научная статья по теме IMPROVEMENT OF FRINGE QUALITY AT LDA MEASURING VOLUME USING COMPACT TWO HOLOLENS IMAGING SYSTEM Физика

Текст научной статьи на тему «IMPROVEMENT OF FRINGE QUALITY AT LDA MEASURING VOLUME USING COMPACT TWO HOLOLENS IMAGING SYSTEM»

ОПТИКА И СПЕКТРОСКОПИЯ, 2015, том 118, № 3, с. 508-515

ГЕОМЕТРИЧЕСКАЯ И ПРИКЛАДНАЯ ОПТИКА

УДК 535.417

IMPROVEMENT OF FRINGE QUALITY AT LDA MEASURING VOLUME USING COMPACT TWO HOLOLENS IMAGING SYSTEM

© 2015 г. Abhijit Ghosh and A. K. Nirala

Biomedical Optics Lab, Department of Applied Physics, Indian School of Mines, Dhanbad 826004, Jharkhand, India

E-mail: abhi.photonics@gmail.com, aknirala@yahoo.com Received July 16, 2014

Design, analysis and construction of an LDA optical setup using conventional as well as compact two holo-lens imaging system have been performed. Fringes formed at measurement volume by both the imaging systems have been recorded. After experimentally analyzing these fringes, it is found that fringes obtained using compact two hololens imaging system get improved both qualitatively and quantitatively compared to that obtained using conventional imaging system. Hence it is concluded that use of the compact two hololens imaging system for making LDA optical setup is a better choice over the conventional one.

DOI: 10.7868/S0030403415030034

INTRODUCTION

Laser Doppler anemometry (LDA) is a well established technique in the field of fluid velocity measurement [1—4]. Since the pioneering work of Yeh and Cummins [5], LDA has become standard in fluid measurement and detailed descriptions of the technique can be found in, for example, Durst et al. [6], Goldstein [7], Albrecht et al. [8], and Zh. Zhang [9]. The most common configuration of the LDA system is the Differential Doppler Technique or Dual Beam Technique. In this technique two beams of equal intensity are generated from a single laser source using an equal path length beam splitter and are then allowed to cross at a point using a converging lens [6] to provide interference fringe pattern in a region where measurement of flow velocity of fluid is performed and is called measurement volume [9].

Several authors [10—12] have investigated the influence of Gaussian beam properties on the frequency spectrum of Doppler signals and the measurement errors caused by the fringe gradients. For increasing measurement accuracy formation of evenly spaced un-distorted interference fringe pattern at measurement volume is highly required. To have undistorted interference fringe pattern at the measurement volume imaging lens should give diffraction limited performance over the two small areas encompassed by two apertures on the lens through which two light beams pass. Monochromatic aberrations (spherical aberration, coma, astigmatism, field curvature and distortion) of the optical system may lead to formation of distorted fringe pattern in LDA measurement volume which causes inaccuracy in measurement of fluid velocity

[13, 14]. Stojanoff et al. [15], Schneider et al. [16], Ghosh et al. [17] and many others [18, 19] have used holographic optical elements in LDA optical system to get credit of replacing costly and bulky conventional optical system by light weight and of low cost holographic optical system. Holographic optical system is not only light weight, compact and of low cost but also gives diffraction limited performance almost free from all monochromatic aberrations under proper recording and play back geometry [20—23]. Elimination of monochromatic aberrations ensures formation of good quality undistorted interference fringe pattern at measurement volume. In present manuscript design and analysis of an LDA optical setup using conventional as well as compact two hololens imaging system has been reported. Fringes formed at measurement volume by both the imaging systems have been recorded using a linear CCD, and analyzed qualitatively as well as quantitatively.

EXPERIMENTAL

Optical Setup Consisting of Conventional Imaging System

In this set up an expanded collimated laser beam is produced using a spatial filtering arrangement and a collimator as shown in Fig. 1. A mask (containing two circular slits of diameter 1.2 mm, separated by a distance 6.7 mm) is placed for producing two identical pencil beam of light in front of achromatic doublet (procured from M/s Newport Corp.) which is used for proper intersection of parallel beam of light at focal

Fig. 1. Schematic of LDA optical set up consisting of conventional imaging system for formation of interference fringe pattern at measurement volume.

Mask contatining two circular slits

£

Laser

Spatial filtering arrangement

Collimator Conventional lens

CCD

-A

Microscopic objective

jJl

Signal processing unit

point and for formation of interference fringe pattern at measurement volume.

For analysis of fringe quality formed at LDA measurement volume, a linear CCD is kept at 40 cm away from a microscopic objective of 20*. CCD sensor is gradually moved step by step to cover-up entire profile picture of measurement volume consists of interference fringe pattern shown in Fig. 2.

Optical Setup Consisting of Holographic Imaging System

In this set up, two hololenses are being used. HL1 is used for producing collimated beam of light and HL2 is used for intersecting two pencil beam of light off ax-ially at its focal point as shown in Fig. 3. The same mask which was used in case of conventional imaging system is placed in between the two hololenses for producing two identical pencil beam of light. However, we have developed a compact hololens imaging system

Fig. 2. Photograph of interference fringe pattern formed at LDA measurement volume.

because it will reduce number of degrees of freedom and therefore the aberration arising out of misalignment will drastically be reduced. Optical assembly (a linear CCD 40 cm apart from a microscopic objective of 20*) is same as that was used in case of conventional imaging system for analyzing fringe quality formed at LDA measurement volume.

Hololens Recording

Hololenses have been recorded using two coherent waves derived from the same laser source. Out of two coherent waves one is spherical wave and other is a plane wave. Schematic of the hololens recording geometry is shown in Fig. 4. To prevent formation of spurious grating, the resulting interference of the two waves should be recorded under index matched condition [22]. Dichromated gelatin have high diffraction efficiency, low noise, high resolution, long life and high refractive index modulation capacity [24], however due to ease of availability and cost effectiveness of source and recording materials, we have recorded ho-lolenses on commercially available high resolution silver halide plate PFG-01 [25] (of film thickness d = = 7 ^m and average refractive index n = 1.61) using He—Ne laser source of power 2 mW.

Angle (6) between plane wave and spherical wave at the time of recording was 17° and hence corresponding fringe spacing formed on hololens is A = V{2n sin (0/2)} = 1.33 ^m and focal length is kept 15 cm. The exposed film was processed using standard procedure [26, 27].

Mask containing two circular slits

unit

Fig. 3. Schematic of LDA optical set up consisting of compact two hololens imaging system for formation of interference fringe pattern at measurement volume.

Fig. 4. Schematic of recording geometry for hololens.

Analysis

The radius of curvature of the wave propagated from HL1 is given by

^ = ±H

rl rt.

J_

V ROi

(1)

■i /

where the subscripts I1, Cb O1, and r1 stand for image, reconstruction, object and reference beams respectively. For the hololens, ^ is the ratio of the wavelengths of light used for reconstructing and recording

the holograms. For He—Ne laser which is source for recording and reconstructing ^ = X C/X r = 1.

The "+" sign normally denotes a virtual image and "—" sign a real image. Hololens HL1 is used in the imaging system as it is recorded in the Fig. 3 and played back by a diverging beam so as to achieve collimated beam of light and hence

J_ R,

1

RCi

+

J_

v roi

1 R

i j

(a)

(b)

(c)

(d)

Fig. 5. CCD images of interference fringe pattern formed at different location of measurement volume by conventional imaging system.

for

Ra = R >

Rl = R0, =

(3)

This shows that under the playback geometry considered, the wave coming out of the hololens HL: will be a plane wave and it will act as a reconstructing wave for HL2.The radius of curvature of the wave propagated from HL2 is given by

1

1

Rr. Rc

1

R02

1

R

Ъ J

Here Rc = Rr = ^ hence

c2 r2

RI2 = -R02

(4)

(5)

This means the radius of curvature of the wave coming out of HL2 is equal in magnitude to the radius of curvature of the object wave. The minus sign shows that outgoing wave is a converging one.

To assess the aberrations introduced by HL1 and HL2 for the conditions under which they are fabricated and used in the system, we have used expressions given by Meier [20] and Champagne [21]. The coefficients of spherical aberration (S ), coma (Cx, Cy ), astigmatism (Ax, Ay, Axy ), curvature of field ( F ) and distortion (Dx, Dy ) can be written as:

^ =

J_

rC

+

Cx

Cy

= Xc.

r3

= Ус

R3

=

r3

+

+

+

f 1 1 Ï 1

V R0 R у rY

f x0 л _ Xz. XI

V R0 R у

r У0 л _ Ул. У1

V R0 Rr у

f 2 x0 2 _ Xr. 2 XI

V R0 Rr у _r3

(6)

(7)

(8) (9)

X

Fig. 5. (Contd.)

A = Zc +

y r3 +

f 2 yo

V Ro

2

Zz.

R3

2

Zl R 3!

(10)

where ( Xc , yC), (, yo), (Xr, yr), and (xi, yi) are coordinates of the constructing, object, reference and reconstructed beams.

A = xcyc +

xy 3 —

RC

Xoyo xryr

RO

R3

xizl

r3 !

F = xC + yC + F r3 +

/ 2 2 Xo + yo

R 3

2 , 2^

Xy + Z

R3

Xi + yi

R 3 '

+

D,

/ 3 2

xo + xoyo

3 ^ 2

= XC + xcyc

= R3

3 , 2\ Xy + xry

+

ro.

3

D = yC + ycxC +

DZ „3 -

R3

r

i 3 , 2 Zo + ZoXo

. ro

3 , 2 yi + yiXi

3 , 2

Xi + xIyi

r3

3 , 2\

yr + yrX

r:

r3

(11)

(12)

(13)

(14)

We see that a plane wave propagates between two hololenses, i.e. RTi = ^ for the first lens. Lens HL1 is constructed under the condition Roi = ^ and played back as it is being recorded. Under condition

Xc — Xy,

yCi = yn, RCi = Rri the coefficients of spherical aberration (S), coma (CX, Cy), astigm

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