Abstract

We investigate second-harmonic generation by an astigmatic partially coherent beam. An explicit expression for the second-order correlation function of the second-harmonic field is obtained. The properties of the generated field and the conversion efficiency for second-harmonic generation are studied numerically. We find that using an astigmatic instead of a stigmatic partially coherent pump beam can increase the conversion efficiency of the second-harmonic generation.

©2007 Optical Society of America

1. Introduction

Partially coherent beams have found wide applications in optical projection, laser scanning, inertial confinement fusion, free space optical communication and imaging applications [1–7]. A partially coherent twisted anisotropic Gaussian Schell-model (TAGSM) beam can be regarded as a general astigmatic partially coherent beam, which was firstly introduced by Simon and collaborators [8, 9]. In the past decades, the properties of partially coherent TAGSM beams have been widely studied [10–18]. Partially coherent TAGSM beams have been demonstrated experimentally through an acousto-optic coherence control technique [10]. Unlike the usual phase curvature, the twist phase is bound in strength and it disappears in the limit of full coherence, and it will rotate the beam spot and increase the beam divergence on propagation [8–15]. It has been found that a partially coherent TAGSM beam possesses high orbitual angular momentum [16]. Recently, Lin and Cai introduced a tensor method instead of the conventional Wigner distribution function to treat the propagation of a partially coherent TAGSM beam [17]. Cai and He investigated the propagation properties of a partially coherent TAGSM beam in turbulent atmosphere [18]. More recently, Wang et al. studied the propagation properties of anisotropic (i.e., astigmatic) electromagnetic partially coherent beam [19].

Second-harmonic generation is a convenient way to generate laser beams at new frequencies that are usually not available and has found wide applications in spectroscopy, optical communications, remote sensing and so on [20, 21]. In the past decades, second-harmonic generations by various types of laser beams, e.g., Gaussian beams, elliptical (i.e., astigmatic) Gaussian beams, flat-topped beams and stigmatic partially coherent beams have been widely studied [22–29]. It has been found that using an astigmatic Gaussian beam or a flat-topped beam or a stigmatic partially coherent beam instead of a stigmatic Gaussian beam can improve the conversion efficiency of second-harmonic generation under certain conditions. In this paper, we study the second-harmonic generation by a general astigmatic partially coherent beam (i.e., partially coherent TAGSM beam). An explicit expression for the second-order correlation function of the second-harmonic field is obtained by using the tensor method. The properties of the second-harmonic field and the conversion efficiency are studied numerically.

2. Second-order correlation function for the second-harmonic field generated by a partially coherent beam with arbitrary beam profile

We consider a lossless nonlinear crystal of length l between the planes z=0 and z = l. A spatially partially coherent pump beam with arbitrary beam profile of frequency w is incident at plane z=0 which gives rise to a second-harmonic beam of frequency of 2w at z = l after interaction with the nonlinear crystal. The transverse dimensions of the crystal are taken to be much larger than the beam width of the pump beam. Within paraxial approximation, the pump and second-harmonic fields satisfy the following two coupled equations in the nonlinear medium [25–28]

2εwx2+2εwy22ik1εwz=K1εw*ε2w,
2ε2wx2+2ε2wy22ik2ε2wz=K2εw2,

where εw and ε 2w are the slowly varying field amplitudes of the pump and the second-harmonic beams, respectively, k 1 and k 2 are the respective wave numbers of the two beams with k 2 = 2k 1, and K 1 = K 2/2 = 4πW 2 d / c, d is the nonlinearity coefficient for the second-harmonic generation. In above derivations, phase-matching is assumed and walk-off effects are ignored [25–28].

In the parametric approximation, the pump field remains undepleted under the assumption of weak nonlinear interaction, hence the right hand side of Eq. (1) becomes zero. Then the solution of Eqs. (1) and (2) becomes [25–28]

ε2w(ρ,l)=∫∫D(ρ,l,s1,s2)εw(s1,0)εw(s2,0)ds1ds2,

with s and ρ are the transverse position vectors at z=0 and z = l , respectively, d s = dsxdsy and

D(ρ,l,s1,s2)=K2k232π2lexp(ik2l)0l1z1exp[ik22lρ2]exp[ik2(s12+s22)8(1l+1z1)]
exp[ik2(ρs1+ρs2)2l]exp[ik2s1s24(1l1z1)]dz1.

then the second-order correlation function of the second-harmonic field at z = l is expressed as [25–28]

Γ(2)(ρ1,ρ2,l)=ε2w(ρ1,l)ε2w*(ρ2,l)=(K2k232π2l)2
D(ρ1,s1,s2)D*(ρ2,s3,s4)Γ(4)(s1,s2,s3,s4)ds1ds2ds3ds4dz1dz2,

with Γ(4)(s 1,s 3,s 3,s 4) = 〈εw(s 1,0)ε3(s 2,0)εw *(s 3,0)εw *(s 4,0)〉 is the fourth-order correlation function of the partially coherent pump beam, which can be approximated as follows [1]

Γ(4)(s1,s2,s3,s4)=Γ(2)(s1,s3,0)Γ(2)(s2,s4,0)+Γ(2)(s1,s4,0)Γ(2)(s2,s3,0).

Eq. (6) (i.e., fourth-order correlation) denotes the intensity correlation of the light, which has been justified by the Hanbury-Brown-Twiss effect [30]. It also forms the basis of ghost imaging [6, 31, 32], which was demonstrated in experiments recently by measuring the photon coincidence rate [33]. Equation (6) is valid both for partially coherent and incoherent sources provided that the source field fluctuations are caused by a Gaussian random process and obey Gaussian statistics [1]. In the derivation of Eq. (6), the coherence time of the source is assumed to be greater than the detector’s response time.

After some arrangement, we can express Eq. (5) in the following tensor form

Γ(2)(ρ1,ρ2,l)=(K2k232π2l)2exp[ik22lρ12ik22lρ22]0l0l1z11z2dz1dz2
[Γ(2)(s1,s3,0)Γ(2)(s2,s4,0)+Γ(2)(s1,s4,0)Γ(2)(s2,s3,0)]exp[ik22s͂TB͂1s͂]exp[ik2s͂TD͂ρ͂]ds,͂

where s͂T = (s T 1,s T 2,s T 3,s T 4) = (S 1x,S 1y,S 2x,S 2y,S 3x,S 3y,S 4x,S 4y), ρ͂T = (ρ T 1 ρ T 1,ρ T 2,ρ T 2) and

B͂1=(B͂lz1100B͂lz21),D͂=12l(I͂00I͂),
Blzi1=(14(1l+1zi)I14(1l1zi)I14(1l1zi)I14(1l+1zi)I),(i=1,2),

where I͂ is a 4×4 unit matrix and I is a 2 × 2 unit matrix. Eq. (7) is suitable for treating the second-harmonic generation by a partially coherent beam with arbitrary beam profile.

3. Second-harmonic generation by an astigmatic partially coherent beam

In this section, we will derive the explicit expression for the second-order correlation function of the second-harmonic field generated by a general astigmatic partially coherent beam and study the properties of the second-harmonic field.

Assuming again a Gaussian type of correlation. The second-order correlation function of a partially coherent TAGSM beam at z=0 is expressed as [8, 9]

Γ(2)(s1,s2,0)=G0exp[14s1T(σI02)1s114s2T(σI02)1s212(s1s2)T(σg02)1(s1s2)]
exp[ik2(s1s2)T(R01+μ0J)(s1+s2)],

where σ I0 2 is the transverse spot width matrix, σ g0 2 is the transverse coherence width matrix and R 0 -1 is the wave front curvature matrix. σ I0 2 , σ g0 2 and R 0 -1 are all symmetric 2×2 matrices, given by:

σI02=(σI0112σI0122σI0122σI0222),σg02=(σg0112σg0122σg0122σg0222),R01=(R0111R0121R0211R0221),

J is an anti-symmetry matrix given by:

J=(0110).

μ 0 is a scalar real-valued twist factor with the dimension of an inverse distance. It is limited by 0 ≤ μ 2 0 ≤ [k 2 det(σ 2 g0]-1 due to the non-negativity requirement of Eq. (10) [13]. G 0 is a normalized factor given by G 0 =1/ȫ -∞ exp[-s 1 T(σ 2 I0)-1 s 1/2]d s 1 = Det[(σ 2 I0)-1]1/2 /2π.

After some tensor operation [17], we can express Γ(2)(s 1,s 3.0)Γ(2) (s 2,s 4,0) and Γ(2) (s 1,s 4,0)Γ(2) (s 2,s 3,0) of a partially coherent TAGSM beam in the following form

Γ(2)(s1,s3,0)Γ(2)(s2,s4,0)=G02exp[ik12s͂TM͂11s͂],
Γ(2)(s1,s4,0)Γ(2)(s2,s3,0)=G02exp[ik12s͂TM͂21s͂],

where M͂-1 1 and M͂-1 2 are 8×8 matrices given by

M͂11=(M͂111M͂121(M͂121)T(M͂111)*),M͂21=(M͂111M͂211(M͂211)T(M͂111)*),

with

M͂111=(R01i2k1(σI02)1ik1(σg02)100R01i2k1(σI02)1ik1(σg02)1),
M͂121=(ik1(σg02)1+μ0J00ik1(σg02)1+μ0J),M͂211=(0ik1(σg02)1+μ0Jik1(σg02)1+μ0J0),

Note that M͂-1 1 and M -1 2 are different from the 4×4 partially coherent complex curvature tensor M -1 of a partially coherent TAGSM beam as shown in Ref. [17].

Substituting Eqs. (13) and (14) into Eq. (7), we obtain (after some vector integration and tensor operation) the following expression for the second-order correlation function of the second-harmonic field generated by a partially coherent TAGSM beam at z = l

Γ(2)(ρ1,ρ2,l)=G02(K28lk2)2exp[ik22lρ12ik22lρ22]0l0l1z11z2dz1dz2
([det(M͂l1)]1/2exp[ik22ρ͂TD͂TM͂l11D͂ρ͂]+[det(M͂l2)]1/2exp[ik22ρ͂TD͂TM͂l21D͂ρ͂]),

where M -1 l1 =(M͂-1 1 /2-B͂-1)-1 ,M l2 -1 = (M͂-1 2/2-B͂-1)-1 .

For the convenience of analysis, we consider two special cases. In the first case, we set σ 2 I0 = σ 2 I0 I, σ 2 g0 = σ 2 g0 I, R -1 0 = 0I and μ 0 = 0 in Eqs. (10) and (17). Then Eq. (10) reduces to the expression for the second-order correlation function of the partially coherent Gaussian Schell-model (GSM) beam [1, 34–36], and Eq. (17) becomes

Γ(2)(ρ1,ρ2,l)=G12(K28lk2)216σg02σI06k22l2σIl20l1bln[1+bla]dz1exp[(ρ12+ρ22)4σIl2(ρ1ρ2)22σgl2],

with G 1 = 1/ ∫ -∞ -∞ exp[-s 2 1 /2σ2 I0]d s 1 = 1 /2πσ2 I0 and

a=σg02σI04k22+σI02k2z1(2iσI02+iσg02),b=(4σI02+σg02)z1σI02k2(2iσI02+iσg02),
σIl2=4σI02l2+σg02l2+σg02σI04k222σg02σI02k22,σgl2=4σI02l2+σg02l2+σg02σI04k222σI04k22,

Equation (18) is the second-order correlation function of the second-harmonic field generated by a partially coherent GSM beam as shown in Refs. [26]–[28]. The second-harmonic field at z = l is also a partially coherent GSM beam with σIl and σgl . being the transverse spot width and coherence width, respectively. From Eq. (19), following relation can be obtained

σIlσgl=σI0σg0,

which means that the ratio of the transverse beam spot width and transverse coherence width of the second-harmonic field at z = l is independent of the crystal’s length and is equal to that of the pump beam. This reciprocity relation was discovered in Refs. [26]–[28] first. A wave front curvature matrix R -1 0 = 0I means that the beam waist of the pump beam is located at the input plane (z=0). As we know the radius of the wave front of a laser beam is infinity at the beam waist. If the beam waist of a pump beam with arbitrary beam profile is also located in the input plane, R -1 0 = 0I also applies to this case.

For the second case, we set

σI02=(σI0200σI02),σg02=(σg0200σg02),R01=(R0100R01),μ0,

in Eqs. (10) and (17). Then Eq. (10) reduces to the expression for the second-order correlation function of the partially coherent twisted Gaussian Schell-model (TGSM) beam of circular symmetry [9], and Eq. (17) reduces to the second-order correlation function of the second-harmonic field generated by a partially coherent TGSM beam as follows

Γ(2)(ρ1,ρ2,l)=G12(K28lk2)216σg02σI06k22l2R02σIl20l1b1ln[1+b1la1]dz1
×exp[(ρ12+ρ22)4σIl2(ρ1ρ2)22σgl2ik221Rl(ρ12ρ22)ik2μlρ1Jρ2],

with G 1 = 1/ 2πσ2 I0 and

a1=σg02σI04k22R02+σI02k2R0z1(2iσI02R0+iσg02R0σg02σI02k2),
b1=(4σI02R02+σg02R02+σg02σI04k22+σg02σI04k22R02μ02)z1σI02k2R0(2iσI02R0+iσg02R0+σg02σI02k2),
σIl2=4σI02l2R02+σg02l2R02+σg02σI04k22(lR0)2+σg02σI04k22μ02l2R022σg02σI02k22R02,
σgl2=4σI02l2R02+σg02l2R02+σg02σI04k22(lR0)2+σg02σI04k22μ02l2R022σI04k22R02,
Rl=4σI02l2R02+σg02l2R02+σg02σI04k22(lR0)2+σg02σI04k22μ02l2R02σg02σI04k22(R0l)4σI02R02lσg02R02lσg02σI04k22μ02R02l,
μl=σI04σg02k22R02μ04σI02l2R02+σg02l2R02+σg02σI04k22(lR0)2+σg02σI04k22μ02l2R02,

One sees from Eq. (22) that the second-harmonic field generated by a partially coherent TGSM beam is also a partially coherent TGSM beam with σIl, σgl, Rl and μl being the corresponding beam parameters at z = l. From Eq. (23), we also obtain following relations that are independent of the crystal’s length

σIl2σgl2=σI02σg02,σgl2μlσg02μ0=12,σIl2μlσI02μ0=12.
 figure: Fig. 1.

Fig. 1. Dependences of the transverse beam spot width σIl and transverse coherence width σgl of the second-harmonic field generated by a partially coherent TGSM beam on the crystal’s length l for different values of the initial coherence width σg0 and of the twist factor μ 0

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 figure: Fig. 2.

Fig. 2. Normalized irradiance distribution (contour graph) of the second-harmonic field generated by a general astigmatic partially coherent beam for different values of crystal’s length l (a) l = 5mm , (b) l = 30mm , (c) l = 50mm , (d) l = 100mm , (e) l = 150mm , (f) pump beam

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 figure: Fig. 3.

Fig. 3. Normalized irradiance distribution (contour graph) of the second-harmonic field generated by a general astigmatic partially coherent beam for different values of the crystal’s length / and of the initial transverse coherence width matrix σ 2 g0 (a) l = 30mm , σ 2 g0 =0.01I(mm)2 , (b) l = 30mm , σ 2 g0 = 0.0025I(mm)2 , (c) l = 30mm , σ 2 g0 = 0.0001I(mm)2, (d) l = 30mm , σ 2 g0 = 0.000025I(mm)2 , (e) l = 300mm , σ 2 g0 = 0.000025I(mm)2

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One finds from Eqs. (19) and (23) that the beam parameters (or beam properties) of the second-harmonic field are related to those of the pump beam and depend on the crystal’s length l critically. Figure 1 shows the dependences of the transverse beam spot width σIl and transverse coherence width σgl of the second-harmonic field (λ = 632.8nm) generated by a partially coherent TGSM beam on the crystal’s length l for different values of the initial coherence width σg0 and of the twist factor μ 0 with σI0 = 0.1mm and Rl = ∞. One finds from Fig. 1 that with decreasing coherence of the pump beam the second-harmonic field spreads more rapidly and its coherence decreases [26–28]. One also finds that the second-harmonic field spreads more rapidly while its coherence increases as the absolute value of the twist factor μ 0 of the partially coherent pump beam increases. In optical coherence theory, it is well-known that the spatial or transverse coherence of a laser beam will increase as the propagation distance increases. Due to diffraction each space point receives field components from different areas of the initial beam. Therefore different sections of the beam become more and more correlated during propagation. Consequently the coherence of the laser beam and its transverse coherence width increases during propagation in free space [1]. For linear filed propagation this phenomenon is quantitatively described by the so-called van Cittert-Zernike theorem [1]. From our numerical analysis we conclude that this phenomenon also exists in nonlinear media.

If σ 2 I0, σ 2 g0, R -1 0 take the form as shown in Eq. (11), then Eq. (17) denotes the most general case (i.e., second-harmonic generation by a general astigmatic partially coherent beam). In this case, it would be very hard to express Eq. (17) in the form of Eq. (10), but we can study the properties of the second-harmonic field numerically using Eq. (17). The irradiance of the second-harmonic field can be obtained by setting ρ 1 = ρ 2 in Eq. (17). Figures 2(a)–2(e) show the normalized irradiance distribution (contour graph) of the second-harmonic field (λ = 632.8nm) generated by a general astigmatic partially coherent beam for different values of the crystal’s length l .T he initial parameters at z=0 are σ 2 g0 =0.01I(mm)2 ,σI02=(0.01000.0025)(mm)2, R -1 0 = 0I μ 0 = 0 . For comparison, the irradiance distribution (contour graph) of the pump beam at z=0 is shown in Fig. 2(f). From Figs. 2(a)–2(e), one sees that beam profile of the second-harmonic field are closely controlled by the crystal’s length. In most cases, the second-harmonic field is astigmatic (or elliptical) beam, and for a small length [see Fig. 2(a) and 2(b)] the elliptical beam spot is similar to that of the pump beam [see Fig. 2(f)], but for long crystal’s length [see Figs. 2(d) and 2(e)], the long axis and short axis of the elliptical beam spot will interchange their positions. For suitable crystal’s length, the second-harmonic field can be a stigmatic (or circular) beam [(see Fig. 2(c)]. Figure 3 shows the normalized irradiance distribution (contour graph) of the second-harmonic field generated by a general astigmatic partially coherent beam for different values of the crystal’s length l and of the initial transverse coherence width matrix σ 2 g0, and other parameters remain as given in Fig. 2. One sees from Fig. 3 that the irradiance of the second-harmonic field is strongly influenced by the initial coherence of the pump beam. When the coherence of the pump beam is high, the irradiance of the second-harmonic field can be astigmatic (or elliptical) [see Fig. 3(a)]. But as the coherence of the pump beam decreases, the second-harmonic field finally becomes stigmatic (or circular) [see Figs. 3(c) and 3(d)]. It remains so even for longer crystal’s length [see Fig. 3(e)], which means the degradation of the coherence of the pump beam will lead to the circularization of the beam spot of the second-harmonic field. Note that the beam spot width of the second-harmonic field also increases as the coherence of the pump beam is degraded, while the irradiance of the pump beam at z=0 is independent of the initial coherence width matrix and twist phase. Figure 4 shows the normalized irradiance distribution (contour graph) of the second-harmonic field generated by a general astigmatic partially coherent beam for different values of the crystal’s length l and of the initial twist factor μ 0 . Other parameters remain as given in Fig. 2. One sees from Figs. 4(a)–4(e) that the twist phase of the pump beam will twist the irradiance distribution of the second-harmonic field, i.e., the elliptical beam spot will rotate anti-clockwise if μ 0 > 0 [see Figs. 4(a)–4(c)]. It will rotate clockwise if μ 0 < 0 [see Figs. 4(d) and 4(e)]. The speed of rotation will increase for larger absolute values of the twist factor μ 0.

 figure: Fig. 4.

Fig. 4. Normalized irradiance distribution (contour graph) of the second-harmonic field generated by a general astigmatic partially coherent beam for different values of the crystal’s length l and of the initial twist factor μ 0 (a) l = 5mm , μ 0 = 0.02mm -1 , (b) l = 30mm , μ 0 = 0.02mm -1 , (c) l = 150mm , μ 0 = 0.02mm -1 , (d) l = 30mm , μ 0 = -0.02mm -l , (e) l = 150mm , μ 0 = -0.02mm -1

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4. Conversion efficiency

In this section, we study the conversion efficiency of the second-harmonic generation. The conversion efficiency is defined as the ratio of the power of the second harmonic field at output plane (z = l) to the power of the pump beam at input plane (z = 0)

η=Γ(2)(ρ,ρ,l)dρxdρyΓ(2)(s,s,0)dsxdsy,

where Γ(2)(s,s,0) and Γ(2)(ρ,ρ,l) are the irradiances of the pump and second-harmonic fields, respectively.

First, we consider the case of second-harmonic generation by a partially coherent GSM beam. By setting σ 2 I0 = σ 2 I0 I , σ 2 g0 = σ 2 g0 I, R -1 0 = 0I and μ 0 = 0 in Eqs. (10), then substituting Eqs. (10) and (18) into Eq. (25), we obtain the following expression for the conversion efficiency of the second-harmonic generation by a partially coherent GSM beam

ηGSM=K22σg02σI028π0l1bln[1+bla]dz1,

where a and b are given by Eq. (19).

In a similar way, by applying Eqs. (21), (22) and (25), we obtain the following expression for the conversion efficiency of the second-harmonic generation by a partially coherent TGSM beam

ηTGSM=K22σg02σI02R028π0l1b1ln[1+b1la1]dz1,

where a 1 and b 1 are given by Eq. (23).

 figure: Fig. 5.

Fig. 5. Dependence of the relative conversion efficiency η = ηTGSM / ηGSM on the crystal’s length for different values of the initial twist factor μ 0 of the partially coherent TGSM beam

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For the most general case (second-harmonic generation by a general astigmatic partially coherent beam), we can calculate the conversion efficiency numerically by applying Eqs. (17) and (25). For the convenience of comparison, we introduce the relative conversion efficiency, which is defined as the ratio of conversion efficiency for the second-harmonic generation by a partially coherent TGSM beam or TAGSM beam to that for the second-harmonic generation by a partially coherent GSM beam with equal power. Figure 5 shows the dependence of the relative conversion efficiency η = ηTGSM /ηGSM on the crystal’s length l for different values of the initial twist factor μ 0 of the partially coherent TGSM beam with σI0 = 0.1mm, σg0 = 0.05mm and R 0 = ∞. One finds from Fig. 5 that the appearance of the twist phase in a partially coherent TGSM beam decreases the conversion efficiency of second-harmonic generation. The relative conversion efficiency decreases also as the crystal’s length l or the absolute value of the twist factor μ 0 increases. Figure 6 shows the dependence of the relative conversion efficiency η = ηTAGSM /ηGSM on the crystal’s length l for different values of the ratio σI011I022 of the partially coherent TAGSM beam with σI0 = 0.01mmg0 = 0.05mm , R -1 0 = 0, μ 0 = 0 ,σI02=(σI011200σI0222) , σI022 = σI0 and σ2 g0 = σ2 g0 I. In this case, σI011 and σI022 are the beam spot widths of the pump beam (TAGSM beam) in x and y direction, respectively, and σI011 / σI022 denotes the astigmatism of the pump beam. One finds from Fig. 6 that using partially coherent TAGSM beam instead of partially coherent GSM beam as pump beam can increase the conversion efficiency under certain conditions. The relative conversion efficiency increases as the crystal’s length or the astigmatism of the pump beam (TAGSM beam) increases.

 figure: Fig. 6.

Fig. 6. Dependence of the relative conversion efficiency η = ηTAGSM / ηGSM on the crystal’s length l for different values of the ratio σI011 / σI022 of th e partially coherent TAGSM beam

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5. Conclusion

We have investigated the second-harmonic generation by a general partially coherent astigmatic beam. We have derived an explicit expression for the second-order correlation function of the second-harmonic field generated by a partially coherent TAGSM beam. We have found that irradiance and coherence properties of the second-harmonic field are closely controlled by the crystal’s length and by the parameters of the partially coherent pump beam. Degradation of the coherence of the astigmatic pump beam can lead to the circularization of the beam spot of the second-harmonic field. The twist phase of the partially coherent pump beam causes the irradiance distribution to twist and enhances the spreading of the second-harmonic field while decreasing the conversion efficiency of second-harmonic generation. The astigmatism of the pump beam leads to the increase of the conversion efficiency of the second-harmonic generation under certain conditions.

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18. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006). [CrossRef]  

19. H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32, 2215–2217 (2007). [CrossRef]   [PubMed]  

20. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961). [CrossRef]  

21. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).

22. J. E. Bjorkholm, “Optical second-harmonic generation using a focused Gaussian laser beam,” Phys. Rev. 142, 126–136 (1966). [CrossRef]  

23. D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966). [CrossRef]  

24. T. Freegarde, J. Coutts, J. Walz, D. Leibfried, and T. W. Hänsch, “General analysis of type I second-harmonic generation with elliptical Gaussian beams,” J. Opt. Soc. Am. B 14, 2010–2016 (1997). [CrossRef]  

25. G. S. Agrawal, “Second-harmonic generation with arbitrary pump-beam profiles,” Phys. Rev. A 23, 1863–1868 (1981). [CrossRef]  

26. M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987). [CrossRef]   [PubMed]  

27. N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986). [CrossRef]  

28. M. Zahid and M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1–7 (1990). [CrossRef]  

29. L. Wang and J. Xue, “Efficiency comparison analysis of second harmonic generation on flattened Gaussian and Gaussian beams through a crystal CsLiB6O10,” Jpn. J. Appl. Phys. 41, 7373–7376 (2002). [CrossRef]  

30. R. Hanbury Brown, The Intensity Interferomenter (Taylor and Francis, London, 1974).

31. B. E. A. Saleh, A. F. Abouraddy, A. V. Sirgienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000). [CrossRef]  

32. Y. Cai and S. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716 (2004) [CrossRef]   [PubMed]  

33. F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005). [CrossRef]   [PubMed]  

34. J. T. Foley and M.S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26 ,297–300 (1978). [CrossRef]  

35. F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1978). [CrossRef]  

36. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun 41, 383–387 (1982). [CrossRef]  

References

  • View by:
  • |
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  • |

  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  2. D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am 65, 887–891 (1975).
    [Crossref]
  3. M. Von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head-mounted retinal projection displays,” Opt. Engineering 43, 1552–1560 (2004).
    [Crossref]
  4. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
    [Crossref]
  5. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002).
    [Crossref]
  6. Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
    [Crossref]
  7. T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004).
    [Crossref] [PubMed]
  8. R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
    [Crossref] [PubMed]
  9. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [Crossref]
  10. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
    [Crossref]
  11. D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
    [Crossref]
  12. R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
    [Crossref]
  13. R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
    [Crossref]
  14. R. Simon and N. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters,” J. Opt. Soc. Am. A 17, 2440–2463 (2000).
    [Crossref]
  15. S. A. Ponomarenko, “Twisted Gaussian Schell-mode solitons,” Phys. Rev. E 64, 036618 (2001).
    [Crossref]
  16. J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26, 405–406 (2001).
    [Crossref]
  17. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27, 216–218 (2002).
    [Crossref]
  18. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
    [Crossref]
  19. H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32, 2215–2217 (2007).
    [Crossref] [PubMed]
  20. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
    [Crossref]
  21. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).
  22. J. E. Bjorkholm, “Optical second-harmonic generation using a focused Gaussian laser beam,” Phys. Rev. 142, 126–136 (1966).
    [Crossref]
  23. D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
    [Crossref]
  24. T. Freegarde, J. Coutts, J. Walz, D. Leibfried, and T. W. Hänsch, “General analysis of type I second-harmonic generation with elliptical Gaussian beams,” J. Opt. Soc. Am. B 14, 2010–2016 (1997).
    [Crossref]
  25. G. S. Agrawal, “Second-harmonic generation with arbitrary pump-beam profiles,” Phys. Rev. A 23, 1863–1868 (1981).
    [Crossref]
  26. M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
    [Crossref] [PubMed]
  27. N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
    [Crossref]
  28. M. Zahid and M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1–7 (1990).
    [Crossref]
  29. L. Wang and J. Xue, “Efficiency comparison analysis of second harmonic generation on flattened Gaussian and Gaussian beams through a crystal CsLiB6O10,” Jpn. J. Appl. Phys. 41, 7373–7376 (2002).
    [Crossref]
  30. R. Hanbury Brown, The Intensity Interferomenter (Taylor and Francis, London, 1974).
  31. B. E. A. Saleh, A. F. Abouraddy, A. V. Sirgienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
    [Crossref]
  32. Y. Cai and S. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716 (2004)
    [Crossref] [PubMed]
  33. F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
    [Crossref] [PubMed]
  34. J. T. Foley and M.S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26 ,297–300 (1978).
    [Crossref]
  35. F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1978).
    [Crossref]
  36. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun 41, 383–387 (1982).
    [Crossref]

2007 (1)

2006 (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

2005 (2)

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[Crossref]

F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref] [PubMed]

2004 (3)

Y. Cai and S. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716 (2004)
[Crossref] [PubMed]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004).
[Crossref] [PubMed]

M. Von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head-mounted retinal projection displays,” Opt. Engineering 43, 1552–1560 (2004).
[Crossref]

2002 (3)

2001 (2)

S. A. Ponomarenko, “Twisted Gaussian Schell-mode solitons,” Phys. Rev. E 64, 036618 (2001).
[Crossref]

J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26, 405–406 (2001).
[Crossref]

2000 (2)

R. Simon and N. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters,” J. Opt. Soc. Am. A 17, 2440–2463 (2000).
[Crossref]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sirgienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

1998 (1)

1997 (1)

1996 (1)

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[Crossref]

1994 (2)

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[Crossref]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

1993 (1)

1990 (1)

M. Zahid and M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1–7 (1990).
[Crossref]

1987 (1)

M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[Crossref] [PubMed]

1986 (1)

N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
[Crossref]

1985 (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[Crossref] [PubMed]

1984 (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

1982 (1)

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun 41, 383–387 (1982).
[Crossref]

1981 (1)

G. S. Agrawal, “Second-harmonic generation with arbitrary pump-beam profiles,” Phys. Rev. A 23, 1863–1868 (1981).
[Crossref]

1978 (2)

J. T. Foley and M.S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26 ,297–300 (1978).
[Crossref]

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1978).
[Crossref]

1975 (1)

D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am 65, 887–891 (1975).
[Crossref]

1966 (2)

J. E. Bjorkholm, “Optical second-harmonic generation using a focused Gaussian laser beam,” Phys. Rev. 142, 126–136 (1966).
[Crossref]

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[Crossref]

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Abouraddy, A. F.

B. E. A. Saleh, A. F. Abouraddy, A. V. Sirgienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

Agrawal, G. S.

G. S. Agrawal, “Second-harmonic generation with arbitrary pump-beam profiles,” Phys. Rev. A 23, 1863–1868 (1981).
[Crossref]

Ambrosini, D.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

Ansari, N. A.

N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
[Crossref]

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Ashkin, A.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[Crossref]

Bache, M.

F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref] [PubMed]

Bagini, V.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

Bjorkholm, J. E.

J. E. Bjorkholm, “Optical second-harmonic generation using a focused Gaussian laser beam,” Phys. Rev. 142, 126–136 (1966).
[Crossref]

Boyd, G. D.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).

Brambilla, E.

F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref] [PubMed]

Cai, Y.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[Crossref]

Y. Cai and S. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716 (2004)
[Crossref] [PubMed]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27, 216–218 (2002).
[Crossref]

Coutts, J.

Davidson, F. M.

Ferri, F.

F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref] [PubMed]

Foley, J. T.

J. T. Foley and M.S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26 ,297–300 (1978).
[Crossref]

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Freegarde, T.

Friberg, A. T.

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[Crossref]

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[Crossref]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun 41, 383–387 (1982).
[Crossref]

Gatti, A.

F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref] [PubMed]

Gori, F.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1978).
[Crossref]

Gureyev, T. E.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004).
[Crossref] [PubMed]

Hanbury Brown, R.

R. Hanbury Brown, The Intensity Interferomenter (Taylor and Francis, London, 1974).

Hänsch, T. W.

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Kato, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Kermisch, D.

D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am 65, 887–891 (1975).
[Crossref]

Kitagawa, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Kleinman, D. A.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[Crossref]

Leibfried, D.

Lin, Q.

Lugiato, L. A.

F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref] [PubMed]

Lukowicz, P.

M. Von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head-mounted retinal projection displays,” Opt. Engineering 43, 1552–1560 (2004).
[Crossref]

Magatti, D.

F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Mayo, S. C.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004).
[Crossref] [PubMed]

McIver, J. K.

M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[Crossref] [PubMed]

Mima, K.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Miyanaga, N.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Movilla, J. M.

Mukunda, N.

Nakatsuka, M.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Paganin, D. M.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004).
[Crossref] [PubMed]

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Ponomarenko, S. A.

S. A. Ponomarenko, “Twisted Gaussian Schell-mode solitons,” Phys. Rev. E 64, 036618 (2001).
[Crossref]

Ricklin, J. C.

Saleh, B. E. A.

B. E. A. Saleh, A. F. Abouraddy, A. V. Sirgienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

Santarsiero, M.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

Serna, J.

Simon, R.

R. Simon and N. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters,” J. Opt. Soc. Am. A 17, 2440–2463 (2000).
[Crossref]

R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
[Crossref]

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[Crossref]

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[Crossref] [PubMed]

Sirgienko, A. V.

B. E. A. Saleh, A. F. Abouraddy, A. V. Sirgienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

Stevenson, A. W.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004).
[Crossref] [PubMed]

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[Crossref] [PubMed]

Sudol, R. J.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun 41, 383–387 (1982).
[Crossref]

Teich, M. C.

B. E. A. Saleh, A. F. Abouraddy, A. V. Sirgienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

Tervonen, E.

Troster, G.

M. Von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head-mounted retinal projection displays,” Opt. Engineering 43, 1552–1560 (2004).
[Crossref]

Turunen, J.

Waldkirch, M. Von

M. Von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head-mounted retinal projection displays,” Opt. Engineering 43, 1552–1560 (2004).
[Crossref]

Walz, J.

Wang, H.

Wang, L.

L. Wang and J. Xue, “Efficiency comparison analysis of second harmonic generation on flattened Gaussian and Gaussian beams through a crystal CsLiB6O10,” Jpn. J. Appl. Phys. 41, 7373–7376 (2002).
[Crossref]

Wang, X.

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Wilkin, S. W.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004).
[Crossref] [PubMed]

Wolf, E.

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Xue, J.

L. Wang and J. Xue, “Efficiency comparison analysis of second harmonic generation on flattened Gaussian and Gaussian beams through a crystal CsLiB6O10,” Jpn. J. Appl. Phys. 41, 7373–7376 (2002).
[Crossref]

Yamanaka, C.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Yang, K.

Zahid, M.

M. Zahid and M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1–7 (1990).
[Crossref]

Zeng, A.

Zhu, S.

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[Crossref]

Y. Cai and S. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716 (2004)
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M. Zahid and M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1–7 (1990).
[Crossref]

M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[Crossref] [PubMed]

N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
[Crossref]

Zubairy, M.S.

J. T. Foley and M.S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26 ,297–300 (1978).
[Crossref]

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

J. Mod. Opt. (1)

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[Crossref]

J. Opt. Soc. Am (1)

D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am 65, 887–891 (1975).
[Crossref]

J. Opt. Soc. Am. A (5)

J. Opt. Soc. Am. B (1)

Jpn. J. Appl. Phys. (1)

L. Wang and J. Xue, “Efficiency comparison analysis of second harmonic generation on flattened Gaussian and Gaussian beams through a crystal CsLiB6O10,” Jpn. J. Appl. Phys. 41, 7373–7376 (2002).
[Crossref]

Opt. Commun (1)

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun 41, 383–387 (1982).
[Crossref]

Opt. Commun. (4)

J. T. Foley and M.S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26 ,297–300 (1978).
[Crossref]

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1978).
[Crossref]

N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
[Crossref]

M. Zahid and M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1–7 (1990).
[Crossref]

Opt. Engineering (1)

M. Von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head-mounted retinal projection displays,” Opt. Engineering 43, 1552–1560 (2004).
[Crossref]

Opt. Lett. (4)

Phys. Rev. (2)

J. E. Bjorkholm, “Optical second-harmonic generation using a focused Gaussian laser beam,” Phys. Rev. 142, 126–136 (1966).
[Crossref]

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[Crossref]

Phys. Rev. A (4)

G. S. Agrawal, “Second-harmonic generation with arbitrary pump-beam profiles,” Phys. Rev. A 23, 1863–1868 (1981).
[Crossref]

M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[Crossref] [PubMed]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sirgienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[Crossref] [PubMed]

Phys. Rev. E (2)

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[Crossref]

S. A. Ponomarenko, “Twisted Gaussian Schell-mode solitons,” Phys. Rev. E 64, 036618 (2001).
[Crossref]

Phys. Rev. Lett. (4)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004).
[Crossref] [PubMed]

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref] [PubMed]

Pure Appl. Opt. (1)

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[Crossref]

Other (3)

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

R. Hanbury Brown, The Intensity Interferomenter (Taylor and Francis, London, 1974).

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Figures (6)

Fig. 1.
Fig. 1. Dependences of the transverse beam spot width σ Il and transverse coherence width σ gl of the second-harmonic field generated by a partially coherent TGSM beam on the crystal’s length l for different values of the initial coherence width σ g0 and of the twist factor μ 0
Fig. 2.
Fig. 2. Normalized irradiance distribution (contour graph) of the second-harmonic field generated by a general astigmatic partially coherent beam for different values of crystal’s length l (a) l = 5mm , (b) l = 30mm , (c) l = 50mm , (d) l = 100mm , (e) l = 150mm , (f) pump beam
Fig. 3.
Fig. 3. Normalized irradiance distribution (contour graph) of the second-harmonic field generated by a general astigmatic partially coherent beam for different values of the crystal’s length / and of the initial transverse coherence width matrix σ 2 g0 (a) l = 30mm , σ 2 g0 =0.01I(mm)2 , (b) l = 30mm , σ 2 g0 = 0.0025I(mm)2 , (c) l = 30mm , σ 2 g0 = 0.0001I(mm)2, (d) l = 30mm , σ 2 g0 = 0.000025I(mm)2 , (e) l = 300mm , σ 2 g0 = 0.000025I(mm)2
Fig. 4.
Fig. 4. Normalized irradiance distribution (contour graph) of the second-harmonic field generated by a general astigmatic partially coherent beam for different values of the crystal’s length l and of the initial twist factor μ 0 (a) l = 5mm , μ 0 = 0.02mm -1 , (b) l = 30mm , μ 0 = 0.02mm -1 , (c) l = 150mm , μ 0 = 0.02mm -1 , (d) l = 30mm , μ 0 = -0.02mm -l , (e) l = 150mm , μ 0 = -0.02mm -1
Fig. 5.
Fig. 5. Dependence of the relative conversion efficiency η = ηTGSM / ηGSM on the crystal’s length for different values of the initial twist factor μ 0 of the partially coherent TGSM beam
Fig. 6.
Fig. 6. Dependence of the relative conversion efficiency η = ηTAGSM / ηGSM on the crystal’s length l for different values of the ratio σ I011 / σ I022 of th e partially coherent TAGSM beam

Equations (40)

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2 ε w x 2 + 2 ε w y 2 2 i k 1 ε w z = K 1 ε w * ε 2 w ,
2 ε 2 w x 2 + 2 ε 2 w y 2 2 i k 2 ε 2 w z = K 2 ε w 2 ,
ε 2 w ( ρ , l ) = ∫∫ D ( ρ , l , s 1 , s 2 ) ε w ( s 1 , 0 ) ε w ( s 2 , 0 ) d s 1 d s 2 ,
D ( ρ , l , s 1 , s 2 ) = K 2 k 2 32 π 2 l exp ( i k 2 l ) 0 l 1 z 1 exp [ i k 2 2 l ρ 2 ] exp [ i k 2 ( s 1 2 + s 2 2 ) 8 ( 1 l + 1 z 1 ) ]
exp [ i k 2 ( ρ s 1 + ρ s 2 ) 2 l ] exp [ i k 2 s 1 s 2 4 ( 1 l 1 z 1 ) ] d z 1 .
Γ ( 2 ) ( ρ 1 , ρ 2 , l ) = ε 2 w ( ρ 1 , l ) ε 2 w * ( ρ 2 , l ) = ( K 2 k 2 32 π 2 l ) 2
D ( ρ 1 , s 1 , s 2 ) D * ( ρ 2 , s 3 , s 4 ) Γ ( 4 ) ( s 1 , s 2 , s 3 , s 4 ) d s 1 d s 2 d s 3 d s 4 d z 1 d z 2 ,
Γ ( 4 ) ( s 1 , s 2 , s 3 , s 4 ) = Γ ( 2 ) ( s 1 , s 3 , 0 ) Γ ( 2 ) ( s 2 , s 4 , 0 ) + Γ ( 2 ) ( s 1 , s 4 , 0 ) Γ ( 2 ) ( s 2 , s 3 , 0 ) .
Γ ( 2 ) ( ρ 1 , ρ 2 , l ) = ( K 2 k 2 32 π 2 l ) 2 exp [ i k 2 2 l ρ 1 2 i k 2 2 l ρ 2 2 ] 0 l 0 l 1 z 1 1 z 2 d z 1 d z 2
[ Γ ( 2 ) ( s 1 , s 3 , 0 ) Γ ( 2 ) ( s 2 , s 4 , 0 ) + Γ ( 2 ) ( s 1 , s 4 , 0 ) Γ ( 2 ) ( s 2 , s 3 , 0 ) ] exp [ i k 2 2 s ͂ T B ͂ 1 s ͂ ] exp [ i k 2 s ͂ T D ͂ ρ ͂ ] d s , ͂
B ͂ 1 = ( B ͂ l z 1 1 0 0 B ͂ l z 2 1 ) , D ͂ = 1 2 l ( I ͂ 0 0 I ͂ ) ,
B l z i 1 = ( 1 4 ( 1 l + 1 z i ) I 1 4 ( 1 l 1 z i ) I 1 4 ( 1 l 1 z i ) I 1 4 ( 1 l + 1 z i ) I ) , ( i = 1,2 ) ,
Γ ( 2 ) ( s 1 , s 2 , 0 ) = G 0 exp [ 1 4 s 1 T ( σ I 0 2 ) 1 s 1 1 4 s 2 T ( σ I 0 2 ) 1 s 2 1 2 ( s 1 s 2 ) T ( σ g 0 2 ) 1 ( s 1 s 2 ) ]
exp [ ik 2 ( s 1 s 2 ) T ( R 0 1 + μ 0 J ) ( s 1 + s 2 ) ] ,
σ I 0 2 = ( σ I 011 2 σ I 012 2 σ I 012 2 σ I 022 2 ) , σ g 0 2 = ( σ g 011 2 σ g 012 2 σ g 012 2 σ g 022 2 ) , R 0 1 = ( R 011 1 R 012 1 R 021 1 R 022 1 ) ,
J = ( 0 1 1 0 ) .
Γ ( 2 ) ( s 1 , s 3 , 0 ) Γ ( 2 ) ( s 2 , s 4 , 0 ) = G 0 2 exp [ i k 1 2 s ͂ T M ͂ 1 1 s ͂ ] ,
Γ ( 2 ) ( s 1 , s 4 , 0 ) Γ ( 2 ) ( s 2 , s 3 , 0 ) = G 0 2 exp [ i k 1 2 s ͂ T M ͂ 2 1 s ͂ ] ,
M ͂ 1 1 = ( M ͂ 11 1 M ͂ 12 1 ( M ͂ 12 1 ) T ( M ͂ 11 1 ) * ) , M ͂ 2 1 = ( M ͂ 11 1 M ͂ 21 1 ( M ͂ 21 1 ) T ( M ͂ 11 1 ) * ) ,
M ͂ 11 1 = ( R 0 1 i 2 k 1 ( σ I 0 2 ) 1 i k 1 ( σ g 0 2 ) 1 0 0 R 0 1 i 2 k 1 ( σ I 0 2 ) 1 i k 1 ( σ g 0 2 ) 1 ) ,
M ͂ 12 1 = ( i k 1 ( σ g 0 2 ) 1 + μ 0 J 0 0 i k 1 ( σ g 0 2 ) 1 + μ 0 J ) , M ͂ 21 1 = ( 0 i k 1 ( σ g 0 2 ) 1 + μ 0 J i k 1 ( σ g 0 2 ) 1 + μ 0 J 0 ) ,
Γ ( 2 ) ( ρ 1 , ρ 2 , l ) = G 0 2 ( K 2 8 l k 2 ) 2 exp [ i k 2 2 l ρ 1 2 i k 2 2 l ρ 2 2 ] 0 l 0 l 1 z 1 1 z 2 d z 1 d z 2
( [ det ( M ͂ l 1 ) ] 1 / 2 exp [ i k 2 2 ρ ͂ T D ͂ T M ͂ l 1 1 D ͂ ρ ͂ ] + [ det ( M ͂ l 2 ) ] 1 / 2 exp [ i k 2 2 ρ ͂ T D ͂ T M ͂ l 2 1 D ͂ ρ ͂ ] ) ,
Γ ( 2 ) ( ρ 1 , ρ 2 , l ) = G 1 2 ( K 2 8 l k 2 ) 2 16 σ g 0 2 σ I 0 6 k 2 2 l 2 σ Il 2 0 l 1 b ln [ 1 + bl a ] dz 1 exp [ ( ρ 1 2 + ρ 2 2 ) 4 σ Il 2 ( ρ 1 ρ 2 ) 2 2 σ gl 2 ] ,
a = σ g 0 2 σ I 0 4 k 2 2 + σ I 0 2 k 2 z 1 ( 2 i σ I 0 2 + i σ g 0 2 ) , b = ( 4 σ I 0 2 + σ g 0 2 ) z 1 σ I 0 2 k 2 ( 2 i σ I 0 2 + i σ g 0 2 ) ,
σ Il 2 = 4 σ I 0 2 l 2 + σ g 0 2 l 2 + σ g 0 2 σ I 0 4 k 2 2 2 σ g 0 2 σ I 0 2 k 2 2 , σ gl 2 = 4 σ I 0 2 l 2 + σ g 0 2 l 2 + σ g 0 2 σ I 0 4 k 2 2 2 σ I 0 4 k 2 2 ,
σ Il σ gl = σ I 0 σ g 0 ,
σ I 0 2 = ( σ I 0 2 0 0 σ I 0 2 ) , σ g 0 2 = ( σ g 0 2 0 0 σ g 0 2 ) , R 0 1 = ( R 0 1 0 0 R 0 1 ) , μ 0 ,
Γ ( 2 ) ( ρ 1 , ρ 2 , l ) = G 1 2 ( K 2 8 l k 2 ) 2 16 σ g 0 2 σ I 0 6 k 2 2 l 2 R 0 2 σ Il 2 0 l 1 b 1 ln [ 1 + b 1 l a 1 ] d z 1
× exp [ ( ρ 1 2 + ρ 2 2 ) 4 σ Il 2 ( ρ 1 ρ 2 ) 2 2 σ gl 2 i k 2 2 1 R l ( ρ 1 2 ρ 2 2 ) i k 2 μ l ρ 1 J ρ 2 ] ,
a 1 = σ g 0 2 σ I 0 4 k 2 2 R 0 2 + σ I 0 2 k 2 R 0 z 1 ( 2 i σ I 0 2 R 0 + i σ g 0 2 R 0 σ g 0 2 σ I 0 2 k 2 ) ,
b 1 = ( 4 σ I 0 2 R 0 2 + σ g 0 2 R 0 2 + σ g 0 2 σ I 0 4 k 2 2 + σ g 0 2 σ I 0 4 k 2 2 R 0 2 μ 0 2 ) z 1 σ I 0 2 k 2 R 0 ( 2 i σ I 0 2 R 0 + i σ g 0 2 R 0 + σ g 0 2 σ I 0 2 k 2 ) ,
σ Il 2 = 4 σ I 0 2 l 2 R 0 2 + σ g 0 2 l 2 R 0 2 + σ g 0 2 σ I 0 4 k 2 2 ( l R 0 ) 2 + σ g 0 2 σ I 0 4 k 2 2 μ 0 2 l 2 R 0 2 2 σ g 0 2 σ I 0 2 k 2 2 R 0 2 ,
σ gl 2 = 4 σ I 0 2 l 2 R 0 2 + σ g 0 2 l 2 R 0 2 + σ g 0 2 σ I 0 4 k 2 2 ( l R 0 ) 2 + σ g 0 2 σ I 0 4 k 2 2 μ 0 2 l 2 R 0 2 2 σ I 0 4 k 2 2 R 0 2 ,
R l = 4 σ I 0 2 l 2 R 0 2 + σ g 0 2 l 2 R 0 2 + σ g 0 2 σ I 0 4 k 2 2 ( l R 0 ) 2 + σ g 0 2 σ I 0 4 k 2 2 μ 0 2 l 2 R 0 2 σ g 0 2 σ I 0 4 k 2 2 ( R 0 l ) 4 σ I 0 2 R 0 2 l σ g 0 2 R 0 2 l σ g 0 2 σ I 0 4 k 2 2 μ 0 2 R 0 2 l ,
μ l = σ I 0 4 σ g 0 2 k 2 2 R 0 2 μ 0 4 σ I 0 2 l 2 R 0 2 + σ g 0 2 l 2 R 0 2 + σ g 0 2 σ I 0 4 k 2 2 ( l R 0 ) 2 + σ g 0 2 σ I 0 4 k 2 2 μ 0 2 l 2 R 0 2 ,
σ Il 2 σ gl 2 = σ I 0 2 σ g 0 2 , σ gl 2 μ l σ g 0 2 μ 0 = 1 2 , σ Il 2 μ l σ I 0 2 μ 0 = 1 2 .
η = Γ ( 2 ) ( ρ , ρ , l ) d ρ x d ρ y Γ ( 2 ) ( s , s , 0 ) d s x d s y ,
η GSM = K 2 2 σ g 0 2 σ I 0 2 8 π 0 l 1 b ln [ 1 + bl a ] d z 1 ,
η TGSM = K 2 2 σ g 0 2 σ I 0 2 R 0 2 8 π 0 l 1 b 1 ln [ 1 + b 1 l a 1 ] d z 1 ,

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