Phase Integrals: Add implementation for 2nd phase integrals + discussion notes + documentation

Author: steindev

src/interfaces/background_field_interface.jl

@@ -216,9 +216,27 @@ Return the first phase integral of the given field, for the given phase space po
 
     ```math
     \\begin{align*}
-        B_1(l, p, p^\\prime)& = \\int \\mathrm{d}\\varphi \\exp[\\imath l \\varphi + \\imath G(\\varphi)] \\\\
+        B_1^\\mu(l, p, p^\\prime)& = \\int \\mathrm{d}\\varphi A^\\mu(\\varphi)\\exp[\\imath l \\varphi + \\imath G(\\varphi)] \\\\
     \\end{align*}
     ```
-    where ``G(\\varphi,p, p^\\prime)`` is the [`phase function`](@ref), ``(p,p^\\prime)`` the given phase space point, and ``l`` the photon number parameter.
+    where ``A^\\mu(\\varphi)`` is the background field, ``G(\\varphi,p, p^\\prime)`` is the [`phase function`](@ref), ``(p,p^\\prime)`` the given phase space point, and ``l`` the photon number parameter.
 """
 function phase_integral_1 end
+
+"""
+
+    phase_integral_2(field::AbstractPulsedPlaneWaveField, pol::AbstractDefinitePolarization, p_in::, p_out::, pnum)
+
+Return the second phase integral of the given field, for the given phase space point `p_in, p_out` and a given photon number parameter `pnum`.
+
+!!! note "Convention"
+
+    The second phase integral is defined as:
+
+    ```math
+    \\begin{align*}
+        B_2(l, p, p^\\prime)& = \\int \\mathrm{d}\\varphi A(\\varphi)^2 \\exp[\\imath l \\varphi + \\imath G(\\varphi)] \\\\
+    \\end{align*}
+    ```
+"""
+function phase_integral_2 end

src/phase_integrals/first_integral.jl

@@ -10,7 +10,7 @@ function phase_integral_1(
     p_out::T,
     pnum::T
 ) where {T<:Real}
-    return quadgk(t -> , endpoints(domain(field))...)[1]
+    return quadgk(t -> amplitude(field, pol, t)*_shared_integrand(field, pol, p_in, p_out, t, pnum), endpoints(domain(field))...)[1]
 end
 
 function phase_integral_1(

src/phase_integrals/phase_integrals.jl

@@ -2,6 +2,25 @@
 # Definitions common to all phase integrals
 ###########################################
 
+# TODO: We need to talk about the handling of field polarization!
+# Since some of the quantities required in phase integral computation are four-vectors, just as the bg-field itself,
+# we need to return these as four-vectors, or their components.
+# However, we do not even provide the field as a four-vector, yet.
+# We just return its "amplitude", which is only the oscillator times the envelope,
+# not even taking the correct maximum value a_0 of the field into account. (TODO!)
+#
+# I think both the polarization and the maximum value of the field should be a user defined quantity,
+# but then these should also be members of the field struct, shouldn't they?
+# Or are these separately set in the Process?
+#
+# All in all, I wonder how we should treat four-vectors in QEDfields.
+# The problem of missing vectorial information appears here in _field_integral(), _kinematic_vector_phase_factor(), and phase_integral_1().
+
+# TODO: The factors and integrals should not depend on the momenta of particles
+# but on the Process and Phase Space Point (the latter of which holds the momenta)
+# Question: Does the process hold a reference to the background field?
+#   If so, the explicit dependence on the background field should be removed to.
+
 # from QEDprocesses.jl/src/constants.jl
 # TODO: we might want to move the constants.jl file to QEDbase? See also TODO in QEDprocesses.jl/src/processes/one_photon_compton/perturbative/cross_section.jl
 const ALPHA = inv(137.035999074)
@@ -51,6 +70,25 @@ end
 end
 
 # non-linear Volkov phase
+"""
+
+    _phase_function(field::AbstractPulsedPlaneWaveField, pol::AbstractPolarization, p_in::, p_out::, phi::)
+
+Return the phase function (or non-linear Volkov phase), for the given phase space point `p_in, p_out` and a given phase value `phi`.
+
+!!! note "Convention"
+
+    The non-linear Volkov phase is defined as:
+
+    ```math
+    \\begin{align*}
+        G(\\varphi,p, p^\\prime)& = \\alpha_1^\\mu \\int\\limits_0^\\varphi \\mathrm{d}\\varphi^\\prime A_\\mu(\\varphi^\\prime)
+            + \\alpha_2 \\int\\limits_0^\\varphi \\mathrm{d}\\varphi^\\prime A^2(\\varphi^\\prime) \\\\
+    \\end{align*}
+    ```
+    where ``A^\\mu(\\varphi)`` is the background field, ``\\alpha_1^\\mu`` is the [`kinematic vector phase factor`](@ref), and ``\\alpha_2`` is the [`kinematic scalar phase factor`](@ref).
+    Both ``\\alpha_1^\\mu`` and ``\\alpha_2`` depend on the given phase space point ``(p,p^\\prime)`` and the field's reference momentum ``k^\\mu`` the photon number parameter.
+"""
 @inline function _phase_function(
     field::AbstractPulsedPlaneWaveField,
     pol::AbstractPolarization,

src/phase_integrals/second_integral.jl

@@ -10,7 +10,7 @@ function phase_integral_2(
     p_out::T,
     pnum::T
 ) where {T<:Real}
-    return quadgk(t -> , endpoints(domain(field))...)[1]
+    return quadgk(t -> amplitude(field, pol, t)^2 * _shared_integrand(field, pol, p_in, p_out, t, pnum), endpoints(domain(field))...)[1]
 end
 
 function phase_integral_2(

src/phase_integrals/zeroth_integral.jl

@@ -2,24 +2,27 @@
 # Zeroth phase integral
 #######################
 
-# Does it need to be the same T for all arguments?
-function phase_integral_0(
-    field::AbstractPulsedPlaneWaveField,
-    pol::AbstractPolarization,
-    p_in::T,
-    p_out::T,
-    pnum::T
-) where {T<:Real}
-    return quadgk(t -> , endpoints(domain(field))...)[1]
-end
+# Is actually diverging, therefore not implmented by now.
+function phase_integral_0 end
+# The expected signature would be
+#phase_integral_0(
+#    field::AbstractPulsedPlaneWaveField,
+#    pol::AbstractPolarization,
+#    p_in::T,
+#    p_out::T,
+#    pnum::T
+#) where {T<:Real}
+# and the questions remains, whether T has to be the same for all arguments.
 
-function phase_integral_0(
-    field::AbstractPulsedPlaneWaveField,
-    pol::AbstractPolarization,
-    p_in::T,
-    p_out::T,
-    photon_number_parameter::AbstractVector{T}
-) where {T<:Real}
-    # TODO: maybe use broadcasting here
-    return map(x -> phase_integral_0(field, p_in, p_out, x), photon_number_parameter)
-end
+
+# Implementation of above function for a vector of photon numbers
+#function phase_integral_0(
+#    field::AbstractPulsedPlaneWaveField,
+#    pol::AbstractPolarization,
+#    p_in::T,
+#    p_out::T,
+#    photon_number_parameter::AbstractVector{T}
+#) where {T<:Real}
+#    # TODO: maybe use broadcasting here
+#    return map(x -> phase_integral_0(field, pol, p_in, p_out, x), photon_number_parameter)
+#end