Analysis of interferograms from a diffractive-lens-based common-path interferometer
Ismo Vartiainen,* Pasi Vahimaa, and Markku KuittinenDepartment of Physics and Mathematics, University of Joensuu, P.O. Box 111, FI-80101 Joensuu, Finland
In the most typically used types of interferometer, re-ference and test wavefronts propagate along separatepaths and are, therefore, differently exposed to me-chanical perturbation and air turbulence [1]. Opticaltesting in nonlaboratory conditions has hastened thedevelopment of common-path interferometers: well-known examples include point diffraction [2] and lat-eral shearing [3,4] interferometers. Also, several com-mon-path interferometers based on scatter plateshave been introduced [5].Perhaps the most popular analysis method in in-
terferometry is the so-called phase shifting method[1,6]. The key idea in the phase shifting interferome-try method is to record multiple interferograms withdifferent phase delays between reference and testwavefronts. Then, by combining them properly, theconstant terms can be eliminated and the phasecan be revealed [1]. This is a very reliable method,since, by combining the wavefronts, noise is reducedautomatically. The use of the phase shiftinginterferometry method requires wavefronts to propa-gate separately in order to introduce the phase de-
lays, which makes the method sensitive toenvironmental conditions.
In shearing interferometers and in interferom-eters based on tilting the wavefronts withrespect to each other, the phase function can be ana-lyzed with the Fourier transform method [7]. Unfor-tunately, selective wavefront manipulation is, ingeneral, impossible, because of the nature of thecommon-path interferometers, in which the testand reference wavefronts experience equal changesin phase when they propagate along the same opticalpath. Therefore, it was necessary to devise new ideason how to analyze these interferograms.
In the beginning of Section 2 we introduce somebasic information about interference between twowaves. We then describe the analysis method usedin this study. The limitations and accuracy issuesare also presented in this section. In Section 3, theresults for tested commercial glass lenses and micro-injection molded plastic lenses with a diameter of4mm are introduced.
2. Theory
A common-path interferometer based on an opti-mized diffractive lens was presented recently [8].The setup consists of a coherent light source, a spa-tial filter, and a diffractive lens; see Fig. 1. The dif-fractive lens is designed to divide the input plane
wave into a converging spherical wave (m ¼ −1) anda plane wave (m ¼ 0) with equal diffraction efficien-cies [8]. Other orders than the desired diffractionorders can be minimized through design and fabrica-tion, but their amplitudes are still nonzero. There-fore, these diffraction orders interfere with thedesired diffraction orders, thus producing interfer-ence with a higher fringe density. These fringes maybe considered as noise.The amplitudes of the interfering fields in the
plane of the inspected lens can be evaluated by usingthe diffraction efficiencies of the diffractive lens [8]and divergence properties of the diffraction orders.Orders other than 0th diffraction orders behave likespherical waves with a specific focal length, whichdepends on the diffraction order. The relative ampli-tudes of the designed diffractive lens are presentedin Table 1. The propagation of the diffraction ordersafter the diffractive lens is presented in Fig. 2.
A. Fundamental Theory on the Interference of Waves
First, we will take a look at some basics on the inter-ference of two waves. The complex presentation ofthe scalar plane wave is of the form
U1 ¼ U01 expðikΔzÞ; ð1Þ
where U01 is the amplitude, k ¼ 2π=λ is the wave-number, where λ is the wavelength and Δz ¼ z − z0is the distance from the source to the inspectionplane. A similar presentation of the spherical waveis of the form
U2 ¼ U02expðikrÞ
r; ð2Þ
where r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2 þΔz2
pis the radius of the curva-
ture. In the case of plane and spherical waves propa-gating along the same axis, the intensity distributionin the inspection plane is
I ¼ ∣U1 þU2∣2
¼ U201 þ
�U02
r
�2þU01U02
rcos½kðr −ΔzÞ�; ð3Þ
where the first two terms are the intensities of theplane and spherical waves and the third term isthe interference term. The desired wavefront is theargument of a cosine function. For two sphericalwaves with radii of curvature of r1 and r2 propagat-ing along the same axis, we get
I ¼ ∣U1 þU2∣2
¼�U01
r
�2þ�U02
r
�2þU01U02
r1r2cos½kðr1 − r2Þ�: ð4Þ
The effect of the other diffraction orders can be pre-dicted by computing the interference of several sphe-rical waves. The higher diffraction orders modulatethe fringes so that within one fringe there will be sev-eral maximums. The modulation can clearly be seenin Fig. 3 which theoretically demonstrates the noisecaused by the −2nd diffraction order. The amplitudeused for diffraction order −2 has to be at least onethird of the amplitude of the 0th and −1st ordersin order to make the effect clearly visible.
As one can see from Eqs. (3) and (4), the intensitydistribution consists of concentric dark and brightfringes. The fringes originate from the periodicityof the cosine function. Therefore, we can define thebright fringe condition for interference between the
Fig. 1. Schematic view of the setup.
Table 1. Theoretical Diffraction Efficiencies
m η [%]
−3 0.3−2 3.7−1 37.70 37.71 1.22 0.2
Fig. 2. Propagation of diffraction orders after the diffractive lens.The diameter of the diffractive lens is 4mm, and focal lengthf 1 ¼ 60mm.
plane and the spherical waves (paraxial approxima-tion):
2πm ¼ kðr −ΔzÞ ≈ kx2 þ y2
2Δz; ð5Þ
where m is an integer and the equation on the right-hand side denotes a paraxial approximation. For thebright fringe condition of the interference of twospherical waves we get
2πm ¼ kðr1 − r2Þ ≈ k
�x21 þ y212Δz1
−x22 þ y222Δz2
þΔz1 −Δz2
�:
ð6Þ
One should also notice that both of these approx-imations are second-order polynomials. In the opti-mal case, where no aberrations are present, theinterference fringes are perfect concentric rings.Aberrations, such as coma and astigmatism, modu-late the fringes so that there will be some deviationfrom the circle, e.g., astigmatism in the inspectedlens causes the circles to become elliptic.
B. Wavefront Computation
As a first step, noise reduction has to be achieved forall original interferograms. We did this by maskingoff the high frequencies in the Fourier domain in boththe x and the y directions in order to reduce thenoise originating mainly from the above-mentioned
Fig. 3. Theoretical effect of an interferogram (a) without noise and (b) with noise created by diffraction order −2. (c), (d) Their crosssections along the positive x axis. The distance between inspected lens and image plane is z ¼ 100mm in both cases. The amplitudesof the interfering fields used for images (b) and (d) were U0 ¼ 1, U−1, and U−2 ¼ 0:3 ×U0.
unwanted higher diffraction orders. The sizes of theoriginal interferograms were 540 pixels in both the xand the y directions. The discrete Fourier transformwas first applied to every column, and then the cutoffin the frequency domain was achieved by giving zerovalue to pixels referring to frequencies higher thanthe cutoff frequency. In the frequency domain, onehas symmetrically positive and negative frequencies[7], and in the masking one can choose which one touse. We used positive frequencies, and we took intoaccount between 40 and 100 pixels, depending on thefringe density in the original interferogram—the big-ger the density the more pixels one needs. The filterhas to be defined separately for each case, since thefringe density varies according to the focal length ofthe lens under inspection. Then we used an inverseFourier transform to get back to the original spacedomain. The same procedure was carried out forevery row of the original interferogram. Alterna-tively, median filtering or some other standard lowpass filtering method could be applied. Median filter-ing was tested too, but it seemed to be harder to findthe maximums after that, since median filteringtends to flatten the peaks. After that we followed
the procedure given below in order to alleviate thefringe counting and wavefront computation:
1. Sample interferograms radially from the centerand form a matrix from the results. Interferogramsconsisting of concentric fringes [Fig. 4(a)] aresampled in a radial direction from the center to ob-tain coordinate transformation to polar coordinates[Fig. 4(b)]. In the coordinate transform, we selectthe pixels along the vectors that start from the centerof the original interferogram and end at the edge pix-els of the original interferogram. The first row in thecoordinate transformation matrix [Fig. 4(b)] isformed from the original interferogram, selectingthe pixels along the vector pointing from the centerto the bottom right corner, and the next rows repre-sent the vectors ending at the next edge point in theoriginal interferogram moving counterclockwise.This means that the coordinate transform matrixhas 270 pixels in the x direction and the amountof edge pixels in the original interferogram in they direction, which is 1080 pixels. In this coordinatesystem, the angle is the same as in the normal polarcoordinate system, but the radius coordinate is notconstant, since it depends on the angle. However,
Fig. 4. (Color online) Procedure of fringe counting and wavefront computation. Interferogram in (a) (x; y) coordinates, (b) polar coordi-nates and wavefronts in (c) polar and (d) (x; y) coordinates.
this coordinate transformation is only used for fringenumbering, and it will be canceled in inversetransformation. The coordinate transformation al-lows us to enumerate fringes from left to right alongevery horizontal line.2. Using Eqs. (5) and (6) and linear fitting form a
continuous phase profile. According to Eqs. (5) and(6), every bright fringe means a 2π phase changein the wavefront. By using this condition and bymak-ing a linear fitting between the points, we get a con-tinuous phase profile in this new artificial coordinatesystem; see Fig. 4(c).3. Construct the phase profile by inversing the co-
ordinate transformation of step 1; see Fig. 4(d).
3. Results
The spherical wavefront of the diffractive lens wasfirst solved by capturing the interferogram withoutthe inspected lens in the setup; see Fig. 1. The origi-nal interferogram captured this way is presented inFig. 5(a). As can be seen, the original interferogramcontains some imaging noise, and it is also modu-lated by the higher diffraction orders.After the noise reduction, the central area of the
interferogram is still modulated by the higher dif-fraction orders. Therefore, the central part up tothe first real maximum was cut off from the interfer-ogram. After this, the procedure described inSubsection 2.A was used with Eq. (5) to solve the wa-vefront. The result is presented in Fig. 5(b).The distance from the focal point of the diffractive
lens to the image plane is denoted by h. This distancecan be evaluated from the wavefront of the diffrac-tive lens by polynomial fitting. The coefficient ofthe second-order polynomial corresponds to Eq. (5),and (Δz) in Eq. (5) is then equal to h. Fitting wasdone to the cross sections of the solved phase func-tions in the x and y directions. Distance h was solvedto be 91:2mm in the x direction and 92:0mm in the ydirection. The difference in h in the x and y directions
is due to a small tilt of the diffractive lens in the set-up. These distances were used afterward to evaluatethe focal lengths of the inspected lenses.
A. Results for Commercial Glass Lenses
The method was tested by measuring the wavefrontsof commercial BK7 lenses with given focal lengths.One of the captured interferograms is presented inFig. 6(a). The effect of the diffraction order −2 isclearly present. The computed wavefront of the dif-fractive lens and glass lens with focal length of48mm as specified by the manufacturer is presentedin Fig. 6(b).
Now the wavefront of the measured glass lens canbe revealed by summing the wavefront of the diffrac-tive lens kr2 and the wavefront kðr1 − r2Þ, ending upwith the desired kr1. This wavefront is presented inFig. 6(c).
The sharp edges in the cross sections near the cen-ter are due to cutting off the central part and linearfitting between the fringes. Focal lengths were calcu-lated from the second-order polynomial fitting, ac-cording to paraxial approximations in Eq. (6). As aresult, we found the focal length to be 49:7mm inthe x direction and 50:5mm in the y direction. Thesevalues are close to the given value of 48mm. The dif-ference in the focal length in the x and y directionsoriginates from the tilt of the diffractive lens inthe setup and is the same as in the computed dis-tance h.
The easiest way to determine how well the compu-tation of the wavefront succeeds is to compare the ori-ginal interferogram to the cosine of the solvedwavefront. According to Eqs. (3) and (4), the intensitymodulation is caused by the cosine function. One cancompare the fringepositions even though theabsolutevalues of the intensities are not comparable. By tak-ing the cosine of the computed wavefront in Fig. 6(b)we get the fringe pattern shown in Fig. 6(d). Bycomparing these figures and partial enlargements
Fig. 5. (Color online) Interferogram of the diffractive lens (a) and computed wavefront of the diffractive lens (b).
of the indicated areas in Figs. 7(a) and 7(b) we noticethe locations and orientations of the fringes are simi-lar in both figures except in the central part, wherethe cutoff was done. The original interferogram andthe computed wavefront are perfectly circularly sym-metric. This indicates that the aberrations of the glasslens are extremely small.
B. Results for Injection Molded Lenses
We also tested our analysis method with injectionmolded plastic lenses known to have severe aberra-tions. The original interferogram of one of the in-spected lenses is presented in Fig. 8(a). Noting thewavefront of the diffractive lens, the wavefront ofthe inspected lens was revealed; see Fig. 8(b). The el-liptic shape of the fringes in the original interfero-gram and in the cosine of the computed wavefrontindicates that there is at least astigmatism in thelens; see Fig. 8(c). Figure 8(c) contains unwanted ar-
tifacts because the noise reduction has not been effi-cient enough. Figures 8(a) and 8(c) have the samefringe orientation. Splitting the fringes due to higherdiffraction orders exists in the original interferogrambut not in the cosine of the computedwavefront. We noticed that discovering the properparameters in filtering is challenging, since fringedensity varies in the interferograms.
4. Conclusions
We have demonstrated a robust method for wave-front evaluation from the interferograms of a diffrac-tion-lens-based common-path interferometer. Themethod is based on fringe counting and on the useof a bright fringe condition. The wavefront of thediffractive lens is extracted from the measured wave-front, which is a sum of the impacts of the diffractiveand inspected lens, to get the desired wavefront ofthe inspected lens. Also, the computed distance from
Fig. 6. (Color online) (a) Original interferogram of the glass lens (f ¼ 48mm), (b) wavefront of the glass lens (f ¼ 48mm) [and diffractivelens kðr1 − r2Þ], (c) wavefront of the glass lens after the extraction of the wavefront of the diffractive lens, and (d) cosine of the wavefront ofthe glass lens and diffractive lens.
Fig. 7. (a) Partial enlargement of the original interferogram in Fig. 6(a) of the cosine of the wavefront in Fig. 6(d).
Fig. 8. (Color online) Original interferogram of the injection molded lens (a), wavefront of the injection molded lens (b), and the cosine ofthe wavefront of the injection molded lens and the diffractive lens (c).
the focal plane of the diffractive lens to the imageplane is used in the evaluation of the focal lengthsof the inspected lenses.We have shown that our phase retrieval method
works quite well. The noise reduction is not very sim-ple when fringe density varies, as it does in this case.Also, if several lenses with different focal lengthshave to be measured, one has to define a filteringwindow each time separately, since the fringe densitydepends on the focal lengths of the inspected lenses.This means that one has to pay special attention inselecting the filtering parameters in order to filterout the noise while saving the crucial information re-garding the fringes. One opportunity to improve theresults presented could be to use adaptive filteringmethods [9].
The authors acknowledge the Finnish FundingAgency for Technology and Innovation (TEKES),the Finnish Ministry of Education (OPM), and theNetwork of Excellence in Micro-Optics (NEMO).
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