:5f1 þ Aðx; yÞ cosð2pfcarrier x /ðx; yÞÞg;
ð9Þ
where fcarrier is the carrier frequency, which is chosen so that overlap in the frequency space is avoided. The normalized transmittance function is continuous and tðx; yÞ has values between 0 and 1. Observing the far-field image of a beam that passed through this hologram reveals a diffraction pattern; diffraction orders are located at angles hm ¼ mkfcarrier . The first diffraction order is shaped as the Fourier transform of Aðx; yÞ expði/ðx; yÞÞ, hence providing the desired wave function. This concept is schematically illustrated in Fig. 2b, where an hologram encoded with Eq. (9) and aimed to generate an Hermite–Gaussian (HG) beam [28], HG11 shown in Fig. 2a, is illuminated with a reference beam. In the far field, one can easily observe the desired beam shape in the first diffraction orders; since the complex conjugate of HG11 has the same amplitude distribution as the beam itself, the same shape is seen in both ±1 orders. Another type of CGH is a binary hologram, where the transmittance function can only have one of two values. Lee [29] developed such a binary coding technique based on hard clipping a sinusoidal function offset by a biasing function. The resulting expression for the transmittance function is 1; cosð2pfcarrier x þ /ðx; yÞÞ cosðpqðx; yÞÞ 0; tðx; yÞ ¼ 0; else; ð10Þ where according to one of the suggested methods in the paper sinðpqðx; yÞÞ ¼ Aðx; yÞ [29]. Examining the firstorder term in the Fourier series decomposition of Eq. (10) shows why choosing the above relation between qðx; yÞand Aðx; yÞ results in shaping the first diffraction order, 1 X sinðmpqðx; yÞÞ tðx; yÞ ¼ mp m¼1 ð11Þ
exp½imð2pfcarrier x þ /ðx; yÞÞ :
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Fig. 2 (Color online) Simulation of the far-field output of two types of computer-generated holograms—b an hologram with continuous amplitude modulation according to Eq. (9) and c an hologram with binary amplitude modulation according to Eq. (10). In both cases, a the holograms were encoded to generate HG11 at the first diffraction order
As with the previous technique, the first diffraction order is shaped as the Fourier transform of Aðx; yÞ expði/ðx; yÞÞ. This concept is schematically demonstrated in Fig. 2c. Another method for binary modulation is the detour phase method [26], in which the hologram is split into small cells having transparent and opaque regions, so that the amplitude in each cell is determined by the transparent area and the phase is determined by its position inside the cell.
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stripe can be used to quasi-phase match a process with a phase mismatch Dk = 2p/K, and since they have the same length, the generated amplitude is the same, but owing to the offset, a phase difference of 2pd/K is created between the waves that are generated in these two stripes. This shows how the phase can be modulated in the process. This method was used, for example, by Ellenbogen et al. [6] to impose a cubic transverse phase of the generated second-harmonic beam, which was converted to an accelerating Airy beam by optical Fourier transformation with a lens. Figure 3b illustrates the ability to modulate the amplitude of the generated beam. Here two stripes with different duty cycles are shown. Since the effective nonlinear coefficient depends on the period, this will change the conversion efficiency and therefore the generated amplitude. The ability to modulate the amplitude and phase of the generated second-harmonic wave, combined with the flexibility of holograms, allows generating arbitrary beam shapes at the output of the nonlinear crystal. In the first demonstration of the concept [11], phase-matching condition was satisfied in a collinear way, as demonstrated in Fig. 1a. But, unlike the conventional one-dimensional periodic structure, an additional modulation was imposed on an axis perpendicular to the direction of wave propagation. In terms of Fig. 1, the X-axis is used for phase matching and the Y-axis for encoding the holographic information. Since only one axis is available, the result is a one-dimensional hologram. A continuous coding technique, a one-dimensional version of Eq. (9), was chosen out of fabrication considerations. The poling pattern of the crystal is given by the following expression, dðx; yÞ ¼ dij Signðcosð2px=KÞ þ ptð yÞÞ;
ð12Þ
when tðyÞ is the one-dimensional version of Eq. (9), tðyÞ ¼ 0:5f1 þ AðyÞ cosð2pfcarrier y /ðyÞÞg:
ð13Þ
3 Nonlinear computer-generated hologram 3.1 Nonlinear holograms for beam shaping 3.1.1 One-dimensional beam shaping Introducing the concepts of computer-generated holograms into the nonlinear optics regime holds the key too many exciting possibilities. The basic idea is to modulate the nonlinear coefficient so that it will simultaneously satisfy phase matching and encode holographic information. Figure 3 illustrates the possibilities of encoding phase and amplitude information in a nonlinear crystal. In Fig. 3a, we compare two stripes having periodic modulation of the nonlinear coefficient with an offset of d between them. Each
Fig. 3 (Color online) a Phase and b amplitude tailoring schemes in QPM
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The suggested setup enables shaping the first diffraction order of the SH beam at the output of the crystal according to the Fourier transform of AðyÞ expði/ðyÞÞ. The concept was demonstrated with the generation of high-order Hermite–Gaussian beams—HG01 and HG02. HG00 was also generated for reference purposes. These beams are unique since their Fourier transform has the same shape as the original beam [28]. Figure 4 shows an illustration of the suggested scheme, simulation results for the SH diffraction pattern and measured beams at the first diffraction order. In addition, the two-dimensional pattern of poling is apparent in the figure. The experiment was conducted in an SLT crystal, where an e-ee second-harmonic generation of an 1,064.5-nm Nd:YAG laser was quasi-phase-matched at 100 °C. This method can be utilized for generating any kind of onedimensional modulation. The experiment was planned for shaping the far-field image through an optical Fourier transform. Equation (12) can also be applied for shaping the near-field result of a nonlinear interaction; in this case, the coded amplitude and phase are those of the desired beam itself. Another way of implementing one-dimensional beam shaping, this time with a simpler poling pattern, relies of a non-collinear interaction [13]. As it was mentioned before, Bragg phase matching can be achieved for the first diffraction order when the crystal is rotated such that the beams propagate along the Y-axis. This is schematically illustrated in Fig. 5a. Phase matching sets the periodic structure seen along the X-axis; additional degree of freedom in choosing the phase-matching work point can be achieved if one breaks the symmetry of the wave-vector diagram. A binary holographic pattern, based on Eq. (10),
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can also be imposed on the same axis. The resulting pattern can be described as follows, dðxÞ ¼ dij Signðcosð2px=K /ðxÞÞ cosðpqðxÞÞÞ;
ð14Þ
where sinðpqðxÞÞ ¼ AðxÞ. In the general case when the symmetry of the diffraction experiment is broken, the period is set by K ¼ 2p cosðhÞ=k2 sinðaÞ, where a is the angle of separation between the FF and SH beams and h is the angle of FF beam propagation inside the crystal with respect to the normal to the crystal facet, as indicated in Fig. 5b. h can be either positive or negative, depending on phase-matching requirements. This was experimentally verified in a KTP crystal, where an o-eo SHG of an 1,064.5-nm Nd:YAG laser was phase-matched with the crystal tilted by 0.206 rad [30] (related to h through Snell’s law). Experimental results are presented in Fig. 5c, d. Microscopic pictures of the poling patterns in a KTP crystal are shown along side with the measured SH beam at the far field outside the crystal. Shaping was demonstrated also for Hermite–Gaussian beams. Compared to the previous scheme here, only a single-shaped beam appears at far field and not a full diffraction pattern. 3.1.2 Two-dimensional beam shaping In order to have a two-dimensional shaping of the output, the two available axes for poling the crystal should be employed for encoding the holographic pattern. The only experimental setup that allows this is when the beams propagate along the Z-axis of the crystal and both X- and Yaxes hold holographic information. Phase matching in this scheme is usually by Raman–Nath. The poling pattern in this case has the general expression [12],
Fig. 4 (Color online) A schematic illustration of a nonlinear interaction based on the modulation introduced in Eqs. (12) and (13). Simulated and measured results at far field for the generation of a HG00, b HG01 and c HG02
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Fig. 5 (Color online) a A schematic illustration of phase matching a nonlinear diffraction Bragg condition. b A tilted configuration where symmetry is broken. Experimental results based on the modulation presented in Eq. (14), for generating c HG10 and d HG20, including microscopic images of the encoded structures in the nonlinear crystal (on the left)
dðx; yÞ ¼ dij Signðcosð2pfcarrier x /ðx; yÞÞ cosðpqðx; yÞÞÞ;
lSH ¼ 2lFF þ mlcrystal ; ð15Þ
based on the binary coding technique described in Eq. (10). This concept is schematically demonstrated in Fig. 6a, where simulation results of the SH beam at far field are presented for three different cases, a periodic structure and two crystals encoded with holographic data. Figure 6b, c corresponds to an experimental demonstration of the concept in an SLT crystal. In this example, in addition to Hermite–Gaussian beams, the demonstration of generating Laguerre–Gaussian [28] beams was also presented. The LG11 mode is a vortex beam [31], carrying a topological charge of l = ?1. The result shows that two-dimensional nonlinear computer-generated holography is a method in which orbital angular momentum can be added through the nonlinear process, so that a non-vortex fundamental beam can be converted into a second-harmonic vortex beam. This type of conversion was theoretically discussed in Ref. [9] and experimentally demonstrated in spiral-shaped and fork-shaped nonlinear photonic crystals in Ref. [32]. Studying the orbital angular momentum relations between the interacting beams allowed formalizing a general orbital angular momentum conservation law,
ð16Þ
where lSH;FF is the topological charge of the beams and lcrystal is the topological charge added by the nonlinear crystal and m is the relevant diffraction order, related to Eq. (11). Experiments in Ref. [32] were performed with Gaussian inputs, and hence lFF ¼ 0, but the general orbital angular momentum conservation law of Eq. (16) was verified and studied in Ref. [33]. The complete experiment of Ref. [33] demonstrated also the radial control available when an FF beam with a topological charge enters a nonlinear crystal encoded with a pattern corresponding to a Laguerre–Gaussian beam. Controlling orbital angular momentum in nonlinear crystals could give rise to new applications of vortex beams in microparticle manipulation and in the field of quantum information [34]. The two-dimensional shaping of a SH beam is not limited only to beams which serve as exact solutions of the paraxial Helmholtz equation, therefore preserving their intensity distribution while propagating after the crystal’s output facet. Furthermore, it is not limited only to selfsimilar shapes which maintain their intensity distribution before and after performing a Fourier transform to the far field, as in the cases of Hermite–Gaussian or Laguerre–
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Fig. 6 (Color online) a A schematic illustration of beam shaping the first order of a Raman–Nath diffraction pattern. b Experimental results for shaping in this scheme, including microscopic pictures of the encoded pattern in the nonlinear crystal and c shaped first diffraction orders
Gaussian beams. Nonlinear Raman–Nath phase-matching condition can be satisfied also when the FF beam enters to the crystal at an angle with respect to the Z-axis, such that the FF and the generated SH beams propagate in a noncollinear fashion in the X–Z plane [19]. The generation of Top-Hat beams at the SH, one with a square cross-section profile and one with a circular cross-section profile, was experimentally demonstrated. The amplitude and the phase of the inverse Fourier transform of a high-order superGaussian function for the first, and of a circ function for the latter, were encoded to poling modulation patterns based on the binary coding scheme of Eq. (15). For a Top-Hat beam with a square cross-section profile, the modulation pattern was calculated according to the inverse Fourier transform of the following transverse spatial distribution: n A0 1 x Aðx; yÞ ¼ exp 1þ if W0n 1 þ if 1 yn exp n ; ð17Þ 1 þ if W0 when A0 is an amplitude constant, W0 is a width parameter, n is an even-order number and fðz; bÞ is defined in terms of a given longitudinal coordinate and the confocal parameter
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[16]. For a Top-Hat beam with a circular cross-section profile and a radius parameter, r0 , the transverse spatial distribution can be expressed as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffi const; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 r0 ; Aðx; yÞ ¼ ð18Þ 0; x2 þ y 2 [ r0 : Thus, the modulation pattern can be calculated according to the term 2J1 ðq=q0 Þ=ðq=q0 Þ, where J1 ðÞ is a Bessel function of order 1 and q0 is a radial constant. In addition, a two-dimensional nonlinear focusing lens, i.e., exploiting a two-dimensional modulation pattern of the nonlinear coefficient in the crystal to focus the generated SH beam after a desired focal length from the crystal’s output facet, is presented. A quadratic phase modulation was encoded to the poling pattern, when based on Eq. (15) the modulation function is given by k2 ðx2 þ y2 Þ dðx; yÞ ¼ dij Sign cos 2pfcarrier x ; ð19Þ 2n2 f where n2 is the refractive index for the SH beam and f is the desired focal length. The experimental setup configuration is similar to the schematic illustration shown in Fig. 6a, with an addition of a tilt angle of the crystal while the structure is illuminated
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by the pump beam. The three poling patterns were implemented in MgO:CLN crystals with a central period length of K ¼ 23 lm. Microscope pictures of the fabricated poling structures that were used for obtaining one of the TopHat beams and the focusing lens are presented in Figs. 7a and 8a, respectively. An e-oo SHG process with a pump wavelength of 1064.5 nm was produced at room temperature, with the crystal tilted by 31° for obtaining the SH Top-Hat shapes and by 9° for obtaining the nonlinear lens shaping. The patterned areas of the squared Top-Hat, the circular Top-Hat and the nonlinear lens were illuminated by a FF beam with waist radii of approximately 500, 780 and 900 lm, respectively, depending on the size of the patterned structure. A comparison between the simulated and the measured results for the intensity profile of the SH Top-Hat beam, with a square cross-section profile, is presented in Fig. 7b.
Fig. 8 (Color online) a Microscope picture of the fabricated poling structure in the crystal that can focus the generated SH beam at the first order of nonlinear Raman–Nath diffraction. b A comparison between the simulated and the measured SH beam radius, both in the horizontal and in the vertical directions, for different propagation distances after the crystal’s output facet. The insets at the bottom of the graph present the evolution of the SH beam’s intensity profile 6 cm before the focus, at the focus, and 6 cm after the focus
Fig. 7 (Color online) a Microscope picture of the fabricated poling structure in the crystal that represents the corresponding pattern of a two-dimensional inverse Fourier transform of a high-order SuperGaussian. The fabricated pattern resembles the main and the first lobes of a two-dimensional sinc function. b A comparison between simulated (left) and measured (right) SH Top-Hat beams, with a square cross-section profile, obtained in Fourier plane at the first order of nonlinear Raman–Nath diffraction. c Three-dimensional representation of the measured intensity profile of the SH Top-Hat beam
A 3D representation of the measured intensity profile is shown in Fig. 7c. The characteristic uniformity of this kind of a beam is reflected, both in the simulation and in the measurement. The SH Top-Hat shapes were obtained at the first order of the nonlinear Raman–Nath diffraction at the far-field Fourier plane after performing an optical Fourier transform. It should be noted that according to this nonlinear beam shaping method, the shapes of the Top-Hat beams can be obtained only in Fourier plane, while outside of it the beam does not maintain its shape. As for the nonlinear lens, a two-dimensional focusing of the output SH beam was obtained at the first order of the nonlinear Raman–Nath diffraction at a distance of approximately 20 cm after the crystal’s output facet, as was originally designed. The focusing of the SH beam is demonstrated in Fig. 8b, with a comparison between the simulated and the measured beam’s radius, in both the horizontal and the vertical directions, for successive propagation distances after the crystal. The only drawback of the concept presented in Figs. 6, 7, and 8 is the low conversion efficiency arising from relying on Raman–Nath phase matching. It turns out that breaking the symmetry of the diffraction experiment in Fig. 6a, in a similar manner as is suggested in Fig. 5b, allows finding work points where the full vectorial phase-
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matching condition is satisfied [13]. Figure 9 demonstrates this idea: In part (a), the concept is illustrated, and in part (b), the feasibility of finding appropriate conditions for satisfying phase matching is examined. It can be seen that a wide range of FF wavelengths can be phase-matched with different values of poling periods and tilting angles. The pattern in the crystal in this case is given by the following expression, dðx; yÞ ¼ dij Signðcosð2px=K /ðx; yÞÞ cosðpqðx; yÞÞÞ; ð20Þ where the period is set by K ¼ 2p cosðhÞ=k2 sinðaÞ, just as in Eq. (14). An experimental verification of this concept was performed in SLT, where an e-ee SHG of a 1,550-nm pump at room temperature was phase-matched with the crystal tilted by 0.86 rad [35] (related to h through Snell’s law). Figure 6c shows microscopic pictures of the fabricated crystal and measured results. Comparing efficiencies with the previous concept presented in Fig. 5 shows an improvement of five orders of magnitude [13], as expected from theory.
When working in a tilted setup, as in Figs. 5b or 9a, the validity of Eq. (10) should be tested, as it was derived assuming an optical beam passes in perpendicular to the hologram. If we strive for a spatial correlation higher than 90 %, the following condition needs to be fulfilled, L 9 tan(h) B 0.45w0, where L is the length of the crystal in the direction of propagation, h is the angle of the pump beam propagation inside the crystal (related to the crystal tilt angle through Snell’s law) and w0 is its waist. The results in Fig. 9c correspond to an FF beam with a wavelength of 1,550 nm. In terms of available input power, it is often desired to work with a wavelength of 1,064 nm; in SLT it is possible, and feasible in terms of fabrication abilities, if one considers the third diffraction order, m = 3 [35]. An additional benefit of the suggested scheme, Fig. 9a, is it allows for an input area of up to approximately 100 cm2 and hence allows entering the nonlinear crystal with high input power avoiding the damage threshold. All the nonlinear spatial holograms demonstrated in this review are based on implementing the concept of planar
Fig. 9 (Color online) a A schematic illustration of shaping a phase-matched non-symmetrical and nonlinear diffraction. b Calculated various phase-matched work points based on the suggested scheme in SLT [35]. Experimental results based on the suggested scheme (Eq. (20)), including microscopic pictures of the structures in the crystal and resulting beams in the SH, c for both HG11 and LG20
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holography. Volume holograms, also referred to as Bragg holograms, can also be introduced to the nonlinear optics regime, as was recently suggested by Hong et al. [36]. 3.2 Functional facets for beam shaping Another option for combining both a nonlinear process and spatial shaping of the beam is patterning the output facet of a nonlinear crystal [37]. In this scheme, the crystal is poled to satisfy a collinear phase-matching condition, as Fig. 1a. A thin layer of gold is evaporated on the output facet of the crystal and patterned with a focused ion beam (FIB). The resulting pattern on the facet serves as a transmittance amplitude mask for both beams, FF and SH, exiting the crystal. The mask can be used to two-dimensionally shape the output beams; the pattern in the gold is given by Eq. (10). Since the diffraction angles of a beam passing through the mask depend on the wavelength of the beam, a spatial separation is achieved between the diffracted FF and SH beams. The first order of the FF beams is located approximately at the second order of the SH beam. The experimental setup and a typical simulation and measured results are shown in Fig. 10. The same idea can be utilized for generating phase masks on the output facet, by using a transparent and conductive material, for example indium tin oxide (ITO) instead of gold. Experimental demonstrations of nonlinear crystals encoded with computer-generated holographic patterns were presented for a limited set of beams, but the general approach apparent from Eqs. (12–15) and (20) enables arbitrary shaping of the output. For example, accelerating beam, such as Airy [7] or parabolic [38] beams, can also be generated using the same schemes. These beams were experimentally generated in the scheme illustrated in Fig. 10 [37].
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3.3 Shaping the phase of the input beam An alternative approach is based on shaping the input beam profile, instead of the poling patterns of the nonlinear crystal. Ellenbogen et al. [39] suggested converting a Gaussian beam into a higher-mode Hermite–Gaussian beam [28] using the cascaded nonlinear processes of sumand difference-frequency generation [16]. This idea relies on the fact that three-wave mixing of a Gaussian beam with an Hermite–Gaussian beam generates an Hermite–Gaussian beam. Laguerre–Gaussian beams were generated in quasi-phase-matched crystals in a sum-frequency generation process by controlling the temperature of the crystal and the wavelength of the interacting beams [40]. Arbitrary shaping the transverse phase of the output beam in a nonlinear process can also be achieved if the phase of the FF beam is tailored correctly [41]. If the crystal is sufficiently short and the fundamental beam is weakly focused, the transverse phase of the second-harmonic output beam is simply twice the transverse phase of the input beam, thereby providing a simple and straightforward manner for controlling the generated beam properties. For longer (and therefore potentially more efficient) crystals, the input phase profile of the fundamental beam can be set, for example, using optimization with genetic algorithms. The key advantage here is the ability to dynamically control the input beam, by using a spatial light modulator or a deformable mirror to shape the fundamental beam’s profile. 3.4 Nonlinear spectral holograms In addition to spatial beam shaping, the output of a nonlinear interaction can also be shaped in the spectral domain [14]. As an example, we will consider the case of sumfrequency generation (SFG) where the result of an interaction between two different input beams, with frequencies x1 and x2, is a new beam with frequency x3 = x1 ? x2, but the same approach can be also used for spectral shaping in a difference-frequency generation process [14] and in some cases may be also extended to second-harmonic generation [42]. Approximating the two input beams as undepleted, the generated beam is the Fourier transform of the pattern along the propagation direction [16], for example the X-axis in Fig. 1a. It can be written as follows, Z1 A3 ðDkÞ / dð xÞeiDkx dx; ð21Þ 1
Fig. 10 (Color online) An illustration of adding functionality to a nonlinear crystal by shaping its output facet. Simulation and experimental results for encoding an LG10 beam
where Dk ¼ k1 þ k2 k3 . Examining the above equation reveals the inverse Fourier relationship between the spatial nonlinear modulation d(x) and the spectral shape of the generated beam. Modulating d(x) is done according to Eq. (10), and the resulting expression is
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dð xÞ ¼ dij SignðcosðDk0 x /ð xÞÞ cosðpqð xÞÞÞ;
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ð22Þ
where Dk0 is the phase mismatch for the central input frequency. An illustration of this scheme and experimental results [15] are presented in Fig. 11. In this example, two poling patterns were used: one that has a spectral response in the form of the HG02 function, and the other one in the form of the Airy function. When a broad input signal is mixed with a narrowband pump in these nonlinear spectral holograms, the spectrum of the generated signal is the product of the signal’s spectrum and the nonlinear crystal’s transmission spectrum. This enables, for example, to select only specific spectral components in the nonlinear mixing process and eliminates the need for additional spectral filters. The method can be also used to control the wavelength dependent phase of the generated signal, thereby enabling temporal shaping of ultrashort pulses in a nonlinear process. It is also possible to combine spectral shaping with spatial shaping, for example, if the Y-axis of the crystal is modulated according to Eq. (13). An interesting recent example of spectral holograms was the realization of ‘‘super-narrow’’ frequency converters [42]. Usually, the width of a frequency converter is inversely proportional to its physical length. This relation originates from the Fourier transform relation between the nonlinearity as a function of space and the efficiency as a function of the phase mismatch. Hence, the only available solution up till now for narrowing the spectral acceptance of a nonlinear converter was by increasing the crystal length. Unfortunately, this is a costly solution, it consumes large physical space and in addition, and the length of the available crystals is usually limited to a few centimeters at most. An alternative new solution is to modulate the
nonlinear coefficient with a super-oscillation [43] function. These are band-limited functions that oscillate locally much faster than their highest Fourier component and, in the context of nonlinear conversion, provide oscillation (in the phase-mismatch coordinates) that is much faster than the inverse of the crystal’s length. A super-narrow converter was experimentally realized, having spectral and thermal response that are narrower by 39 % and 69 % compared to the side lobes and main lobe of the sinc function response of a standard frequency doubling crystal with the same length [42].
4 Summary and Conclusions Different approaches for spatially shaping the output of nonlinear interactions were presented in this review. Shaping one-dimension of the output is possible with both a one- and two-dimensionally patterned nonlinear crystals. Two-dimensional shaping is also possible in two-dimensional structures, allowing for both shaping and high conversion efficiencies. Introducing holographic techniques is also possible in the spectral domain, resulting with generated signals with arbitrary shaped spectrum. These methods save the need for additional external elements such as lenses, splitters and filters by utilizing available fabrication capabilities of nonlinear crystals to incorporate these tasks into the nonlinear device itself. Moreover, the nonlinear process enables all-optical control of the parameters of the generated waves such as the orbital angular momentum in the case of vortex beam or the acceleration in the case of Airy beam. Complete control of the output of a nonlinear
Fig. 11 (Color online) Demonstration of spectral shaping, a spectral input on the left passes through a modulated crystal with two different modulation patterns. Corresponding spectral outputs—on the right
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interaction opens exciting options in the fields of particle manipulation, spectroscopy and optical communications and quantum information. Acknowledgments This work was supported by the Israel Science Foundation (1310/13) and by the Israeli Ministry of Science, Technology and Space in the framework of the Israel–Italy bi-national collaboration program. The authors would like to acknowledge HC Photonics Corporation for the manufacture of the poled MgO:CLN crystals according to our customized designs. Conflict of interest of interest.
The authors declare that they have no conflict
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1415 18. Boyd GD, Kleinman DA (1968) Parametric interaction of focused Gaussian light beams. J Appl Phys 39:3597 19. Saltiel SM, Neshev DN, Krolikowski W et al (2009) Mulitorder nonlinear diffraction in frequency doubling processes. Opt Lett 34:848–850 20. Saltiel SM, Sheng Y, Voloch-Bloch N et al (2009) Cerenkov-type second-harmonic generation in two-dimensional nonlinear photonic structures. IEEE J Quantum Electron 45:1465–1472 21. Freund I (1968) Nonlinear diffraction. Phys Rev Lett 21:1404 22. Saltiel SM, Neshev DN, Fischer R et al (2008) Generation of the second-harmonic conical waves via nonlinear Bragg diffraction. Phys Rev Lett 100:103902 23. Saltiel SM, Neshev DN, Krolikowski W et al (2010) Nonlinear diffraction from a virtual beam. Phys Rev Lett 104:083902 24. Shapira A, Arie A (2011) Phase-matched nonlinear diffraction. Opt Lett 36:1933–1935 25. Gabor DA (1948) A new microscopic principle. Nature 161:777–778 26. Brown BR, Lohmann AW (1966) Complex spatial filtering with binary masks. Appl Opt 5:967–969 27. Burch JJ (1967) A computer algorithm for the synthesis of spatial frequency filters. IEEE Proc 55:599–601 28. Saleh BEA, Teich MC (1991) Fundamentals of photonics. Wiley, New York 29. Lee WH (1979) Binary computer-generated holograms. Appl Opt 18:3661–3669 30. Shoji I, Kondo T, Kitamoto A et al (1997) Absolute scale of second-order nonlinear-optical coefficients. J Opt Soc Am B: Opt Phys 14:2268–2294 31. Arlt J, Dholakia K, Allen L et al (1998) The production of multiringed Laguerre–Gaussian modes by computer-generated holograms. J Mod Opt 45:1231–1237 32. Voloch-Bloch N, Shemer K, Shapira A et al (2012) Twisting of light by nonlinear photonic crystals. Phys Rev Lett 108:233902 33. Shemer K, Volovh-Bloch N, Shapira A et al (2013) Azimuthal and radial shaping of vortex beams generated in twisted nonlinear photonic crystals. Opt Lett 38:5470–5473 34. Lu LL, Xu P, Zhong ML et al (2015) Orbital angular momentum entanglement via fork-poling nonlinear photonic crystals. Opt Express 23:1203–1212 35. Dolev I, Ganany-Padowicz A, Gayer O et al (2009) Linear and nonlinear optical properties of MgO:LiTaO3. Appl Phys B 96:423–432 36. Hong XH, Yang B, Zhang C et al (2014) Nonlinear volume holography for wave-front engineering. Phys Rev Lett 113:163902 37. Shapira A, Libster A, Lilach Y et al (2013) Functional facets for nonlinear crystals. Opt Comm 300:244–248 38. Bandres MA (2008) Accelerating parabolic beams. Opt Lett 33:1678–1680 39. Ellenbogen T, Dolev I, Arie A (2008) Mode conversion in quadratic nonlinear crystals. Opt Lett 33:1207–1209 40. Zhou ZY, Li Y, Ding DS et al (2014) Generation of light with controllable spatial patterns via the sum frequency in quasi-phase matching crystals. Sci Rep 4:5650 41. Libster-Hershko A, Trajtenberg-Mills S, Arie A (2015) Dynamic control of light beams in second harmonic generation. Opt Lett 40:1944–1947 42. Remez R, Arie A (2015) Super-narrow frequency conversion. Optica 2:472–475 43. Berry M (1994) Quantum coherence and reality. World Scientific, Singapore, pp 55–65
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www.scibull.com www.springer.com/scp
Review
Physics & Astronomy
Nonlinear optical holograms for spatial and spectral shaping of light waves Asia Shapira • Liran Naor • Ady Arie
Received: 8 June 2015 / Accepted: 8 July 2015 / Published online: 30 July 2015 Ó Science China Press and Springer-Verlag Berlin Heidelberg 2015
Abstract Shaping either the spatial or the spectral output of a nonlinear interaction is accomplished by introducing basic concepts of computer-generated holography into the nonlinear optics regime. The possibilities of arbitrarily spatially shaping the result of a nonlinear interaction are presented for different phase-matching schemes allowing for both one- and two-dimensional shaping. Shaping the spectrum of a beam in nonlinear interaction is also possible by utilizing similar holographic techniques. The novel and complete control of the output of a nonlinear interaction opens exciting options in the fields of particle manipulation, optical communications, spectroscopy and quantum information. Keywords Nonlinear optics Beam shaping Spectral shaping Computer-generated holograms
1 Introduction The scheme for nonlinear wave mixing is based on interaction between laser beams having a Gaussian profile. The generated beam is then shaped by various additional elements such as lenses, filters and holograms. However, as we will describe in this paper, it is now possible to realize these shaping tasks within the nonlinear converter, by suitable modulation of its nonlinear coefficient. The motivation for beam shaping comes from the fact it can save both cost and space compared with the alternative approach
A. Shapira (&) L. Naor A. Arie Department of Physical Electronics, School of Electrical Engineering, Tel Aviv University, 69978 Tel Aviv, Israel e-mail: [email protected]
of first frequency converting the beam and then manipulating it. In addition, such shaping techniques open new possibilities for all-optical control of beam parameters that cannot be achieved in linear optics [1]. Nonlinear optical shaping can be done not only in the spatial domain but also in the spectral domain, and in some cases, both spatial shaping and spectral shaping are possible, thereby enabling the realization of interesting wave functions such as light bullets [2]. One-dimensional shaping of the generated beam in a nonlinear process was first suggested by Imeshev et al. [3]. The generation of a flat top beam at the output of the crystal, i.e., near field, was achieved by changing the interaction length along the profile of the incoming Gaussian beam. This technique can be used for various onedimensional manipulations on the amplitude of the generated beam. Ellenbogen et al. [4] proposed a two-dimensional structure satisfying non-collinear phase matching, implementing an all-optical deflector. At the output of the crystal, two Gaussian beams were generated and their relative location could be controlled by different parameters in the experiment. Manipulating the phase of the output was suggested by Kurtz et al. [5] and demonstrated for linear phase arrays and lenses. In another study, Ellenbogen et al. [6] proposed using the transverse axis to the propagation direction to impose a cubic phase on the generated beam, when such a beam undergoes an optical Fourier transform and the resulting beam is a self-accelerating Airy beam [7]. This is a unique feature of this specific beam since other beams Fourier transform cannot necessarily be expressed in terms of a Gaussian beam multiplied by some defined phase function. A nonlinear structure that generated multiple focal points at the converted frequency was presented by Qin et al. [8].
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Two common characteristics exist in all of the abovementioned devices. First, they all allow manipulating the output beam of a nonlinear interaction only in a single dimension. A theoretical device for two-dimensional manipulation—in order to nonlinearly generate vortex beams—was previously proposed by Bahabad and Arie [9], but due to the fact that the main fabrication process for modulating the nonlinear coefficient, electric field poling of ferroelectric crystals [10], is a planar technique, it cannot yet be realized in practice. The second, and very important, common feature is the fact that none of the mentioned studies proposed a general approach for arbitrary shaping of the output beam; i.e., each study suggested a specific ad hoc type of solution for a specific type of desired beam. In recent years, arbitrary shaping of beams in a nonlinear process was demonstrated, by bringing the concept of computer-generated holography into nonlinear optics, first for one-dimensional beam shaping [11] and later for two-dimensional beam shaping [12, 13]. This concept was further extended for spectral shaping of the generated signal in a nonlinear three-wave mixing process [14, 15]. In this review, we present an overview of new research in the field of spatial and spectral shaping based on exploiting holographic techniques in nonlinear crystals. First, a theoretical background of nonlinear interactions is presented, followed by an introduction to basic concepts of holography. Next, different shaping schemes are described and evaluated.
2 Theoretical background 2.1 Three-wave mixing process and quasi-phase matching In a three-wave mixing process, two input beams of frequencies, x1 and x2, can generate a new wave of frequency x3. The generated wave may be equal to the sum or the difference of the two input waves, or the second harmonic of each one of them. In this review, we will mainly focus on the nonlinear process of second-harmonic generation SHG, where both incoming beams have the same fundamental frequency (FF), x1, and the generated second-harmonic (SH) beam has a frequency of x2 = x1 ? x1, although all the concepts we present here can also be utilized with other nonlinear optical interactions. The interaction taking place inside a nonlinear crystal between the two frequencies in SHG can be described in terms of two coupled-wave equations. These equations are derived from the wave equation using the slowly varying amplitude approximation [16]. The resulting equations are,
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dA1 2ix21 deff ¼ A2 A1 eiDkz ; dz k1 c2
ð1Þ
dA2 ix22 deff 2 iDkz ¼ A e ; dz k2 c2 1
ð2Þ
where A1 and A2 are the amplitudes of the two waves, k1 and k2 are the wave vectors, deff is the nonlinear susceptibility coefficient and Dk = 2k1 - k2. In the process of SHG, the annihilation of two photons with energy hx1 is followed by a generation of a single photon with energy hx2; energy is conserved. Achieving momentum conservation is more complicated due to dispersion. The parameter Dk is defined to asses this phenomena and is termed phase mismatch. As a result of phase mismatch, Dk = 0, different dipoles in the nonlinear crystal oscillate in different phases and this destructive interference results with a low conversion efficiency of the SHG. Under the assumption of an un-depleted pump beam, which occurs if A2 A1 throughout the entire process, an exact solution to the coupled equations, for a crystal with length l, is given by [16], ix1 deff 2 sin Dkl A2 ðz ¼ lÞ ¼ A l Dkl2 eiDkl=2 : ð3Þ n2 c 1 2 The effect of phase mismatch is clearly seen in the above expression. The two well established possible solutions to the phasemismatch problem are using the natural birefringence existing in many nonlinear crystals or quasi-phase matching (QPM) [17]. Birefringence is the dependence of the refractive index on the direction of polarization of the optical radiation. Finding the polarization combination allowing for phase matching is sometimes possible with angle tuning, i.e., setting the angle of beam propagation inside the crystal with regard to the different axes of the crystal. This method has several serious drawbacks: Since the method relies on given material properties of dispersion and birefringence, it is not always possible to find conditions in which phase matching is possible. Moreover, when the beams do not propagate along primary axes of the crystal, the effect of walk-off is observed [16]. In addition, this technique does not allow working with the more efficient diagonal components of the nonlinear susceptibility tensor, for example d33, which is accessed only when all beams are polarized along the Z-axis. In QPM [17], the basic idea is to overcome the phase mismatch by modulating the sign of the nonlinear coefficient. In ferroelectric crystals, this can be done by inverting the ferroelectric domain orientation. In the specific case of a periodic modulation in the sign of the nonlinear coupling coefficient, the nonlinear coefficient is dðzÞ ¼ dij Signðcosð2pz=KÞÞ;
ð4Þ
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where K is the period. For an infinitely long crystal, the nonlinear coefficient can also be described in terms of a Fourier series, 1 mp X 2 2pm dðzÞ ¼ dij sin z : ð5Þ exp i mp 2 K m¼1 When the above expression is substituted into Eqs. (1) and (2), it can be seen that choosing the period such that K = 2pm/Dk compensates for the phase mismatch. It should be noted that the above analysis is for plane waves, whereas tightly focused Gaussian beams are discussed in Ref. [18]. 2.2 Nonlinear diffraction If both beams, FF and SH, propagate in the same direction and the reciprocal wave vector G owing to the periodic alteration of the structure is also in this direction, the process is collinear. This is illustrated in Fig. 1a. However, the most general case is to consider the vectorial nature of ! ! ! the phase-matching condition, i.e., Dk ¼ 2k1 k2 ; where the vectors of the interacting waves and the crystal’s vector are at arbitrary directions. Phase matching can also be achieved if there is an angle between the FF and SH wave vectors. The vector corresponding to the periodic structure
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in Fourier space can have a component along two axes; this is schematically illustrated in Fig. 1b. The physical process by which spatial patterning is achieved in ferroelectric crystals is electric field poling [10], where a strong applied electric field is used to invert the internal polarization vector. This technique is planar, enabling to modulate the nonlinear coefficient only in the X and Y crystallographic coordinates of the crystal. The mask for defining the electrodes that induce the poling process and the subsequent poled structure are defined by standard lithography procedures. A specific case of interest is when the poled pattern in the crystal is one-dimensional and the FF propagates in perpendicular to it. This case is referred to as nonlinear diffraction, because the SH beam is diffracted due the FF beam passing through the crystal, in a similar manner to the linear diffraction of a light beam from a periodic amplitude or phase grating. The diffraction pattern, a symmetrical one in the case described here, is attained in the SH. A case where the FF propagates along the Z-axis of the crystal is illustrated in Fig. 1c. In this case, three different diffraction processes can be seen: Raman–Nath [19], Cerenkov [20] and Bragg [21]; insets (i), (ii) and (iii) show the wavevector diagram corresponding to the three processes. In Raman–Nath, the angles of diffraction are given by
Fig. 1 (Color online) Different schemes for phase-matching SHG. a A collinear scheme and b a general non-collinear scheme. c Nonlinear diffraction for three different cases: i - Raman–Nath, ii - Cerenkov and iii - Bragg
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sinðam Þ ¼
mG0 ; k2
ð6Þ
where m is the order of diffraction and G0 is the shortest reciprocal vector. In Cerenkov, the angle of diffraction is defined by cosðaÞ ¼
2k1 : k2
ð7Þ
In Bragg diffraction, both Eqs. (6) and (7) are satisfied. In both Raman–Nath and Cerenkov, only part of the vectorial condition for phase matching is satisfied. Satisfying the Bragg condition with small values of m, from Eq. (5), is very difficult in most materials and for wavelengths in the visible and near infrared because it requires submicron patterning of the crystal. A possible solution may be to work with high QPM orders, but this will sufficiently reduce the efficiency of the process, as suggested by Eq. (5) and demonstrated in Refs. [22, 23]. An additional limitation of an experimental setup based on light propagating along the Zaxis is a short interaction length, typically around 0.5–1 mm. Nonlinear diffraction can also be observed if the FF propagates along the Y-axis of the crystal. It has been shown that in this scheme Bragg diffraction can be observed for low QPM orders, with relatively long poling periods that can be easily fabricated [24]. 2.3 Computer-generated holography In conventional holography [25], information of an object is stored in a light-sensitive medium (e.g., a photographic plate) by recording the interference pattern between a reference beam and a beam reflected from the object. This information can later be restored and the object reconstructed by illuminating this recording with a reference beam. There are no limitations on the used object as long as its size is larger than the wavelength. The interference pattern can be described in terms of a two-dimensional intensity pattern, I ðx; yÞ ¼jR expði2px=T Þ þ Aðx; yÞ expði/ðx; yÞÞj2 ¼R2 þ Aðx; yÞ2 þ2RAðx; yÞ cosð2px=T þ /ðx; yÞÞ; ð8Þ where R expði2px=T Þ is the reference beam and Aðx; yÞ expði/ðx; yÞÞ is the object and the third term in the above expression is the key for reconstructing the recorded information about the object. Holograms can be used to reconstruct the object either in the near field or in the far field, depending on the relationship between the object and the complex wave front recorded in the hologram. In computer-generated holography [26], the first step of optically recording the hologram is replaced with a
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numerical computation. This is possible if the amplitude and phase of the object are known, and hence, the third term in Eq. (8) can be computed. The result of this computation is an amplitude transmittance function. In the case of a Fourier hologram, the object is formed in the Fourier space after the reference beam passes through the hologram and the encoded information in the hologram is the Fourier transform of the object. Intensive study was conducted for finding the optimized encoding technique for computergenerated holograms (CGH). Burch [27] found that instead of using the third term in Eq. (8), it is more beneficial to use the following expression for the amplitude transmittance of the hologram, tðx; yÞ ¼ 0:5f1 þ Aðx; yÞ cosð2pfcarrier x /ðx; yÞÞg;
ð9Þ
where fcarrier is the carrier frequency, which is chosen so that overlap in the frequency space is avoided. The normalized transmittance function is continuous and tðx; yÞ has values between 0 and 1. Observing the far-field image of a beam that passed through this hologram reveals a diffraction pattern; diffraction orders are located at angles hm ¼ mkfcarrier . The first diffraction order is shaped as the Fourier transform of Aðx; yÞ expði/ðx; yÞÞ, hence providing the desired wave function. This concept is schematically illustrated in Fig. 2b, where an hologram encoded with Eq. (9) and aimed to generate an Hermite–Gaussian (HG) beam [28], HG11 shown in Fig. 2a, is illuminated with a reference beam. In the far field, one can easily observe the desired beam shape in the first diffraction orders; since the complex conjugate of HG11 has the same amplitude distribution as the beam itself, the same shape is seen in both ±1 orders. Another type of CGH is a binary hologram, where the transmittance function can only have one of two values. Lee [29] developed such a binary coding technique based on hard clipping a sinusoidal function offset by a biasing function. The resulting expression for the transmittance function is 1; cosð2pfcarrier x þ /ðx; yÞÞ cosðpqðx; yÞÞ 0; tðx; yÞ ¼ 0; else; ð10Þ where according to one of the suggested methods in the paper sinðpqðx; yÞÞ ¼ Aðx; yÞ [29]. Examining the firstorder term in the Fourier series decomposition of Eq. (10) shows why choosing the above relation between qðx; yÞand Aðx; yÞ results in shaping the first diffraction order, 1 X sinðmpqðx; yÞÞ tðx; yÞ ¼ mp m¼1 ð11Þ
exp½imð2pfcarrier x þ /ðx; yÞÞ :
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Fig. 2 (Color online) Simulation of the far-field output of two types of computer-generated holograms—b an hologram with continuous amplitude modulation according to Eq. (9) and c an hologram with binary amplitude modulation according to Eq. (10). In both cases, a the holograms were encoded to generate HG11 at the first diffraction order
As with the previous technique, the first diffraction order is shaped as the Fourier transform of Aðx; yÞ expði/ðx; yÞÞ. This concept is schematically demonstrated in Fig. 2c. Another method for binary modulation is the detour phase method [26], in which the hologram is split into small cells having transparent and opaque regions, so that the amplitude in each cell is determined by the transparent area and the phase is determined by its position inside the cell.
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stripe can be used to quasi-phase match a process with a phase mismatch Dk = 2p/K, and since they have the same length, the generated amplitude is the same, but owing to the offset, a phase difference of 2pd/K is created between the waves that are generated in these two stripes. This shows how the phase can be modulated in the process. This method was used, for example, by Ellenbogen et al. [6] to impose a cubic transverse phase of the generated second-harmonic beam, which was converted to an accelerating Airy beam by optical Fourier transformation with a lens. Figure 3b illustrates the ability to modulate the amplitude of the generated beam. Here two stripes with different duty cycles are shown. Since the effective nonlinear coefficient depends on the period, this will change the conversion efficiency and therefore the generated amplitude. The ability to modulate the amplitude and phase of the generated second-harmonic wave, combined with the flexibility of holograms, allows generating arbitrary beam shapes at the output of the nonlinear crystal. In the first demonstration of the concept [11], phase-matching condition was satisfied in a collinear way, as demonstrated in Fig. 1a. But, unlike the conventional one-dimensional periodic structure, an additional modulation was imposed on an axis perpendicular to the direction of wave propagation. In terms of Fig. 1, the X-axis is used for phase matching and the Y-axis for encoding the holographic information. Since only one axis is available, the result is a one-dimensional hologram. A continuous coding technique, a one-dimensional version of Eq. (9), was chosen out of fabrication considerations. The poling pattern of the crystal is given by the following expression, dðx; yÞ ¼ dij Signðcosð2px=KÞ þ ptð yÞÞ;
ð12Þ
when tðyÞ is the one-dimensional version of Eq. (9), tðyÞ ¼ 0:5f1 þ AðyÞ cosð2pfcarrier y /ðyÞÞg:
ð13Þ
3 Nonlinear computer-generated hologram 3.1 Nonlinear holograms for beam shaping 3.1.1 One-dimensional beam shaping Introducing the concepts of computer-generated holograms into the nonlinear optics regime holds the key too many exciting possibilities. The basic idea is to modulate the nonlinear coefficient so that it will simultaneously satisfy phase matching and encode holographic information. Figure 3 illustrates the possibilities of encoding phase and amplitude information in a nonlinear crystal. In Fig. 3a, we compare two stripes having periodic modulation of the nonlinear coefficient with an offset of d between them. Each
Fig. 3 (Color online) a Phase and b amplitude tailoring schemes in QPM
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The suggested setup enables shaping the first diffraction order of the SH beam at the output of the crystal according to the Fourier transform of AðyÞ expði/ðyÞÞ. The concept was demonstrated with the generation of high-order Hermite–Gaussian beams—HG01 and HG02. HG00 was also generated for reference purposes. These beams are unique since their Fourier transform has the same shape as the original beam [28]. Figure 4 shows an illustration of the suggested scheme, simulation results for the SH diffraction pattern and measured beams at the first diffraction order. In addition, the two-dimensional pattern of poling is apparent in the figure. The experiment was conducted in an SLT crystal, where an e-ee second-harmonic generation of an 1,064.5-nm Nd:YAG laser was quasi-phase-matched at 100 °C. This method can be utilized for generating any kind of onedimensional modulation. The experiment was planned for shaping the far-field image through an optical Fourier transform. Equation (12) can also be applied for shaping the near-field result of a nonlinear interaction; in this case, the coded amplitude and phase are those of the desired beam itself. Another way of implementing one-dimensional beam shaping, this time with a simpler poling pattern, relies of a non-collinear interaction [13]. As it was mentioned before, Bragg phase matching can be achieved for the first diffraction order when the crystal is rotated such that the beams propagate along the Y-axis. This is schematically illustrated in Fig. 5a. Phase matching sets the periodic structure seen along the X-axis; additional degree of freedom in choosing the phase-matching work point can be achieved if one breaks the symmetry of the wave-vector diagram. A binary holographic pattern, based on Eq. (10),
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can also be imposed on the same axis. The resulting pattern can be described as follows, dðxÞ ¼ dij Signðcosð2px=K /ðxÞÞ cosðpqðxÞÞÞ;
ð14Þ
where sinðpqðxÞÞ ¼ AðxÞ. In the general case when the symmetry of the diffraction experiment is broken, the period is set by K ¼ 2p cosðhÞ=k2 sinðaÞ, where a is the angle of separation between the FF and SH beams and h is the angle of FF beam propagation inside the crystal with respect to the normal to the crystal facet, as indicated in Fig. 5b. h can be either positive or negative, depending on phase-matching requirements. This was experimentally verified in a KTP crystal, where an o-eo SHG of an 1,064.5-nm Nd:YAG laser was phase-matched with the crystal tilted by 0.206 rad [30] (related to h through Snell’s law). Experimental results are presented in Fig. 5c, d. Microscopic pictures of the poling patterns in a KTP crystal are shown along side with the measured SH beam at the far field outside the crystal. Shaping was demonstrated also for Hermite–Gaussian beams. Compared to the previous scheme here, only a single-shaped beam appears at far field and not a full diffraction pattern. 3.1.2 Two-dimensional beam shaping In order to have a two-dimensional shaping of the output, the two available axes for poling the crystal should be employed for encoding the holographic pattern. The only experimental setup that allows this is when the beams propagate along the Z-axis of the crystal and both X- and Yaxes hold holographic information. Phase matching in this scheme is usually by Raman–Nath. The poling pattern in this case has the general expression [12],
Fig. 4 (Color online) A schematic illustration of a nonlinear interaction based on the modulation introduced in Eqs. (12) and (13). Simulated and measured results at far field for the generation of a HG00, b HG01 and c HG02
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Fig. 5 (Color online) a A schematic illustration of phase matching a nonlinear diffraction Bragg condition. b A tilted configuration where symmetry is broken. Experimental results based on the modulation presented in Eq. (14), for generating c HG10 and d HG20, including microscopic images of the encoded structures in the nonlinear crystal (on the left)
dðx; yÞ ¼ dij Signðcosð2pfcarrier x /ðx; yÞÞ cosðpqðx; yÞÞÞ;
lSH ¼ 2lFF þ mlcrystal ; ð15Þ
based on the binary coding technique described in Eq. (10). This concept is schematically demonstrated in Fig. 6a, where simulation results of the SH beam at far field are presented for three different cases, a periodic structure and two crystals encoded with holographic data. Figure 6b, c corresponds to an experimental demonstration of the concept in an SLT crystal. In this example, in addition to Hermite–Gaussian beams, the demonstration of generating Laguerre–Gaussian [28] beams was also presented. The LG11 mode is a vortex beam [31], carrying a topological charge of l = ?1. The result shows that two-dimensional nonlinear computer-generated holography is a method in which orbital angular momentum can be added through the nonlinear process, so that a non-vortex fundamental beam can be converted into a second-harmonic vortex beam. This type of conversion was theoretically discussed in Ref. [9] and experimentally demonstrated in spiral-shaped and fork-shaped nonlinear photonic crystals in Ref. [32]. Studying the orbital angular momentum relations between the interacting beams allowed formalizing a general orbital angular momentum conservation law,
ð16Þ
where lSH;FF is the topological charge of the beams and lcrystal is the topological charge added by the nonlinear crystal and m is the relevant diffraction order, related to Eq. (11). Experiments in Ref. [32] were performed with Gaussian inputs, and hence lFF ¼ 0, but the general orbital angular momentum conservation law of Eq. (16) was verified and studied in Ref. [33]. The complete experiment of Ref. [33] demonstrated also the radial control available when an FF beam with a topological charge enters a nonlinear crystal encoded with a pattern corresponding to a Laguerre–Gaussian beam. Controlling orbital angular momentum in nonlinear crystals could give rise to new applications of vortex beams in microparticle manipulation and in the field of quantum information [34]. The two-dimensional shaping of a SH beam is not limited only to beams which serve as exact solutions of the paraxial Helmholtz equation, therefore preserving their intensity distribution while propagating after the crystal’s output facet. Furthermore, it is not limited only to selfsimilar shapes which maintain their intensity distribution before and after performing a Fourier transform to the far field, as in the cases of Hermite–Gaussian or Laguerre–
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Fig. 6 (Color online) a A schematic illustration of beam shaping the first order of a Raman–Nath diffraction pattern. b Experimental results for shaping in this scheme, including microscopic pictures of the encoded pattern in the nonlinear crystal and c shaped first diffraction orders
Gaussian beams. Nonlinear Raman–Nath phase-matching condition can be satisfied also when the FF beam enters to the crystal at an angle with respect to the Z-axis, such that the FF and the generated SH beams propagate in a noncollinear fashion in the X–Z plane [19]. The generation of Top-Hat beams at the SH, one with a square cross-section profile and one with a circular cross-section profile, was experimentally demonstrated. The amplitude and the phase of the inverse Fourier transform of a high-order superGaussian function for the first, and of a circ function for the latter, were encoded to poling modulation patterns based on the binary coding scheme of Eq. (15). For a Top-Hat beam with a square cross-section profile, the modulation pattern was calculated according to the inverse Fourier transform of the following transverse spatial distribution: n A0 1 x Aðx; yÞ ¼ exp 1þ if W0n 1 þ if 1 yn exp n ; ð17Þ 1 þ if W0 when A0 is an amplitude constant, W0 is a width parameter, n is an even-order number and fðz; bÞ is defined in terms of a given longitudinal coordinate and the confocal parameter
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[16]. For a Top-Hat beam with a circular cross-section profile and a radius parameter, r0 , the transverse spatial distribution can be expressed as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffi const; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 r0 ; Aðx; yÞ ¼ ð18Þ 0; x2 þ y 2 [ r0 : Thus, the modulation pattern can be calculated according to the term 2J1 ðq=q0 Þ=ðq=q0 Þ, where J1 ðÞ is a Bessel function of order 1 and q0 is a radial constant. In addition, a two-dimensional nonlinear focusing lens, i.e., exploiting a two-dimensional modulation pattern of the nonlinear coefficient in the crystal to focus the generated SH beam after a desired focal length from the crystal’s output facet, is presented. A quadratic phase modulation was encoded to the poling pattern, when based on Eq. (15) the modulation function is given by k2 ðx2 þ y2 Þ dðx; yÞ ¼ dij Sign cos 2pfcarrier x ; ð19Þ 2n2 f where n2 is the refractive index for the SH beam and f is the desired focal length. The experimental setup configuration is similar to the schematic illustration shown in Fig. 6a, with an addition of a tilt angle of the crystal while the structure is illuminated
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by the pump beam. The three poling patterns were implemented in MgO:CLN crystals with a central period length of K ¼ 23 lm. Microscope pictures of the fabricated poling structures that were used for obtaining one of the TopHat beams and the focusing lens are presented in Figs. 7a and 8a, respectively. An e-oo SHG process with a pump wavelength of 1064.5 nm was produced at room temperature, with the crystal tilted by 31° for obtaining the SH Top-Hat shapes and by 9° for obtaining the nonlinear lens shaping. The patterned areas of the squared Top-Hat, the circular Top-Hat and the nonlinear lens were illuminated by a FF beam with waist radii of approximately 500, 780 and 900 lm, respectively, depending on the size of the patterned structure. A comparison between the simulated and the measured results for the intensity profile of the SH Top-Hat beam, with a square cross-section profile, is presented in Fig. 7b.
Fig. 8 (Color online) a Microscope picture of the fabricated poling structure in the crystal that can focus the generated SH beam at the first order of nonlinear Raman–Nath diffraction. b A comparison between the simulated and the measured SH beam radius, both in the horizontal and in the vertical directions, for different propagation distances after the crystal’s output facet. The insets at the bottom of the graph present the evolution of the SH beam’s intensity profile 6 cm before the focus, at the focus, and 6 cm after the focus
Fig. 7 (Color online) a Microscope picture of the fabricated poling structure in the crystal that represents the corresponding pattern of a two-dimensional inverse Fourier transform of a high-order SuperGaussian. The fabricated pattern resembles the main and the first lobes of a two-dimensional sinc function. b A comparison between simulated (left) and measured (right) SH Top-Hat beams, with a square cross-section profile, obtained in Fourier plane at the first order of nonlinear Raman–Nath diffraction. c Three-dimensional representation of the measured intensity profile of the SH Top-Hat beam
A 3D representation of the measured intensity profile is shown in Fig. 7c. The characteristic uniformity of this kind of a beam is reflected, both in the simulation and in the measurement. The SH Top-Hat shapes were obtained at the first order of the nonlinear Raman–Nath diffraction at the far-field Fourier plane after performing an optical Fourier transform. It should be noted that according to this nonlinear beam shaping method, the shapes of the Top-Hat beams can be obtained only in Fourier plane, while outside of it the beam does not maintain its shape. As for the nonlinear lens, a two-dimensional focusing of the output SH beam was obtained at the first order of the nonlinear Raman–Nath diffraction at a distance of approximately 20 cm after the crystal’s output facet, as was originally designed. The focusing of the SH beam is demonstrated in Fig. 8b, with a comparison between the simulated and the measured beam’s radius, in both the horizontal and the vertical directions, for successive propagation distances after the crystal. The only drawback of the concept presented in Figs. 6, 7, and 8 is the low conversion efficiency arising from relying on Raman–Nath phase matching. It turns out that breaking the symmetry of the diffraction experiment in Fig. 6a, in a similar manner as is suggested in Fig. 5b, allows finding work points where the full vectorial phase-
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matching condition is satisfied [13]. Figure 9 demonstrates this idea: In part (a), the concept is illustrated, and in part (b), the feasibility of finding appropriate conditions for satisfying phase matching is examined. It can be seen that a wide range of FF wavelengths can be phase-matched with different values of poling periods and tilting angles. The pattern in the crystal in this case is given by the following expression, dðx; yÞ ¼ dij Signðcosð2px=K /ðx; yÞÞ cosðpqðx; yÞÞÞ; ð20Þ where the period is set by K ¼ 2p cosðhÞ=k2 sinðaÞ, just as in Eq. (14). An experimental verification of this concept was performed in SLT, where an e-ee SHG of a 1,550-nm pump at room temperature was phase-matched with the crystal tilted by 0.86 rad [35] (related to h through Snell’s law). Figure 6c shows microscopic pictures of the fabricated crystal and measured results. Comparing efficiencies with the previous concept presented in Fig. 5 shows an improvement of five orders of magnitude [13], as expected from theory.
When working in a tilted setup, as in Figs. 5b or 9a, the validity of Eq. (10) should be tested, as it was derived assuming an optical beam passes in perpendicular to the hologram. If we strive for a spatial correlation higher than 90 %, the following condition needs to be fulfilled, L 9 tan(h) B 0.45w0, where L is the length of the crystal in the direction of propagation, h is the angle of the pump beam propagation inside the crystal (related to the crystal tilt angle through Snell’s law) and w0 is its waist. The results in Fig. 9c correspond to an FF beam with a wavelength of 1,550 nm. In terms of available input power, it is often desired to work with a wavelength of 1,064 nm; in SLT it is possible, and feasible in terms of fabrication abilities, if one considers the third diffraction order, m = 3 [35]. An additional benefit of the suggested scheme, Fig. 9a, is it allows for an input area of up to approximately 100 cm2 and hence allows entering the nonlinear crystal with high input power avoiding the damage threshold. All the nonlinear spatial holograms demonstrated in this review are based on implementing the concept of planar
Fig. 9 (Color online) a A schematic illustration of shaping a phase-matched non-symmetrical and nonlinear diffraction. b Calculated various phase-matched work points based on the suggested scheme in SLT [35]. Experimental results based on the suggested scheme (Eq. (20)), including microscopic pictures of the structures in the crystal and resulting beams in the SH, c for both HG11 and LG20
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holography. Volume holograms, also referred to as Bragg holograms, can also be introduced to the nonlinear optics regime, as was recently suggested by Hong et al. [36]. 3.2 Functional facets for beam shaping Another option for combining both a nonlinear process and spatial shaping of the beam is patterning the output facet of a nonlinear crystal [37]. In this scheme, the crystal is poled to satisfy a collinear phase-matching condition, as Fig. 1a. A thin layer of gold is evaporated on the output facet of the crystal and patterned with a focused ion beam (FIB). The resulting pattern on the facet serves as a transmittance amplitude mask for both beams, FF and SH, exiting the crystal. The mask can be used to two-dimensionally shape the output beams; the pattern in the gold is given by Eq. (10). Since the diffraction angles of a beam passing through the mask depend on the wavelength of the beam, a spatial separation is achieved between the diffracted FF and SH beams. The first order of the FF beams is located approximately at the second order of the SH beam. The experimental setup and a typical simulation and measured results are shown in Fig. 10. The same idea can be utilized for generating phase masks on the output facet, by using a transparent and conductive material, for example indium tin oxide (ITO) instead of gold. Experimental demonstrations of nonlinear crystals encoded with computer-generated holographic patterns were presented for a limited set of beams, but the general approach apparent from Eqs. (12–15) and (20) enables arbitrary shaping of the output. For example, accelerating beam, such as Airy [7] or parabolic [38] beams, can also be generated using the same schemes. These beams were experimentally generated in the scheme illustrated in Fig. 10 [37].
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3.3 Shaping the phase of the input beam An alternative approach is based on shaping the input beam profile, instead of the poling patterns of the nonlinear crystal. Ellenbogen et al. [39] suggested converting a Gaussian beam into a higher-mode Hermite–Gaussian beam [28] using the cascaded nonlinear processes of sumand difference-frequency generation [16]. This idea relies on the fact that three-wave mixing of a Gaussian beam with an Hermite–Gaussian beam generates an Hermite–Gaussian beam. Laguerre–Gaussian beams were generated in quasi-phase-matched crystals in a sum-frequency generation process by controlling the temperature of the crystal and the wavelength of the interacting beams [40]. Arbitrary shaping the transverse phase of the output beam in a nonlinear process can also be achieved if the phase of the FF beam is tailored correctly [41]. If the crystal is sufficiently short and the fundamental beam is weakly focused, the transverse phase of the second-harmonic output beam is simply twice the transverse phase of the input beam, thereby providing a simple and straightforward manner for controlling the generated beam properties. For longer (and therefore potentially more efficient) crystals, the input phase profile of the fundamental beam can be set, for example, using optimization with genetic algorithms. The key advantage here is the ability to dynamically control the input beam, by using a spatial light modulator or a deformable mirror to shape the fundamental beam’s profile. 3.4 Nonlinear spectral holograms In addition to spatial beam shaping, the output of a nonlinear interaction can also be shaped in the spectral domain [14]. As an example, we will consider the case of sumfrequency generation (SFG) where the result of an interaction between two different input beams, with frequencies x1 and x2, is a new beam with frequency x3 = x1 ? x2, but the same approach can be also used for spectral shaping in a difference-frequency generation process [14] and in some cases may be also extended to second-harmonic generation [42]. Approximating the two input beams as undepleted, the generated beam is the Fourier transform of the pattern along the propagation direction [16], for example the X-axis in Fig. 1a. It can be written as follows, Z1 A3 ðDkÞ / dð xÞeiDkx dx; ð21Þ 1
Fig. 10 (Color online) An illustration of adding functionality to a nonlinear crystal by shaping its output facet. Simulation and experimental results for encoding an LG10 beam
where Dk ¼ k1 þ k2 k3 . Examining the above equation reveals the inverse Fourier relationship between the spatial nonlinear modulation d(x) and the spectral shape of the generated beam. Modulating d(x) is done according to Eq. (10), and the resulting expression is
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dð xÞ ¼ dij SignðcosðDk0 x /ð xÞÞ cosðpqð xÞÞÞ;
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ð22Þ
where Dk0 is the phase mismatch for the central input frequency. An illustration of this scheme and experimental results [15] are presented in Fig. 11. In this example, two poling patterns were used: one that has a spectral response in the form of the HG02 function, and the other one in the form of the Airy function. When a broad input signal is mixed with a narrowband pump in these nonlinear spectral holograms, the spectrum of the generated signal is the product of the signal’s spectrum and the nonlinear crystal’s transmission spectrum. This enables, for example, to select only specific spectral components in the nonlinear mixing process and eliminates the need for additional spectral filters. The method can be also used to control the wavelength dependent phase of the generated signal, thereby enabling temporal shaping of ultrashort pulses in a nonlinear process. It is also possible to combine spectral shaping with spatial shaping, for example, if the Y-axis of the crystal is modulated according to Eq. (13). An interesting recent example of spectral holograms was the realization of ‘‘super-narrow’’ frequency converters [42]. Usually, the width of a frequency converter is inversely proportional to its physical length. This relation originates from the Fourier transform relation between the nonlinearity as a function of space and the efficiency as a function of the phase mismatch. Hence, the only available solution up till now for narrowing the spectral acceptance of a nonlinear converter was by increasing the crystal length. Unfortunately, this is a costly solution, it consumes large physical space and in addition, and the length of the available crystals is usually limited to a few centimeters at most. An alternative new solution is to modulate the
nonlinear coefficient with a super-oscillation [43] function. These are band-limited functions that oscillate locally much faster than their highest Fourier component and, in the context of nonlinear conversion, provide oscillation (in the phase-mismatch coordinates) that is much faster than the inverse of the crystal’s length. A super-narrow converter was experimentally realized, having spectral and thermal response that are narrower by 39 % and 69 % compared to the side lobes and main lobe of the sinc function response of a standard frequency doubling crystal with the same length [42].
4 Summary and Conclusions Different approaches for spatially shaping the output of nonlinear interactions were presented in this review. Shaping one-dimension of the output is possible with both a one- and two-dimensionally patterned nonlinear crystals. Two-dimensional shaping is also possible in two-dimensional structures, allowing for both shaping and high conversion efficiencies. Introducing holographic techniques is also possible in the spectral domain, resulting with generated signals with arbitrary shaped spectrum. These methods save the need for additional external elements such as lenses, splitters and filters by utilizing available fabrication capabilities of nonlinear crystals to incorporate these tasks into the nonlinear device itself. Moreover, the nonlinear process enables all-optical control of the parameters of the generated waves such as the orbital angular momentum in the case of vortex beam or the acceleration in the case of Airy beam. Complete control of the output of a nonlinear
Fig. 11 (Color online) Demonstration of spectral shaping, a spectral input on the left passes through a modulated crystal with two different modulation patterns. Corresponding spectral outputs—on the right
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interaction opens exciting options in the fields of particle manipulation, spectroscopy and optical communications and quantum information. Acknowledgments This work was supported by the Israel Science Foundation (1310/13) and by the Israeli Ministry of Science, Technology and Space in the framework of the Israel–Italy bi-national collaboration program. The authors would like to acknowledge HC Photonics Corporation for the manufacture of the poled MgO:CLN crystals according to our customized designs. Conflict of interest of interest.
The authors declare that they have no conflict
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