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ARTICLE IN PRESS Optik xxx (2014) xxx–xxx

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Fourth order correction to a Gaussian beam focused with a parabolic index lens Soo Chang Department of Physics, Hannam University, 133 Ojungdong, Taejon 306-791, Republic of Korea

a r t i c l e

i n f o

Article history: Received 20 June 2013 Accepted 15 December 2013 Available online xxx PACS: 42.30.Va Keywords: Gaussian beam Spherical aberration Beam quality factor Gradient index lens

a b s t r a c t We formulate the fourth order correction to a paraxial Gaussian beam propagated along the axis of symmetry of a parabolic index lens. First we examine the evolution of a complex-source-point spherical wave (equivalent paraxially to a Gaussian beam) through the lens in a two-dimensional xz plane. Taking into account the terms of up to fourth order in aperture variables, we ﬁnd a ray-optical solution to the exit beam that is represented in terms of aberration function. We also analyze the effect of the lens aberration exerted on the degradation in the quality of a Gaussian beam. The fourth order-corrected wave function derived here may be used to evaluate the quality of a Gaussian beam focused with a parabolic index lens. Further it may be applied to the case of an orthogonal system in which the index variations are different in the xz and yz planes. © 2014 Elsevier GmbH. All rights reserved.

1. Introduction A laser beam is generally highly directional and does not have a spatially uniform intensity distribution. The laser beam of fundamental mode conﬁned to a region near the optic axis is well-described as a Gaussian beam [1]. It is also known that the sum of all the higher order corrections to the paraxial Gaussian beam is equivalent to a spherical wave emanating from a complex source point [2,3]. Recently we have examined the evolution of a complex-source-point spherical wave (CSPSW) through a rotationally symmetric optical system which is composed of various refracting (or reﬂecting) surfaces such as spheres, aspherics, and zone plates [4–8]. As a result, fourth order corrections have been made in terms of Seidel-type aberration coefﬁcients to the Gaussian beam passing through the optical system. We have also analyzed what aberrations of the system degrade the quality of the Gaussian beam. Today inhomogeneous or gradient index (GRIN) materials are ﬁnding a variety of applications as lenses in micro-optical systems and a substitute for aspherics in conventional imaging systems [9]. In particular, the parabolic index proﬁle is worth considering in the context of GRIN lenses since it yields analytic results which facilitate the understanding of more general forms of proﬁle gradient [10]. However, we have not discussed the quality of a Gaussian beam focused with a parabolic index lens.

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In this article, we discuss the Seidel-type aberration of a parabolic index lens which degrades the quality of a Gaussian beam propagated along the axis of symmetry of the lens. First we examine the evolution of the equivalent CSPSW through the lens in a two-dimensional xz plane. Taking into account the terms of up to fourth order in aperture variables, we ﬁnd a ray-optical solution to the exit beam that is represented in terms of aberration function. The real part of the aberration function has an analogy to the spherical wavefront aberration in ordinary ray optics [11], while its imaginary part gives the fourth order correction to the amplitude variation of the Gaussian beam. Using the derived formula, we also analyze the effect of the lens aberration exerted on the quality of a Gaussian beam. The fourth order-corrected wave function derived here may be used to evaluate the quality of a Gaussian beam focused with a parabolic index lens. Further it may be applied to the case of an orthogonal system in which the index variations are different in the xz and yz planes. 2. Evolution of a complex-source-point spherical wave through a parabolic index lens Fig. 1 shows (a) a paraxial Gaussian beam of vacuum wavelength which is focused with a parabolic index lens and (b) a complex-source-point spherical wave (CSPSW) which is equivalent paraxially to the Gaussian beam. w0 (or w0 ) is the minimum spot size of the incident (or exit) Gaussian beam. The lens of thickness ı separates two media of refractive indices n and n . The coordinate system is referenced to the front surface of the lens which is

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Please cite this article in press as: S. Chang, Fourth order correction to a Gaussian beam focused with a parabolic index lens, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.012

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12 =

2

ı (n20 + ˇ12 ) 2ˇ 1 1 √ √ 2(x1 ) − n20 − ˇ12 nx21 2n0 ı B n0 ı B 2 + sin − , sin √ r ˇ ˇ1 4n0 B 1 (5)

(x)[(dx)2 + (dz)2 ] =

in terms of the constant of motion in the z-direction

ˇ1 =

(x1 ) − (

nx1 2 ) . r

(6)

On the one hand, in the paraxial regions such that x12 |z0 +

ib0 |2 and Bx21 1, the light disturbance transferred from the source point O through the lens to an observation point Q can be evaluated using the Fresnel–Kirchhoff diffraction integral [12]. If the terms of up to second order in the aperture variable x1 are taken into account, the amplitude of the light at Q of coordinates (x , z ) is given by

Fig. 1. (a) A paraxial Gaussian beam of vacuum wavelength is propagated along the axis of symmetry of a parabolic index lens. The lens of thickness ı separates two media of refractive indices n and n . The coordinate system is referenced to the front surface of the lens. w0 (or w0 ) is the minimum spot size of the incident (or exit) beam. (b) The Gaussian beam is equivalent paraxially to the spherical wave with a center at a complex location (0, z0 + ib0 ) or (0, z0 + ib0 ) in the xz plane. The Rayleigh range of the incident (or exit) beam is given by b0 = nw02 / (or b0 = n w02 /). The path of the complex ray is determined by applying Fermat’s principle.

parallel to the x-and y-axes, and the axis of symmetry of the lens is taken as the z-axis. A ray of light starts from a source point O on the z-axis, passes through the points P1 and P2 on the front and rear surfaces of the lens, and goes to an image point O on the z-axis. By rotational symmetry of the lens under consideration, the path of the ray from O through P1 and P2 to O must always lie on one plane. Therefore, we may treat this problem in a two-dimensional space, chosen as the xz plane. If the source point O is denoted by coordinates (0, z0 + ib0 ), the points P1 and P2 by coordinates (x1 , 0) and (x2 , ı), and the image point O by coordinates (0, z0 + ib0 ), the light disturbance arriving at P1 from O may be represented by (x1 , 0) =

C exp (iknr − iωt) , r

2 1/2

.

(2)

In the above we choose the branch of r such that its real part is equal to −z0 when it is large. By so doing, the wave function in Eq. (1) is equivalent to the sum of all the higher order corrections to the paraxial Gaussian beam of Rayleigh range b0 , where b0 = nw02 / [2,3]. If the medium of the lens has a dielectric function of parabolic type

(x) = n20 (1 − Bx2 ),

(3)

in the xz plane [10], where n0 is the index of refraction on the zaxis and B (> 0) is the parameter governing the index variation, the transmission coefﬁcient for a ray of light impinging on P1 may be expressed as follows: 12 0 exp(ik12 )

(x , z ) C

(4)

where 0 is the constant factor and 12 is the optical path length of the ray from P1 to P2 , evaluated as

dx1 exp[ik(nr + 12 + n r )],

(7)

aperture

where C is the factor independent of x1 and the path lengths are written as

r −(z0 + ib0 ) −

x12 2(z0 + ib0 )

,

(x − x2 )2 (|z − ı| |x − x2 |), 2(z − ı) √

√ nsin2 (ı B) sin(2ı B) n2 2 n0 ı + − n0 B + x12 , √ 2 z0 + ib0 4n0 B (z0 + ib0 ) (8)

r (z − ı) + 12

in a quadric approximation. It should be noted here that a timeharmonic factor exp(−iωt) has been dropped from Eq. (7) for simplicity. Assuming the size of aperture is large enough to accept the paraxial Gaussian beam, we analytically solve the diffraction integral (7) to get

(1)

√ where C is the normalization constant, i(= −1) is the imaginary symbol, k(= 2/) is the magnitude of the wave vector in vacuum, ω is the angular frequency of the light, and r = x12 + (z0 + ib0 )

(x , z ) A exp in k

x 2

,

2(z − z0 − ib0 )

(9)

where A is the factor independent of x and the paraxial beam parameters are deﬁned by √ √ ⎤ √ ⎛ ⎞ ⎡ cos ı B n0 B sin ı B n x2

⎝ z0 − ı + ib0 ⎠ = ⎢ ⎣ x2

√ 1 − √ sin ı B n0 B

×

nx1 z0 + ib0

√

cos ı B

⎥ ⎦

,

(10)

x1 with the equivalent power of the lens system √ √ K = n0 B sin ı B .

(11)

The wave function in Eq. (9) is equivalent paraxially to a spher ical wave with a center at a complex location (0, z0 + ib0 ). It also represents the exit Gaussian beam of Rayleigh range b0 , where b0 = n w02 /. On the other hand, if we regard the equivalent CSPSW in Eq. (1) as a bundle of rays originating from a complex source point O, the formula in Eq. (9) can be derived from Fermat’s principle. The path

Please cite this article in press as: S. Chang, Fourth order correction to a Gaussian beam focused with a parabolic index lens, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.012

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of the physically possible ray propagated from O through P1 and P2 to Q must be stationary by varying P1

∂ (nr + 12 + n r ) = 0. ∂x1

(12)

Hence the path of the complex ray obeys the formal equation.

√ √ n sin ı B + cos ı B x2 , √ n0 B z0 − ı + ib0

x1 =

=

x2

z0 − ı + ib0

x ,

z0 − z + ib0

(13)

where we have used the paraxial relations in Eqs. (8) and (10). Eliminating the integration variable x1 by applying Eq. (13), we obtain the ray-optical solution of the integral (7) which corresponds to the wave function in Eq. (9), except for the constant factor. To ﬁnd the function of spherical aberration, we need again to solve Eq. (7) by considering the terms of up to fourth order in the aperture variable x1 on the paraxial path of the complex ray in Eq. (13). As a result, the fourth order correction has been made to the amplitude of the light at Q

(x , z ) A exp

ik

n x 2 2(z

−

z0

−

ib0 )

−

n x 4 8(z − z0 −

3 ib0 )

− V (x , z )

(14)

,

where A is the factor independent of x and the aberration function is given by V (x , z ) = SI

4

(z0 − ı + ib0 )

8(z0 − z + ib0 )

with the coefﬁcient of

n

1 SI = − 2 3 n (z0 − ı + ib0 )

×

√ cos ı B −

+

ı n30

n20 B

−

x , 4 4

3n 2

√ √ − n0 B sin ı B

2 z0 − ı + ib0

n

+ ib0

+

n 2

2 z0 − ı + ib0

n + n0 ıB. z0 + ib0

n 3

(z0 + ib0 )

−

. (16)

3

(x , z0 ) exp −i2fx x ,

=

x | m

(19)

(x , z0 )|2 ,

dx |

(x , z0 )|2

2

df x fxm (fx , z0 )

(20)

2 ,

df x (fx , z0 )

+ n0 ıB2 .

M2 =

n W ,

(21)

where we let

(17)

n (z0 + ib0 )

respectively, the M2 factor is given by

Hence the aberration coefﬁcient in Eq. (16) is also reduced to a simple form SI

dx

fxm =

The aberration function V(x , z ) depends upon the fourth power of the ray height x but it is independent of an inclination of the principal ray. The real part of the aberration function has an analogy to the spherical wavefront aberration in ordinary ray optics [11], while its imaginary part gives the fourth order correction to the amplitude variation of the Gaussian beam.√ In case of a thin GRIN element where ı B approaches zero, the relations in Eq. (10) may be approximated as z0

sin ı B √ n0 B n20 B

dx

where fx is the component of spatial frequency in the x-direction. If we deﬁne the mth moments of the irradiance distribution in space and in spatial frequency as

3

√

(fx , z0 ) =

x m

z0 − ı + ib0 3n

departure from a Gaussian is related to the second moments of the irradiance distribution of the beam in the near and far ﬁelds [13]. The far-ﬁeld angular spread of the beam in Eq. (14) may be represented by a Fourier integral

(15)

√ n cos ı B

z0 − ı + ib0

Fig. 2. The equivalent power of a parabolic index lens which is plotted as a function of the thickness of the lens ı. The dielectric function in the medium of the lens is deﬁned by (x) = n20 (1 − Bx2 ) in the xz plane. The solid, dashed, and dotted curves are of n0 B1/2 = 1.0 m−1 , 1.5 m−1 , and 2.0 m−1 , respectively.

(18)

3. Quality factor of the fourth order-corrected beam Based on Eqs. (14)–(16), we have numerically evaluated the M2 factor of the fourth order-corrected CSPSW passing through a system of parabolic index lens. The M2 factor which measures the

W =2

x 2 − x 2 ,

=

2 2 fx − fx 2 . n

(22)

It is worthy of notice that if the fourth order correction in Eq. (14) is negligible, W = w0 and = / n w0 so that M2 = 1. Fig. 2 shows the equivalent power of a parabolic index lens which is plotted as a function of the thickness of the lens ı. The √ solid, dashed, and dotted curves are of n0 B = 1.0 m−1 , 1.5 m−1 , and 2.0 m−1√, respectively. The equivalent power of the lens √is equal to√zero at ı B = 0 or , while it reaches a maximum of n0 B when ı B = /2. In Fig. 3 we plot the M2 factor of a Gaussian beam focused with a parabolic index lens, where the thickness of the lens ı is taken as a variable. The solid, dashed, dotted, and dash-dotted curves are of w0 = 10 mm, 15√ mm, 18 mm, and 20 mm, respectively. We also let = 632.8 nm, n0 B = 1.5 m−1 , n = n = 1, and z0 = 0. The M2 factor is√shown to oscillate with the increase of ı. We ﬁnd that M2 1 at ı B 0, 0.22, and 0.84, where the real part of the aberrtion function in Eq. (15) disappears. Fig. 4 also shows the M2 factor of the exit beam which is plotted beam w0 . as a function of the minimum spot size of the incident √ The solid, dashed, dotted, and dash-dotted curves are of ı B = /3, /2, 2/3, and 5/6, respectively. Other parameters are the same as

Please cite this article in press as: S. Chang, Fourth order correction to a Gaussian beam focused with a parabolic index lens, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.012

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ARTICLE IN PRESS S. Chang / Optik xxx (2014) xxx–xxx

Fig. 3. The M2 factor of a Gaussian beam focused with a parabolic index lens, which is plotted as a function of the thickness of the lens ı. The solid, dashed, dotted, and dash-dotted curves are of w0 = 10 mm, 15 mm, 18 mm, and 20 mm, respectively. We also let = 632.8 nm, n0 B1/2 = 1.5 m−1 , n = n = 1, and z0 = 0.

Fig. 4. The M2 factor of the exit beam which is plotted as a function of the minimum spot size of the incident beam w0 . The solid, dashed, dotted, and dash-dotted curves are of ıB1/2 = /3, /2, 2/3, and 5/6, respectively. Other parameters are the same as those in Fig. 3.

the position of the incident √ beam waist z0 . We take the param= 15 mm. eters as = 632.8 nm, n0 B = 1.5 m−1 , n = n = 1, and w0√ The solid, dashed, dotted, and dash-dotted curves are of ı B = /3, /2, 2/3, and 5/6, respectively. The M2 factor of the exit beam is found to increase by shifting the lens away from the waist plane of the incident beam, because the height of the ray impinging on the lens surface increases with the magnitude of z0 . In general, the fourth order-corrected wave function in Eq. (14) may be used to evaluate the degradation in the quality of a Gaussian beam propagated along the axis of symmetry of a parabolic index lens. Further if we deﬁne the paraxial beam parameters separately in the xz and yz planes, it may be applied to the case of an orthogonal system in which the index variations are different in each azimuthal plane. 4. Conclusions In this article, we have discussed the Seidel-type aberration of a parabolic index lens which degrades the quality of a Gaussian beam propagated along the axis of symmetry of the lens. First we have employed a complex-source-point spherical wave (CSPSW) which is equivalent paraxially to the Gaussian beam. Then we have analyzed the evolution of the equivalent CSPSW through the lens in a two-dimensional xz plane. Taking into account the terms of up to fourth order in aperture variables, we have found a ray-optical solution to the exit beam that is represented in terms of aberration function. The real part of the aberration function has an analogy to the spherical wavefront aberration in ordinary ray optics, while its imaginary part gives the fourth order correction to the amplitude variation of the Gaussian beam. Using the derived formula, We have also analyzed the effect of the lens aberration exerted on the degradation in the quality of a Gaussian beam. The fourth ordercorrected wave function derived here may be used to evaluate the quality of a Gaussian beam focused with a parabolic index lens. Further if we deﬁne the paraxial beam parameters separately in the xz and yz planes, it may be applied to the case of an orthogonal system in which the index variations are different in each azimuthal plane. References

Fig. 5. The M2 factor of the exit beam which is plotted as a function of the position of the incident beam waist z0 . The solid, dashed, dotted, and dash-dotted curves are of ıB1/2 = /3, /2, 2/3, and 5/6, respectively. Other parameters are given by = 632.8 nm, n0 B1/2 = 1.5 m−1 , n = n = 1, and w0 = 15 mm.

those in Fig. 3. We ﬁnd that M2 = 1 at w0 0, and M2 changes to the value greater than 1 as w0 increases. If w0 0, the height of the ray impinging on the lens surface becomes zero so that the inﬂuence of spherical aberration can be neglected. In Fig. 5 we show the M2 factor of a Gaussian beam passing through a parabolic index lens, which is plotted as a function of

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Please cite this article in press as: S. Chang, Fourth order correction to a Gaussian beam focused with a parabolic index lens, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.012