9. Quasi-phase-matching. Generalize the discussion of the text leading from Eq. (2.4.1) to Eq. (2.4.6) by...

  
 

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9. Quasi-phase-matching. Generalize the discussion of the text leading from Eq. (2.4.1) to Eq. (2.4.6) by allowing the lengths of the inverted and non- inverted sections of nonlinear optical material to be different. Let A be the period of the structure and I be the length of the inverted region. Show how each of the equations in this range is modified by this different as- sumption, and comment explicitly on the resulting modification to the value of do and to the condition for the establishment of quasi-phase- matching. On the topic: square-wave function which can be represented as d(z) = deff sign[cos (27z/A)]; (2.4.1) more complicated spatial variations are also possible. In this equation, deff denotes the nonlinear coefficient of the homogeneous material. The spatial variation of the nonlinear coefficient leads to a modification of the coupled amplitude equations describing the nonlinear optical interaction. The nature of the modification can be deduced by noting that, in the derivation of the cou- pled amplitude equations, the constant quantity deff appearing in Eq. (2.2.6) must be replaced by the spatially varying quantity d (z). It is useful to describe the spatial variation of d(z) in terms of a Fourier series as d(z)=deff Gm exp(ikmz), (2.4.2) where km = 2лm/A is the magnitude of the grating vector associated with the mth Fourier component of d (z). For the form of modulation given in the example of Eq. (2.4.1), the coefficients Gm are readily shown to be given by (2/mx) sin(mл/2). Gm (2.4.3) from which it follows that the fundamental amplitude G₁ is given by G₁ = 2/π. itude countiens. Secti one assumes that one particular Fourier component of d(z) provides the dominant coupling among the interacting waves. After making the slowly varying amplitude approximation, one obtains the set of equations d A₁ dz d A2 dz d A3 dz = = 2iw.do nic 2iwzdo n2c 2iw3dQ n3c = A3A2e-i(Akq-2km)z A3A e-i(Akq-2km)z (2.4.4a) A1A₂e Akoz (2.4.4c) where do is the nonlinear coupling coefficient which depends on the Fourier order m according to do = deff Gm and where the wavevector mismatch for order m is given by Ako = k₁+k₂-k3 +km. (2.4.4b) (2.4.5) (2.4.6) The amplitude of the nonlinear polarization can then be written as P3=4e0deff A1 A2e (k₁+k₂)² = p3e¹(k₁+k₂)z (2.2.6) The question is as follows LI 9. Quasi-phase-matching. Generalize the discussion of the text leading from Eq. (2.4.1) to Eq. (2.4.6) by allowing the lengths of the inverted and non- inverted sections of nonlinear optical material to be different. Let A be the period of the structure and / be the length of the inverted region. Show how each of the equations in this range is modified by this different as- sumption, and comment explicitly on the resulting modification to the value of do and to the condition for the establishment of quasi-phase- matching. 9. Quasi-phase-matching. Generalize the discussion of the text leading from Eq. (2.4.1) to Eq. (2.4.6) by allowing the lengths of the inverted and non- inverted sections of nonlinear optical material to be different. Let A be the period of the structure and I be the length of the inverted region. Show how each of the equations in this range is modified by this different as- sumption, and comment explicitly on the resulting modification to the value of do and to the condition for the establishment of quasi-phase- matching. On the topic: square-wave function which can be represented as d(z) = deff sign[cos (27z/A)]; (2.4.1) more complicated spatial variations are also possible. In this equation, deff denotes the nonlinear coefficient of the homogeneous material. The spatial variation of the nonlinear coefficient leads to a modification of the coupled amplitude equations describing the nonlinear optical interaction. The nature of the modification can be deduced by noting that, in the derivation of the cou- pled amplitude equations, the constant quantity deff appearing in Eq. (2.2.6) must be replaced by the spatially varying quantity d (z). It is useful to describe the spatial variation of d(z) in terms of a Fourier series as d(z)=deff Gm exp(ikmz), (2.4.2) where km = 2лm/A is the magnitude of the grating vector associated with the mth Fourier component of d (z). For the form of modulation given in the example of Eq. (2.4.1), the coefficients Gm are readily shown to be given by (2/mx) sin(mл/2). Gm (2.4.3) from which it follows that the fundamental amplitude G₁ is given by G₁ = 2/π. itude countiens. Secti one assumes that one particular Fourier component of d(z) provides the dominant coupling among the interacting waves. After making the slowly varying amplitude approximation, one obtains the set of equations d A₁ dz d A2 dz d A3 dz = = 2iw.do nic 2iwzdo n2c 2iw3dQ n3c = A3A2e-i(Akq-2km)z A3A e-i(Akq-2km)z (2.4.4a) A1A₂e Akoz (2.4.4c) where do is the nonlinear coupling coefficient which depends on the Fourier order m according to do = deff Gm and where the wavevector mismatch for order m is given by Ako = k₁+k₂-k3 +km. (2.4.4b) (2.4.5) (2.4.6) The amplitude of the nonlinear polarization can then be written as P3=4e0deff A1 A2e (k₁+k₂)² = p3e¹(k₁+k₂)z (2.2.6) The question is as follows LI 9. Quasi-phase-matching. Generalize the discussion of the text leading from Eq. (2.4.1) to Eq. (2.4.6) by allowing the lengths of the inverted and non- inverted sections of nonlinear optical material to be different. Let A be the period of the structure and / be the length of the inverted region. Show how each of the equations in this range is modified by this different as- sumption, and comment explicitly on the resulting modification to the value of do and to the condition for the establishment of quasi-phase- matching.

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