Transmitting volume Bragg grating with longitudinal moir&#x00E9; apodization

Sergiy Mokhov

doi:10.1364/OSAC.2.003471

1. Introduction

Volume Bragg gratings (VBGs) in photo-thermo-refractive (PTR) glass are used for various laser applications because of their robustness and tolerance to high laser power owing to their large apertures. These gratings are fabricated in a photosensitive glass by holographically recording the interference fringe pattern formed by two overlapping coherent waves formed by splitting a flattop UV laser beam [1]. The thermal treatment of the exposed glass sample produces a permanent refractive index modulation (RIM) inside it. VBGs are manufactured by properly cutting a sample followed by polishing it and applying an anti-reflection coating.

VBGs operate in the two basic geometries of reflection and transmission. Reflective VBGs with uniform RIM demonstrate narrow reflection bandwidths. While transmitting volume Bragg gratings (TVBGs), the focus of this study, are characterized by a narrow angular selectivity in a plane of diffraction. Depending on the type of selectivity, spectral or angular required in a particular laser application, reflective or transmitting VBG is used. TVBGs can be used for the angular filtering of laser beams [2,3] and beam steering [4]. An efficient 2D angular filtering requires two TVBGs [5]. A single TVBG combined with a pair of surface transmitting gratings can provide angular filtering in a wide spectral range [6]. Reflective VBGs are typically considered for spectral beam combining (SBC) of lasers [7]; however, SBC can also be realized with TVBGs [8]. The diffraction of laser pulses on TVBGs has been studied in [9,10]. The operation of the solid-state laser at fundamental mode can be supported with a TVBG placed inside the cavity [11]. TVBGs with encoded phase profiles can perform complex manipulations with wavefronts of laser beams [12,13]. TVBGs holographically recorded in other photosensitive materials have a more extensive range of applications, e.g., TVBGs in photorefractive crystals have been utilized for generating Bessel beams [14,15]. The presented study follows from our previous work on high power laser applications, which can be effectively realized with VBGs in PTR glass.

Volume Bragg diffraction of laser beams is described by the coupled wave theory originally formulated by Kogelnik [16]. Operation of a reflective VBG is similar to a fiber Bragg grating (FBG). It is known that apodization of RIM inside the FBG leads to suppression of the secondary lobes in the reflection spectrum [17]. Apodization is indicated by a smooth decrease in the amplitude of RIM to zero at both sides of the grating. Recently the apodized reflective VBG [18] displaying a high agreement between the measured and the modeled spectra was demonstrated. In this study, the design of the holographic fabrication of the apodized TVBG is presented. The apodization method is based on the sequential recording of two uniform interference patterns with the same modulation amplitudes and periods and a small rotation angle between their directions of modulation.

2. Refractive index modulation and volume Bragg diffraction

Bragg modulation of refractive index is considered as

(1)$$n({\textbf r}) = {n_0} + {n_1}({\textbf r})\cos ({\textbf Qr} + \gamma ),\quad |{{n_1}} |< < {n_0}.$$

Where n₀ is the background refractive index of the glass medium, Q is the vector of Bragg modulation, n₁ is the slowly varying amplitude of RIM, and it is a constant for uniform VBG, γ is the phase assumed to be a constant.

Generally, the phase γ could be varying. In particular, its quadratic spatial dependence provides a linear chirp of local resonant wavelength. Additionally, a small slow variation of the background refractive index can be added to n(r). This however is beyond the scope of this study.

Propagation of monochromatic electromagnetic waves with frequency ω in a dielectric medium with spatially varying permittivity ɛ_r(r) = n²(r) is described by the Maxwell equations. This leads to the stationary equations for spatially dependent amplitudes of the electric and magnetic fields as follows:

(2)$$\nabla \times {\textbf E} = i\omega {\mu _0}{\textbf H},\quad \nabla \times {\textbf H} ={-} i\omega {\varepsilon _0}{\varepsilon _\textrm{r}}({\textbf r}){\textbf E},\quad {\varepsilon _\textrm{r}}({\textbf r}) = n_0^2 + {n_0}{n_1}({\textbf r})({e^{i{\textbf Qr} + i\gamma }} + {e^{ - i{\textbf Qr} - i\gamma }}).$$

In the last equation, the negligible term proportional to the square of n₁ has been omitted.

Modulation of the refractive index provides the coupling of two electromagnetic waves causing the exchange of power between them when their propagation wave vectors k_A and k_B are close to the satisfaction of the resonant Bragg condition, k_A − k_B ≈ Q. Two vectors Q and k_A form the plane of Bragg diffraction. Near the Bragg condition, when the efficiency of diffraction is noticeable, the vector k_B can be assumed as belonging to this plane. The amplitude E(r) of the TE polarized electric field, perpendicular to the plane of diffraction, satisfies the wave equation following from Eq. (2) as

(3)$${\nabla ^2}E + {k^2}(1 + n_0^{ - 1}{n_1}({\textbf r})({e^{i{\textbf Qr} + i\gamma }} + {e^{ - i{\textbf Qr} - i\gamma }}))E = 0,\quad k = {{{n_0}\omega } \mathord{\left/ {\vphantom {{{n_0}\omega } c}} \right.} c} = {{2\pi {n_0}} \mathord{\left/ {\vphantom {{2\pi {n_0}} \lambda }} \right.} \lambda }.$$

Here λ is the vacuum wavelength.

The electric field amplitude can be presented as a sum of the two waves subject to coupling as

(4)$$E({\textbf r}) = A({\textbf r}){e^{i{{\textbf k}_\textrm{A}}{\textbf r}}} + B({\textbf r}){e^{i{{\textbf k}_\textrm{B}}{\textbf r}}},\quad {{\textbf k}_{\textrm{A,B}}} = {{\textbf u}_{\textrm{A,B}}}k,\quad |{{{\textbf u}_{\textrm{A,B}}}} |= 1.$$

Here u_A,B are the unit vectors corresponding to the wave vectors. A(r) and B(r) are the slowly varying envelope amplitudes so that their second derivatives can be neglected, which leads to

(5)$${\nabla ^2}(A({\textbf r}){e^{i{{\textbf k}_\textrm{A}}{\textbf r}}}) = (2i({{\textbf k}_\textrm{A}}\nabla A) - {k^2}A){e^{i{{\textbf k}_\textrm{A}}{\textbf r}}}.$$

A similar expression for the amplitude B can be written down.

After substituting Eq. (4) with Eq. (5), and the expression for B into Eq. (3), the terms corresponding to the standard wave equation without Bragg modulation cancel each other. The remaining terms describing the slowly varying envelope approximation (SVEA) are considered. Collecting the terms with the phase factor of A-wave, exp(ik_Ar), approximately equal to the phase factor of the B-wave coupled with the RIM near assumed Bragg condition, exp(ik_Br + iQr), the differential equation for the envelope amplitude A is obtained as

(6)$$2i({{\textbf k}_\textrm{A}}\nabla A) + {k^2}n_0^{ - 1}{n_1}({\textbf r}){e^{i\gamma }}{e^{i({\textbf Q} + {{\textbf k}_\textrm{B}} - {{\textbf k}_\textrm{A}}){\textbf r}}}B = 0.$$

With a similar equation for the amplitude B, the system of the coupled wave equations is

(7)$$\left\{ {\begin{array}{{c}} {({{\textbf u}_\textrm{A}}\nabla A) = i\kappa ({\textbf r}){e^{ - 2i{\textbf Dr}}}B({\textbf r}),\;}\\ {({{\textbf u}_\textrm{B}}\nabla B) = i{\kappa^\ast }({\textbf r}){e^{2i{\textbf Dr}}}A({\textbf r}),\;} \end{array}} \right.\quad \quad \begin{array}{{c}} {{\kappa _{\textrm{TE}}}({\textbf r}) = {e^{i\gamma }}{{\pi {n_1}({\textbf r})} \mathord{\left/ {\vphantom {{\pi {n_1}({\textbf r})} \lambda }} \right.} \lambda },}\\ {{\textbf D} = {\textstyle{1 \over 2}}({{\textbf k}_\textrm{A}} - {{\textbf k}_\textrm{B}} - {\textbf Q}).} \end{array}$$

Here κ_TE(r) is a coupling coefficient in the case of TE polarization, and D is the detuning vector. At the exact Bragg condition, D = 0, the coupling process between the waves has the highest efficiency.

In TM polarization of the coupled waves, when their polarization vectors are in the plane of diffraction, the coupling coefficient is reduced by factor cos(2θ), where θ is the angle between the vectors k_A and Q so

(8)$${\kappa _{\textrm{TM}}}({\textbf r}) = {\kappa _{\textrm{TE}}}({\textbf r})\cos (2\theta ),\quad \theta = \angle ({{\textbf k}_\textrm{A}},{\textbf Q}).$$

The origination of the factor cos(2θ) for TM polarization is discussed in Appendix A in details.

The mathematical problem of volume Bragg diffraction of incident plane A-wave by particular VBG is described by the system of partial differential Eq. (7) with established boundary conditions implying no incident B-wave. The amplitude of the outgoing B-wave founded after solving the diffraction problem determines the efficiency of diffraction.

Basic VBGs are plates with a uniform RIM and parallel surfaces. VBG is considered as a reflective grating if a diffracted wave is outgoing through the same surface from where the incident wave is coming. If the diffracted wave is outgoing through the opposite surface, then such VBG is transmitting. These two geometries of VBGs are presented in Fig. 1.

Fig. 1. (a) Uniform reflecting VBG; (b) Uniform transmitting VBG.

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For clarity, unslanted VBGs without a tilt of orientation of the RIM pattern are considered, so the Bragg vector Q is either perpendicular or parallel to both the surfaces. Generally, the RIM can be nonuniform. In this study, however, Q is prescribed to be a constant vector, so that the modulation period Λ = 2π/Q does not change, and the amplitude of RIM n₁(r) will be considered as a constant or sinusoidally varying.

Apodized VBGs are characterized by the amplitude of RIM significantly reduced, preferably dropped to zero, at both the surfaces z = 0 and z = l. Apodization of a holographically recorded interference pattern can be realized in various ways, such as with the use of overlapping beams having Gaussian transverse profiles instead of uniform beams [19]. However, this approach does not lead to the complete dropping to zero of the recorded amplitude of RIM.

Our presented method of apodization is based on the sequential recording of two uniform patterns with the same amplitudes of RIM and slightly different Bragg vectors Q₁ and Q₂. Mathematical foundation of this approach is expressed by the well-known trigonometric equation

(9)$$\cos ({{\textbf Q}_1}{\textbf r}) + \cos ({{\textbf Q}_2}{\textbf r}) = 2\cos ({\textbf {Mr}})\cos ({\textbf {Qr}}),\quad {\textbf Q} = {\textstyle{1 \over 2}}({{\textbf Q}_1} + {{\textbf Q}_2}),\quad {\textbf M} = {\textstyle{1 \over 2}}({{\textbf Q}_1} - {{\textbf Q}_2}).$$

Value M of the moiré vector is designed to be relatively small, M<<Q, which produces a slow variation of the amplitude of RIM with resulting Bragg vector Q. Figure 2 shows two primary moiré patterns for producing apodized reflective and transmitting VBGs according to Eq. (9).

Fig. 2. (a) Moiré modulation along the Bragg modulation for creation of an apodized reflecting VBG; (b) Moiré modulation perpendicular to the Bragg modulation for creation of an apodized transmitting VBG.

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The cutting of a specimen at planes of two closest zeros of moiré envelope will produce apodized VBGs with the amplitude of RIM completely dropped to zero at grating surfaces.

Initially, we studied the properties of reflective VBGs fabricated with moiré patterns in [20,21], where the cuttings of specimens were made for one full period of moiré patterns. Such a reflective VBG can be considered as a combination of two adjusted reflective VBGs with an intermediate phase π-shift in RIM. This explains its operation as a resonant Fabry-Perot cavity with a narrow transmission peak at the resonant Bragg wavelength. Recently in our paper cited in Section 1, the moiré apodized reflective VBG was reported. In this paper, an apodized TVBG designed according to Fig. 2b is studied.

3. Transmitting volume Bragg grating with moiré apodization

Consider the orientation of the coordinate system for the problem of wave diffraction by transmitting VBG, as shown in Fig. 2b. The Bragg vector Q is directed along the x-axis, and diffraction occurs in the (x,z)-plane. Refractive index is modulated as

(10)$$n({\textbf r}) = {n_0} + {n_1}(z)\cos (Qx),\quad Q = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } \Lambda }} \right.} \Lambda },\quad 0 \le z \le l.$$

Here l is the thickness of a grating, and Λ is the spatial period of RIM along the x-axis.

For the angle θ_air of incidence in the air at the plane z = 0, the refracted angle θ of propagation wave vector k_A in a glass is defined by Snell’s law. The coupled wave generated through Bragg diffraction propagates with the wave vector k_B at an angle with absolute value θ_B. The unit vectors introduced in Eq. (4) are the following

(11)$$\begin{array}{{cc}} \begin{array}{l} {n_0}\sin \theta = \sin {\theta _{\textrm{air}}},\\ \sin {\theta _{\textrm{B,air}}} = {n_0}\sin {\theta _\textrm{B}}, \end{array}&\begin{array}{l} {{\textbf u}_\textrm{A}} = {{{{\textbf k}_\textrm{A}}} \mathord{\left/ {\vphantom {{{{\textbf k}_\textrm{A}}} k}} \right.} k} = ({u_{\textrm{A},x}},{u_{\textrm{A},z}}) = (\cos \theta ,\sin \theta ),\\ {{\textbf u}_\textrm{B}} = {{{{\textbf k}_\textrm{B}}} \mathord{\left/ {\vphantom {{{{\textbf k}_\textrm{B}}} k}} \right.} k} = ({u_{\textrm{B},x}},{u_{\textrm{B},z}}) = ( - \cos {\theta _\textrm{B}},\sin {\theta _\textrm{B}}). \end{array} \end{array}$$

Here θ_B,air is the angle of propagation of the diffracted wave in the air. The absolute value k of the wave vector in a glass is mentioned in Eq. (3).

The most efficient Bragg coupling between the two waves occurs at the exact Bragg condition determined by zero detuning D in Eq. (7)

(12)$${\textbf D} = 0\quad \to \quad {{\textbf k}_\textrm{B}} = {{\textbf k}_\textrm{A}} - {\textbf Q}.$$

With non-zero detuning, oscillating phase factors presented in Eq. (7) deteriorate the wave coupling leading to a reduction in the total diffraction efficiency.

Suppose the transmitting VBG is designed to provide the strongest coupling of waves with wavelength λ_res at an angle θ₀ of wave propagation inside the grating. Then, according to Eqs. (10−12), the Bragg vector Q and the spatial period Λ needed to be recorded are

(13)$$Q = 2{k_{\textrm{res}}}\sin {\theta _0},\quad {k_{\textrm{res}}} = {{2\pi {n_0}} \mathord{\left/ {\vphantom {{2\pi {n_0}} {{\lambda_{\textrm{res}}}}}} \right.} {{\lambda _{\textrm{res}}}}},\quad \Lambda = {{{\lambda _{\textrm{res}}}} \mathord{\left/ {\vphantom {{{\lambda_{\textrm{res}}}} {(2{n_0}\sin {\theta_0})}}} \right.} {(2{n_0}\sin {\theta _0})}}.$$

Transverse phase matching condition D_x = 0 eliminates the explicit x-dependence in the system of coupled Eqs. (7) and, as a result, amplitudes A and B can be considered as dependent only on the z-coordinate. According to Eq. (11) and Eq. (12), this phase matching condition provides the actual propagation direction of the diffracted wave B inside the transmitting VBG as

(14)$${D_x} = 0\quad \to \quad \sin {\theta _\textrm{B}} = {Q \mathord{\left/ {\vphantom {Q k}} \right.} k} - \sin \theta .$$

After satisfying the transverse phase matching condition, amplitudes A and B are sought as z-dependent functions, and the system of coupled Eqs. (7) is reduced to

(15)$$\left\{ {\begin{array}{{c}} {\cos \theta {\textstyle{d \over {dz}}}A = i\kappa (z){e^{ - 2i{D_z}z}}B(z),\;}\\ {\cos {\theta_\textrm{B}}{\textstyle{d \over {dz}}}B = i{\kappa^\ast }(z){e^{2i{D_z}z}}A(z),\;} \end{array}} \right.\quad \quad \begin{array}{{c}} {{\kappa _{\textrm{TE}}}(z) = {e^{i\gamma }}{{\pi {n_1}(z)} \mathord{\left/ {\vphantom {{\pi {n_1}(z)} \lambda }} \right.} \lambda },}\\ {{D_z} = \pi {n_0}{\lambda ^{ - 1}}(\cos \theta - \cos {\theta _\textrm{B}}).} \end{array}$$

Obtained system of ordinary differential Eq. (15) can be rewritten in a simplified form with redefining the envelope amplitudes after a proper phase adjustment

(16)$$\begin{array}{l} \begin{array}{{c}} {A = a{e^{ - i{D_z}z + i{\gamma \mathord{\left/ {\vphantom {\gamma 2}} \right.} 2}}},}\\ {B = b{e^{i{D_z}z - i{\gamma \mathord{\left/ {\vphantom {\gamma 2}} \right.} 2}}},} \end{array}\quad \to \quad \frac{d}{{dz}}\left( {\begin{array}{{c}} a\\ b \end{array}} \right) = \left( {\begin{array}{{cc}} {i{D_z}}&{{{i\kappa } \mathord{\left/ {\vphantom {{i\kappa } {\cos {\theta_0}}}} \right.} {\cos {\theta_0}}}}\\ {{{i\kappa } \mathord{\left/ {\vphantom {{i\kappa } {\cos {\theta_0}}}} \right.} {\cos {\theta_0}}}}&{ - i{D_z}} \end{array}} \right)\left( {\begin{array}{{c}} a\\ b \end{array}} \right),\\ {\kappa _{\textrm{TE}}}(z) = {{\pi {n_1}(z)} \mathord{\left/ {\vphantom {{\pi {n_1}(z)} {{\lambda_{\textrm{res}}}}}} \right.} {{\lambda _{\textrm{res}}}}},\quad {\kappa _{\textrm{TM}}} = {\kappa _{\textrm{TE}}}\cos (2{\theta _0}). \end{array}$$

Here constant parameters λ_res and θ₀ have been used in off-diagonal matrix elements because varying values of the wavelength and the angle are near the exact Bragg condition, where noticeable diffraction efficiency is achievable.

Firstly, the uniform TVBG with the constant amplitude of RIM can be considered for instructive purposes. In this case, the solution of Eq. (16) in the analytic form is

(17)$$\begin{array}{{cc}} \begin{array}{l} {n_1}(z) = {{\bar{n}}_1} = \textrm{const},\\ \left( {\begin{array}{{c}} {a(l)}\\ {b(l)} \end{array}} \right) = \left( {\begin{array}{{cc}} p&q\\ q&{{p^\ast }} \end{array}} \right)\left( {\begin{array}{{c}} {a(0)}\\ {b(0)} \end{array}} \right), \end{array}&\begin{array}{l} p = \cos G + i\Phi {G^{ - 1}}\sin G,\quad q = i{S_\textrm{u}}{G^{ - 1}}\sin G,\\ {|p |^2} + {|q |^2} = 1,\\ G = \sqrt {S_\textrm{u}^2 + {\Phi ^2}} ,\quad \Phi = {D_z}l = \pi {n_0}{\lambda ^{ - 1}}(\cos \theta - \cos {\theta _\textrm{B}})l,\\ {S_{\textrm{u,TE}}} = {{\pi {{\bar{n}}_1}l} \mathord{\left/ {\vphantom {{\pi {{\bar{n}}_1}l} {({\lambda_{\textrm{res}}}\cos {\theta_0})}}} \right.} {({\lambda _{\textrm{res}}}\cos {\theta _0})}},\quad {S_{\textrm{u,TM}}} = {S_{\textrm{u,TE}}}\cos (2{\theta _0}). \end{array} \end{array}$$

Here subscript u indicates uniform RIM. Two dimensionless parameters have been introduced, namely S the strength of diffraction, and Φ the phase detuning, which can be presented as a sum of the two contributions from the varying wavelength and the incident angle in air as

(18)$$\Phi = {D_z}l = \frac{{2\pi l}}{{{\lambda _{\textrm{res}}}}}\tan {\theta _0}\left( {\frac{{\sin {\theta_{\textrm{air},0}}}}{{{\lambda_{\textrm{res}}}}}\Delta \lambda - \cos {\theta_{\textrm{air},0}}\Delta {\theta_{\textrm{air}}}} \right),\quad \Delta \lambda = \lambda - {\lambda _{\textrm{res}}},\quad \Delta {\theta _{\textrm{air}}} = {\theta _{\textrm{air}}} - {\theta _{\textrm{air},0}}.$$

It was derived taking into account Eq. (11), Eq. (13), and Eq. (14).

Amplitude coefficient of diffraction of the A-wave into the B-wave and corresponding intensity diffraction efficiency η are determined using the boundary condition B(0) = 0 at the surface of incidence at z = 0 as

(19)$$\frac{{B(l)}}{{A(0)}} = {e^{i\Phi - i\gamma }}q,\quad {\eta _\textrm{u}} = \frac{{{{|{B(l)} |}^2}}}{{{{|{A(0)} |}^2}}} = {|q |^2} = \frac{{S_\textrm{u}^2}}{{S_\textrm{u}^2 + {\Phi ^2}}}{\sin ^2}\sqrt {S_\textrm{u}^2 + {\Phi ^2}} .$$

A diffraction efficiency of 100% is achievable at zero detuning Φ = 0 and strength S equal to π/2 with possibly added integer numbers of π. As a result, according to Eq. (17), the lowest amplitude of the uniform RIM required for the full-wave diffraction is determined as

(20)$${\eta _\textrm{u}} = 1\quad \to \quad {S_\textrm{u}} = {\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}\quad \to \quad {\bar{n}_{1,\textrm{TE}}} = {{{\lambda _{\textrm{res}}}\cos {\theta _0}} \mathord{\left/ {\vphantom {{{\lambda_{\textrm{res}}}\cos {\theta_0}} {(2l)}}} \right.} {(2l)}}.$$

The diffraction properties of TVBG with longitudinal moiré apodization were studied as follows. Moiré pattern was generated as shown in Fig. 2b, using a sequential holographic recording of two uniform Bragg patterns with a small angle between the two Bragg vectors. It produces slow oscillation of the RIM inside the glass wafer in the direction perpendicular to the resulting Bragg vector. After fabrication of the wafer, the cutting of TVBG with a thickness l equal half of moiré sinusoidal period provides a reduction of the amplitude of RIM to zero at surfaces as

(21)$${n_1}(z) = {N_1}\sin ({{\pi z} \mathord{\left/ {\vphantom {{\pi z} l}} \right.} l}).$$

With the z-dependent coupling coefficient κ, the system of Eq. (16) has an analytic solution only at the exact Bragg resonance at zero detuning as

(22)$$\begin{array}{l} {D_z} = 0:\quad \quad \left( {\begin{array}{{c}} {a(l)}\\ {b(l)} \end{array}} \right) = \left( {\begin{array}{{cc}} {\cos S}&{i\sin S}\\ {i\sin S}&{\cos S} \end{array}} \right)\left( {\begin{array}{{c}} {a(0)}\\ {b(0)} \end{array}} \right),\\ {S_{\textrm{TE}}} = \frac{\pi }{{{\lambda _{\textrm{res}}}\cos {\theta _0}}}\int_0^l {{n_1}(z)dz} = \frac{{2{N_1}l}}{{{\lambda _{\textrm{res}}}\cos {\theta _0}}},\quad {S_{\textrm{TM}}} = {S_{\textrm{TE}}}\cos (2{\theta _0}). \end{array}$$

Here the strength of diffraction S is proportional to the integral of varying amplitude of RIM over the length of grating, and it was calculated for n₁(z) from Eq. (21). For uniform RIM, this integral has a simple value S_u mentioned in Eq. (17).

The diffraction efficiency of incident A-wave into B-wave is determined using the previously mentioned initial boundary condition B(0) = 0, meaning no B-wave is incident on TVBG. At the exact Bragg condition, efficiency of the resonant diffraction follows from Eq. (22) with notations from Eq. (16)

(23)$$\eta (S,\Phi ) = {{{{|{B(l)} |}^2}} \mathord{\left/ {\vphantom {{{{|{B(l)} |}^2}} {{{|{A(0)} |}^2}}}} \right.} {{{|{A(0)} |}^2}}};\quad \quad \Phi = 0:\quad B(l) = i\sin {S_0}{e^{ - i\gamma }}A(0),\quad {\eta _\textrm{0}} = {\sin ^2}S_0^2.$$

A 100% resonant diffraction efficiency η₀ is achievable at S₀ = π/2 with optional additional integer numbers of π. According to Eq. (22), the corresponding amplitude N₁ of the sinusoidal RIM in Eq. (21) required for full-wave diffraction is

(24)$${\eta _0} = 1\quad \to \quad {S_0} = {\pi \mathord{\left/ {\vphantom {\pi 2}} \right.} 2}\quad \to \quad {N_{1,\textrm{TE}}} = {{\pi {\lambda _{\textrm{res}}}\cos {\theta _0}} \mathord{\left/ {\vphantom {{\pi {\lambda_{\textrm{res}}}\cos {\theta_0}} {(4l),\quad }}} \right.} {(4l),\quad }}{N_{1,\textrm{TM}}} = {{{N_{1,\textrm{TE}}}} \mathord{\left/ {\vphantom {{{N_{1,\textrm{TE}}}} {\cos (2{\theta_0})}}} \right.} {\cos (2{\theta _0})}}.$$

For arbitrary detuning Φ, the diffraction efficiency η in Eq. (23) can be found after numerical solving the system of coupled Eq. (16) with z-dependent coupling coefficient κ based on the apodizing profile of amplitude of RIM n₁(z) in Eq. (21). Figure 4 presents the numerically calculated diffraction efficiency η of apodized TVBG and the analytically known diffraction efficiency η_u of uniform TVBG from Eq. (19). The strengths of diffraction S₀ = π/2 provides 100% peaks.

Fig. 3. Diffraction efficiencies: η of moiré apodized TVBG (solid line) and η_u of uniform TVBG (dashed line).

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The full width at half maximum of the diffraction efficiency peak of apodized TVBG is ΔΦ_FWHM = 3.426, and, in uniform TVBG, ΔΦ_u,FWHM = 2.509, as shown in Fig. 3. In apodized TVBG, the peak is wider because of apodization, causing the effective thickness equal to thickness of the equivalent uniform grating to be smaller than the nominal thickness. First zeros of the diffraction efficiency η are located at the absolute values Φ₁ = 4.068, Φ₂ = 7.644, Φ₃ = 10.85; major secondary peaks are η₁ = 1.801% at Φ = ± 5.485 and η₂ = 0.233% at Φ = ± 9.018. For reference, in the diffraction efficiency η_u of uniform TVBG, first zeros are located at the absolute values Φ_u1 = 2.721, Φ_u2 = 6.084, Φ_u3 = 9.293, and major secondary peaks are η_u1 = 11.64% at Φ = ± 4.210 and η_u2 = 4.066% at Φ = ± 7.564. The dependence of diffraction efficiency of a particular TVBG on angular or wavelength detunings from the parameters of Bragg resonance can be presented using Eq. (18) for Φ through numerical parameters of the actual grating.

As a practical example, consider an apodized TVBG with thickness l = 2 mm, made from glass with refractive index n₀ = 1.5, operating at resonant wavelength λ_res = 1064 nm and incident angle θ_air,0 = 15°. According to Eq. (11), the refracted angle inside TVBG is θ₀ = 9.936°. To provide 100% diffraction efficiency for TE wave, the required maximum amplitude of the apodized profile of RIM is N_1,TE = 412 ppm, according to Eq. (24), and N_1,TM = 438 ppm for TM polarization. From Eq. (20), in a uniform TVBG with the same l, n₀, λ_res, θ_air,0, the required amplitudes of RIM are $\bar{n}$_1,TE = 262 ppm and $\bar{n}$_1,TM = 279 ppm. The dimensionless detuning is expressed through wavelength and angular detunings, according to Eq. (18), as Φ = K_λ·Δλ + K_θ·Δθ_air, K_λ = 0.503 nm⁻¹, K_θ = −1998 rad⁻¹. As a result, the spectral bandwidth of apodized TVBG is Δλ_FWHM = ΔΦ_FWHM/K_λ = 6.81 nm, and the angular bandwidth is Δθ_air,FWHM = ΔΦ_FWHM/|K_θ| = 1.71 mrad. Bandwidths of uniform VBGs with similar parameters would be narrower, because ΔΦ_u,FWHM mentioned above is smaller than ΔΦ_FWHM by factor 0.732.

Angular dispersion δθ_B,air/δλ is expressed as a variation of the diffracted angle θ_B,air = θ_0,air+δθ_B,air with a variation of tunable wavelength λ = λ_res+δλ of A-wave incident on TVBG at fixed angle θ_0,air. In this study, unslanted TVBGs are considered, so θ_B,air = θ_0,air at the exact Bragg condition. Angular dispersion can be calculated from Eq. (14) multiplied by n₀ and taking into account Eq. (13)

(25)$$\sin ({\theta _{\textrm{air},0}} + \delta {\theta _{\textrm{B,air}}}) = 2\sin {\theta _{\textrm{air},0}}(1 + {{\delta \lambda } \mathord{\left/ {\vphantom {{\delta \lambda } {{\lambda_{\textrm{res}}}}}} \right.} {{\lambda _{\textrm{res}}}}}) - \sin {\theta _{\textrm{air},0}}\quad \to \quad {{\delta {\theta _{\textrm{B,air}}}} \mathord{\left/ {\vphantom {{\delta {\theta_{\textrm{B,air}}}} {\delta \lambda }}} \right.} {\delta \lambda }} = {{2\tan {\theta _{\textrm{air},0}}} \mathord{\left/ {\vphantom {{2\tan {\theta_{\textrm{air},0}}} {{\lambda_{\textrm{res}}}}}} \right.} {{\lambda _{\textrm{res}}}}}.$$

For θ_0,air and λ_res mentioned above, angular dispersion is equal to 0.504 mrad/nm.

The plots presented in Fig. 3 are replotted in the logarithmic scale in Fig. 4. This figure clearly shows the enhanced suppression of secondary lobes in the diffraction efficiency of apodized TVBG in comparison with the lobes in diffraction efficiency of similar uniform TVBG.

Fig. 4. Dependence of diffraction efficiency η on dimensionless detuning Φ for different types of TVBGs: apodized (thick line), uniform (thin line), and apodized with 5% phase mismatch in apodization profile (dashed line).

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As shown in Fig. 2, precise cutting of the specimen with moiré pattern is vital for producing an accurate apodization. If the cut is not made exactly at the zeros of the moiré envelope, the suppression of the diffraction sidelobes will not be efficient. Figure 4 shows a curve of the diffraction efficiency of the apodized TVBG with 5% relative phase shift in the sinusoidal semi-period envelope of RIM to illustrate the importance of precise fabrication of apodization profile.

Asymptotic decrease of sidelobes with increasing detuning can be expressed in an analytic form. At large absolute values of Φ, the diffraction efficiency is decreasing to zero, and the amplitude of the incident wave can be considered constant, as A(z) = const. Thus, the process of diffraction can be described in the Born approximation. By finding the amplitude of the diffracted wave B(l) with the assumption of the constant amplitude of A-wave from the second differential equation in Eq. (15), an asymptotic expression can be derived for the diffraction efficiency of the apodized TVBG at large detunings.

(26)$$\begin{array}{l} A(z) = A(0):\quad {\textstyle{d \over {dz}}}B = {\textstyle{i \over 2}}{e^{ - i\gamma }}\pi {l^{ - 1}}S\sin ({{\pi z} \mathord{\left/ {\vphantom {{\pi z} l}} \right.} l}){e^{2i{D_z}z}}A(0),\quad \xi = {z \mathord{\left/ {\vphantom {z l}} \right.} l}:\\ {{B(l)} \mathord{\left/ {\vphantom {{B(l)} {A(0)}}} \right.} {A(0)}} = {\textstyle{i \over 2}}{e^{ - i\gamma }}\pi S\int_0^1 {\sin (\pi \xi ){e^{2i\Phi \xi }}d\xi } ={-} i{e^{ - i\gamma }}{\pi ^2}S{e^{i\Phi }}{{\cos \Phi } \mathord{\left/ {\vphantom {{\cos \Phi } {(4{\Phi ^2} - {\pi^2})}}} \right.} {(4{\Phi ^2} - {\pi ^2})}},\\ \eta = {{{{{|{B(l)} |}^2}} \mathord{\left/ {\vphantom {{{{|{B(l)} |}^2}} {|{A(0)} |}}} \right.} {|{A(0)} |}}^2} = {\pi ^4}{S^2}{{{{\cos }^2}\Phi } \mathord{\left/ {\vphantom {{{{\cos }^2}\Phi } {{{(4{\Phi ^2} - {\pi^2})}^2}}}} \right.} {{{(4{\Phi ^2} - {\pi ^2})}^2}}},\quad \left\langle \eta \right\rangle = {{{\textstyle{1 \over 2}}{\pi ^4}{S^2}} \mathord{\left/ {\vphantom {{{\textstyle{1 \over 2}}{\pi^4}{S^2}} {{{(4{\Phi ^2} - {\pi^2})}^2}}}} \right. } {{{(4{\Phi ^2} - {\pi ^2})}^2}}}. \end{array}$$

Here S is the strength of diffraction from Eq. (22). $\left\langle \eta \right\rangle$ is the diffraction efficiency averaged over several periods of Φ. It was obtained by substituting the periodic factor cos²Φ with its average value ½.

An asymptotic expression for the diffraction efficiency η_u of uniform TVBG in the Born approximation at a large detuning can be derived from the analytic result achieved in Eq. (19)

(27)$$\Phi > > {S_\textrm{u}}:\quad {\eta _\textrm{u}} = {{S_\textrm{u}^2} \mathord{\left/ {\vphantom {{S_\textrm{u}^2} {{\Phi ^2}}}} \right.} {{\Phi ^2}}} \cdot {\sin ^2}\Phi ,\quad \left\langle {{\eta_\textrm{u}}} \right\rangle = {\textstyle{1 \over 2}}{{S_\textrm{u}^2} \mathord{\left/ {\vphantom {{S_\textrm{u}^2} {{\Phi ^2}}}} \right.} {{\Phi ^2}}}.$$

The strength of diffraction S_u for uniform TVBG is defined in Eq. (17). After averaging the periodic factor sin²Φ, $\left\langle {{\eta_\textrm{u}}} \right\rangle$ is obtained. Thus, in an apodized TVBG according to Eq. (26), the asymptotic dependence of the diffraction efficiency on detuning is proportional to Φ⁻⁴. This power dependence provides a very fast reduction of the secondary lobes in comparison with the dependence Φ⁻² for conventional uniform TVBG.

Successive diffraction of a laser beam by a pair of uniform TVBGs is defined by the product of two diffraction efficiencies of individual gratings, and it leads to the suppression of diffraction sidelobes [22]. The total efficiency of double diffraction is described by the asymptotic dependence Φ⁻⁴ similar to the diffraction with an apodized TVBG. The enhancement of diffraction selectivity by utilizing a single apodized TVBG instead of consecutive implementation of two uniform TVBGs preserves the robustness of the optical system.

4. Summary

Moiré apodization of RIM along the thickness of TVBG was proposed. Sinusoidal semi-period profile of apodization can be created by sequential recording two identical holographic patterns slightly rotated relatively to each other and cutting the developed specimen at the closest parallel planes of zero modulation. Coupled wave equations were derived for modeling of diffraction properties of apodized TVBG. Expressions for parameters of apodized TVBG providing 100% efficiencies of wave diffraction with two polarizations have been presented. Simulation results show essential improvement in the selective properties of apodized TVBG due to the significant suppression of diffraction sidelobes. According to asymptotic formulas, the diffraction efficiency of an apodized grating decreases inversely proportional to the fourth power of the angular or the wavelength detuning, while the diffraction efficiency of the uniform grating decreases inversely proportional to the square of detuning. To conclude, the apodized TVBG is a compact and a robust optical element characterized by narrow angular selectivity with enhanced suppression of the diffractive sidelobes.

Appendix: Coupling of TM waves

According to the notations used previously, consider the Bragg vector Q directed along the x-axis and the diffraction occurred in the (x,z)-plane. Then electromagnetic waves with TM polarization have a y-oriented magnetic field. Therefore

(28)$${\textbf H} = {{\textbf e}_y}H,\quad {\textbf H} \bot {{\textbf k}_\textrm{A}},{{\textbf k}_\textrm{B}},{\textbf Q},\quad {\textbf Q} = {{\textbf e}_x}Q.$$

Maxwell equations presented in Eq. (2) lead to the second-order equation on H

(29)$$- {\nabla ^2}{\textbf H} = \nabla {\varepsilon _\textrm{r}} \times ( - i\omega {\varepsilon _0}{\textbf E}) - i\omega {\varepsilon _0}{\varepsilon _\textrm{r}}\nabla \times {\textbf E} = {\textbf g} \times (\nabla \times {\textbf H}) + {\omega ^2}{c^{ - 2}}{\varepsilon _\textrm{r}}{\textbf H},\quad {\textbf g} = \varepsilon _\textrm{r}^{ - 1}\nabla {\varepsilon _\textrm{r}}.$$

Introduced vector function g based on ɛ_r from Eq. (2) can be written explicitly as

(30)$${\textbf g} = {{\textbf e}_x}g,\quad g = in_0^{ - 1}{n_1}({\textbf r})({e^{i{\textbf Qr} + i\gamma }} - {e^{ - i{\textbf Qr} - i\gamma }})Q.$$

Here slow variation of n₁ has been neglected.

Taking into account Eq. (28), the double vector product in Eq. (29) is equal to

(31)$${\textbf g} \times (\nabla \times {\textbf H}) ={-} ({\textbf g}\nabla ){\textbf H} ={-} {{\textbf e}_y}g{{\partial H} \mathord{\left/ {\vphantom {{\partial H} {\partial x}}} \right.} {\partial x}}.$$

Now the Eq. (29) can be presented in a scalar form similar to Eq. (3)

(32)$${\nabla ^2}H + {k^2}(1 + n_0^{ - 1}{n_1}({\textbf r})({e^{i{\textbf Qr} + i\gamma }} + {e^{ - i{\textbf Qr} - i\gamma }}))H - g{{\partial H} \mathord{\left/ {\vphantom {{\partial H} {\partial x}}} \right.} {\partial x}} = 0.$$

The magnetic field amplitude presented as the sum of two waves is

(33)$$H({\textbf r}) = {A_\textrm{m}}({\textbf r}){e^{i{{\textbf k}_\textrm{A}}{\textbf r}}} + {B_\textrm{m}}({\textbf r}){e^{i{{\textbf k}_\textrm{B}}{\textbf r}}}.$$

The subscript m stands for the magnetic field.

Near the resonant Bragg condition, x-components of the vectors are approximately equal to

(34)$${{\textbf k}_\textrm{A}} - {{\textbf k}_\textrm{B}} - {\textbf Q} \approx 0:\quad {k_{\textrm{A},x}} = k\cos \theta,\quad \theta = \angle ({{\textbf k}_\textrm{A}},{\textbf Q}),\quad Q \approx 2k\cos \theta,\quad {k_{\textrm{B},x}} ={-} k\cos {\theta _\textrm{B}},\quad {\theta _\textrm{B}} \approx \theta .$$

As g is proportional to the small amplitude n₁ of RIM, the last derivative in Eq. (32) can be calculated considering only the fast spatial dependencies described by the wave vectors from Eq. (34)

(35)$${{\partial H} \mathord{\left/ {\vphantom {{\partial H} {\partial x}}} \right.} {\partial x}} = ik\cos \theta ({A_\textrm{m}}({\textbf r}){e^{i{{\textbf k}_\textrm{A}}{\textbf r}}} - {B_\textrm{m}}({\textbf r}){e^{i{{\textbf k}_\textrm{B}}{\textbf r}}}).$$

After substituting Eqs. (30, 33, and 35) into Eq. (32), and collecting terms at the oscillating phase of A-wave exp(ik_Ar), one coupled wave equation is

(36)$$2i({{\textbf k}_\textrm{A}}\nabla {A_\textrm{m}}) - \cos (2\theta ){k^2}n_0^{ - 1}{n_1}({\textbf r}){e^{i\gamma }}{B_\textrm{m}}({\textbf r}){e^{i({{\textbf k}_\textrm{B}} + {\textbf Q} - {{\textbf k}_\textrm{A}}){\textbf r}}} = 0.$$

A similar differential equation follows for another coupled wave. To simplify, the trigonometric identity 2cos²θ – 1 ≡ cos(2θ) was used.

Equation (36) and another for B-wave, after final transformations, are represented as a system of coupled wave equations

(37)$$\left\{ {\begin{array}{{c}} {({{\textbf u}_\textrm{A}}\nabla {A_\textrm{m}}) ={-} i\kappa ({\textbf r}){e^{ - 2i{\textbf Dr}}}{B_\textrm{m}}({\textbf r}),\;}\\ {({{\textbf u}_\textrm{B}}\nabla {B_\textrm{m}}) ={-} i{\kappa^\ast }({\textbf r}){e^{2i{\textbf Dr}}}{A_\textrm{m}}({\textbf r}),\;} \end{array}} \right.\quad \quad \begin{array}{{c}} {{\kappa _{\textrm{TM}}}({\textbf r}) = {e^{i\gamma }}\cos (2\theta ){{\pi {n_1}({\textbf r})} \mathord{\left/ {\vphantom {{\pi {n_1}({\textbf r})} \lambda }} \right.} \lambda },}\\ {{\textbf D} = {\textstyle{1 \over 2}}({{\textbf k}_\textrm{A}} - {{\textbf k}_\textrm{B}} - {\textbf Q}).} \end{array}$$

Where unit vectors u_A,B are defined in Eq. (4).

The coupled wave Eq. (37) were proved equivalent to Eq. (7) with an additional factor cos(2θ) in the coupling for TM polarization. The negative signs in Eq. (37) are the phase factors appeared due to the derivations performed for amplitudes A_m and B_m of magnetic field instead of the electric field amplitudes A and B. These negative signs can be eliminated with substitution, A_m = cA, B_m = − cB, in Eq. (37), where c is a constant of proper dimensionality, and Eq. (7) are reproduced with the coupling coefficient presented in Eq. (8).

Disclosures

The author declares no conflicts of interest.

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Transmitting volume Bragg grating with longitudinal moiré apodization

Abstract

1. Introduction

2. Refractive index modulation and volume Bragg diffraction

3. Transmitting volume Bragg grating with moiré apodization

4. Summary

Appendix: Coupling of TM waves

Disclosures

References

Cited By

Figures (4)

Equations (37)

OSA Continuum