Spectral Shift of an Electromagnetic Gaussian Schell ...

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Spectral Shift of an Electromagnetic Gaussian Schell-model Beam Shijun Zhu and Yangjian Cai School of Physical Science and Technology, Soochow University, Suzhou 215006, China

Abstract— By using a tensor method, spectral changes of a focused electromagnetic Gaussian Schell-model (EGSM) beam with twist phase. Our results show that the spectral shift of twisted EGSM beam is closely related to the degree of polarization, twist phase and correlation coefficients of the initial beam.

1. INTRODUCTION

Gaussian Schell-model (GSM) beam is a typical scalar partially coherent beam whose spectral degree of coherence and the intensity distribution are Gaussian functions [1–3]. A more general partially coherent beam can possess a twist phase. Twisted GSM beam was introduced by Simon and Mukunda [4]. In the past several years, Electromagnetic Gaussin Schell-model (EGSM) beam (i.e., vectorial GSM beam) has attracted more and more attention owing to its importance in theories of coherence and polarization of light and in some applications, e.g., free-space optical communications [5–8]. More recently, Cai and Korotkova introduced twisted EGSM beam, and studied its propagation in free space [9] and its radiation force on a Rayleigh dielectric particle [10]. The spectral changes of scalar GSM and twisted GSM beams focused by a thin lens were studied in [11, 12]. To our knowledge no results have been reported up until now on spectral changes of an electromagnetic GSM beam with or without twist phase focused by a thin lens. This paper is aimed to investigate spectral shift of a focused twisted EGSM beam. 2. THEORY

The second-order statistical properties of a twisted EGSM beam can be characterized by the 2 × 2 ↔ cross-spectral density matrix W (r1 , r2 ; 0) specified at any two points with position vectors r1 and r2 in the source plane with elements [9] · ¸ Aα Aβ Bαβ Γ20 iω T −1 exp − ˜ (1) r M0αβ ˜ r , (α = x, y; β = x, y), Wαβ (˜ r, ω, 0) = 2c (ω − ω0 )2 + Γ20 where Aα is the square root of the spectral density of electric field component Eα , Bαβ = |Bαβ | exp(iφ) is the correlation coefficient between the Ex and Ey field components, satisfying the relation ∗ , M−1 is a 4 × 4 matrix of the form [9] Bαβ = Bβα 0αβ   M−1 0αβ =

1 ik

³

1 2 2σaβ

i 2 I kδαβ

+

+

1

´

2 δαβ µαβ JT

I

i 2 I + µαβ J kδαβ ´ ³ 1 1 1 + 2 2 ik 2σαβ δαβ I

 ,

(2)

where σαβ and δαβ denote the widths of the spectral density and correlation coefficient, respectively. 4 ) if α = β due to the non-negativity µαβ represents the twist factor and is limited by µ2αβ ≤ 1/(k 2 δαβ requirement of the cross-spectral density (Eq. (1)) [9]. Aα , Bαβ , σαβ , δαβ and µαβ are independent of position but, in general, depend on the frequency. Here we have assumed that the initial spectrum is of the Lorentz type with ω0 being the central frequency and Γ0 being the half-width at halfmaximum, ω is the angular frequency. k = ω/c is the wave number with c being the velocity of light in vacuum. After propagating through an axially nonsymmetric ABCD optical system, the elements of the cross-spectral density matrix can be expressed in the following tensor form [9, 13] · ¸ Aα Aβ Bαβ Γ20 h ³ ¯ ¯ −1 ´i−1/2 iω T −1 exp − ρ˜ M1αβ ρ˜ , (α = x, y; β = x, y) (3) det A+ BM0αβ Wαβ (˜ ρ, ω, z) = 2c (ω−ω0 )2 +Γ20

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where det stands for the determinant of a matrix, ρ˜ ≡ (ρ1 ρ2 ) with ρ1 and ρ2 being the transverse −1 position vectors in the output plane, while M−1 1αβ and M0αβ are related by the following known tensor ABCD law [13] ³ ´³ ´ −1 −1 ¯ + DM ¯ −1 ¯ ¯ M−1 = C A + BM . (4) 0αβ 0αβ 1αβ

¯ B, ¯ C ¯ and D ¯ are 4 × 4 matrices of the form: Here A, µ ¶ µ ¶ µ ¶ A 0I B 0I C 0I ¯ ¯ ¯ A= , B= , C= , 0I A∗ 0I −B∗ 0I −C∗

µ ¯ = D

D 0I 0I D∗

¶ .

(5)

where A, B, C and D are the 2 × 2 sub-matrices of the ABCD optical system, and “*” in Eq. (9) is required for an general optical system with loss or gain. 3. NUMERICAL RESULTS

The focusing geometry is shown in Fig. 1, where the thin lens with focal length f is located at z = z1 and the exit plane is located at z. Here the transformation matrix of the total optical system between the source plane and the exit plane has the form µ ¶ µ ¶µ ¶µ ¶ µ ¶ A B I (z − z1 )I I 0I I z1 I (1 − Z)I (z − z1 Z)I = = . (6) C D 0 I − (1/f ) I I 0I I − (1/f ) I (1 − z1 /f ) I with Z = (z − z1 )/f . The spectral density at point ρ is then given by the formula ↔

S (ρ, z, ω) ≡ TrW (ρ, ρ, z, ω) = Wxx (ρ, ρ, z, ω) + Wyy (ρ, ρ, z, ω) .

(7)

We can calculate the spectral shift of the focused twisted EGSM beam focused by a thin lens by using Eqs. (3)–(7). In this paper, we only consider the twisted EGSM beam that is generated by a stochastic electromagnetic source whose cross-spectral density matrix is diagonal, i.e., of the form ¶ µ ↔ Wxx (r1 , r2 , 0, ω) 0 . (8) W(r1 , r2 , 0, ω) = 0 Wyy (r1 , r2 , 0, ω) The degree of polarization of the initial source beam at point r can be expressed as follows [5] v u ↔ u 4 det W(r, r, 0, ω) t . (9) P0 (r, 0, ω) = 1 − ↔ [TrW(r, r, 0, ω)]2 Under the condition of Wxx (r1 , r2 , 0, ω) = 0 or Wyy (r1 , r2 , 0, ω) = 0, the twisted electromagnetic GSM beam reduces to a scalar twisted GSM beam with P0 (r, 0, ω) = 1. In the following numerical examples, we set σx = σy = 3 mm, Ax = 1, Bxx = Byy = 1, f = 100 mm and z1 = 200 mm. In this A2 −A2

case, the polarization properties are uniform across the source plane with P0 (r, 0, ω) = | A2x +A2y |, x

Figure 1: Focusing geometry.

y

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Figure 2: On-axis normalized spectrum S(0, z, ω)/S(0, z, ω)max of a focused electromagnetic GSM beam without twist phase (γxx = γyy = 0) at several propagation distances.

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Figure 3: On-axis relative spectral shift η of a focused electromagnetic GSM beam without twist phase (γxx = γyy = 0) versus propagation distance for different values of the degree of polarization of the initial beam with δxx = 0.01 mm and δyy = 0.02 mm.

and the degree of polarization at source plane varies as the value of Ay varies. The parameters used in all numerical calculations are set to ω0 = 3.2 × 1015 rad/s and Γ0 = 0.6 × 1015 rad/s. We calculate in Fig. 2 the on-axis normalized spectrum S(0, z, ω)/S(0, z, ω)max of a focused electromagnetic GSM beam without twist phase (γxx = γyy = 0) at several propagation distances with P0 = 0.6, δxx = 0.01 mm, δyy = 0.02 mm. For the convenience of comparison, the source spectrum is also calculated (see solid line of Fig. 2). One finds from Fig. 2 that spectrum profile of the on-axis normalized spectrum the focused electromagnetic GSM beam behind the thin lens is similar to the source spectrum, but its peak position is blue-shifted. This phenomenon also occurs for a focused scalar GSM beam as shown in [11, 12]. Now we discuss the dependence of the relative spectral shift of a focused electromagnetic GSM beam on its initial degree of coherence and other beam parameters. The spectral shift ∆ω is the difference between the peak frequency ωm of the spectrum of the field after propagation and the peak frequency ω0 of the source spectrum. A positive value of ∆ω denotes a blue shift, while a negative value represents a red shift. We will study the variation of the relative spectral shift, which is defined as η = (ωm − ω0 ) /ω0 . (10) Figure 3 shows the on-axis relative spectral shift η of a focused electromagnetic GSM beam without twist phase (γxx = γyy = 0) versus propagation distance for different values of the degree of polarization of the initial beam with δxx = 0.01 mm and δyy = 0.02 mm. One finds from Fig. 3 that the maximum relative spectral shift of a focused electromagnetic GSM beam occurs at the back focal plane, while the minimum relative spectral shift (i.e., no spectral shift) occurs at the image plane (z = 400 mm). The results are in agreement with the results derived for the case of scalar GSM beam [11, 12]. It is also evident that the degree of polarization of the initial beam affects the relative spectral shift strongly. The relative spectral shift increases as the degree of polarization increases, while we should note that the position of the maximum relative spectral shift and the position of the zero spectral shift are independent of the degree of polarization. Figure 4 shows the on-axis relative spectral shift η of a focused twisted electromagnetic GSM beam versus propagation distance for different values of the twist factors γxx and γyy with P0 = 0.6, δxx = 0.01 mm and δyy = 0.02 mm. From Fig. 4, we find that the twist phase has significant influence on the relative spectral shift on propagation. The relative spectral shift on propagation and the maximum relative spectral shift at the back focal plane increase as the absolute values of the twist factors decrease, while zero spectral shift still occurs at the image plane. All above discussions are confined to the on-axis spectrum and relative spectral shift. In Fig. 5, we calculate the relative spectral shift of a focused twisted electromagnetic GSM beam versus the

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Figure 4: On-axis relative spectral shift η of a focused twisted electromagnetic GSM beam versus propagation distance for different values of the twist factors γxx and γyy with P0 = 0.6, δxx = 0.01 mm and δyy = 0.02 mm.

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Figure 5: Relative spectral shift of a focused twisted electromagnetic GSM beam versus the transverse coordinate x for different values of the degree of polarization of the initial beam at z = 350 mm with γxx = 0.003 (mm)−1 , γyy = 0.0002 (mm)−1 , δxx = 0.01 mm and δyy = 0.02 mm.

transverse coordinate x for different values of the degree of polarization of the initial beam at z = 350 mm with γxx = 0.003 (mm)−1 , γyy = 0.0002 (mm)−1 , δxx = 0.01 mm and δyy = 0.02 mm. One finds from Fig. 5 that the relative spectral shift gradually decreases as the transverse coordinate x increases, red shift can be observed when x is enough large. For certain value of transverse coordinate x, no spectral shift occurs. Furthermore, the absolute value of the relative spectral shift increases as the initial degree of polarization increases. Numerical results (not present here to save space) also show that the absolute value of the relative spectral shift decreases as the absolute values of initial twist factors (γxx and γyy ) increases. 4. CONCLUSION

The spectral shift of a focused twisted EGSM beam focused has been studied in detail. Dependence of the spectral shifts on the degree of polarization, twist phase and correlation coefficients of the initial beam is explored numerically. It is shown that the relative spectral shift of focused twisted EGSM beam is related to the initial beam’s parameters, the blue shift occurs at on-axis points, while the red shift can occur at off-axis points. ACKNOWLEDGMENT

The author gratefully acknowledges the support of K. C. Wong Education Foundation, Hong Kong. REFERENCES

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