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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 12 — Jun. 9, 2008
  • pp: 9132–9143
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Spectral synthesis using an array of micro gratings

Guangya Zhou and Fook Siong Chau  »View Author Affiliations


Optics Express, Vol. 16, Issue 12, pp. 9132-9143 (2008)
http://dx.doi.org/10.1364/OE.16.009132


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Abstract

In this paper, we present the design of a novel spectral synthesis system that is based on an array of in-plane moving micro gratings, instead of using a conventional out-of-plane micromirror array. Utilizing the unique nondispersive characteristic of the optical phase modulation mechanism based on in-plane movable micro gratings, we demonstrate that the synthetic spectra problem can be greatly simplified and effectively reduced to that of conventional phase retrieval.

© 2008 Optical Society of America

1. Introduction

Spectral synthesis [1–3

1. M. B. Sinclair, M. A. Butler, S. H. Kravitz, W. J. Zubrzycki, and A. J. Ricco, “Synthetic infrared spectra,” Opt. Lett. 22, 1036–1038 (1997). [CrossRef] [PubMed]

] using an array of micromirrors has been previously proposed for remote sensing of chemicals based on correlation spectroscopy. In this spectral synthesis system, the relative height of each micromirror in the array is accurately controlled to generate a diffractive surface relief. When illuminated with a collimated broadband light, the diffractive surface relief produces a desired intensity spectrum at a predetermined diffraction angle. This artificially-generated intensity spectrum - emulating an absorption spectrum of a reference material - is then used in correlation spectroscopy for chemical sensing applications, where it is correlated with the spectral pattern of a light beam transmitted through a sample. Using the microelectromechanical systems (MEMS) technology, it is possible to design and fabricate a single programmable micromirror array to create the spectra of a large number of materials [4–7

4. M. Lacolle, R. Belikov, H. Sagberg, O. Solgaard, and A. S. Sudbø, “Algorithms for the synthesis of complex-value spectral filters with an array of micromechanical mirrors,” Opt. Express 14, 12590–12612 (2006). [CrossRef] [PubMed]

]. As a result, the correlation spectrometer based on synthetic spectra has the potential to form a new generation of compact multi-component chemical sensing system for field use.

Fig. 1. Spectral synthesis using a programmable micromirror array.

As shown in Fig. 1, considering a broadband collimated electromagnetic wave that strikes a one-dimensional micromirror array at normal incidence, and working in the Fraunhofer approximation, it is easy to show that the diffracted field U(u) at a fixed diffraction angle θ is described by the following equation: [8]

U(u)=C(u)m=1Mexp(i4πdmu)exp[i2πmwsinθu],
(1)

where u is the wavenumber of the light, C(u) is a wavenumber dependent constant, dm is the out-of-plane displacement of the m th micromirror, w is the width of a micromirror element, and M is the total number of elements in the array. The synthetic spectra problem is equivalent to solving the above equation for micromirror displacement dm for a given desired intensity spectrum of U(u). This is a challenging problem and typically, complicated optimization algorithms have to be employed to obtain an approximate solution [2

2. M. B. Sinclair, M. A. Butler, A. J. Ricco, and S. D. Senturia, “Synthetic spectra: a tool for correlation spectroscopy,” Appl. Opt. 36, 3342–3348 (1997). [CrossRef] [PubMed]

,4

4. M. Lacolle, R. Belikov, H. Sagberg, O. Solgaard, and A. S. Sudbø, “Algorithms for the synthesis of complex-value spectral filters with an array of micromechanical mirrors,” Opt. Express 14, 12590–12612 (2006). [CrossRef] [PubMed]

,8

8. G. Zhou, F. E. H. Tay, and F. S. Chau, “Design of the diffractive optical elements for synthetic spectra,” Opt. Express 11, 1392–1399 (2003). [CrossRef] [PubMed]

]. The above-mentioned problem results from the fact that the phase modulation mechanism of a micromirror array is dispersive in nature (the second term of the right-hand side of the equation). The phase profile imposed on the incident light by the micromirror array is wavelength-dependent, thus complicating the problem. In this paper, instead of using a dispersive micromirror array, we report a novel spectral synthesis method using a micro grating array. Using the nondispersive characteristic of the optical phase modulation mechanism based on in-plane movable micro gratings, we demonstrate that the synthetic spectra problem can be greatly simplified and reduced to a conventional phase retrieval problem [9

9. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]

].

2. Spectral synthesis using micro gratings

2.1. Operation principle

Figures 2(a) and (b) show schematically the 3D and top views of the proposed synthetic spectra system using a MEMS driven in-plane movable micro grating array. A compensation grating of the same period is used in conjunction with the micro grating array to cancel out the angular dispersion of an individual grating element. As shown in Fig. 2(b), consider a collimated light beam that illuminates the compensation grating at an incident angle of θi. The grating lines are aligned along the z-axis. The incident light wave is then:

Fig. 2. (a) 3D view and (b) top view of a spectral synthesis system using a micro grating array.
Uin(u)=U0(u)exp[i2πu(sinθixcosθiy)],
(2)

where U 0(u) is the amplitude of the incident wave and u is the wavenumber. At y=0, the compensation grating bestows a spatial phase modulation exp[i2πuh(x)] on the incident light wave, which can be expanded in a Fourier series due to its periodicity along the x-axis as:

exp[i2πuh(x)]=kCk(u)exp[i2πkxp],
(3)

where h(x) is a function describing the grating surface profile, k represents the diffraction order and p is the grating period. If we select k=-1order beam and direct it to the micro grating array, the light wave diffracted from the compensation grating is then:

U1(u)=U0(u)η(u)exp{i2π[(usinθi1p)x+ucosθdy]},
(4)

U2,m(u)=U0(u)η(u)exp{i2π[(usinθi1p)x+ucosθdl]}exp[i2πuh(xdm)],
(5)

where dm is the in-plane displacement of the grating element driven by its MEMS actuator for phase modulation. Using the Fourier expansion:

exp[i2πuh(xdm)]=kCk(u)exp[i2πk(xdm)p],
(6)

and again taking k=-1, we get:

U2,m(u)=U0(u)η2(u)exp(i2πusinθix)exp(i2πucosθdl)exp(i2πdmp).
(7)

The above equation implies that parallel beams of different wavelengths incident on the first grating are still parallel when they leave the second grating. Additionally, the phase shift introduced to the output light beam by the in-plane movement of the grating element along a direction perpendicular to the grating lines is dependent only on the ratio of the in-plane displacement to the grating pitch and is independent of wavelength. The total diffracted field that just leaves the micro grating array can then be expressed by:

U2(u)=U0(u)η2(u)exp(i2πusinθix)exp(i2πucosθdl)m=MMexp(i2πdmp)rect(zmΔΔ).
(8)

where Δ is the width of the element along the z’ axis, and 2M+1 is the total number of grating elements. Again, working in the Fraunhofer approximation, the light wave diffracted to a predetermined direction specified by the direction cosines (-sinθi, cosθi, sinα) in the x’y’z’ coordinate system as shown in Fig. 2(a) is given by [11

11. J. W. Goodman, Introduction to Fourier Optics, 2nd Edition, (McGraw-Hill, New York, 1996), Chap. 4, 63–90.

]:

U3(u)=AuU2(u)exp[i2πu(sinαzsinθix)]dxdz,
(9)

where A is a constant that does not depend on wavenumber u. Substituting Eq. (8) into Eq. (9), we obtain:

U3(u)=ΔABU0(u)uη2(u)sinc(uΔsinα)exp(i2πucosθdl)
×m=MMexp(i2πdmp)exp[i2πmuΔsinα],
(10)

We define a set of discrete sampling wavenumbers un within the spectral range of interest [umin, umax]:

un=n(2M+1)Δsinα+lΔsinα,
(11)

where n=0,±1, …, ±M, and l is an integer. We further define:

ϕm=2πdmp,
(12)
U(u)=U3(u)ΔABU0(u)uη2(u)sinc(uΔsinα).
(13)

Taking the modulus and then squaring both sides of Eq. (10) and utilizing Eqs. (11), (12), and (13), we obtain:

In=U(un)2=m=MMexp(iϕm)exp(i2πnm2M+1)2=DFT{exp(iϕm)}2,
(14)

where DFT stands for the operation of discrete Fourier transform. The synthetic spectra problem is now equivalent to solving the above equation for phase modulation ϕm for a given desired intensity spectrum In or |U(un)|2. The synthetic spectra problem using an array of nondispersive phase modulators has therefore now been reduced to a conventional phase retrieval problem [12–13

12. Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Comm. 30, 304–308 (1979). [CrossRef]

] - more specifically, recovering phases from two known intensity constraints, i.e. |U(un)|2 and 1. A number of approaches have been proposed for this latter problem, including the Gerchberg-Saxton algorithm [14–15

14. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

], simulated annealing [16

16. J. Turunen, A. Vasara, and J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).

] and genetic algorithm [17

17. G. Zhou, Y. Chen, Z. Wang, and H. Song, “Genetic local search algorithm for optimization design of diffractive optical elements,” Appl. Opt. 38, 4281–4290 (1999). [CrossRef]

].

2.2. Spectral resolution and bandwidth

Fig. 3. Spectral resolution and bandwidth vs. number of grating elements and element width.

According to Eq. (11), the spectral resolution of the proposed spectral synthesis system is inversely proportional to (2M+1)Δsinα, which is actually the total length of the grating array along the z’ axis projected onto the propagation direction of the outgoing light of interest. In addition, from Eqs. (10), (11) and (13), it is clear that the amplitude spectrum |U(u)| is periodic with a period of 1/(Δsinα). Therefore, for practical design considerations of the proposed system, one can simply determine the width of the grating element in the array based on the reciprocal of the spectral bandwidth of interest, i.e.,

Δ=1(umaxumin)sinα,
(15)

and determine the total length of the array and consequently, the total number of elements, from the reciprocal of the required spectral resolution δu,

(2M+1)Δ=1δusinα.
(16)

The relationships are displayed in Fig. 3.

In a practical spectral synthesis system shown in Fig. 2, a lens together with a pinhole is employed to select the light beam diffracted from the array to the desired direction. It is noted that the use of a lens and pinhole will affect the spectral resolution of the synthesized spectrum. The effect of the lens and pinhole however may be minimized to a negligible level when a long-focal-length lens coupled with a small pinhole (for example smaller than the diffraction-limited spot size) is used.

2.3. Efficiency considerations

Using Parseval’s theorem on Eq. (14), we obtain:

n=MMU(un)2=(2M+1)2.
(17)

This equation implies that the loss of the system is proportional to the complexity of the desired spectrum. For example, if the desired spectrum contains only a single non-zero value at a certain wavenumber u 1, then the efficiency of the system can be potentially very high. This is true because we separate the input beam into multiple identical channels and recombine them to form a single output. If we configure the system such that the subdivided light beams emerging from all channels are in phase at wavenumber u 1, then the light of this particular wavenumber can be completely transmitted through the system, thereby achieving high efficiency. However, if the desired spectrum contains a set of non-zero values at different wavenumbers, then the in-phase condition has to be compromised to achieve the desired intensity spectrum. According to Eq. (17), the values of |U(un)|2 at these targeted wavenumbers will generally reduce since the total amount is fixed. As a result, at each wavenumber, the light is attenuated by the system.

Fig. 4. (a) Schematic showing grating elements in the array’s local coordinate system (b) shift of the sinc function values using an auxiliary grating profile.

exp[i2πuh(xdm)]exp[i2πuf(zmΔ)]={kCk(u)exp[i2πk(xdm)p]}
×{kDk(u)exp[i2πk(zmΔ)g]},
(18)

where h(x’) and f(z’) are the functions describing the grating surface profile. Here, we assume that the surface profiles of the grating elements are identical and the micro gratings are confined to move only along the x’ axis to introduce phase shifts. Selecting the negative and positive first-order terms from the first and second Fourier expansions respectively on the right-hand side of Eq. (18) and assuming Δ/g is an integer (each element has exact multiple full periods of the auxiliary grating profile), we find that instead of Eq. (10), the outgoing light beam from the grating array to the predetermined direction is now given by:

U3(u)=ΔABU0(u)uη2(u)ξ(u)sinc[Δ(usinα1g)]exp(i2πucosθdl)
×m=MMexp(i2πdmp)exp[i2πmuΔsinα],
(19)

where ξ(u)=D 1(u). Following the same procedure described in section 2.1, we define:

U(u)=U3(u)ΔABU0(u)uη2(u)ξ(u)sinc[Δ(usinα1g)].
(20)

We find that the main equation for spectral synthesis, ie. Eq. (14) remains unchanged. Comparing Eq. (20) with Eq. (13), it is noted that in addition to a new term ξ(u), the maximum of the sinc function in Eq. (20) is shifted. We can utilize this to minimize the loss of the system, for example, by setting the period g of the auxiliary grating to fulfill the following equation:

u0sinα=1g,
(21)

where u 0 is a chosen wavenumber, typically the center of the spectral band u 0=(umax+umin)/2, while at the same time, choosing a blazed profile at wavenumber u 0 for the auxiliary grating so as to maximize the value of ξ(u). From Eq. (21), we see that the function of the auxiliary grating profile is to diffract the light of wavenumber u 0 with high efficiency from normal incidence to the predetermined angle α. The increase in the sinc function values using an auxiliary grating profile is illustrated in Fig. 4(b).

In conclusion, in order to achieve high efficiency, the first compensation grating must be blazed at a certain wavenumber within the spectral range of interest, usually at the center of the band. In addition, each grating element in the second phase-shifting array must be a 2D crossed-grating that can be decomposed into two orthogonal grating profiles with one identical to that of the compensation grating and the other blazed at a selected wavenumber (for example the center of the spectral band) to the predetermined direction.

2.4. Design method

We have shown in the above sections that the synthetic spectra problem using an array of MEMS-driven in-plane-moving gratings is reduced to a conventional phase retrieval problem. In this section, we will develop a gradient research method to optimize the design of the proposed spectral synthesis system.

Based on the system requirements including the spectral range of interest, spectral resolution, and diffraction angle α, we can determine the width of each grating element as well as the total number of required elements in the array using Eqs. (15) and (16). The period of the primary grating for phase shifting can be determined by considering the maximum stroke of the driving MEMS microactuator. The period of the auxiliary grating is determined by the required diffraction angle α and a chosen wavenumber. Next, the detailed grating profiles and consequently, the efficiency terms η(u) and ξ(u), can be obtained by properly selecting blazing wavenumbers within the spectral range of interest. Based on the desired intensity spectrum of the outgoing light wave |U 3 d(u)|2, and using Eq. (20), we can then determine the targeted intensity constrain Idn=|Ud(un)|2 for the Fourier transform Eq. (14).

To evaluate the system performance, we define an error function to describe the difference between the calculated intensity spectrum In (from the phase shifts ϕm using DFT) and the desired intensity Idn:

E=n=MM(IndγIn)2,
(22)

where the parameter γ is a scaling factor that scales In for a minimum E fit to Idn. It can be derived by setting the partial derivative of E with respect to γ equal to 0, resulting in:

γ=n=MMInIndn=MMIn2.
(23)

After the error function is defined, the phase retrieval problem is equivalent to the following optimization problem:

minϕm{E(ϕm)}and0ϕm<2π.
(24)

Using the method reported in Ref. [8

8. G. Zhou, F. E. H. Tay, and F. S. Chau, “Design of the diffractive optical elements for synthetic spectra,” Opt. Express 11, 1392–1399 (2003). [CrossRef] [PubMed]

], we find:

dEdϕm=2Im[exp(iϕm)·k=MM2γ(γIkIkd)U(uk)exp(i2πkm2M+1)],
(25)

Fig. 5. Block diagram showing the proposed gradient search method for spectral synthesis.

2.5. Alternative setup

Fig. 6. Alternative spectral synthesis system using in-plane moving gratings.

Alternatively, the synthetic spectra using micro gratings can also be configured as shown in Fig. 6. Two lenses in a 4-f configuration are inserted between the parallely aligned compensation grating and micro grating array. The 4-f setup reverses the sign of the angles so that the beams will hit the second grating at the same angle as they leave the first one. This can also effectively cancel out the angular dispersion of the MEMS grating elements.

3. Design examples and results

Here, we will investigate designs reported in Ref. [8

8. G. Zhou, F. E. H. Tay, and F. S. Chau, “Design of the diffractive optical elements for synthetic spectra,” Opt. Express 11, 1392–1399 (2003). [CrossRef] [PubMed]

]. Our spectral range of interest is from 3600 cm-1 to 4200 cm-1. The spectrum of the incident light beam is assumed to be uniform over the spectral range of interest. First, we set our spectral resolution to be around 2.3 cm-1, which corresponds to about 256 points in the spectral range of interest. In this work, instead of using 2048 micromirrors, we use only 256 MEMS-driven in-plane moving micro-grating elements. The design goal is to determine the phase shift or equivalently, the in-plane displacement of each grating element in the array so that the diffraction intensity spectrum of the light that passes through the spectral synthesis system shown in Fig. 2 and diffracts to a predetermined direction (α=15°) is identical to a desired intensity spectrum. Based on the spectral range of interest and diffraction angle α, we determine the width of each grating element to be 60 µm. The total length of the array is hence around 15.4 mm. The periods of the primary and auxiliary gratings are both set at 10 µm. All gratings are blazed at the center wavenumber u 0 of the spectral band of interest, i.e. around 3922 cm-1. Under thin grating assumption, the efficiency terms in Eq. (20) are approximately given by: η 2(u)=ξ 2(u)=sinc2(1-u/u 0) [21

21. B. Kress and P. Meyrueis, Digital diffractive optics: an introduction to planar diffractive optics and related technology, (John Wiley & Sons. Ltd, New York, 2000), Chap. 1, 35–42.

].

In the first design example, the targeted intensity spectrum is shown in Fig. 7(a). The design started from a randomly-chosen initial point and terminated when the error defined by Eq. (22) converged or a predetermined number of iterations is reached. The design took only a few seconds on an ordinary PC, and the results are shown in Fig. 7(b). To investigate the efficiency of the spectral synthesis system, the following parameter ζ is defined:

ζ=I(up)I0(up)×100%,
(26)

where I(up) is the intensity of the light for a specific wave number up (typically the wave number corresponding to the peak of the target spectrum) passes through the grating-array-based spectral synthesis system and diffracts to the predetermined direction, and I 0(up) is the intensity of the same incident light that passes through a double-mirror system and propagates to a direction with 00 diffraction angle. I 0(up) can be estimated using Eq. (10) by setting η(u) to 1, diffraction angle α to 0, and at the same time all displacements dm to 0. The value of ζ in this case is about 1.1%.

Fig. 7. (a) Targeted intensity spectrum of the first design example (b) calculated intensity spectrum of the outgoing light wave from the optimized spectral synthesis system.

The target intensity spectrum of the second design example is shown in Fig. 8(a). It consisted of three extremely sharp lines located at 3800, 3900, and 4000 cm-1. We used the same spectral synthesis system to produce this spectrum and again, the design again took only a few seconds to be completed. The results shown in Fig. 8(b) give a ζ value of 33.5%. From these two design examples, we noted that, as compared with the conventional micromirror-based spectral synthesis systems, the proposed system using an array of in-plane moving gratings leads to improved results with fewer elements needed in the array. In addition, with reduced memory requirements and shorter computational time, the optimization algorithm can easily handle a large number of sampling points within the spectral range of interest, hence making high-resolution spectral synthesis possible.

To demonstrate this, we re-investigated the second design. We now set the total number of sampling points in the spectral range of interest to be 1024 instead of 256. Since the spectral range of interest and diffraction angle α are kept the same, the system parameters including the width of grating element, grating periods, and blazing wavelength remain unchanged. We simply increased the number of grating elements to 1024, resulting in a spectral resolution of about 0.6 cm-1. The optimization process took less than a minute and results are shown in Fig. 9. The value of ζ obtained in this case is around 40%.

Fig. 8. (a) Targeted intensity spectrum of the second design example (b) calculated intensity spectrum of the outgoing light wave from the optimized spectral synthesis system.
Fig. 9. Calculated intensity spectrum of the outgoing light wave from the optimized spectral synthesis system with 1024 grating elements for the desired spectrum shown in Fig. 8(a).

4. Conclusion

In this paper, we have presented a novel design for a spectral synthesis system using an array of in-plane moving micro gratings wherein the phase modulation mechanism is nondispersive. Hence, it greatly facilitated the solution of the synthetic spectra problem. The proposed system has been shown theoretically to perform well and has high potential of being used in many applications including correlation spectroscopy and optical wavelength filtering.

Acknowledgments

Financial support by the Ministry of Education’s AcRF Tier 1 funding under Grant No. R-265-000-235-112 and R-265-000-211-112/133 is gratefully acknowledged.

References and links

1.

M. B. Sinclair, M. A. Butler, S. H. Kravitz, W. J. Zubrzycki, and A. J. Ricco, “Synthetic infrared spectra,” Opt. Lett. 22, 1036–1038 (1997). [CrossRef] [PubMed]

2.

M. B. Sinclair, M. A. Butler, A. J. Ricco, and S. D. Senturia, “Synthetic spectra: a tool for correlation spectroscopy,” Appl. Opt. 36, 3342–3348 (1997). [CrossRef] [PubMed]

3.

T. A. Leskova and A. A. Maradudin, “Synthetic spectra from rough-surface scattering,” Opt. Lett. 30, 2784–2786 (2005). [CrossRef] [PubMed]

4.

M. Lacolle, R. Belikov, H. Sagberg, O. Solgaard, and A. S. Sudbø, “Algorithms for the synthesis of complex-value spectral filters with an array of micromechanical mirrors,” Opt. Express 14, 12590–12612 (2006). [CrossRef] [PubMed]

5.

H. Sagberg, M. Lacolle, I. R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard, and A. S. Sudbø, “Micromechanical gratings for visible and near-infrared spectroscopy,” IEEE J. Sel. Top. Quantum. Electron. 10, 604–613 (2004). [CrossRef]

6.

M. Lacolle, H. Sagberg, I. R. Johansen, O. Løvhaugen, O. Solgaard, and A. S. Sudbø, “Reconfigurable near-infrared optical filter with a micromechanical diffractive Fresnel lens,” IEEE Photon. Technol. Lett. 17, 2622–2624 (2005). [CrossRef]

7.

R. Belikov and O. Solgaard, “Optical wavelength filtering by diffraction from a surface relief,” Opt. Lett. 28, 447–449 (2003). [CrossRef] [PubMed]

8.

G. Zhou, F. E. H. Tay, and F. S. Chau, “Design of the diffractive optical elements for synthetic spectra,” Opt. Express 11, 1392–1399 (2003). [CrossRef] [PubMed]

9.

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]

10.

G. Zhou and F. S. Chau, “Nondispersive optical phase shifter array using microelectromechanical systems based gratings,” Opt. Express 15, 10958–10963 (2007). [CrossRef] [PubMed]

11.

J. W. Goodman, Introduction to Fourier Optics, 2nd Edition, (McGraw-Hill, New York, 1996), Chap. 4, 63–90.

12.

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Comm. 30, 304–308 (1979). [CrossRef]

13.

M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22, 907–909 (1997). [CrossRef] [PubMed]

14.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

15.

F. Wyrowski and O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography” J. Opt. Soc. Am. A5, 1058–1065 (1988). [CrossRef]

16.

J. Turunen, A. Vasara, and J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).

17.

G. Zhou, Y. Chen, Z. Wang, and H. Song, “Genetic local search algorithm for optimization design of diffractive optical elements,” Appl. Opt. 38, 4281–4290 (1999). [CrossRef]

18.

T. J. Suleski and D. C. O’Shea, “Gray-scale masks for diffractive-optics fabrication: I. Commercial slide imagers,” Appl. Opt. 34, 7507–7517 (1995). [CrossRef] [PubMed]

19.

G. J. Swanson and W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).

20.

M. A. Wolfe, Numerical Methods for Unconstrained Optimization: an Introduction, (Van Nostrand Reinhold Company Ltd., New York, 1978), Chap. 6, 161–167.

21.

B. Kress and P. Meyrueis, Digital diffractive optics: an introduction to planar diffractive optics and related technology, (John Wiley & Sons. Ltd, New York, 2000), Chap. 1, 35–42.

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(050.1970) Diffraction and gratings : Diffractive optics
(230.3990) Optical devices : Micro-optical devices

ToC Category:
Diffraction and Gratings

History
Original Manuscript: January 23, 2008
Revised Manuscript: April 4, 2008
Manuscript Accepted: April 10, 2008
Published: June 5, 2008

Citation
Guangya Zhou and Fook Siong Chau, "Spectral synthesis using an array of micro gratings," Opt. Express 16, 9132-9143 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-9132


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References

  1. M. B. Sinclair, M. A. Butler, S. H. Kravitz, W. J. Zubrzycki, and A. J. Ricco, "Synthetic infrared spectra," Opt. Lett. 22, 1036-1038 (1997). [CrossRef] [PubMed]
  2. M. B. Sinclair, M. A. Butler, A. J. Ricco, and S. D. Senturia, "Synthetic spectra: a tool for correlation spectroscopy," Appl. Opt. 36, 3342-3348 (1997). [CrossRef] [PubMed]
  3. T. A. Leskova and A. A. Maradudin, "Synthetic spectra from rough-surface scattering," Opt. Lett. 30, 2784-2786 (2005). [CrossRef] [PubMed]
  4. M. Lacolle, R. Belikov, H. Sagberg, O. Solgaard, and A. S. Sudbø, "Algorithms for the synthesis of complex-value spectral filters with an array of micromechanical mirrors," Opt. Express 14, 12590-12612 (2006). [CrossRef] [PubMed]
  5. H. Sagberg, M. Lacolle, I. R. Johansen, O. Løvhaugen, R. Belikov, O. Solgaard, and A. S. Sudbø, "Micromechanical gratings for visible and near-infrared spectroscopy," IEEE J. Sel. Top. Quantum. Electron. 10, 604-613 (2004). [CrossRef]
  6. M. Lacolle, H. Sagberg, I. R. Johansen, O. Løvhaugen, O. Solgaard, and A. S. Sudbø, "Reconfigurable near-infrared optical filter with a micromechanical diffractive Fresnel lens," IEEE Photon. Technol. Lett. 17, 2622-2624 (2005). [CrossRef]
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