Patrones

In the previous sections, we studied the parametric interaction of plane waves.

Although many of the observed phenomena can be explained with this simple model, some experiments presented in this thesis can only be explained if one extends the theory to the interaction of Gaussian fields. Since focussing the pump beam has a strong impact on the brightness of the photon-pair source, absolute photon-pair generation rates can only be calculated if the beam profiles of the interacting fields are taken into account.

The parametric interaction of Gaussian beams in classical nonlinear optics was studied in great detail in the seminal work of Boyd and Kleinman¹⁷⁴ in the

nonlinear crystal

cluster

𝜔_s/FSRs

ω_s/FSR_s a

b

c

Figure 1.4 | Spectrumofphotonsgeneratedbycavity-enhancedparametricdown-conversion.a, Single-crystalcavity-enhanced parametric down-conversion: If there is no compensation crystal inside the cavity, the free spec-tral range of the signal photons is not identical to the free specspec-tral range of the idler photons, in general.

E.g. for type-II phase matching, signal and idler photons experience a different refractive index inside the nonlinear crystal. Therefore, the spectral structure is different to the one shown in Fig.1.1. b, Cluster effect in cavity-enhanced parametric down-conversion: Here, the signal spectral density

^𝜓(^𝜔s) 

2( ) for un-equal free spectral range of the signal and idler photons is plotted. For this plot, the signal/idler finesse is F_s/i = 5 and the ratio of the free spectral ranges is 1 : 1.09. Note, that the maximum of the parametric gain envelope

^𝜓_sp(^𝜔s) 

2P(^𝜔s,𝜔_p −^𝜔s)( ) does not in general coincide with the center of a clus-ter. c,Detail of the plot shown in a: Each cluster consists of several joint resonances where signal ( ) and idler ( ) resonances partially overlap. To guide the eye, each of the resonances defined by A_s(^𝜔s)^and A_i(^𝜔p−^𝜔s)is scaled by

^𝜓(^𝜔s) 

2. For larger values of the signal/idler finesse, the overlap of adjacent signal/idler resonances is reduced, resulting in fewer modes per cluster.

late 1960ies. Their work covers a broad range of parametric interactions and does not solely target difference-frequency generation. With the advent of bright sources of non-classical light based on spontaneous parametric down-conversion, the need to provide a more comprehensive theory targeting photon-pair generation and absolute generation rates arose. Numerous works addressed various configurations under different conditions.^189–194The discussion in this section is based on the theory of Bennink⁵⁹since it is specifically targeted at collinear spontaneous parametric down-conversion. It is therefore the most suitable theory which can be extended to describe cavity-enhanced photon-pair generation, and it is thoroughly tested and verified experimentally.¹⁹⁵

To provide an intuitive approach, this section will start from classical (non-resonant) nonlinear optics to discuss the effect of focussing on the phase-matching condition and the brightness of the source. The modified phase matching results in a modified spectrum of the generated photons, which we will discuss in detail. Subsequently, we will analyze the parametric interaction inside a cavity, taking into account the Gaussian nature of the fundamental mode of the cavity. To account e.g. for additional compensation crystals inside the cavity, this section studies cavities of arbitrary geometry and with an arbi-trary number of optical media. For triply-resonant standing-wave cavities, the beam profile of the three fields cannot be chosen independently. In the final part of this section, we will discuss how this affects the brightness of the photon-pair source.

Gouy phase shift in classical nonlinear optics. In the case of focused Gaussian beams interacting with a nonlinear medium, the so called Gouy phase shift has to be taken into account as it affects the phase matching between the electrical fields and the nonlinear polarizations.¹⁷⁵Upon propagation through the focus with beam waist 𝑤0, an incident wave acquires a phase shift 𝜙Gouy of 𝜋

𝜙_Gouy(^𝑧) = −^arctan

 𝑧 𝑧_R



, (1.2.1)

where 𝑧Ris the Rayleigh length 𝑧R =(^𝜋𝑤₀²)/^𝜆. One can show that the parametric interaction is most efficient if the Rayleigh length of the generated fields is identical to the Rayleigh length of the pump field.^59,174 Then, according to Eq.1.2.1, the Gouy phase shift for all involved fields is the same.

Since the Gouy phase is an entirely classical phenomenon, the notation in-troduced in section 1.1.1can be reused. The field amplitudes are replaced by:

𝐴_𝑛(^𝑧) → ^𝐴0𝑛(^𝑧)^{𝑒𝑖𝜙Gouy(𝑧)}, 𝑛 =_{p, s, i. (1.2.2)}

nonlinear crystal filter cavity

Figure 1.5 | Filtering with a cavity after the generation of photon pairs.Instead of placing the nonlinear crystal inside the cavity, a cavity can be used to spectrally filter the photon pairs generated by spontaneous parametric down-conversion. The two-photon state can be post-selected for the case that both photons are transmit-ted by the filter cavity. Then, the state cannot be distinguished from the two-photon state generatransmit-ted by (non pump resonant) cavity-enhanced parametric down-conversion, given that the free spectral range and linewidth of the cavity are identical. The key difference between these two photon-pair sources is, that a cavity-enhanced source can be much brighter in terms of photon-pair generation rate per pump power.

nonlinear crystal

l b

a b

Figure 1.6 | Compensation of the Gouy phase shift.a, Simulation of the pump ( ) and signal/idler ( ) beam profile with a nonlinear crystal of length l and identical confocal parameters b =2zR. At degeneracy, the waist of the signal/idler field is a factor of√

2 larger than the waist of the pump field. The confocal parameter is chosen so that the ratio𝜉 = l/^b = 2.838 and the conversion efficiency is maximum.¹⁷⁴ b, Partial compensation of the Gouy phase shift for a nonzero phase mismatch ∆k > 0. The Gouy phase causes a spatially varying phase mismatch between the electrical fields and the corresponding nonlinear polar-ization. At nominal phase matching (∆k =0), a phase mismatch of𝜙_Gouy(^z)( ) would occur, which reduces the conversion efficiency. A positive phase mismatch yields a spatial phase of ∆kz ( ). At the optimum phase mismatch ∆k =3.254/^{land when𝜉}=2.838, the phase mismatch ΦGouy−∆kz ( ) is minimum and the spectral density is maximized.

In the case of degenerate parametric down-conversion with perfect phase match-ing, the nonlinear polarization at the pump frequency acquires an additional phase shift of 2𝜙Gouy:

˜

𝑃_p(^{𝑧, 𝑡}) =^4𝜖0𝑑_eff𝐴⁰

s𝐴⁰

i𝑒^{−𝑖2𝜙Gouy(𝑧)}𝑒^{−𝑖[𝜔p𝑡+𝑘p𝑧]}, for∆𝑘=0. (1.2.3) Therefore, there is a phase difference of 𝜙Gouy between the nonlinear polariza-tion ˜𝑃_p(^{𝑧, 𝑡})and the electrical field

˜

𝐸_p(^{𝑧, 𝑡}) = ^𝐴p⁰𝑒^{−𝑖𝜙Gouy(𝑧)}𝑒^{−𝑖[𝜔p𝑡+𝑘p𝑧]}, (1.2.4) which results in a reduced conversion efficiency for∆𝑘=0. The same is true for the signal and idler fields. But, for a positive wave-vector mismatch∆𝑘>_0, there is a partial compensation of this phase difference between the nonlinear polarization and the corresponding electrical field. The effective phase mismatch is shown in Fig.1.6. It is given by:

𝜙_Gouy−^∆𝑘𝑧. (1.2.5)

Therefore, the largest values of the joint spectral density can no longer be found for∆𝑘=_{0 but for}∆𝑘> _0.

Intuitively, one would assume that by focussing the pump beam more tightly one would obtain a higher pump intensity and thus a higher photon-pair generation rate. The overall brightness does indeed increase monotonically (up to a point where the Rayleigh length is much smaller than the crystal length, i.e. 𝜉 &10)⁵⁹ But, due to the Gouy phase shift, the spectral width of the generated photon pairs gets larger and the spectral brightness is reduced. Therefore, there is an optimum beam waist (and a corresponding phase mismatch∆𝑘>0) for which the parametric gain envelope reaches its maximum value.

Boyd-Kleinmanfactor. For focussed Gaussian beams, the mode function (Eq.1.1.12) takes the form:⁵⁹

E(^{𝑘, 𝒓, 𝑤}0) = p¹ 𝜋/²

𝑤₀ 𝑞

𝑒⁻

𝑥 2+^{𝑦 2} 𝑞 +^𝑖𝑘𝑧

. (1.2.6)

Here, 𝑤0is the beam waist and

𝑞=^𝑤2₀+2𝑖𝑧/^{𝑘. (1.2.7)}

If all fields have the same confocal parameter 𝑏=_2𝑧Rand there is negligible walkoff (no double refraction of the pump beam), the mode overlap for a

Gaussian beam ^OGaussian (Eq. 1.1.23) is proportional to the Boyd-Kleinman focus position parameter 𝜇is defined by:

𝜇=₁−2𝑧f/^𝑙, (1.2.10)

where 𝑧f is the distance between the focus and the crystal’s front facet.

1.2.1 | Spectral features and focussing parameter

For a monochromatic pump, the spectral density of the signal photons is pro-portional to the Boyd-Kleinman factor:



^𝜓_s(^𝜔s) 

2 ∝^ℎ(^𝜎(^𝜔s, 𝜔_i)^{, 𝜉, 𝜇}) =^ℎ(^∆𝑘(^𝜔s, 𝜔_p−^𝜔s)^{, 𝑙, 𝜉, 𝜇})^{. (1.2.11)}

The integral appearing in Eq.1.2.8cannot be solved analytically. This section subsequently shows that in the limit of small values of 𝜉 the Boyd-Kleinman factor converges to the sinc²shape known from the interaction of plane waves (Eq.1.1.30). For very tight focussing, i.e. larger values of 𝜉, the maximum of

the spectral density shifts to larger values of∆𝑘𝑙as expected.

Weakly focussed pump. For weakly focussed pump beams (small values of 𝜉), ℎ is proportional to the familiar⁵⁹sinc² ∆𝑘𝑙/²

term (Eq.1.1.11), as can be shown

by a variable substitution (𝜏→^𝜉(^𝑡+^𝜇)) and Taylor expansion: As expected for small values of 𝜉, the conversion efficiency is independent of the focus position 𝜇 and is proportional to the pump intensity 𝐼,¹⁹⁶since

𝜉 ∝¹/^𝑧R ∝ ^{𝐼. (1.2.16)}

Since 𝜉 = ^𝑙/^{𝑏, Eq. 1.2.15} shows that for weakly focussed pump beams the photon-pair generation rate is proportional to the length of the nonlinear crys-tal.

Strongly focussed pump. As discussed earlier, for larger values of 𝜉, ℎ reaches its maximum value only for a phase mismatch∆𝑘 > 0, resulting in a focus-dependent wavelength shift of the generated signal and idler photons.¹⁹⁷The spectrum of the generated photons also deviates from the sinc² shape and becomes more and more skewed for larger values of 𝜉. The Boyd-Kleinman factor is shown for different values of 𝜉 in Fig.1.7.

The largest value of parametric gain envelope ℎ(^𝜉, 𝜎, 𝜇) is found for 𝜉max ≈ 2.838 and 𝜎max ≈^0.576:174

ℎ_max =^ℎ(^𝜉max, 𝜎max, 0) ≈^{1.06. (1.2.17)} For larger values 𝜉>^𝜉_max, the maximum of the gain envelope decreases. But, with increasing values of 𝜉, the gain bandwidth increases due to the decreased effective length of phase-matched parametric interaction.

The overall photon-pair generation rate summed over all frequencies increases monotonically with 𝜉 up to 𝜉 ≈ 10: The generation rate is proportional to arctan 𝜉

to good approximation.⁵⁹

0 0.5 1 a

b

Figure 1.7 | Boyd-Kleinman factor. a, For different values of the focussing parameter𝜉the Boyd-Kleinman factor h(∆kl,𝜉_{, 0})^(Eq.1.2.12) is plotted as a function of the (dimensionless) phase mismatch ∆kl. The maxi-mum shifts to ∆k > 0 as𝜉increases. The global maximum is found at𝜉 = 2.84. While the spectral distribution is almost symmetric for small values of𝜉, it becomes increasingly skewed and broader for 𝜉 > 1. b, Sections through a for different values of𝜉(indicated in the plot). For small values of𝜉_{, h is} proportional to sinc² ∆kl/²
. Although the maximum over all values of ∆kl of h decreases for values 𝜉>2.838, the overall brightness (which is proportional to the area under the curve) is still monotonically increasing if the pump is focussed more strongly.⁵⁹If the focus of the pump is not in the center of the non-linear crystal (𝜇 ,0), the spectrum shows qualitatively the same behavior, but h(^{∆k, l,𝜉,𝜇})^{is always} smaller than h(∆k, l,𝜉_{, 0})^.174

If the photon-pair generation rate summed over all frequencies is the main figure of merit the pump focus should be as small as possible. If, on the other hand, the aim is a large number of photons generated into a specific spectral mode of a cavity-enhanced source, the Boyd-Kleinman factor of the cavity should be maximized for optimum performance. But, this kind of brightness optimization could affect the heralding ratio of the photon-pair source. We will discuss this reduction in heralding ratio in section1.2.4.

1.2.2 | Gouy phase inside a resonator

The spectral properties of photon pairs generated by spontaneous parametric down-conversion in a triply-resonant cavity are affected by the relative phase acquired between the nonlinear crystal and the first mirror (see section1.1.4).

For the interaction of Gaussian beams, the additional Gouy phase has to be taken into account to estimate this relative phase. If there are, besides the nonlinear crystal, additional elements (e.g. compensation crystals) placed inside the cavity, the expression for the optical phase at any point and the resonance conditions become more involved. The Gouy phase at any point in the cavity

depends on the position of each optical element and the radii of curvature of the mirrors. Here, we will provide an expression for the position-dependent confocal parameter. With the help of this expression, the total Gouy phase acquired in a single pass through the cavity and the Gouy phase at any point can be calculated. In the literature, comparable expressions can commonly be found for empty cavities.^198–200In contrast to this, in this section, we will discuss the case of a linear cavity consisting of an arbitrary number of optical elements and an arbitrary position of the nonlinear crystal.

Confocal parameter and effective cavity length. The confocal parameter 𝑏(^𝑧)^{of a beam} matched to the fundamental Gaussian mode of the cavity, depends on the refractive index 𝑛(^𝑧) inside the resonator. For a cavity with mirror radii of curvature 𝑅1 and 𝑅2, it is:¹⁹⁸ If the resonator consists of 𝑚 elements of length 𝑙^𝑖and refractive index 𝑛^𝑖(see Fig.1.8a), the effective length 𝑙eff in Eq.1.2.18is:

𝑙_eff =

𝑚

Õ

𝑖=₁

𝑙_𝑖/^{𝑛𝑖. (1.2.19)}

The effective cavity length is smaller than the distance of the mirrors.²⁰⁰ Total Gouy phase and resonance condition. Assuming the two mirrors are placed at positions 𝑧1and 𝑧2, the total Gouy phase acquired in a single pass through the cavity is:¹⁹⁹ The resonance condition given in Eq.1.1.80now can be extended to:

𝜙_rt(^𝜔𝑞) =^{2𝜋𝑞, 𝑞} ∈ N^{, (1.2.21)}

M1 M2

| {z }

∆kl | {z }

∆𝜙2

| {z }

∆𝜙1

l₁ l₂ l₃ ^{. . .} l_j-1 l_j l_j+1 ^{. . .} l_m-2 l_m-1 lm nonlinear crystal

M1 M2

zeff(z)(a.u.) 𝜙Gouy(z)

n=^{2.4 n}=^{1.7 n}=^{2.0 n}=^{2.2 n}=^2.3

M1 M2

n=^{2.4 n}=^{1.7 n}=^{2.0 n}=^{2.2 n}=^2.3

z₀ z₀

z₀^eff a

b ^c

Figure 1.8 | Linear resonator with arbitrary number of optical elements.a, Here, a linear resonator with m elements (e.g. free space or compensation crystals) of length liand refractive index niis shown. The nonlinear crystal with refractive index njis placed at position j. The focus position as well as the Gouy phase at each point in the resonator depend on the mirror radii of curvature, as well as the position and ordering of the optical elements. b, For a cavity with five crystals (each with an arbitrarily chosen refractive index), the effective position zeff(^z)^(Eq.1.2.26) is shown. Above the plot, the position and refractive index of each crystal is indicated. The effective position is needed to find an expression for the Gouy phase𝜙_Gouy(^z)at any point in the cavity, which is shown in c. The position of the focus z₀and the effective focal position z₀^effare indicated with dotted lines.

where the round-trip phase is given by:

Here, for completeness, the poling periodΛ(Eq.1.1.8) must be included. The corresponding resonance frequencies are:

𝜔_𝑞 =^{𝑐2𝜋𝑞}−^𝛿1−^𝛿2+2ΦGouy−^4𝜋𝑙_Λ^𝑗 2Í^𝑚

𝑖=1𝑙_𝑖𝑛_𝑖 . (1.2.23)

Position dependent Gouy phase. If the first mirror is placed at position 𝑧1 =_{0, the} effective position 𝑧^eff₀ of the focal point is given by:¹⁹⁸

𝑧^eff

To compute the Gouy phase at any point in the cavity, Eq.1.2.1needs to be modified:

𝜙_Gouy(^𝑧) =arctan 𝑧_eff(^𝑧) −^𝑧eff₀ 𝑧_R

!

, (1.2.25)

where the effective position 𝑧eff(^𝑧)is given by:

𝑧_eff(^𝑧) =^Lin {^𝐿}^,{^𝐿eff}^{, 𝑧}

. (1.2.26)

The function Lin {^𝐿},{^𝐿eff}, 𝑧

is the linear interpolation between the points:

(^𝐿0, 𝐿0,eff)^, (^𝐿1, 𝐿1,eff)^{, . . . ,} (^{𝐿𝑚, 𝐿𝑚},eff) ^(1.2.27) evaluated at the position 𝑧, where the following accumulated lengths are intro-duced:

The effective position 𝑧eff(^𝑧)can also be expressed as an integral: where 𝑛(^𝑧)is the refractive index at position 𝑧 in the cavity. The effective position 𝑧_eff(^𝑧) and the Gouy phase 𝜙Gouy(^𝑧) are shown in Fig. 1.8 for an exemplary cavity.

The position 𝑧₀ of the focal point measured from the first mirror can be found by:

𝑧₀ =Lin

{^𝐿eff}^,{^𝐿}^{, 𝑧eff}₀ 

. (1.2.30)

Boyd-Kleinman factor with arbitrary focus position. If the focal position lies outside of the nonlinear crystal, the focus position parameter (Eq.1.2.10) in the Boyd-Kleinman integral (Eq. 1.2.8) has to be modified to take into account the different refractive indices outside the crystal. After some algebra, one finds:

𝜇_eff =1−^2𝑛𝑗

If the focal position is inside the nonlinear crystal, 𝜇eff = ^𝜇. If it is outside, 

^𝜇_eff   >_1.

1.2.3 | Triply-resonant cavity with Gaussian beams

Gaussian mode functions. For the cavity depicted in Fig.1.8with 𝑚 elements of length 𝑙^𝑖and refractive index 𝑛^𝑖, the mode function for photons generated in the forward direction is:

where^T(^𝜔), the cavity transmission function defined in Eq.1.1.61, now depends on the round-trip phase given in Eq.1.2.22. The spatial phase factor is:

𝜙_opt(^𝜔, 𝑧) = ^𝜔 where the accumulated optical path lengths are:

𝐿_𝜈,opt =

𝜈

Õ

𝑖=₁

𝑙_𝑖𝑛_𝑖, with 𝐿0,opt =0. (1.2.34)

This definition implies that the first mirror 𝑀1is placed at the position 𝑧1 =0.

Then, the beam parameter 𝑞eff(^𝑧)is given by Eq.1.2.7, modified according to Eq.1.2.25:

𝑞_eff(^𝑧) =^𝑤0²+ 2𝑖h

𝑧_eff(^𝑧) −^𝑧eff₀ i

𝑘 . (1.2.35)

The mode function for photons generated in the backward direction is given by:

Phase-balancing factor. Now, the steps leading to the derivation of state ^𝜓_cavity

 (Eqs. 1.1.72and 1.1.79) can be repeated with the Gaussian mode functions.

The only difference is the spatial phase-factor 𝜙optand that the overlap integral 

O(^𝜔s, 𝜔i) 

2 has to be replaced by the Boyd-Kleinman factor:



O_Gaussian(^𝜔s, 𝜔i) 

2 ∝^ℎ(^𝜎(^𝜔s, 𝜔i), 𝜉, 𝜇eff), (1.2.37) where 𝜇effis given by Eq.1.2.31. We will discuss the proportionality constant in Eq.1.2.37in section1.3.

Utilizing the resonance condition (Eq. 1.2.21), we find the phase-balancing amplitude (see Eq.1.1.73) to be:

𝑝_Gaussian(^𝜔s, 𝜔i) =1+^𝑟_1p^𝑟_2s^𝑟_2i^𝑡_sp,p^𝑡_sp,s^𝑡_sp,i^{𝑒𝑖[∆𝑘qpm𝑙+∆ΦGaussian]}_{, (1.2.38)} where∆𝑘_qpmis the phase mismatch including the quasi-phase-matching term:

∆𝑘_qpm =∆𝑘+^{2𝜋𝑙𝑗}

Λ . (1.2.39)

The relative phase∆Φ_Gaussian is given by:

∆Φ_Gaussian =∆𝛿₁−2∆𝜙^eff

Gouy is the relative Gouy phase acquired between the center of the nonlinear crystal at position 𝑧cand the first mirror at position 𝑧1:

∆𝜙^eff

Gouy =∆𝜙_Gouy(^𝑧c) −^∆𝜙Gouy(^𝑧1)^{. (1.2.41)} We will discuss the expression for joint spectral density for Gaussian beams in section1.3.1.

1.2.4 | Heralding ratio

If the photon-pair source is used as a source of heralded single photons, the heralding ratio^59,195 is an important figure of merit. It is defined by:

𝜂_herald = ^𝑃si

𝑃_i , (1.2.42)

where 𝑃_si is the probability to detect both photons from a pair and 𝑃_i is the probability to detect just the (heralding) idler photon. If both signal and idler photons are collected into Gaussian modes, Bennink⁵⁹showed that the heralding ratio decreases with increasing values of 𝜉. This decrease in the heralding ratio was verified by Guerreiro et al.²⁰¹ and Dixon et al.¹⁹⁵It is caused by the spatial entanglement of the photons in a photon pair. The detection of the heralding idler photon in a specific Gaussian collection mode (e.g. an optical fiber) projects the signal photon into a superposition of Laguerre-Gaussian modes which has a reduced overlap with a Gaussian collection mode. This results in a reduction of the signal photon collection efficiency of up to 25 % for 𝜉 &1.

To the best of the author’s knowledge, it has not been studied in the literature whether this phenomenon also applies to cavity-enhanced parametric down-conversion. At least for a cavity with a low finesse it still must apply. But, for a higher finesse, the presence of a cavity should suppress the emission into non-Gaussian modes since the higher-order Laguerre-Gaussian modes have different resonance frequencies than the fundamental mode and are therefore suppressed. The experiments performed within the scope of this thesis were using a cavity with a small value of 𝜉≈0.2, where Bennink’s theory predicts only a small reduction in the heralding ratio. Therefore, a quantitative experimental evaluation of the impact of the cavity on the heralding rate is not within the scope of this thesis.

If the photon-pair generation rate is the most important figure of merit, the Boyd-Kleinman factor needs to be optimized. In a triply-resonant cavity, the spatial modes of all three fields are determined by the cavity geometry and cannot be optimized independently. This constraint will be discussed in the following.

1.2.5 | Boyd-Kleinman-theory for unequal confocal parameters

Since, in a triply-resonant cavity, the confocal parameters of the three interacting fields cannot be chosen independently (Fig.1.9), the Boyd-Kleinman factor has to be modified to account for unequal confocal parameters.

For this case, Bennink⁵⁹introduced the effective confocal parameter ¯𝑏, which is

Bennink showed that, for unequal confocal parameters, the Boyd-Kleinman factor has to be modified:

ℎ(^{𝜎, 𝜉, 0}) → ^ℎ¯(^{¯𝜎, ¯𝜉, 0}) = ⁴ 𝛼𝛽

ℎ(^{¯𝜎, ¯𝜉, 0})^{. (1.2.46)} The effective Boyd-Kleinman factor ¯ℎ can be expressed in terms of the con-ventional Boyd-Kleinman factor ℎ and effective parameters ¯𝜉= ^𝑙/^¯𝑏and ¯𝜎 =

∆𝑘𝑏¯/^2.

Boyd and Kleinman¹⁷⁴also derived an expression for ¯ℎ. Although they have different definitions for ¯𝑏, 𝛼 and 𝛽 due to different approximations, numerically the values of the effective Boyd-Kleinman factor agree within the range of a few percent with the values obtained based on Bennink’s theory.

Boyd and Kleinman¹⁷⁴also derived an expression for ¯ℎ. Although they have different definitions for ¯𝑏, 𝛼 and 𝛽 due to different approximations, numerically the values of the effective Boyd-Kleinman factor agree within the range of a few percent with the values obtained based on Bennink’s theory.

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