Imaging Characteristics of the Axicon Imaging System

Zhongsheng Zhai

1

, Qinghua Lv

2, 3

, Xuanze Wang

1

, Liangen Yang

1

, Zhongbao Xu

1

and He Tao

1

1

School of Mechanics and Engineering, Hubei University of Technology, Lizhi Road, Wuhan, Hubei, China

2

Hubei Collaborative Innovation Center for High-efficient Utilization of Solar Energy,

Hubei University of Technology, Lizhi Road, Wuhan, Hubei, China

3

School of Science, Hubei University of Technology, Lizhi Road, Wuhan, Hubei, China

Keywords: Depth of Field, Non-diffracting Beams, Axicon, Imaging Characteristics.

Abstract: The depth of an image system can be extended by an axicon which can generate line focus. According to

physical optical theory, the diffracting patterns of the defocus point spread function (PSF) for the imaging

system with axicon are analyzed through the generalize pupil function. The expressions of the PSF for the

imaging system illuminated by white light are described as the superposition of the intensities in individual

monochromatic patterns. Experimental results show that the central portion contains the most energy of the

diffraction pattern from the PSF produced by the white light, and the contrast of secondary outside circular

rings decreased rapidly. Furthermore，the central spot radius varied slowly with the increase of defocus

parameter, and the depth of field of the imaging system is effectively extended with a shortcoming that the

images need further processing.

1 INTRODUCTION

In 1987, the team of J. Durnin found the solution of

the Maxwell's wave equation, which was given as

zero-order Bessel function forms, and first put

forward the concept of nondiffracting beams (Durnin,

1987). Nondiffracting beams which has

characteristics that the size of central spot and shape

does not change significantly over a propagation

distance. In 1992, G. Scott and other researchers used

the axicon to generate nondiffracting beams (Scott

and McArdie, 1992).

The ‘axicon’, first introduced by McLeod in 1954,

can form an extended focal segment which has the

zero-order Bessel distribution, and it can also

generate annular beams in the far region. (McLeod,

1954). Axicons have been incorporated in numerous

applications. Zhai using the axicon, simulated by a

spatial light modulator to to improved the efficiency

of laser processing (Zhai and Kuang, 2014).

Guillaume Druart demonstrated the diffractive axcion

has image-zooming capability, and they realized a x2

linear system (Druart et al., 2008). Zeng et al. used a

refractive axicon to transform an input Gaussian laser

beam into a collimated annular beam, which refer to

as optical trepanning. (Zeng et al., 2006).

In recent years the ability of annular linear axicons

for extending the depth of field of imaging system has

proposed by some researchers (Mikula et al., 2005).

They fixed the distance between the input object and

the diffractive elements, and obtained output images

in different output planes behind the axicon. In an

early publication (Zhai and Zhao, 2007), we have

derived the diffraction patterns of an axicon

illuminated by a red high brightness LED, and the

patterns were looked as the PSFs of the axicon

imaging system. However, the expressions of the

PSFs were calculated by the point light source in

different positions，not by the defocus aberration.

The aim of this paper is to analyze the imaging

principle of axicon, and to analyze the relationship

between the PSF and the defocus parameter. Imaging

results proved that the axicon can extend the depth of

field. The images created by this system can be

observed in real time, but they exhibit a very poor

contrast. For getting clearer images, digital

processing method is required.

Zhai, Z., Lv, Q., Wang, X., Yang, L., Xu, Z. and Tao, H.

Imaging Characteristics of the Axicon Imaging System.

DOI: 10.5220/0005742201350139

In Proceedings of the 4th International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS 2016), pages 137-141

ISBN: 978-989-758-174-8

Copyright

c

2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

137

2 THEORIES

2.1 Defocus Aberration

It is well known that defocus aberration manifests

itself by a quadratic phase at the imaging system

pupil, i.e.

)](exp[),,(

22

vuivuG +=

ϕϕ

(1)

where (u, v) are the normalized coordinates of the

pupil plane, and the defocus parameter φ is defined

by the following expression (Eliezer et al., 2008):

)

111

(

2

'

2

20

fl

l

d

W −−==

λ

π

λ

π

ϕ

(2)

where W

20

is Hopkins defocus factor, d is the pupil

radius, λ is the wavelength, f, l and l’ are the lens focal

length, the distances from the object and the image to

the lens respectively.

Obviously, when imaging condition is fulfilled:

0

111

'

=−−

fl

l

(3)

From equation 2, we can observe that the defocus

parameter φ equals zero. In the large DOF imaging

system, when a focus aberration occurs, as shown in

figure 1, the wavefront is deformed and it can use the

generalize pupil function to describe the defocus

error. The defocus factor is given as:

2

20

)

11

(

2

1

r

zz

W

la

⋅−=

(4)

where r is the radius of the pupil. z

l

is the distance

from the ideal image plane to the lens, and z

a

is the

distance between the lens and the defocus image

plane. In the common imaging system, the defocus

factor is changed with the increase of z

a

.

Figure 1: Scheme of defocus system.

2.2 Imaging System Design

An imaging system was arranged as shown in figure

2. It mainly includes a light source, battery of lens, an

axicon, a CCD detector and a computer. The axicon

is placed at the back of lens, and a CCD camera is

used to capture intermediate images when the object

moves within a specified region before the lens. The

object is illuminated with incoherent light. The

quality of the intermediate images is so poor that an

image processing portion is needed. For better

considering the depth of field, in the designed

imaging system, the distances between the lens,

axicon and CCD camera are fixed. It requires post-

processing steps in order to achieve a high quality

image, so it is a two step imaging system.

Figure 2: Schematic diagram of non- diffractive beam

imaging system.

2.3 PSF of Defocus

Consider a thin axicon with a transmittance function

given by

+

−−

=

0

2

)1(

exp

)(

22

vu

Dnik

rt

θ

1

1

22

22

>+

≤+

vu

vu

(5)

where u, v are the normalized pupil coordinates, n is

the refractive index, D is the diameter of the axicon,

k is the wave number 2π/λ, and θ is refracting angle.

After putting the axicon in the pupil of the

imaging system, when aberrations are introduced, the

generalized pupil function may be described as

)](exp[]

2

)1(

exp[

),,(),(),,(

2222

vuivu

Dnik

vuGvutvuQ

++

−−

=

=

ϕ

θ

ϕ

ϕ

(6)

where φ is the defocus parameter.

According to the theory of Fourier optics, in the

diffraction-limited imaging system, the point spread

function is the Fraunhofer diffraction pattern of pupil

function. Therefore, we can obtain the PSF of the

system with the axicon as:

Defocus

image

z

a

Objec

t

Pupil Image

Z

l

Compute

r

Axicon

CCD

Objec

t

Image

PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology

138

dudvyvxu

z

i

vuivu

Dnik

dud

v

yvxu

z

ivuQ

z

A

yxh

i

ii

]}[

2

exp{

](exp[]

2

)1(

exp[

]}[

2

exp{),,(),,(

2222

+−⋅

++

−−

=

+−=

λ

π

ϕ

θ

λ

π

ϕ

λ

ϕ

(7)

where A is the amplitude of the incident beam, Z

i

is

the distance from the axicon to the image plane.

The optical system is circular symmetry, and for

the sake of simplicity, we make a transformation to

polar coordinates in both the (u, v) and the (x, y)

planes as follows:

γργρ

γ

ρ

βρβρ

β

ρ

sin,cos

)arctan(

sin,cos

)arctan(

11

22

1

22

==

=

+=

==

=

+=

yx

x

y

yx

vu

u

v

vu

(8)

Applying the coordinate transforms, the PSF can

be written as

−−⋅

+

−−

=

π

βγβρρ

λ

π

ρϕρρ

θ

ρϕρ

2

0

1

2

1

)]cos(

2

exp[

]

2

)1(

exp[),(

d

z

i

di

Dnik

h

i

(9)

Taking into account the circular symmetry of the

second integral in Eq (9), we use the Bessel function

identity

)(2)]cos(exp[

0

2

0

aJdia

=−

π

πβγβ

(10)

where J

0

is a Bessel function of the first kind, zero

order. Substituting (10) in Eq.(9), we can obtain

ρϕρρ

θ

ρρ

λ

π

ρπϕρ

di

Dnik

z

Jh

i

]

2

)1(

exp[

)

2

(2),(

2

101

+

−−

⋅

=

(11)

With the stationary phase method (Goodman,

1996), Eq. (11) can be approximated by following:

]

2

)1(

exp[

)

2

(2),(

2

101

ss

s

i

s

i

Dnik

z

Jh

ϕρρ

θ

ρρ

λ

π

ρ

ϕ

π

πϕρ

+

−−

⋅

≈

(12)

where ρ

s

is the stationary point,

ϕ

θ

ρ

4

)1( Dnk

s

−

=

(13)

In an incoherent imaging system, the intensity of

point spread function can be given by

22

)},({),,(),,(

ϕρϕϕ

QFyxhyxh

I

==

(14)

For the sake of simplicity, 1-D analysis of Eq.(14)

is performed as:

)

)1(

(

4

)

2

(

4

),(

1

20

2

0

20

3

1

2

0

3

1I

ρ

θ

ρ

π

ρρ

λ

π

ρ

ϕ

π

ϕρ

Wz

Dnk

J

kW

z

Jh

i

s

s

i

s

−

=

≈

(15)

Note that Eq.(15) is the expression of the PSF of

the axicon imaging system under monochromatic

light. If the light source is polychromatic light or

white light, the PSFs for the these source are different

from Eq.(15). The white light can be represented by a

combination of mutually incoherent monochromatic

components extending over a range of frequencies.

Each component produces a diffraction pattern as

described above, and the total intensity is everywhere

the sum of the intensities in these monochromatic

patterns. If the wavelength bandwidth of the light

source is λ

∈

[λ

beg

, λ

end

], the PSF of imaging system

with axicon illuminated with white light can be

expressed as

λρ

θ

ρ

ϕ

π

ϕρ

λ

λ

d

Wz

Dnk

Jh

end

beg

i

s

−

= )

)1(

(

4

),(

1

20

2

0

3

1I

(16)

3 SIMULATION

AND EXPERIMENT

To demonstrate the above theoretical analysis, we have

carried out numerical simulations and experiments.

Numerical simulations for the PSFs at different

defocus aberration conditions φ =0.5π, π, 2π, 4π, were

done by use of Eq. (16), as shown in figure 3.

Imaging Characteristics of the Axicon Imaging System

139

Some parameters were given by θ = 0.05, n =1.5,

z =120mm, f =120mm, D = 10mm and λ

∈

[380, 780]

nm. From figure 3, it follows that the central spot

radius varied with the change of defocus parameter φ.

Figure 3: Simulation of PSFs for different φ.

An experimental imaging system was constructed

based on figure 2, and some parameters of this

imaging system are the refractive index of the axicon

n = 1.5, the refracting angle of the axicon θ = 0.01rad.

Figure 4 gives the experiment result of the normalized

PSFs for the system under incoherent illumination in

case of l =160, 210, 260, 310mm. Obviously, the

central portion contains the most energy of the

diffraction pattern from the PSF produced by the

white light, and the contrast of secondary outside

circular rings decreased rapidly. Therefore, the

imaging characteristics of the system with the axicon

are mainly determined by the central spot.

Figure 4: Experiment results of the PSFs for different l. (a)

160mm, (b) 210mm, (c) 260mm, (d) 310mm.

Figure 5: Structural diagram of a candlestick.

Figure 6: Imaging results by the system (a) (b) without the

axicon, (c) with the axicon.

An imaging experiment for a candlestick with two

different diameter parts was exhibited using the same

system. Figure 5 presents the structure diagram. The

images in figure 6(a) and 6(b) were taken by the

standard imaging system without the axicon, and the

image in figure 6(c) was acquired in the case of the

system with the axicon. From figure 6(a) and 6(b), it

is obvious that only one part of the candlestick is clear

which is in the focus plane, and the other part is

blurring for the reason of defocus.

In figure 6(c), we also can observe that even the

φ =0.5π φ =π

φ =2π φ =4π

（a） （

b

）

（c） （

d

）

Φ12

10

Φ48

b

lured

b

lure

d

clear

clear

(a)

(b)

(c)

not clea

r

no

t

clear

PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology

140

images of the both parts of the candlestick are not as

clear as the images in focus plane, but they are more

clear than those in the out of focus place. Therefore

the image acquired by the imaging system with the

axicon is insensitive to defocus, and the depth of field

of the imaging system is effectively extended.

However, it has a shortcoming that the images have

low contrast and low resolution. The quality of the

intermediate images is so poor that an image

processing portion is needed.

4 CONCLUSIONS

We have investigated the imaging feature of the

axicon for extending depth of field. The diffraction

intensity distribution of PSFs of the imaging system

illuminated with monochromatic and white light are

clearly derived based on the generalize pupil

function. The experimental results proved that the

PSF for white light could be obtained by the

superposition of the intensities in individual

monochromatic patterns, and imaging results showed

that the axion can extend the depth of the field with

unclear images which need further processing.

ACKNOWLEDGEMENTS

This work was supported by the National Natural

Science Foundation of China (Nos. 51405143,

51575164, 5125157, 51275158), the Doctoral

Scientific Research Foundation of Hubei University

of Technology (No. BSQ13048) and Research

Foundation of Hubei Collaborative Innovation Center

for High-efficient Utilization of Solar Energy (No.

HBSKFZD2014007).

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Imaging Characteristics of the Axicon Imaging System

141