Tutorial : general - quality factor

The Raman-Nath effect applies for a thin acoustic wave. The diffracted wave is then very similar to the one diffracted by a diffraction grating (see "diffraction" tutorial).

Let consider a transmissive grating with a refraction index and a thickness e which is small enough to neglect any propagation effect in the grating. An incident plane wave, is diffracted in several orders (several direction) as shows the diffracted amplitude :

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λ₀ is the wavelength of the optical wave in the air and θ_i is the incidence angle (in the material) and Λ is the grating pitch (see the scheme on the left side).

The diffracted wave is the superposition of plane waves propagating in the directions θ_p verifying :

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p is the diffration order.

The intensity diffracted in the order p is proportionnal to .

The normalized intensity for the different orders 0 (blue), 1 (red), 2 (green) and 3 (yellow) depending on Φ are represented on the left-side curves. The maximum diffraction efficiency is obtained for the order 1 and can not exceed 33%.

In an acousto optic material, the acoustic wave is travelling. Therefore, the grating is travelling with the same velocity. The refraction index is then :

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The amplitude of the diffracted wave is :

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Because of the conservation of the cinematic moment, the pulsations and thus the wavelengths of the diffracted orders are all different. The pulsation is given by :

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If the thickness of the acoustic wave increase, the diffracted wave is the result of intereference from diffracted waves at different positions in the thickness of the material.

Let consider a plane wave illuminating a grating with an incident angle θ_i = 0. The optical wavelength is considered much smaller than the grating pitch and the diffraction angle in the order p is considered small as well as the grating thickness d. The phase shift between the wave diffracted in the order p at the grating entrance and the one diffracted at the grating output is :

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Consequently, the interference become destructive in the order p for a thickness :

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For a thickness larger than L_c, the efficiency in the order p is significantly decreasing. The formula shows that L_c decreases with the order. Thus, increasing the grating thickness makes the diffraction orders disappear starting from the higher order.

Let us take advantage of this paragraph to mention that for acousto optics, the thickness is generally considered as the critical interaction length. It corresponds to the distance covered by a wave propagating perpendiculary to the acoustic wave for running through half of an acoustic wavelength in the transverse plane ( plane perpendicular to the light wave propagation ) because of the diffraction. Indeed, at a given time, the acousto optic can be schematized as a succession of low index and high index zones of thickness . The optical wave can be therefore considered as diffracted by FORMULA thick slits. The Raman-Nath regime can only occure if the interaction length is far below L_c.

Back to gratings, for an incidence angle θ_i, the amplitude of the wave diffracted in the order p is :

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A₀ ( θ_p ) is the amplitude of the wave diffracted in the order p per unit length.

This shows that if , the grating is considered thin and several orders can be diffracted (see "Raman-Nath" section). If the thickness increases, the diffraction efficiency decreases for all orders except for the one verifying the following equation :

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This applies for the order 0 and the order p where . This condition is called the Bragg condition. The absolute value of the incident ( and diffracted ) wave is then :

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In an acousto optic, the Bragg configuration applies for orders -1 and therefore :

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As the grating is moving at the acoustic wave velocity, there is a frequency shift of the diffracted wave given by : . The efficiency in the bragg conditions is :

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A maximum efficiency of almost 100% is obtained for .

The acoustic intensity is then :

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For an incidence angle close to the Bragg angle, the efficiency of the diffracted wave is :

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The angular acceptance Δ θ of the acousto optic is defined as the angle variation referred to the Bragg angle for which the efficiency drops to 50% (3 dB). It is :

*

Also, the frequency bandwidth Δ F_a is defined as the frequency variation referred to the exact frequency in Bragg configuration inducing an efficiency drop of 50% (3 dB). Δ F_a is given by the formula :

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(*) the angles are calculated in the acousto optic material. The Fresnel law must be applied to them to calculate the corresponding values in the air.

For an acousto optic, the Quality factor is defined as follows :

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The acousto optic can be considered as thin and its optical behavior can be modellized according to the Raman-Nath regime if : .

Q is then much smaller than 1.

The Bragg regime model is applicable when Q > 1.

In the Bragg regime, the efficiencies in the order 0 and 1 are then respectively :


and
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