1. Introduction

Bragg mirror is one of most widely used optical components in many applications. One of these applications is the reflection mirror in a laser cavity. Reflection hologram made of PTR glass as Bragg mirror has great advantage of resisting the damage under the working condition of high power laser beam [1–3]. The good spectral selectivity of reflection hologram indeed gives a sharp frequency spectrum in the output laser beam. However, the conventional Bragg mirror doesn’t feature good angular selectivity. In this paper, we propose a new scheme based on double-recorded crossed gratings inside the volume hologram, which shows great improvement of angular selectivity.

A scheme considered earlier by Jacob et al. [4] uses surface hologram with the grooves placed at top of a planar waveguide. It selects beam with normal incident angle, which beam then propagates transversely inside the waveguide, and then is normally reflected back. Our scheme here is functionally analogous to the one from [4] but with reflection and transverse propagation at the same time. The low tolerance of surface gratings to strong laser power in their design will possibly be replaced by rather strong durability of holographic gratings.

For beams propagation inside the transmission hologram of double-recorded-gratings, some results have been shown in previous works [5] and [6]. The early work of Alferness [7] in detail explored the angular selectivity of transmission hologram of crossed gratings but not with reflection hologram. Here our new type of cross-gratings results a high reflectivity would considerably act as a reflection hologram. With respect to such reflection hologram, especially in the case of symmetric crossed-gratings, nontrivial behavior of four-wave coupling arises, which will be presented and analyzed in the following sections.

2. Four-waves coupling in the reflection hologram

Considering the derivation of the equations of coupled-wave theory [8–10], non-resonant terms are dropped off in the phase matching of Bragg wavevector. Along the same way, in our crossed-hologram, we can consider the wavevector matching condition of the crossed-gratings in Fig. 1(a), and have the beam propagation directions in the following diagram, Fig. 1(b).

 

Fig. 1. (a). Two crossed-gratings are to be recorded throughout the whole PTR glass. (b) Propagation of waves in two-crossed-grating hologram, where four waves arise and couple to each other. They are A ₊, A _-, B ₊, B _-, respectively, and A ₊ is the incident wave.

Download Full Size | PDF

Here four waves are appearing in the hologram, with A ₊, A _- as the incident and reflected waves, and B ₊, B _- as the excited intermediate waves. Due to the symmetry of our crossed-gratings, the waves A ₊ and A _- propagate oppositely in the ±z-directions, while B ₊ and B _-propagate oppositely in the ±x-directions. This specific wave coupling actually requires that the two gratings should be mutually perpendicular to each other and be oriented at 45° with respect to incident and reflected beams. To illustrate how it works, we should discuss the total field of TE polarized beam (E∝e _y) within the Maxwell equations. When the recording exposure is weak, the modulation of refractive index δn is proportional to the linear sum of these two independent interference profiles. The refractive index thus can be written as,

Here I _1,2(r) are the respective intensity profiles of exposure during holographic recording, n ₀ is the average index of refraction, µ₁, µ₂ are the values of grating modulation strength, and Q ₁, Q ₂ are the grating vectors. If the recording process is strongly nonlinear, for example with saturation as a function of exposure, additional “inter-modulation” gratings may arise, such as the ones with Q ₃=Q ₁+Q ₂ or Q ₄=Q ₁ - Q ₂ etc. Since here we assume the exposure is sufficiently weak, no such inter-modulation gratings will appear. Therefore, the resultant Helmholtz wave equation is

with the total field E,

For volume hologram with thickness L λ/2π, and very small modulation of refractive index, |δn|≤10^-3, the Slowly Varying Envelope Approximation (SVEA) is valid for interacting waves, for example, in the visible/near infrared range of λ and for L≅2.5 mm. Substituting of Eq. (3) into Eq. (2) and using SVEA, we arrive to the following set of four equations including the amplitude absorption coefficient β=n _I ω/c [1/meter] (n _I is the imaginary part of refractive index), and the first order frequency detuning γ=n ₀ δω/c, also [1/meter]:

Here κ ₁=µ ₁ ω/2c, κ2=µ ₂ ω/2c. We further consider the effect of angular detuning by taking Fourier components of all the waves with respect to the normal incidence angle, i.e. B _±≈e ^iqx with q=δk_x=δθ _air ω/c. Obviously from the last two equations in Eq. (4), this leads to,

Next, we substitute Eq. (5) back into the first two equations in Eq. (4) to obtain the coupled-wave equations for A-waves only. For the case κ ₁=κ ₂=κ=κ*, we get

While we have come to the coupled-wave Eqs. (6) for A-waves, originally there was no direct coupling between A ₊ and A _-. If we carefully take a look at Eqs. (4), it is the intermediate waves B _± are directly coupled with A _±. This means that two Bragg conditions should be satisfied simultaneously for each coupling in order to have good diffraction efficiency. And this is the main feature of our new design of hologram, which results in improvement of angular selectivity. In reality, for the finite size of incident beams, the angular detuning q actually plays a critical role in our scheme. As Eq. (7) indicates, since P grows together with Q at small values of angular detuning q, the coupling strength between A ₊ and A _- waves is mostly characterized by Q. Indeed, if we ignore the absorption and rewrite it as Q=iκ ²[1/(q-γ)-1/(q+γ)]=2iκ ² γ/(q ²-γ ²), then Q∝1/q ², i.e. the coupling strength is inverse proportional to the square of angular detuning q. As a consequence, this relation illustrates why our new type of reflection hologram has very good angular selectivity.

3. Results of reflection efficiency with proper boundary condition

Now we seek the solution of Eq. (6) with proper boundary condition. The solution in terms of initial condition has the following matrix form:

With definition $\rho =\sqrt{P^2-Q^2}$, and $\hat{M}=\left[\begin{array}{cc}P& Q\\ -Q& -P\end{array}\right]$, the matrix elements are

Consider the boundary condition for A _±(z) with incident beam A ₊(z=0)=1, and the reflected beam at the bottom is zero, A _-(z=L)=0, here L being the thickness of hologram. Accordingly, the reflection and transmission efficiency coefficients are respectively defined as η _ref=|A _-(0)/A ₊(0) |² and η _tran=|A ₊(L)/A ₊(0)|². After calculations, the results could be easily found,

From the above solution, we are ready to see some characteristics of our hologram. In Fig. 2, we compare both the spectral and angular selectivity of our crossed-gratings hologram with those of one-grating reflection hologram. We see that in Figs. 2(a) and 2(b), the maximum reflection efficiency are depicted as η _ref=0.8 for reflection grating and η _ref=0.81 for our crossed gratings for comparison. It shows both gratings all possess good spectral selectivity. At first glance, the diffraction efficiency curve of typical reflection hologram seems sharper. But the reflection hologram should pay the price in increasing its thickness, i.e. larger hologram strength to catch up the same reflection efficiency η _ref of our crossed hologram. The curve will broaden if the hologram strength decreases and the η _ref will also reduce dramatically.

 

Fig. 2. The reflection efficiency η for conventional reflection hologram with thickness L=2.5·10^-3 m and for our crossed hologram with thickness L=1·10^-3 m, for wavelength λ _vac=1.06·10^-6 m, refraction index n ₀=1.5, amplitude absorption β=10 m ^-1, and the grating amplitude κ varies from 0 to 1000 m ^-1 at normal incidence. (a) Conventional reflection hologram, thickness L=2.5·10^-3 m, with no angular detuning, Δθ=0, Δλ from -3.58·10^-10 m to 3.58·10^-10 m, Δλ _HWHM=1.48·10^-10 m (b) Our crossed hologram, thickness L=10^-3 m, with other parameters same as (a), Δλ _HWHM=2.34·10^-10 m; (c) Reflection hologram, thickness L=2.5·10^-3 m, with no spectral detuning Δλ=0, the angular detuning Δθ _air from -3° to 3°, Δθ_HWHM=1.44° (d) Our crossed hologram, thickness L=2.5·10^-3 m, the angular detuning Δθ_air from -0.03° to 0.03°, Δθ_HWHM=0.0028°; (e) The compensation between spectral detuning and angular detuning for reflection hologram with thickness L=2·10^-3 m, Δλ from -3.58·10^-10 m to 3.58·10^-10 m and Δθ_air from -3° to 3° (f) The compensation for our crossed hologram with parameters same as (e) except that Δθ_air is from -0.03° to 0.03°.

Download Full Size | PDF

In Figs. 2(c) and 2(d) we keep the same maximum reflection efficiency and other parameters as in Figs. 2(a) and 2(b) except there is no spectral detuning, which is replaced by angular detuning. We can see that our crossed grating has a much better angular selectivity. The reflection grating has a small but pretty flat region of high diffraction efficiency. In comparison, our angular selectivity is very sharp. Although people may argue that this crossed hologram has a longer interacting length, since the B _± waves are propagating horizontally. This interacting thickness would be diminished as long as we include the absorption in our calculation.

We also explored the case that both spectral and angular detunings are present. From our Fig. 2(e) and 2(f), it indicates the compensation between spectral and angular detuning in our crossed-grating are more in a symmetric manner. As deviated from the central region, this compensation gradually fades away. On the contrary, this is not the case for reflection hologram. We can see that, along the top bent cure in the two diagonal regions, significant compensation is observed and continuously keeps the effect. Thus it is more unpleasant to obtain the unexpected higher reflection efficiency for typical reflection hologram.

4. Conclusion

In conclusion, we have suggested a new type of crossed-reflection hologram. With four-wave interaction inside the volume, the incident and reflected wave indirectly coupled to each other through two excited intermediate waves, which propagate along the transverse direction with respect to the incident wave. Our analysis shows that the coupling behavior is inverse square to the detuning angle. Therefore, extraordinary high angular selectivity is achieved by our crossed-reflection hologram. With regard to the poor beam quality of high divergence angle of semiconductor lasers, our hologram may be used to improve this divergence due to its high angular selectivity as used for the mirror of external laser cavity. This work is supported by DARPA under the contract number H0011-06-1-0010.