Generation of Atomic Optical Lattices by Dammann Gratings

We first calculated the diffraction intensity distributions of the Dammann gratings illuminated by Gaussian light wave. The empirical equations were deduced by numerical calculations to calculate the parameters, such as the spatial period, the maximum intensity, and the maximum intensity gradient, of the optical trap array composed by a set of Dammann gratings and a focus lens. Thus, a novel type of optical trap array for trapping cold atoms (or molecules) was proposed and its features were discussed. The results showed the optical trap array with very short period could be generated. High optical dipole potential could be presented so as to have strong attractive force to the atoms to form atomic optical lattices of high lattice density. Compared with the optical lattices formed by standing wave interferences of CO2 laser, there are many unique advantages of which are formed by Dammann gratings.


Introduction
In early 1970s, the concept of "Dammann grating" was first proposed by Dammann and Görtler [1] when they studied the copy of multiple imaging.The Dammann grating is a kind of phase-type grating with several groups of phase jump points in each period.The positions of the phase jump points can be computed by optimal designing.When the Dammann grating is illuminated by a coherent monochromatic light wave, many beams of diffraction light wave with equal intensity can be generated without the nonuniformity of beam intensity caused by sinc functions in common gratings.Many types [2][3][4][5] of Dammann gratings with high efficiency and large splitting ratio are developed in several decades to satisfy their applications.The orthogonal binary-phase gratings completing 64 × 64 splitting ratio have been successfully fabricated.Due to their advantages, such as the simplicity to be designed [6], the mature production process, and the easy calibration, they have been widely paid attentions to.In 1980s, Dammann grating was applied to the field of optical computing and optical interconnection.In 1990s, Tooley realized the image process of optical cells by using Dammann grating [7].
In recent years, a series of important developments and fruitful results have been obtained both in theory and in experiment on the cooling, trapping, and manipulating the neutral atoms (or molecules).As the deeply developing of the quantum information process and optical lattices [8][9][10], the surface microtrap array for trapping the cold atoms (or molecules) has become the research frontier in the field of atomic, molecular, and optical physics.Schemes of generating surface magnetic lattices, magneto-optical lattices, and Bose-Einstein condensation (BEC) array have been proposed [11][12][13].Birkl and his coworkers realized all-optical array of trapping neutral cold atoms by using micro-trap array generated by a microlens array [14,15].The all-optical array has many advantages over the magnetotrap array.However, many disadvantages exist in the method of using the microlens array to generate the micro-trap array.Producing a large number of the sublens array requires high techniques.For example, the deficiency of the production of the micro-lens array and the nonuniformity of the illuminating light beam will directly affect the uniformity of the intensity distribution of the optical trap array, and additionally, the accompanying stray light will heat the atoms (or molecules) trapped.The difficulty can be effectively overcome by replacing the microlens array with a Dammann grating and a focusing Fourier lens to form the optical trap array.In Section 2, we calculated the intensity distributions of the optical trap array generated by Dammann grating.The disadvantages about  using Dammann grating to form the optical trap array were analyzed, and thus, the scheme of generating the optical trap array with high density of the optical traps was proposed in Section 3. In Section 4, we discussed the potential applications of this optical trap array in atomic and molecular optics.And a summary was stated in the final section.

The Generation and the Intensity Distribution Computation of the Optical Trap Array with Dammann Grating
We only discuss the orthogonal phase-type Dammann grating with the phase value of 0 or π in the present work.The splitting ratio of the grating is K ×K , and the period is T.
The coordinates of the phase jump points of 1D grating in each period are a l , b l (l = 1, 2, . . .L). Here, L is the number of the phase jump points.The phase distribution is shown in Figure 1.The amplitude of the corresponding transmittance function can be written as where rect(•) is the rectangular function [16].To the phasetype Dammann gratings with the splitting ratio of 15 × 15 and 25 × 25, the values [17] of a l and b l are listed in Table 1 (the period of the grating is normalized to T = 1).The transmittance function of the 2D Dammann grating can be written as ( Assuming the number of the period of the grating is N × N, the diffraction light is focused by a lens with a focal length f , the lens and the grating are superposed on x o oy o (z = 0) plane, and the light axis of the lens is along z axis, as is shown in Figure 2. The grating is perpendicularly illuminated by TEM 00 mode of a Gaussian light wave, and the function of the incident light is where P is the output power of the laser, k = 2π/λ l , and λ l is the wavelength of the laser.When z = 0, w(0) = w o , where w o is the waist radius of the light, R(0) → ∞, and ζ(0) = 0.According to the Fresnel diffraction theory, the complex amplitude of light field on the xoy plane, which is located in the back of the lens and has a distance z to the lens, is given by (omitting the constant phase factors) where n and m are integers, 0 ≤ n, m ≤ (N − 1).F(x − x o , y − y o ) and t(x o , y o ), the phase propagation factor of the Fresnel diffraction and the phase transmittance function of the lens, respectively, are as Then, the intensity distribution on the back of lens is I(x, y, z) = |U(x, y, z)| 2 .On the focal plane (z = f ), the complex amplitude can be written as It is shown that U f (x, y, z) is the Fourier transform of the product of the transmission function of the Dammann grating and the Gaussian wave function if the quadratic phase factor before the sum operators is omitted.Calculation shows that the intensity distribution on the focal plane number of principal maximum bright spots, which we call as the optical trap array and the optical trap array centers at (p f λ/T, q f λ/T), where p and q are integers with −K/2 ≤ p, q ≤ K/2.The period of the optical trap array is given by Figure 3 shows the density distribution of the intensity on the focal plane of a 15 × 15 optical trap array.The spatial intensity distributions of each optical trap in the array are basically identical.We numerically calculated the parameters that characterize the intensity distribution of the single optical trap.The results show that the intensity of the optical trap is related with several parameters of the optical system, such as the width of the grating (NT), the focal length of the lens f , the laser power P, and the waist radius w o of the Gaussian light wave.To a given laser power, when the waist radius is set to (0.50 ± 0.01) NT, each optical trap of the array will reach their maximum intensity simultaneously.Additionally, the intensity of each optical trap has a good uniformity when the number of the grating periosd satisfies N ≥ 3. To Dammann gratings with the splitting ratio of 15 and 25, taking w o = NT/2, we numerically calculated many groups of the optical systems with different parameters and got an empirical formulae.The maximum intensity of the optical trap I max can be written as where β = (NT/2)/ f is the effective relative aperture of the lens.Take 1/e 2 of the maximum intensity as the boundary, the transverse and longitudinal widths of the optical trap are given by respectively.Regarding the optical trap as a rotating ellipsoid, the trapping volume can be written as The transverse and longitudinal gradients of the optical trap are given by respectively.Equations ( 8)- (13) show that demanded optical trap arrays can be obtained by varying the parameters of the optical system, including relative aperture of the lens β, the period of the grating T, the number of the period N, the splitting ratio K, and the laser power P. If the power and wavelength of the illuminating light are fixed, the intensity and its gradient of the optical trap increase when we properly increase the relative aperture β, while the trapping volume increases when we decrease β.Note that there is a certain degree of error in ( 9)- (13), which is relative to the number of the period N and the relative aperture β.When N is large while β is small, the error will be little.Equations ( 9)-( 13) are valid when N > 3 and β < 0.1.Furthermore, the maximum intensity of each optical trap in the array is not exactly identical.A factor Uni is defined to scale the maximum intensity uniformity of the optical traps: Table 2 lists the array of optical trap formed by choosing different parameters of the optical system.Here, the intensity distribution parameters listed in rows of the 2nd to 5th are the results numerically calculated after (7).It is coincident with ( 8)- (13).
Taking T = 2.0 mm, N = 20, f = 400 mm, w o = 20 mm, λ = 1.06 μm, and P = 20 W, which correspond to the 2nd row in Table 2, the intensity distribution of a single optical trap in the array is shown in Figure 4. Figure 4(a) gives the intensity distribution contours on the focal plane, and Figure 4(b) shows the corresponding 2D intensity distribution.The maximum intensity of the optical trap is I max = 1.55 × 10 8 Wm −2 .The intensity distribution on the xoy plane is close to Gaussian distribution, and the intensity gradient along x (or y) direction is reached to a magnitude of 10 13 Wm −3 .Also, the intensity distribution along z direction is close to Gaussian distribution.The corresponding volume of the optical trap V 1/e 2 is about 10 −7 cm 3 .The spatial period of the optical array is about 0.2 mm.
Through analyzing the geometrical parameters of the optical trap Δx 1/e 2 , Δy 1/e 2 , Δz 1/e 2 , and V 1/e 2 , we can find that each optical trap in the array generated by Dammnn grating is quite similar with that generated by a single square lens.It implies that the combination of Dammann grating and a lens is equivalent to the lens array composed by K × K number of microlens.However, generating the optical trap array by Dammnn grating has its unique advantages.The assembling is very simple; only one lens is needed to obtain the number of K 2 optical traps.Furthermore, the intensity distributions of the optical trap in the array are basically uniform when the intensity distribution of the illuminating light is inhomogeneous as the Gaussian light wave, while if adopting microlens array, the parameters of each micro-lens and the light wave illuminating each lens are required to be uniform.

Generation of High-Density Optical Trap Array by Dammann Grating
It is well known that Dammann grating is an optical element of far field diffraction.Calculating intensity distribution of diffraction by using (4) requires that the relative aperture, β = NT/(2 f ), of the lens should be small.It can be known from ( 8) that the distance between two optical traps in the array generated by Dammann grating is Δl = f λ/T = Nλ/2β.Clearly, N, the number of the period of the grating, can not be small.So, when the wavelength of the illuminating light wave is not very short or the diffraction light is focused by a common spherical lens, the period of the optical trap evidently should be large.This will result in low optical trap density number in the array.In practical applications, however, we need high-density optical trap arrays.For example, when employing the array to form an optical lattice to trap cold atoms (molecules), we want the optical lattice constant, Δl, to be small.It is well known that the wavelength of the illuminating light λ l should be longer than the resonant wavelength λ a of the cold atoms when the atoms are trapped at strong-field-seeking state.Consequently, large relative aperture of the focal lens is required to generate high-density optical trap array by Dammann grating.But the approximate condition of Fresnel diffraction is not valid any more in this case; that is, (5) or (7) or their deduced (8)-( 13) are not valid any more.Sommerfeld diffraction theory is now needed to calculate the intensity distribution.Equation ( 5) should be corrected as and 1/z term in (4) now should also be replaced as If the phase transmittance function of the lens is still as described in ( 6), there would be large error of the diffraction intensity calculated by Fresnel diffraction theory when β > 0.1, while identical intensity and identical spaced optical array cannot be obtained through calculations by Sommerfeld diffraction theory.When the splitting ratio K of the grating is not very large and the distance between two adjacent optical traps is small,  the diffraction intensity mainly distributes near the focal point.According to aplanatic theory, ( 6) can be corrected as where α 1 and α 2 are two constants that can be determined by λ l and β.Choosing λ = 0.53 μm, which can be obtained by frequency doubling of YAG laser with wavelength of 1.06 μm, numerical calculation shows that, when β 0.2, we can get that α 1 = 0.9990 and α 2 = 0.9299.Substituting ( 14) and ( 16) into (7) and calculating the diffraction intensity distribution focused by lens with large aperture show that relatively uniform optical trap array can still be obtained.
Taking β = 0.2, two kinds of optical trap arrays were calculated and the results are listed at rows of the 6th and 7th in Table 2.In the 6th row, T = 40 mm, N = 4, the space between two optical traps is Δl = 5.3 μm; this is the same with the ones generated by CO 2 laser standing wave (Δl = λ CO2 /2 = 5.3 μm). Figure 5 gives the intensity distributions of the optical traps on the focal plane.Figures 5(a) and 5(b) show the intensity distributions of a single trap.Comparing with Figure 4, it can be found that the intensity distributions of the two kinds of optical traps are almost the same.Figure 5(d) gives the 1D intensity distribution passing through the center of the optical trap.It can be found that the intensity distributions of the optical trap array are still approximately the same as Fraunhofer diffraction if the system is focused by the modified lens, only the uniformity of the intensity distribution slightly decreases, which will affect little the formation of the optical lattices.The intensity I max = 1.34 × 10 10 Wm −2 in the table is the average value of the maximum light intensities of all optical traps in the array.In the 7th row where T = 53.3mm and N = 3, the space between two optical traps is Δl = 3.97 μm; other parameters are almost the same as those in 6th row.

Potential Applications of the Optical Trap
Array Generated by Dammann Grating 4.1.Optical Trapping of Cold Atoms.It is well known a two-level atom moved in an inhomogeneous light field will undergo dipole force owing to the ac Stark effect.When the light field is red detuned; that is, the laser frequency ω l is less than the atom resonant frequency ω a and the atom will experience gravitational effect of the light field and be attracted to the position of maximum intensity.When the absolute value of the detuning δ (δ = ω l − ω a < 0) is large, the interactional dipole potential is given by [14] On the other hand, the spontaneous emission will take place when a two-level atom moves in the light field.The scattering rate can be estimated as where r is the spatial position, Γ is the natural line width of the atoms, and I( r) is the spatial distribution of the intensity.
The attractive force, F ( r) = −∇U D ( r), of the light field to the atoms is directly relative to the intensity gradient.
The dipole potential and spontaneous emission rate of the trapped cold atoms can be calculated after (17) and (18) with the values of Γ and ω a reported in references of atomic optics.Here, some atoms were chosen; and the results were calculated as listed in Table 2.It can be found that although the illuminating lights are the low-power laser in common use, the dipole potential is far higher than the temperature of the cold atoms in optical molasses (∼20 μK), and the spontaneous emission rates are all low.Consequently, the optical trap can be used to trap cold atoms under this condition.Properly increasing the laser power, the array of optical trap can also be used to trap cold molecules.

Fabrication of Optical Lattices.
Optical lattice, similar to the crystal structure in solid-state physics, is formed by ensemble cold atoms (or molecules) into the periodical optical potentials generated by optical trap arrays to form a periodical arrangement in spatial of cold atoms (or molecules) [18][19][20].The optical lattice, on the one hand, can be used to study the dynamics process of cold atoms in the lattice, such as Bloch oscillations, quantum tunneling effect, coherent manipulation and control of atomic wave packet, and coherent transfer and control quantum states [21][22][23].On the other hand, it can be used to experimentally study of quantum tunneling effect of BEC atoms and their macroscopic quantum interferences.In addition, it may have wide and  important applications in the field of quantum calculation and quantum information processing [24][25][26].
Different kinds of optical lattices can be fabricated by using the optical trap arrays listed in Table 2. On the one hand, when the relative aperture β of the optical system is relatively small and the wavelength of the illuminating light is relatively long, the lattice constant and the volume of the obtained optical trap are relatively large, as to the arrays listed in 2nd row.The lattice constant is about 0.2 mm, and the volume of a single optical trap can reach 1.65 × 10 −7 cm 3 .To the BEC sodium atomic gas with a density of 10 13 ∼ 10 14 /cm 3 [27], the trapped atoms of a single trap are more than 10 6 .On the other hand, when the relative aperture β is relatively large and the wavelength of the illuminating light is relatively short, the lattice constant and the volume of the optical trap are very small.To the arrays listed in 6th and 7th row, the volume of a single optical trap is as small as 10 −11 cm 3 .For the atomic gas with thin density, each optical trap can ensemble one or two atoms.Atomic lattices with high density can be formed with small lattice constant (5.3 μm and 3.97 μm).
In recent years, researchers have proposed various schemes [28,29] of generating optical lattices with standing wave interferences of CO 2 laser and studied them both theoretically and experimentally.Figure 6 shows the 2D array of optical trap generated by four CO 2 lasers, where two of them propagate reversely to each other.The lattice constants of the array of optical traps showed in Figures 5 and 6 are same.Comparing Figures 5(c) and 5(d) with Figures 6(a) and 6(b), one can easily find their differences.Firstly, the intensity surrounding each optical trap is zero in the optical trap array generated by Dammann grating.Though there have secondary maximum intensity between two neighbored optical trap, it is far lower than the maximum intensity.However, in the array of optical trap formed by standing wave interferences of CO 2 laser, the secondary maximum intensity is half of the maximum intensity to result in the effective depth of the optical trap which is half of the maximum dipole potential.The secondary maximum intensity will heat up cold atoms.Secondly, the intensity of the optical trap array formed by standing wave interferences of CO 2 laser changes by a rule of sine function.This will result in that, on the one hand, the intensity gradient is small and the attractive force used to trap cold atoms is correspondently small, on the other hand, the width of the optical trap is almost equal to the lattice constant resulting in thatthe condense ratio of the intensity distribution, which is the ratio between the total area of the array of optical trap and the area of the optical trap, is too small to condense the atoms together.However, in the optical trap array formed by Dammann grating, the transverse width of the optical trap is far smaller than the lattice constant.So, the condense ratio of the intensity distribution and the intensity gradient are large.Correspondingly, the attractive force of the light field to the cold atoms is large.It is easier to condense the cold atoms to a centralized space to form a lattice structure.Therefore, there are many advantages to generate array of optical trap by using Dammann grating over using standing wave interferences of CO 2 laser.

Conclusion
In this paper, we calculated the diffraction intensity distributions of the Dammann gratings illuminated by Gaussian light wave.Also, we deduced the empirical equations for calculating the parameters of the optical trap array, such as the spatial period, the maximum intensity, the maximum intensity gradient, and the trap width, by numerical calculations.Different optical trap array can be obtained by adjusting the parameters of the optical system according to these equations.The optical trap array generated by Dammann grating is similar to that formed by micro-lens array however, the former has many advantages over the latter.Nevertheless, we analyzed the disadvantages of the scheme that generates the optical trap array by Dammann grating and proposed a novel experimental scheme to generate high-density optical trap array by using aplanatic lens to focus the diffraction light of Dammann grating.The result shows that the present array of optical trap with small period can be formed.Comparing with the optical lattices formed by standing wave interferences of CO 2 laser, there are many unique advantages.For instance, the optical lattices have high optical dipole potential, and the optical traps have strong attractive force to the cold atoms.

Figure 2 :
Figure 2: Light path of the optical trap array employing a set of a Dammann grating and a confus lens.

Figure 5 :
Figure 5: Intensity distribution of the array of optical trap with high density generated by Dammann grating (β = 0.2, N = 4, K = 15, P = 10 W, λ l = 0.53 μm).(a) Intensity isoline of a single optical trap; (b) 2D intensity distribution of a single optical trap; (c) density distribution of the intensity for the array of optical trap (fraction); (d) 1D intensity distribution along x direction that passes through the center of optical trap.

Figure 6 :
Figure 6: Array of optical trap generated by standing wave interferences of CO 2 laser (λ = 10.6 μm).(a) Intensity distribution of 2D array of optical trap (fraction); (b) 1D intensity distribution along x direction that passes through the center of optical trap.

Table 1 :
Coordinates of the phase jump points of two Dammann gratings.

Table 2 :
Optical trap arrays generated by different optical system (w