All-optical Fresnel lens in coherent media: controlling image with image

L. Zhao; Wenhui Duan; S. F. Yelin

doi:10.1364/OE.19.000981

1. Introduction

Fresnel lenses, as a well-known diffractive element in optics, play an important role in many areas such as optical communications and information processing, optical imaging systems, and vision correction devices. Recently, substantial progress has been made in developing Fresnel lenses based on electrically controllable liquid crystal (LC) materials [1–6]. These LC lenses with Fresnel zone structures can produce tunable focusing and imaging for incident light fields by diffraction and thus dramatically extend the functionality of the lenses. However, the LC Fresnel lenses also present several drawbacks. In a binary-phase LC Fresnel lens, the theoretical limit of the maximum diffraction efficiency (i.e., the ratio of energy obtained at the primary focus to incident energy) is very low (only 41%) at a phase difference of π between the neighboring zones [1–3]. Multilevel-phase LC Fresnel lenses usually entail complicated microfabrication processes to create discrete phase levels in each zone [4–6]. Furthermore, the inertia of LC molecules leads to slow response times of the lenses (> millisecond), and thus one may encounter serious difficulties in high-speed and large-capacity optical signal processing. In order to overcome these obstacles, all-optical imaging systems become an attractive option because they eliminate the intermediate steps of optical-electrical-optical signal conversion, which means simple device structures and high processing speed and efficiency.

In recent decades coherent media based on electromagnetically induced transparency (EIT) have been intensively studied [7, 8]. In EIT media, the dispersion and absorption of a weak probe light field can be coherently and dynamically modified by a strong coupling light field when both fields interact with appropriate atomic transitions. This is the basis for EIT’s usefulness for all-optical systems. In previous work, due to the inhomogeneous transverse intensity profiles of the coupling fields in EIT media, a phenomenon known as electromagnetically induced focusing (EIF) was observed on the probe fields in both hot and cold atomic ensembles [9–11]. However, because the transverse variation of the induced refractive index in EIF is far from quadratic, it is hard to accurately converge the probe fields, which may restrict the applicability of EIF in all-optical imaging. Very recently, we have suggested that optical intensity images in coupling fields can pattern EIT media in a controlled way and thus efficiently and rapidly diffract probe fields into desired optical vortex modes [12]. Based on this proposal, it is notable that different coupling images can generate a variety of spatial patterns to diffract probe fields in different manners. This effect can greatly enhance the flexibility and controllability to construct all-optical diffractive devices in EIT media, thereby satisfying the increasing demand for diverse optical performance and functionality.

In this paper, our goal is to introduce an all-optical model for constructing diffractive Fresnel lenses based on the space-dependent coherent optical effects in EIT media. We will analyze how to accurately manipulate the modulo-2π quadratic phase profiles to create Fresnel lenses in EIT media using intensity-modulated concentric ring images of the coupling fields. Then, we will present numerical simulations of particular Fresnel lenses and study the focusing and imaging properties for a probe light field. In detail, using the fast Fourier transform algorithm, we show that the images in coupling fields can flexibly control the images in probe fields by diffraction. Furthermore, due to the fast response of atoms to light fields in EIT media, image-bearing square pulse trains can be employed in coupling fields to rapidly modulate the lenses. Our results show that the EIT-based all-optical lenses can be generated with simple techniques (e.g., the concentric ring images of the coupling fields can be easily produced using amplitude masks and glass lenses.), have large focal length tunability from 1 m to infinity, and can operate with high modulation speed (∼ megahertz), high transmission and diffraction efficiency (> 88%). Additionally, we will also discuss how well present EIT experimental techniques could realize our proposal. As a consequence, those lenses could potentially improve functional optical devices in the applications of image processing, optical correlation, remote sensing, optical communications and quantum information, and others.

2. Theoretical model

In order to create an ideal Fresnel lens for incident light fields, a modulo-2π quadratic phase profile should be imprinted in optical materials, which results in multiple zones [Fig. 1(a)]. Assuming that the desired Fresnel lens has a focal length of f_F, the outer radius of the mth zone is then given by [5]

\begin{array}{l} r_{m} = \sqrt{2 m f_{F} λ}, & m = 1, 2, \dots, M, \end{array}

where λ is the wavelength of the incident field and M is the total number of the zones.

Fig. 1 (Color online) (a) Modulo-2π quadratic phase profile for generating a Fresnel lens to focus and image incident light fields by diffraction. For simplicity, we only draw four zones in the phase profile. (b) Energy-level scheme of a Λ-type EIT system with two copropagating light fields interacting with three atomic energy levels. Intensity-modulated images can be adopted in the strong coupling field to pattern the EIT system and modulate the probe field.

Download Full Size | PDF

To generate a Fresnel lens in EIT media, let us first consider a Λ-type EIT system where two continuous-wave (cw) light fields interact with three atomic energy levels [Fig. 1(b)]. The weak probe light field E_p is resonant with the transition |1〉 ↔ |3〉 (ω_p = ω₃₁). The strong coupling light field E_c drives the transition |2〉 ↔ |3〉 with a small single photon detuning Δ = ω_c − ω₃₂. Accordingly, the Rabi frequencies of the probe and coupling fields are given by Ω_p = μ₁₃E_p/h̄ and Ω_c = μ₂₃E_c/h̄, where μ₁₃ and μ₂₃ are the dipole moments of the corresponding transitions, and Ω_p ≪ Ω_c under the EIT condition. If hot atomic vapor is adopted, we need to additionally assume that the two light fields are copropagating and the coupling light field is sufficiently strong (i.e., $Ω_{c}^{2} ≫ Δ ω_{D} δ ω_{D}$ , where Δω_D is the single-photon Doppler broadening and δω_D is the two-photon Doppler broadening) in order to suppress the influence of Doppler effect [13, 14]. This problem will be discussed in detail in Section 4. According to the derivations in Ref. [7], the linear susceptibility for the cw probe field is given by

χ = χ' + i χ ″ = K \frac{Δ Ω_{c}^{2} + i [2 Δ^{2} Γ + γ (Ω_{c}^{2} + Γ γ) / 2]}{{| Ω_{c}^{2} + Γ (γ + i 2 Δ) |}^{2}},

where the coefficient

K = 4 μ_{13}^{2} N / (ɛ_{0} \bar{h})

, N is the atomic number density, Γ represents the decay rate from |3〉 to |1〉, and γ represents the nonradiative decoherence rate between |1〉 and |2〉. Owing to the strong coupling field in the EIT system, i.e.,

Ω_{c} ≫ {Γ, Δ} ≫ γ,

Eq. (2) can be rewritten as

χ' = \frac{K Δ}{Ω_{c}^{2}} and χ ″ = K \frac{2 Δ^{2} Γ + γ Ω_{c}^{2} / 2}{Ω_{c}^{4}},

which tell us that, for a constant single-photon detuning Δ, transverse intensity images in the coupling field (i.e., the spatial variation of Ω_c) can pattern the EIT system to induce position-dependent variations in the dispersion (χ′) and absorption (χ″) of the probe field (i.e., the spatial variations in χ′ and χ″). This effect can simultaneously cause spatial phase and amplitude modulation for the probe light. In addition, it is known that, with appropriate parameters in the EIT transparency window, χ″ can usually be two or three orders of magnitude smaller than χ′ (e.g., χ′ ≈ 10⁻³ and χ″ ≈ 10⁻⁶, which will be shown later in the numerical simulations of our scheme). This suggests that the phase modulation should dominate over the amplitude modulation under some conditions in the patterned EIT system.

Based on Eqs. (4) and the geometry shown in Eq. (1), one should design the Rabi frequency of the coupling field in the piecewise form of

Ω_{c} (r) = Γ {[ξ mod (- \frac{r^{2}}{r_{1}^{2}}, 1) + ζ]}^{- 1 / 2},

where “mod” denotes the modulo operation, r₁ = (2f_Fλ_p)^1/2 is the outer radius of the first zone, and λ_p is the wavelength of the probe light. The parameters ξ and ζ are constants that can be chosen freely as long as the strong coupling condition Eq. (3) is satisfied. Accordingly, the intensity-modulated profile shown as a concentric ring image is thus given by

I_{c} (r) = \frac{1}{2} ɛ_{0} c E_{c}^{2} = \frac{ɛ_{0} c {\bar{h}}^{2} Ω_{c}^{2} (r)}{2 μ_{23}^{2}} = \frac{ɛ_{0} c {\bar{h}}^{2} Γ^{2}}{2 μ_{23}^{2}} {⌈ ξ mod (- \frac{r^{2}}{r_{1}^{2}}, 1) + ζ ⌉}^{- 1},

where c is the speed of light in vacuum. Furthermore, the dispersion χ′ in Eqs. (4) can be rewritten as

χ' (r) = \frac{K Δ}{Γ^{2}} [ξ mod (- \frac{r^{2}}{r_{1}^{2}}, 1) + ζ] .

Thus χ′ is a quadratic function of the radius r and continuous in each zone. To achieve the 2π phase shift in each zone, the optical path difference should be

[χ' (r \to r_{m}) - χ' (r = r_{m - 1})] d / 2 = λ_{p},

where d is the thickness of the EIT medium.

On the other hand, accompanying the phase modulation in the EIT system, there exists the amplitude modulation induced by the spatial variation of χ″ in Eqs. (4). That is

χ ″ (r) = \frac{2 K Δ^{2}}{Γ^{3}} {[ξ mod (- \frac{r^{2}}{r_{1}^{2}}, 1) + ζ]}^{2} + \frac{K γ}{2 Γ^{2}} [ξ mod (- \frac{r^{2}}{r_{1}^{2}}, 1) + ζ],

which is a quartic function of the radius r. Within the EIT transparency window, because χ″ is usually much smaller than unity, low absorption losses can be achieved for the modulation of the probe light.

In the design of the EIT-based Fresnel lens, the thickness d of the lens (and thus the EIT medium) is an important parameter. Intuitively, the lens cannot be very thick. Otherwise, volume (i.e., Bragg) diffraction effects will make it experimentally much harder to satisfy the path difference condition of Eq. (8). In order to determine the limit of d for Eq. (8) we discuss now the propagation of the probe light inside the EIT medium. For simplicity, we consider here a cylindrical Fresnel lens instead of the circular Fresnel lens mentioned above, where the zones are multiple slits with different widths instead of multiple rings. The geometrical parameter r_m in Eq. (1) should be replaced by x_m redefined as the distance from the lens center line to the outer edge of the mth zone (slit). Under the paraxial approximation (i.e., k_p,x ≪ k_p,z ≈ k_p and |∂²E_p/∂z²| ≪ k_p|∂E_p/∂z|), we obtain the paraxial propagation equation of the probe field

\frac{\partial^{2} E_{p}}{\partial x^{2}} + 2 i k_{p} \frac{\partial E_{p}}{\partial z} = - k_{p}^{2} χ E_{p},

where k_p = 2π/λ_p is the wave vector of the probe field. As an example, we simulate the propagation in only one slit. Also, we take into account the relationship χ″ ≪ χ′ ≪ 1. Therefore, the paraxial equation can be analogous to the well-known Schrödinger equation describing harmonic oscillator. By utilizing the path integral propagator [15], and assuming that the incident probe field is a uniform plane wave, the simplified formal solution of the transmitted probe field at the exit of the EIT system is

E_{p}^{z} (d) = E_{p}^{z} (0) exp (\frac{i}{2} k_{p} χ^{z} d) \times \frac{1}{\sqrt{i λ_{p} d}} \int_{x_{m - 1}}^{x_{m}} exp [\frac{i k_{p}}{2 d} {(x - x')}^{2}] d x',

where the superscript [^z] denotes the quantities in the single zone (slit), d is the thickness of the EIT system, E_p(0) is the incident probe field, and E_p(d) is the transmitted probe field immediately behind the EIT system. Note that the integral term in Eq. (11) stands for the diffraction effect of the probe field, and it is the well-known Fresnel’s integral [16]. If we set the Fresnel number

N_{F} = \frac{w_{m}^{2}}{4 λ_{p} d} ≫ 1,

where w_m = x_m − x_m₋₁ is the width of the mth zone, then we can find

\frac{1}{\sqrt{i λ_{p} d}} \int_{x_{m - 1}}^{x_{m}} exp [\frac{i k_{p}}{2 d} {(x - x')}^{2}] d x' \approx 1

within the zone, which means that the diffraction of the probe field inside the EIT system can be ignored, and the geometrical shape of the zone can be preserved in the transmitted probe field at the exit of the patterned EIT medium [16]. The transmitted probe field thus takes the regular expression of

E_{p} (d) = E_{p} (0) exp (i k_{p} χ d / 2),

which ensures the validity of Eq. (8) for the optical path difference. This confirms scalar diffraction theory [16], where a large Fresnel number means that geometrical optics gives the correct result. Therefore, the modification for the transmitted probe field at the exit of the EIT medium predominantly results from the phase and amplitude modulation directly imposed by the EIT system. Equation (12) gives an additional motivation for d to be small, i.e.,

d ≪ w_{m}^{2} / (4 λ_{p})

. This result is also consistent with the diffraction criterion in the Raman-Nath regime for thin acousto-optic spatial light modulators and thin holographic gratings (refer to Chapters 8 and 9 in Ref. [16]). In addition, according to Eq. (14), the transmission rate of the probe field is

T_{p} = exp (- α d),

where α = k_pχ″ is the absorption coefficient.

In order to simulate an all-optical Fresnel lens, we assume ⁸⁷Rb D₁ line (5S_1/2 ↔ 5P_1/2) to establish the Λ-type EIT system in a hot vapor cell. For example, the atomic states |5S_1/2, F = 1〉, |5S_1/2, F = 2〉, and |5P_1/2, F′ = 2〉 can correspond to the energy levels |1〉, |2〉, and |3〉, respectively, where the probe and coupling fields are orthogonally linearly polarized and λ_p ≈ λ_c ≈ 795 nm [17]. The atomic parameters can be given by μ₁₃ = μ₂₃ = 2.54 × 10⁻²⁹ C m, Γ = 36.13 × 10⁶/s, γ = 2π × 3000/s, Δ = 0.5Γ, and N = 3 × 10¹³/cm³ [18]. We also choose the parameters ξ = 3/2116 and ζ = 1/2116 in Eq. (5), which yield 23Γ ≤ Ω_c(x) ≤ 46Γ to satisfy the strong-coupling-field condition. Furthermore, supposing that the focal length of the induced Fresnel lens is f_F = 1 m, Eq. (1) can give the outer radiuses of r₁ = 1.26 mm, r₂ = 1.78 mm, r₃ = 2.18 mm, and r₄ = 2.52 mm for the first four zones, where the width of the narrowest (4th) zone is w₄ = r₄ − r₃ = 0.34 mm. Additionally, the thickness d of the EIT system can be calculated using Eq. (8). Based on the above parameters, the value is given by d = 977.5 m. The minimum Fresnel number given by the 4th zone is $N_{F} = w_{4}^{2} / (4 λ_{p} d) = 36.7 ≫ 1$ . Therefore, a thin Fresnel lens can be effectively constructed in the EIT system. The characteristics of this EIT system, including the coupling intensity, the susceptibility, and the transmission rate of the probe field, are numerically depicted in Fig. 2. Note that the thin EIT system also simultaneously offers some other benefits for our scheme. For example, in such a short propagation distance, the diffraction of the coupling field is negligible, which can produce high-resolution coupling images to pattern the EIT system. Besides, the transmission rate of the probe field is so high (> 96%) that the amplitude modulation can be further suppressed.

Fig. 2 (Color online) (a) Numerical illustration of the intensity profile of the image in the coupling field based on Eq. (6), where the image gives four zones (i.e., M = 4) and their outer radiuses are r₁ = 1.26 mm, r₂ = 1.78 mm, r₃ = 2.18 mm, and r₄ = 2.52 mm, respectively. For example, in the first zone, we have I_c(0) = 1.58 W/cm² and I_c(r → r₁) = 6.33 W/cm². (b) The induced r-dependent susceptibility χ′ (solid curve) and χ″ (dotted curve). Therefore, both phase and amplitude modulation can be induced for the probe field. (c) High transmission rate (T > 96%) of the probe field, which suggests that the amplitude modulation should be weak.

Download Full Size | PDF

3. Optical performance

To evaluate the performance of the induced Fresnel lens, we numerically investigate its focusing and imaging properties which may be influenced by the amplitude modulation (absorption) given by Eq. (9) and the finite aperture shown in Fig. 2.

At first we calculate the intensity distribution in the focal plane including both the absorption and the finite aperture of the induced lens. For simplicity, we assume the normally incident probe field is a unit-amplitude plane wave [i.e., E_p(0) = 1] and the lens (aperture) diameter is 2r₄ = 5.04 mm. The electric field in the focal plane is given by [19]

E_{f}^{N} (r_{f}) = c_{0} \int_{0}^{2 π} \int_{0}^{r_{4}} exp [- k_{p} χ ″ d / 2 - i k_{p} r r_{f} cos (ϕ - ϕ_{f}) / f_{F}] d r d ϕ,

where (r, ϕ) are the polar coordinates in the lens plane, (r_f, ϕ_f) are the polar coordinates in the focal plane, c₀ is the normalization coefficient in the absence of the absorption, and

E_{f}^{N}

is independent of ϕ_f according to the azimuthal symmetry. Notice that Eq. (16) can also be treated as the point spread function in the focal plane (f_F = 1 m), which gives a measure for the imaging quality of the induced lens [16]. The normalized intensity is given by

I_{f}^{N} (r_{f}) = {| E_{f}^{N} (r_{f}) |}^{2}

(Fig. 3). It can be clearly seen that the size of the focal point described by the full width at half maximum (FWHM) is predominantly determined by the diffraction of the finite aperture. The amplitude modulation only slightly lowers the intensity maximum to ≈ 98%, and thus almost has no influence on the size (FWHM = 0.162 mm). This fact also tells us that about 98% of the incident intensity can be focused near the focal point in the focal plane, which means that the induced lens has a high transmission efficiency. Therefore, the amplitude modulation (absorption) for the probe field can be ignored throughout the induced lens.

Fig. 3 (Color online) (a) Intensity profiles in the focal plane with (solid curve) and without (dotted curve) the amplitude modulation in the induced lens. The amplitude modulation (i.e., absorption) only slightly lower the principal maximum to ≈ 98% (blue box, also shown in (b) for detailed information) and almost has no influence on the FWHM.

Download Full Size | PDF

In order to indicate the image formation through the induced Fresnel lens, we redesign the system by putting a thin glass lens L_G with a focal length of f_G = f_F = 1 m immediately behind the induced lens (vapor cell). Thus these two lenses can constitute an all-optically tunable compound lens. The motivation for such design is that this type of hybrid diffractive-refractive lens could be, in practice, a simple and useful configuration to suppress image aberrations and improve image quality [20]. The focal length of the compound lens is given by f = f_Ff_G/(f_F + f_G) = 0.5 m [19]. The letters “TH” representing the initials of the institute “Tsing Hua” are used as the object. It is normally illuminated by the unit-amplitude plane-wave probe field and placed one meter before the compound lens (i.e., the object distance p = 2 f = 1 m) [Fig. 4(a)]. We also assume that the aperture of the glass lens is larger than that of the induced Fresnel lens. Thus, the aperture of the compound lens is determined by that of the Fresnel lens with the diameter of 2r₄ = 5.04 mm [Fig. 4(b)]. According to the lensmaker’s formula, a photodetector can be placed one meter behind the lens to measure the image (i.e., t = q = 2 f = 1 m, where t is the photodetector distance and q is the image distance). Ideally, the image should be inverted and have the same size as the object. However, considering the finite aperture of the compound lens, we numerically calculate the image formation [Fig. 4(c)] using the fast Fourier transform (FFT) algorithm [21], where the weak absorption of the EIT system is ignored.

Fig. 4 (Color online) (a) Two letters “TH” as the original object in the probe field. (b) Image in the coupling field to induce the four-zone Fresnel lens in the EIT system, where its intensity and diameter are numerically illustrated in Fig. 2(a). Note that, in our scheme, a glass lens is put right behind the EIT system to constitute a compound lens for imaging. Moreover, using the FFT algorithm, we can numerically calculate the image formation through the compound lens. (c) Image of the letters “TH” in the probe field generated by the compound lens and measured by the photodetector. The size of this picture (and all the following pictures of the images) is 9.5 × 9.5 mm². (d) Image with a uniform intensity in the coupling field, which actually induces an aperture and leads to an infinite focal length for the EIT system. Consequently, a new compound lens can be produced for imaging. (e) Frauhofer diffraction pattern of the letters “TH” in the probe field generated by the new compound lens and measured by the photodetector. (f) Image in the coupling field to induce the three-zone Fresnel lens in the EIT system. (g) Corresponding fuzzy image received by the photodetector. The color bar on the right gives the normalized intensity of the probe images.

Download Full Size | PDF

To clearly demonstrate the tunability of the lens in our scheme, we can change the image in the coupling field from the modulated intensity [Fig. 4(b)] to, for example, a uniform intensity [Fig. 4(d)], whereas other trivial tunable parameters such as the positions of the object “TH” and the photodetector are kept unchanged (i.e., p = t = 1 m). Also, we assume that the uniform intensity is equal to the maximum value of the modulated intensity [e.g. Ω_uni = Ω_c(r → r₁) = 46Γ]. Thus the EIT system works as a finite aperture and the focal length of the EIT system becomes infinite (i.e., f_F = +∞). Accordingly, the focal length of the compound lens is changed from f = 0.5 m to f′ = f_G = 1 m, while its aperture is still determined by the coupling image with the same diameter of 2r₄ = 5.04 mm. In this case, the object “TH” and the photodetector are actually located at the front and back focal planes of this new compound lens. The photodetector will receive the Fraunhofer diffraction pattern of the object “TH” instead of its image. Note that this Fraunhofer diffraction pattern is also limited by the finite aperture of the compound lens. Again, we use the FFT algorithm to numerically illustrate the Fraunhofer pattern [Fig. 4(e)]. As a consequence, one can clearly see the large focal length tunability of the EIT-based Fresnel lens from 1 m to infinity and, correspondingly, the focal length of the compound lens can be doubled from 0.5 m to 1 m. The probe field at the photodetector can be substantially converted between the inverted image of “TH” [Fig. 4(c)] and the Fraunhofer pattern of “TH” [Fig. 4(e)], thereby indicating a significant spatial switching effect.

In principle, by choosing other coupling images with proper intensity distributions, different Fresnel lenses can be induced in the EIT system. Accordingly, the focal length of the compound lens can be tuned flexibly and thus one can obtain different intensity distributions of probe fields. For example, we can assume the focal length of the induced Fresnel lens to be 1.33 m. Using the derivations in Section 2, we can generate a three-zone Fresnel lens having the same diameter of 5.04 mm as the four-zone Fresnel lens mentioned above. For simplicity, we here do not show the coupling intensity, the susceptibility, and the transmission of the corresponding EIT system for the three-zone Fresnel lens because they are similar to the properties of the four-zone Fresnel lens (see Fig. 2) except the number of zones. In this case, the focal length of the compound lens becomes f″ = 0.57 m. If the object distance and the photodetector distance are still fixed at 1 m (i.e., p = t = 1 m). Considering the finite aperture of the compound lens [Fig. 4(f)], we again use the FFT algorithm to numerically calculate the intensity distribution of the probe field (i.e., a fuzzy image of “TH”) at the photodetector [Fig. 4(g)]. This distribution clearly gives an intermediate stage of the evolution of the probe field from the inverted image of “TH” to the Fraunhofer pattern of “TH”. Therefore, the images in the probe field can be flexibly controlled by the images in the coupling field.

To practically realize the switching between different coupling images [e.g., Figs. 4(b) and 4(d)] and thus modulate the induced Fresnel lenses rapidly, image-bearing square pulse trains could be employed (refer to the proposed experimental setup in Ref. [12]). However, these pulse trains impose an additional frequency broadening (δF = 1/T) on the coupling field, where T is the repetition period of the pulse trains. The modulation speed is thus limited by this broadening effect. To evaluate the speed of our scheme, we analyze the atomic response to the light fields using the parameters mentioned earlier in Section 2. Considering the similarity between the parameters in the different zones, we here use the first zone as an example to do the evaluation. First, the switching of the images results in the re-establishment of the atomic coherence in the EIT system. Assuming that, initially, the coupling field carries the image shown in Fig. 4(b), then it is switched to the uniform illumination in Fig. 4(d). The re-establishment time can be estimated by $τ_{r} = α d Γ / Ω_{uni}^{2},$ where α is the absorption coefficient given by Eq. (15), and Ω_uni = 46Γ. Because the strongest absorption occurs at r = 0, the longest re-establishment time is τ_r = 3.32 ps [22, 23]. Second, adiabatic manipulation is often required in EIT systems to maintain the atomic coherence. The time scale for the adiabatic approximation can be given by $τ_{a} = Γ / Ω_{uni}^{2} = 82.2 ps$ [24]. Third, we should also take into account the transparency window of the EIT system given by $δ F_{trans} = Ω_{uni}^{2} / (2 π \sqrt{N σ d} Γ) = 129.3 MHz$ and the dispersion window given by $δ F_{dis} = Ω_{uni}^{2} / (N σ d Γ) = 8.64 MHz$ , where $σ = 3 λ_{p}^{2} / (2 π)$ is the absorption cross section of one atom [7]. The modulation speed should be limited by δF ≪ {1/τ_r, 1/τ_a, δF_trans, δF_dis}, which means that megahertz modulation can be realized in our scheme. Note that the optical depth of our system is also given by OD = Nσd = 8849, which is a moderate value for usual slow light and light storage experiments in EIT media [7, 8]. In addition, it is also seen that the coupling intensity is not completely turned off during the rapid modulation [12], which ensures the high transmission of the probe field. Generally, the fast response and wide bandwidth induced by small thickness of atomic ensembles and strong coupling Rabi frequency in EIT systems can facilitate the realization of all-optical modulation with high speed [7].

Additionally, the diversity of the coupling images can induce different Fresnel lenses in the EIT system to image the probe field. Figure 5 gives the coupling images with different aperture diameters and the corresponding probe images measured by the photodetector using the same parameters and experimental setup as those in Fig. 4, which again clearly shows the flexibility of our scheme.

Fig. 5 (Color online) (a)-(c) Coupling images with different aperture diameters, where the intensity distributions and the aperture diameters are numerically illustrated in Fig. 2(a). (d)-(f) Corresponding probe images measured by the photodetector. With the increase of the aperture diameters of the coupling images, the quality of the probe images can be improved. The color bar on the right gives the normalized intensity of the probe images.

Download Full Size | PDF

4. Discussion of potential error sources

Considering the realistic setup in experiments, we should discuss our scheme in more detail.

Because the piecewise intensity profile of the coupling image, in practice, cannot have so sharp boundaries between the different zones as shown in Fig. 2, we should consider the influence of the blur of the coupling image on the diffraction efficiency of the induced Fresnel lens. The sharpness of the boundary can be estimated by the spatial resolution of the coupling image. Assuming that the coupling lens for constructing the coupling images has an aperture of diameter D = 5 cm and its focal length is f_c = 1 m, we can find that its spatial resolution limit under coherent illumination is given by R = 1.62λ_cf_c/D = 25.8 μm [19]. Analogous to the fringing field effect in LC phased arrays [25], the diffraction efficiency of a particular Fresnel zone can be simply estimated by $η_{m} = {(1 - \frac{R}{w_{m}})}^{2},$ where the resolution limit R can be analogous to the width of the flyback region in LC phased arrays and w_m is the width of the mth Fresnel zone. Here, the main difference between the Fresnel lens and the phased array is that the Fresnel zones in the lens have different widths. Thus, we need to calculate the average of the diffraction efficiencies of all Fresnel zones. According to the radiuses shown earlier in Section 2, we can obtain that w₁ = r₁ = 1.26 mm, w₂ = r₂ − r₁ = 0.52 mm, w₃ = r₃ − r₂ = 0.4 mm, and w₄ = r₄ − r₃ = 0.34 mm. The corresponding diffraction efficiencies are η₁ = (1 − R/w₁)² = 95.9%, η₂ = 90.3%, η₃ = 87.5%, and η₄ = 85.4%. The resulting diffraction efficiency of the Fresnel lens with four zones can be estimated by the average η̄ = 89.8%. Moreover, considering the low absorption (≈ 98% transmission rate) shown in Fig. 3, the overall diffraction efficiency is given by η = 89.8% × 98% = 88%. Furthermore, it also can be seen that the Fresnel lenses with smaller aperture diameters such as those shown in Figs. 5(a)–5(c) should have higher diffraction efficiencies, which is also consistent with the experimental measurements for the LC Fresnel lenses in Ref. [6]. Additionally, to further improve the diffraction efficiency, more advanced imaging systems can be used to enhance the sharpness of the coupling images [16].
Sub-millimeter-thin vapor cells have been fabricated in experiments [26, 27]. Also, antirelaxation wall coatings of octadecyltrichlorosilane have been successfully prepared, thus suppressing the collisional decoherence between the two lower levels. Meanwhile, high Rb vapor density up to 9 × 10¹³/cm³ can be obtained [28, 29]. Combining these two existing techniques, the thin vapor cell desired in our scheme can be realized. In addition, in the numerical simulations in Section 2, for convenience, we assume the atomic number density N = 3 × 10¹³/cm³ and find the thickness d = 977.5 μm. In practice, it is difficult to fabricate a cell with so precise thickness and atomic density. However, note that the coupling intensity is totally tunable in our case. Based on Eqs. (6)–(8), if the thickness or the density slightly deviates from our suggested values, the coupling intensity can be proportionally tuned to compensate the deviations. Therefore, in principle, we can adaptively adjust the parameters to make the 2π optical path difference hold, thus calibrating our system at the beginning of experiments.
For the hot atomic vapor, the Doppler effect should be analyzed. Assuming that the vapor temperature is about 170 °C [28, 29], the most probable thermal velocity of the Rb atoms is given by v_p = 291 m/s, and thus the single-photon Doppler broadening is $Δ ω_{D} = 2 \sqrt{ln 2} k_{p} ν_{p} = 3.83 \times 10^{9} / s .$ The angular misalignment of the probe field can be estimated by the F-number of the induced Fresnel lens [19], which can give the maximum misalignment Δθ = 1/F-number = 2r₄/f_F = 5.04 mrad. Thus the two-photon Doppler broadening is δω_D = ΔθΔω_D = 19.3 × 10⁶/s. Under the strong-coupling-field condition (i.e., 23Γ ≤ Ω_c = 46Γ), we have $Ω_{c}^{2} ≫ Δ ω_{D} δ ω_{D}$ which means that the Doppler effect can be suppressed in our scheme, and thus Eq. (2) is a reasonable approximation to calculate the susceptibility of the probe field [13, 14]. Apparently, if even smaller apertures are used (see Fig. 5), the Doppler effect can also be suppressed. To further suppress the Doppler effect, light-induced atomic desorption technique could be employed, where high atomic density can be obtained and fully controlled at relatively low temperature [30,31]. Furthermore, the motion-induced decoherence for atoms should be considered in the hot vapor. The decoherence length of the moving atoms can be estimated by L_d = 2πv_p/γ ≈ 10 cm [27]. Note the nonradiative decoherence rate γ = 3000 Hz was previously assumed in Section 2, which is a typical value in the EIT experiments [7, 8]. This length is much larger than both the possible dimensions of the cell (e.g., ∼ 5 × 5 × 1 mm³) and the size of the coupling images (∼ 5 mm). Because the antirelaxation wall coatings are suggested in our system, the atomic coherence can be preserved for a long time during light-atom interaction [28, 29]. Therefore, the EIT phenomenon required in our scheme can be well established. Moreover, another possible method to preserve the atomic coherence in hot vapor is using high-pressure buffer gases which slow the diffusion of atoms to the walls of the thin cell [26]. However, it should be stressed that high buffer gas pressure can considerably broaden the optical transitions of EIT systems and thus weaken the stability of the performance of our scheme [28, 29]. Besides the hot atomic vapor, cold atomic ensemble with high density is also a good choice to establish the EIT system [11, 32]. In this case, the atomic motion is negligible and thus the Doppler effect can be nearly eliminated. The transparency condition of our system becomes Ω_c ≫ (Γγ)^1/2 ≈ 0.023Γ which means that the coupling intensity (or Rabi frequency) can be dramatically reduced [7].
Note that we use ⁸⁷Rb D₁ line (5S_1/2 ↔ 5P_1/2) as an example to show the generation of all-optical Fresnel lenses, where other off-resonant levels are ignored. It is known that the interaction between the light fields and the off-resonant levels (i.e., ac Stark shifts) may greatly degrade the EIT effect. First, we consider the totally off-resonant levels (e.g., 5P_3/2). Because they are far detuned (∼ 7.26 THz) from the light frequencies, these levels really do not play any role. Moreover, for other hyperfine levels, such as |5P_1/2, F′ = 1〉, it has experimentally proven possible that proper level schemes can be used to cause the compensation between these shifts and maintain the transparency [33]. Additionally, other EIT schemes, such as the ladder-type system in Refs. [9, 10, 14], may also be employed to generate the lenses.
Commercial pulse generators can be used to generate the square pulse trains with megahertz repetition rates in our scheme. As long as the rise (fall) time of the pulse is much shorter that the pulse duration, the overlap between the two pulse trains is negligible. Hence the losses due to the switching between different coupling images can be minimized.
We must emphasize here that our scheme is only an example to show the ability of EIT systems to construct all-optical diffractive imaging devices. Based on the principles in our scheme, all-optical binary-phase or binary-amplitude zone plates could be generated as well. Furthermore, considering the multiple energy levels of atoms, it is also possible to realize polarization-dependent multifocal Fresnel lenses or nonlinear image processing at low light levels in tripod or N-type EIT systems [34].

5. Conclusion

In summary, we have shown the feasibilities of constructing all-optically tunable Fresnel lenses in an EIT system using intensity-modulated images in coupling fields. The induced lenses have large focal length tunability from 1 m to infinity, and can focus and image probe fields by diffraction with high transmission and diffraction efficiency (> 88%), whereby image controls image. Due to the fast atomic response to light fields, a modulation rate up to megahertz can be implemented to tune the induced lenses. Our scheme significantly enhances the applicability and practicality of EIT-based all-optical devices to spatial signal processing of multimode light fields, which may find various potential applications, including optical information processing with massive parallelism, optical imaging systems, and optical multichannel communications. Additionally, due to the low absorption and high efficiency of our scheme, simultaneous manipulation of multiple weak optical signals even at the single-photon level may also be realized.

Acknowledgments

L.Z. thanks R. Zhou and F. Peng for helpful discussions. L.Z. and W.D. acknowledge the financial support from the Ministry of Science and Technology of China (Grants No. 2011CB921901 and 2011CB606405) and China Postdoctoral Science Foundation (Grant No. 20100470362). S.F.Y. acknowledges funding from NSF through award PHY-0970055.

References and links

1. J. S. Patel and K. Rastani, “Electrically controlled polarization-independent liquid-crystal Fresnel lens arrays,” Opt. Lett. 16, 532–534 (1991). [CrossRef] [PubMed]

2. Y.-H. Fan, H. Ren, and S.-T. Wu, “Switchable Fresnel lens using polymer-stabilized liquid crystals,” Opt. Express11, 3080–3086 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-23-3080. [CrossRef] [PubMed]

3. L.-C. Lin, H.-C. Jau, T.-H. Lin, and A. Y.-G. Fuh, “Highly efficient and polarization-independent Fresnel lens based on dye-doped liquid crystal,” Opt. Express15, 2900–2906 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-6-2900. [CrossRef] [PubMed]

4. M. Ferstl and A.-M. Frisch, “Static and dynamic Fresnel zone lenses for optical interconnections,” J. Mod. Opt. 43, 1451–1462 (1996). [CrossRef]

5. G. Li, D. L. Mathine, P. Valley, P. Äyräs, J. N. Haddock, M. S. Giridhar, G. Williby, J. Schwiegerling, G. R. Meredith, B. Kippelen, S. Honkanen, and N. Peyghambarian, “Switchable electro-optic diffractive lens with high efficiency for ophthalmic applications,” Proc. Natl. Acad. Sci. U.S.A. 103, 6100–6104 (2006). [CrossRef] [PubMed]

6. P. Valley, D. L. Mathine, M. R. Dodge, J. Schwiegerling, G. Peyman, and N. Peyghambarian, “Tunable-focus flat liquid-crystal diffractive lens,” Opt. Lett. 35, 336–338 (2010). [CrossRef] [PubMed]

7. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]

8. M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457–472 (2003). [CrossRef]

9. R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, “Spatial Consequences of Electromagnetically Induced Transparency: Observation of Electromagnetically Induced Focusing,” Phys. Rev. Lett. 74, 670–673 (1995). [CrossRef] [PubMed]

10. R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, “Electromagnetically-induced focusing,” Phys. Rev. A 53, 408–415 (1996). [CrossRef] [PubMed]

11. M. Mitsunaga, M. Yamashita, and H. Inoue, “Absorption imaging of electromagnetically induced transparency in cold sodium atoms,” Phys. Rev. A 62, 013817 (2000). [CrossRef]

12. L. Zhao, T. Wang, and S. F. Yelin, “Two-dimensional all-optical spatial light modulation with high speed in coherent media,” Opt. Lett. 34, 1930–1932 (2009). [CrossRef] [PubMed]

13. M. Fleischhauer and M. O. Scully, “Quantum sensitivity limits of an optical magnetometer based on atomic phase coherence,” Phys. Rev. A 49, 1973–1986 (1994). [CrossRef] [PubMed]

14. J. Gea-Banacloche, Y.-Q. Li, S. Jin, and M. Xiao, “Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A 51, 576–584 (1995). [CrossRef] [PubMed]

15. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

16. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

17. Y.-Q. Ling and M. Xiao, “Electromagnetically induced transparency in a three-level Λ-type system in rubidium atoms,” Phys. Rev. A 51, R2703–R2706 (1995). [CrossRef]

18. D. A. Steck, “Rubidium 87 D Line Data,” available online at http://steck.us/alkalidata (revision 2.1.2, 12 August 2009).

19. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, UK, 1999).

20. T. Stone and N. George, “Hybrid diffractive-refractive lenses and achromats,” Appl. Opt. 27, 2960–2971 (1988). [CrossRef] [PubMed]

21. V. K. Ingle and J. G. Proakis, Digital Signal Processing Using MATLAB V.4 (PWS Publishing Company, Boston, MA, 1997).

22. S. E. Harris and Z.-F. Luo, “Preparation energy for electromagnetically induced transparency,” Phys. Rev. A 52, R928–R931 (1995). [CrossRef] [PubMed]

23. H. Schmidt and R. J. Ram, “All-optical wavelength converter and switch based on electromagnetically induced transparency,” Appl. Phys. Lett. 76, 3173–3175 (2000). [CrossRef]

24. M. Fleischhauer and A. S. Manka, “Propagation of laser pulses and coherent population transfer in dissipative three-level systems: An adiabatic dressed-state picture,” Phys. Rev. A 54, 794–803 (1996). [CrossRef] [PubMed]

25. P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Lieberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE84, 268–298 (1996). [CrossRef]

26. S. Knappe, L. Hollberg, and J. Kitching, “Dark-line atomic resonances in submillimeter structures,” Opt. Lett. 29, 388–390 (2004). [CrossRef] [PubMed]

27. A. Sargsyan, D. Sarkisyan, and A. Papoyan, “Dark-line atomic resonances in a submicron-thin Rb vapor layer,” Phys. Rev. A 73, 033803 (2006). [CrossRef]

28. Y. W. Yi, H. G. Robinson, S. Knappe, J. E. MacLennan, C. D. Jones, C. Zhu, N. A. Clark, and J. Kitching, “Method for characterizing self-assembled monolayers as antirelaxation wall coatings for alkali vapor cells,” J. Appl. Phys. 104, 023534 (2008). [CrossRef]

29. S. J. Seltzera and M. V. Romalis, “High-temperature alkali vapor cells with antirelaxation surface coatings,” J. Appl. Phys. 106, 114905 (2009). [CrossRef]

30. M. Meucci, E. Mariotti, P. Bicchi, C. Marinelli, and L. Moi, “Light-Induced Atom Desorption,” Europhys. Lett. 25, 639–643 (1994). [CrossRef]

31. A. Bogi, C. Marinelli, A. Burchianti, E. Mariotti, L. Moi, S. Gozzini, L. Marmugi, and A. Lucchesini, “Full control of sodium vapor density in siloxane-coated cells using blue LED light-induced atomic desorption,” Opt. Lett. 34, 2643–2645 (2009). [CrossRef] [PubMed]

32. W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993). [CrossRef] [PubMed]

33. I. Novikova, A. B. Matsko, V. L. Velichansky, M. O. Scully, and G. R. Welch, “Compensation of ac Stark shifts in optical magnetometry,” Phys. Rev. A 63, 063802 (2001). [CrossRef]

34. R. Drampyan, S. Pustelny, and W. Gawlik, “Electromagnetically induced transparency versus nonlinear Faraday effect: Coherent control of light-beam polarization,” Phys. Rev. A 80, 033815 (2009). [CrossRef]

All-optical Fresnel lens in coherent media: controlling image with image

Abstract

1. Introduction

2. Theoretical model

3. Optical performance

4. Discussion of potential error sources

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (17)

Optics Express