## Novel microwave photonic fractional Hilbert transformer using a ring resonator-based optical all-pass filter |

Optics Express, Vol. 20, Issue 24, pp. 26499-26510 (2012)

http://dx.doi.org/10.1364/OE.20.026499

Acrobat PDF (2444 KB)

### Abstract

We propose and demonstrate a novel wideband microwave photonic fractional Hilbert transformer
implemented using a ring resonator-based optical all-pass filter. The full programmability of
the ring resonator allows variable and arbitrary fractional order of the Hilbert transformer.
The performance analysis in both frequency and time domain validates that the proposed
implementation provides a good approximation to an ideal fractional Hilbert transformer. This
is also experimentally verified by an electrical S_{21} response characterization
performed on a waveguide realization of a ring resonator. The waveguide-based structure allows
the proposed Hilbert transformer to be integrated together with other building blocks on a
photonic integrated circuit to create various system-level functionalities for on-chip
microwave photonic signal processors. As an example, a circuit consisting of a splitter and a
ring resonator has been realized which can perform on-chip phase control of microwave signals
generated by means of optical heterodyning, and simultaneous generation of in-phase and
quadrature microwave signals for a wide frequency range. For these functionalities, this simple
and on-chip solution is considered to be practical, particularly when operating together with a
dual-frequency laser. To our best knowledge, this is the first-time on-chip demonstration where
ring resonators are employed to perform phase control functionalities for optical generation of
microwave signals by means of optical heterodyning.

© 2012 OSA

## 1. Introduction

1. J. Capmany and D. Novak,
“Microwave photonics combines two worlds,” Nat.
Photonics **1**(6), 319–330
(2007). [CrossRef]

2. J. Yao,
“Microwave photonics,” J. Lightwave Technol. **27**(3), 314–335
(2009). [CrossRef]

3. F. Liu, T. Wang, L. Qiang, T. Ye, Z. Zhang, M. Qiu, and Y. Su,
“Compact optical temporal differentiator based on silicon microring
resonator,” Opt. Express **16**(20), 15880–15886
(2008). [CrossRef] [PubMed]

9. M. Burla, D.
A.
I. Marpaung, L. Zhuang, C.
G. H. Roeloffzen, M.
R. Khan, A. Leinse, M. Hoekman, and R.
G. Heideman, “On-chip CMOS compatible
reconfigurable optical delay line with separate carrier tuning for microwave photonic signal
processing,” Opt. Express **19**(22), 21475–21484
(2011). [CrossRef] [PubMed]

11. R. Ashrafi and J. Azaña,
“Terahertz bandwidth all-optical Hilbert transformers based on long-period
gratings,” Opt. Lett. **37**(13), 2604–2606
(2012). [CrossRef] [PubMed]

14. M. Li and J. Yao,
“Experimental demonstration of a wideband photonic temporal Hilbert transformer
based on a single fiber bragg grating,” Photon. Technol.
Letters **22**(21), 1559–1561
(2010). [CrossRef]

15. H. Emami, N. Sarkhosh, L.
A. Bui, and A. Mitchell,
“Wideband RF photonic in-phase and quadrature-phase
generation,” Opt. Lett. **33**(2), 98–100
(2008). [CrossRef] [PubMed]

17. T. X.
H. Huang, X. Yi, and R.
A. Minasian, “Microwave photonic quadrature filter
based on an all-optical programmable Hilbert transformer,” Opt.
Lett. **36**(22), 4440–4442
(2011). [CrossRef] [PubMed]

18. N.
Q. Ngo, Y. Song, and B. Lin,
“Design of Hilbert transformers with tunable THz bandwidths using a
reconfigurable integrated optical FIR filter,” Opt. Commun. **284**(3), 787–794
(2011). [CrossRef]

19. A.
W. Lohmann, D. Mendlovic, and Z. Zalevsky,
“Fractional Hilbert transform,” Opt. Lett. **21**(4), 281–283
(1996). [CrossRef] [PubMed]

_{21}response characterization performed on a waveguide realization of the proposed FHT. Section 4 demonstrates two system-level functionalities achieved using the FHT, namely on-chip phase control of microwave signals generated by means of optical heterodyning, and simultaneous generation of in-phase and quadrature microwave signals. The conclusions of this paper are formulated in Section 5.

## 2. Principle and performance analysis

### 2.1 Implementation of an FHT using waveguide-based optical filters

*t*) is the temporal complex envelop of an input optical signal

_{o}(

*t*) =

*t*)·exp(j2π

*f*

_{o}

*t*) to a conventional Hilbert transformer (HT) (

*f*

_{o}is the frequency of the optical carrier), the corresponding envelop

*t*) of the output signal

_{o}(

*t*) =

*t*)·exp(j2π

*f*

_{o}

*t*) can be expressed bywhere P.V. means principle value [10]. Thus, the HT can be implemented using a linear optical filter giving an impulse response

_{HT,o}(

*t*) =

*h*

_{HT}(

*t*)·exp(j2π

*f*

_{o}

*t*) with an envelope

*h*

_{HT}(

*t*) = 1/(π

*t*) and a corresponding baseband frequency response

*H*

_{HT}(

*f*) = -jsgn(

*f*), where

*f*is the frequency variable. When a conventional HT is generalized into an FHT [19

19. A.
W. Lohmann, D. Mendlovic, and Z. Zalevsky,
“Fractional Hilbert transform,” Opt. Lett. **21**(4), 281–283
(1996). [CrossRef] [PubMed]

*f·T*represents the angular frequency normalized to the defined unit delay

*T*of the filter,

*ϕ*=

*ρ·*π/2 and

*ρ*indicates the fractional order. Equation (2) signifies that an FHT is an ideal phase shifter to the input signals. A FHT becomes the conventional HT when

*ρ =*1 (

*H*

_{FHT}(

*ρ =*0 (

*H*

_{FHT}(

*H*

_{FHT}(

*ρ*, since

*H*

_{FHT}(

*=*

_{ρ}*H*

_{FHT}(

_{ρ}_{± 4}holds. Take the inverse Fourier transform of

*H*

_{FHT}(

*h*

_{FHT}(

*nT*) is zero when

*n*is a nonzero even integer, and its negative taps have opposite-valued coefficients to the corresponding positive taps, forming an anti-symmetric profile with respect to

*n*= 0. An illustration of the frequency and impulse response of an ideal FHT with

*ρ =*1 is depicted in Fig. 1 . Apparently, a theoretical FHT is not realizable using FIR and IIR filter implementations, since

*h*

_{FHT}(

*nT*) in Eq. (3) is non-causal and has critical taps in negative time. To solve this problem, one can approximate the modified FHT specification

*H*

_{FHT,mod}(

*H*

_{FHT}(

*e*

^{-}^{j}

*with*

^{n0Ω}*n*

_{0}being a prescribed delay (normalized to the defined unit delay). This is expressed as below:As illustrated in Fig. 1, the additional delay shifts the impulse response such that the critical taps have their presence in the positive time and truncation can be applied to obtain causality.

*G*(

*H*

_{FHT,mod}(

*G*(

_{FHT}(

*e*

^{-}^{j}

*.*

^{n0Ω}*H*

_{FHT,mod}(

*z*-transform (

*z*= exp(-j

*N*

_{th}order all-pass filter can be described bywhere

*a*

_{1},…,

*a*are the filter coefficients. Then, the goal is to optimize those filter coefficients such that the frequency response of the filter

_{N}*G*(

_{a,N}*H*

_{FHT,mod}(

21. M. Lang and T.
I. Laakso, “Simple and robust method for the
design of allpass filters using least squares phase error criterion,”
IEEE Trans. Circuits Syst. II **41**(1), 40–48
(1994). [CrossRef]

### 2.2 Optical ring resonator

*z*-transform of an ORR is described bywhere

*c*= (1 -

*κ*)

^{1/2}with

*κ*being the power coupling coefficient,

*r*indicates the roundtrip amplitude transmission coefficient which in practice is determined by the roundtrip loss, and

*θ*represents an additional roundtrip phase shift which can be introduced intentionally to achieve a shift of the resonance frequency of the ORR [20]. This signifies that an ideal lossless ORR is a first-order all-pass filter, and a cascade of

*N*ORRs can therefore be used to implement

*G*(

_{a,N}*z*) in Eq. (6).

### 2.3 Frequency-domain analysis

*θ*is neglected for simplicity. As an all-pass filter, these frequency responses are able to approximate to the modified FHT specification

*H*

_{FHT,mod}(

*r =*0.97 (for a typical roundtrip loss of 0.12 dB) is used for the calculation. Evidently, by varying the power coupling coefficient

*κ,*different values of the fractional order

*ρ*are achieved. In principle, the phase response of an ORR has a constant span of 2π across one frequency period, or in other words one free spectral range (FSR) [20]. Using this as a condition for Eq. (4), a relation

*n*

_{0}= 1-

*ρ*/2 results, which is a property of the proposed FHT and can be used to determine the additional delay

*n*

_{0}for a given value of

*ρ.*

*ρ*approaches 2, corresponding to a high-Q state of the ORR. This will result in strong optical carrier suppression, making direct optical detection inapplicable. To solve this problem, one possible approach is to apply coherent optical detection instead, using an optical carrier reinsertion circuitry as explained in [6

6. L. Zhuang, C.
G.
H. Roeloffzen, A. Meijerink, M. Burla, D.
A.
I. Marpaung, A. Leinse, M. Hoekman, R.
G. Heideman, and W. van
Etten, “Novel
ring resonator-based integrated photonic beamformer for broadband phased-array antennas-Part
II: experimental prototype,” J. Lightwave Technol. **28**(1), 19–31
(2010). [CrossRef]

*N*ORRs can be described bywhere \

*H*

_{C,}

*(*

_{N}

^{N}_{n = 1}\

*H*

_{ORR,}

*(*

_{n}*Ψ*

_{C,}

*(*

_{N}

^{N}_{n = 1}

*Ψ*

_{ORR,}

*(*

_{n}*ρ*can be obtained by letting each ORR in the cascade provide only a fraction of

*ρ,*such that the high-Q state of ORR is avoided. Consequently, a significant reduction of the optical carrier suppression will be achieved. To demonstrate this, the frequency responses of a cascade of two ORRs and those of a single ORR are compared in Fig. 5 for

*ρ =*2 and

*r*= 0.97.

### 2.4 Time-domain analysis

## 3. Device characterization

^{TM}waveguide technology, a proprietary technology of LioniX B.V [22

22. R.
G. Heideman, A. Leinse, W. Hoving, R. Dekker, D. Geuzebroek, E. Klein, R. Stoffer, C.
G.
H. Roeloffzen, L. Zhuang, and A. Meijerink,
“Large-scale integrated optics using TriPleX waveguide technology:from UV to
IR,” Proc. SPIE **7221**, 72210R, 72210R-15
(2009). [CrossRef]

23. L. Zhuang, D.
A.
I. Marpaung, M. Burla, W.
P. Beeker, A. Leinse, and C.
G. H. Roeloffzen, “Low-loss, high-index-contrast
Si₃N₄/SiO₂ optical waveguides for optical delay lines in microwave photonics signal
processing,” Opt. Express **19**(23), 23162–23170
(2011). [CrossRef] [PubMed]

_{3}N

_{4}with a thickness of 170 nm, forming an “=” shape in its cross-section. Between the two strips is an intermediate layer of SiO

_{2}with a thickness of 500 nm, which is also the material of surrounding cladding. The use of this double-strip geometry increases the effective index of the optical mode as compared to a single strip geometry, thus increasing the confinement of the mode and thereby reducing the bend loss. The width of the strips is optimized to result in a high effective index of the mode while the waveguide only supports a single (TE) mode at a wavelength around 1550 nm. This waveguide is considered to be an enabling platform technology for the realization of low loss, compact on-chip MWP signal processors. The ORRs used for the experiments of this paper have the layout as depicted in Fig. 2(a). The roundtrip length and the FSR are 13 mm and 15 GHz, respectively. In both the ring path and the lower arm of the MZI, an optical phase shifter is implemented using a resistor-based heater placed on top of a length of the waveguide. Therefore, the ORRs are programmable with tunability in both the resonance frequency and the power coupling coefficient by means of the thermo-optical tuning mechanism.

23. L. Zhuang, D.
A.
I. Marpaung, M. Burla, W.
P. Beeker, A. Leinse, and C.
G. H. Roeloffzen, “Low-loss, high-index-contrast
Si₃N₄/SiO₂ optical waveguides for optical delay lines in microwave photonics signal
processing,” Opt. Express **19**(23), 23162–23170
(2011). [CrossRef] [PubMed]

_{21}response to the state where the ORR is decoupled (

*κ*= 0). This removes the effects of the other components in the setup from the measurement results. As expected, the S

_{21}measurements of the ORR are in good agreement with the calculations in Fig. 3, approximating flat magnitude responses and linear phase responses in the bandwidth of interest. In Fig. 7(a) , the measured phase responses for different values of the power coupling coefficient

*κ*are depicted and are fitted to the design target specifications described in Eq. (4). In Fig. 7(b), the resulting FHT phase shifts are demonstrated by means of removing the delay effect (linear phase) from the measured phase responses as explained in Eq. (5). Evidently, a wideband FHT is achieved, which is characterized by an operation bandwidth from 3 GHz to 12 GHz and a maximum phase ripple of 5 degree.

## 4. System-level functionality demonstration

### 4.1 Functionality description

2. J. Yao,
“Microwave photonics,” J. Lightwave Technol. **27**(3), 314–335
(2009). [CrossRef]

*Ψ*

_{RF}(

*f*

_{RF}) is equal to the phase difference between the two beating optical signals Δ

*Ψ*

_{o}(

*f*

_{1},

*f*

_{2}) with

*f*

_{1}and

*f*

_{2}denoting the frequencies of the two beating optical signals. Here, the proposed FHT can be employed before the detector to introduce an additional phase difference Δ

*Ψ*

_{ORR}(

*f*

_{1},

*f*

_{2}) in Δ

*Ψ*

_{o}(

*f*

_{1},

*f*

_{2}), which serves as a simple and on-chip solution to provide the phase control functionality on Δ

*Ψ*

_{o}(

*f*

_{1},

*f*

_{2}), or equivalently, the phase control functionality on

*Ψ*

_{RF}(

*f*

_{RF}), since

*Ψ*

_{RF}(

*f*

_{RF}) = Δ

*Ψ*

_{o}(

*f*

_{1},

*f*

_{2}).

*ρ*of the FHT (implemented by tuning the power coupling coefficient

*κ*of the ORR), Δ

*Ψ*

_{ORR}(

*f*

_{1},

*f*

_{2}) is varied accordingly; and by shifting the FHT phase response to different operation positions (implemented by tuning the additional roundtrip phase shift

*θ*of the ORR), a full 2π changing range of Δ

*Ψ*

_{ORR}(

*f*

_{1},

*f*

_{2}) can be achieved. However, to allow the variability of Δ

*Ψ*

_{ORR}(

*f*

_{1},

*f*

_{2}), the frequency period of the FHT, or equivalently, the FSR of the ORR Δ

*f*

_{FSR}must fulfill the condition

*n·*Δ

*f*

_{FSR}≠Δ

*f*

_{o}=

*f*

_{2}−

*f*

_{1}=

*f*

_{RF}with

*n*being an positive integer. The reason for this is that when

*n*·Δ

*f*

_{FSR}= Δ

*f*

_{o}, Δ

*Ψ*

_{ORR}(

*f*

_{1},

*f*

_{2}) becomes a constant Δ

*Ψ*

_{ORR}(

*f*

_{1},

*f*

_{2}) = 2πΔ

*f*

_{o}/Δ

*f*

_{FSR}=

*n*·2π which is independent of

*κ*and

*θ*of the ORR. In this case, the FHT loses its capability to modify Δ

*Ψ*

_{ORR}(

*f*

_{1},

*f*

_{2}) and therefore is unable to perform the phase control functionality. Moreover, it is preferable that the Δ

*f*

_{FSR}of the ORR is such that both beating optical signals can be accommodated in the FHT operation bandwidth. This way, the possible strong amplitude suppression around the resonance frequency of the ORR can be avoided.

### 4.2 Experimental demonstration

*f*

_{1}and

*f*

_{2}, microwave signals with frequencies from 7 GHz to 10 GHz were generated. For the phase measurement, an oscilloscope (Agilent Infiniium 54854A) was employed which has a bandwidth limit of 4 GHz. Due to this bandwidth limit, electrical frequency downconversion was performed before the phase measurement. Here, to guarantee accurate measurement results, a phase-calibrated LO signal was used for the signal mixing.

*Ψ*

_{RF}(

*f*

_{RF}) was varied using the principle illustrated in Fig. 8. Figure 10(a) depicts the generated microwave signal at output 2, where output 1 was used as the phase reference. Further, based on the same principle, the simultaneous generation of in-phase and quadrature microwave signals using both outputs of the splitter circuit was demonstrated. Figure 10(b) depicts the generated microwave signals of both I and Q channels, where a constant quadrature phase relation is achieved for the signal frequency varying from 7 GHz to 10 GHz. To achieve this functionality, the delay effects (linear phase) of the I and Q channel need to be equalized as illustrated in the inset of Fig. 9, so that the phase difference between the two outputs Δ

*Ψ*

_{RF,I-Q}is only determined by the FHT phase shift

*ϕ*of the ORR with the relation Δ

*Ψ*

_{RF,I-Q}= 2

*ϕ*. In this demonstration, the power coupling coefficient

*κ*of the ORR was set to 0.52 corresponding to a FHT phase shift

*ϕ*= −135° (see Fig. 7(b)). This results in the desired quadrature phase relation Δ

*Ψ*

_{RF,I-Q}= −270°, or equivalently, Δ

*Ψ*

_{RF,I-Q}= 90°. In principle, the frequency periodicity of the ORR allows this quadrature phase relation to be available for multiple ranges of frequencies. However, constrained by the speed limitation of the photodetectors in this setup, this functionality was only demonstrated for a maximum frequency of 10 GHz.

## 5. Conclusions

## Acknowledgment

## References and links

1. | J. Capmany and D. Novak,
“Microwave photonics combines two worlds,” Nat.
Photonics |

2. | J. Yao,
“Microwave photonics,” J. Lightwave Technol. |

3. | F. Liu, T. Wang, L. Qiang, T. Ye, Z. Zhang, M. Qiu, and Y. Su,
“Compact optical temporal differentiator based on silicon microring
resonator,” Opt. Express |

4. | M. Ferrera, Y. Park, L. Razzari, B.
E. Little, S.
T. Chu, R. Morandotti, D.
J. Moss, and J. Azaña,
“On-chip CMOS-compatible all-optical integrator,” Nat.
Commun. |

5. | A. Meijerink, C.
G.
H. Roeloffzen, R. Meijerink, D. A.
I. Leimeng
Zhuang, M.
J. Marpaung, M. Bentum, J. Burla, P. Verpoorte, A. Jorna, Hulzinga, and W. van
Etten, “Novel
ring resonator-based integrated photonic beamformer for broadband phased-array antennas-Part
I: design and performance analysis,” J. Lightwave Technol. |

6. | L. Zhuang, C.
G.
H. Roeloffzen, A. Meijerink, M. Burla, D.
A.
I. Marpaung, A. Leinse, M. Hoekman, R.
G. Heideman, and W. van
Etten, “Novel
ring resonator-based integrated photonic beamformer for broadband phased-array antennas-Part
II: experimental prototype,” J. Lightwave Technol. |

7. | D. A.
I. Marpaung, C. G.
H. Roeloffzen, A. Leinse, and M. Hoekman,
“A photonic chip based frequency discriminator for a high performance microwave
photonic link,” Opt. Express |

8. | D. A.
I. Marpaung, L. Chevalier, M. Burla, and C.
G. H. Roeloffzen, “Impulse radio ultrawideband pulse
shaper based on a programmable photonic chip frequency discriminator,”
Opt. Express |

9. | M. Burla, D.
A.
I. Marpaung, L. Zhuang, C.
G. H. Roeloffzen, M.
R. Khan, A. Leinse, M. Hoekman, and R.
G. Heideman, “On-chip CMOS compatible
reconfigurable optical delay line with separate carrier tuning for microwave photonic signal
processing,” Opt. Express |

10. | S. L. Hahn, |

11. | R. Ashrafi and J. Azaña,
“Terahertz bandwidth all-optical Hilbert transformers based on long-period
gratings,” Opt. Lett. |

12. | M.
H. Asghari and J. Azaña,
“All-optical hilbert transformer based on a single phase-shifted fiber bragg
grating: design and analysis,” Opt. Lett. |

13. | M. Li and J. Yao,
“All-fiber temporal photonic fractional hilbert transformer based on a directly
designed fiber bragg grating,” Opt. Lett. |

14. | M. Li and J. Yao,
“Experimental demonstration of a wideband photonic temporal Hilbert transformer
based on a single fiber bragg grating,” Photon. Technol.
Letters |

15. | H. Emami, N. Sarkhosh, L.
A. Bui, and A. Mitchell,
“Wideband RF photonic in-phase and quadrature-phase
generation,” Opt. Lett. |

16. | Z. Li, Y. Han, H. Chi, X. Zhang, and J. Yao,
“A continously tunable microwave fractional hilbert transformer based on a
nonuniformly spaced photonic microwave delay-line filter,” J.
Lightwave Technol. |

17. | T. X.
H. Huang, X. Yi, and R.
A. Minasian, “Microwave photonic quadrature filter
based on an all-optical programmable Hilbert transformer,” Opt.
Lett. |

18. | N.
Q. Ngo, Y. Song, and B. Lin,
“Design of Hilbert transformers with tunable THz bandwidths using a
reconfigurable integrated optical FIR filter,” Opt. Commun. |

19. | A.
W. Lohmann, D. Mendlovic, and Z. Zalevsky,
“Fractional Hilbert transform,” Opt. Lett. |

20. | C. K. Madsen and J. H. Zhao, |

21. | M. Lang and T.
I. Laakso, “Simple and robust method for the
design of allpass filters using least squares phase error criterion,”
IEEE Trans. Circuits Syst. II |

22. | R.
G. Heideman, A. Leinse, W. Hoving, R. Dekker, D. Geuzebroek, E. Klein, R. Stoffer, C.
G.
H. Roeloffzen, L. Zhuang, and A. Meijerink,
“Large-scale integrated optics using TriPleX waveguide technology:from UV to
IR,” Proc. SPIE |

23. | L. Zhuang, D.
A.
I. Marpaung, M. Burla, W.
P. Beeker, A. Leinse, and C.
G. H. Roeloffzen, “Low-loss, high-index-contrast
Si₃N₄/SiO₂ optical waveguides for optical delay lines in microwave photonics signal
processing,” Opt. Express |

**OCIS Codes**

(060.2360) Fiber optics and optical communications : Fiber optics links and subsystems

(070.6020) Fourier optics and signal processing : Continuous optical signal processing

(130.3120) Integrated optics : Integrated optics devices

(350.4010) Other areas of optics : Microwaves

(060.5625) Fiber optics and optical communications : Radio frequency photonics

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: August 1, 2012

Revised Manuscript: October 26, 2012

Manuscript Accepted: November 1, 2012

Published: November 9, 2012

**Citation**

Leimeng Zhuang, Muhammad Rezaul Khan, Willem Beeker, Arne Leinse, René Heideman, and Chris Roeloffzen, "Novel microwave photonic fractional Hilbert transformer using a ring resonator-based optical all-pass filter," Opt. Express **20**, 26499-26510 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-26499

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### References

- J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007). [CrossRef]
- J. Yao, “Microwave photonics,” J. Lightwave Technol. 27(3), 314–335 (2009). [CrossRef]
- F. Liu, T. Wang, L. Qiang, T. Ye, Z. Zhang, M. Qiu, and Y. Su, “Compact optical temporal differentiator based on silicon microring resonator,” Opt. Express 16(20), 15880–15886 (2008). [CrossRef] [PubMed]
- M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun. 1(29) (2010).
- A. Meijerink, C. G. H. Roeloffzen, R. Meijerink, D. A. I. Leimeng Zhuang, M. J. Marpaung, M. Bentum, J. Burla, P. Verpoorte, A. Jorna, Hulzinga, and W. van Etten, “Novel ring resonator-based integrated photonic beamformer for broadband phased-array antennas-Part I: design and performance analysis,” J. Lightwave Technol. 28(1), 3–18 (2010). [CrossRef]
- L. Zhuang, C. G. H. Roeloffzen, A. Meijerink, M. Burla, D. A. I. Marpaung, A. Leinse, M. Hoekman, R. G. Heideman, and W. van Etten, “Novel ring resonator-based integrated photonic beamformer for broadband phased-array antennas-Part II: experimental prototype,” J. Lightwave Technol. 28(1), 19–31 (2010). [CrossRef]
- D. A. I. Marpaung, C. G. H. Roeloffzen, A. Leinse, and M. Hoekman, “A photonic chip based frequency discriminator for a high performance microwave photonic link,” Opt. Express 18(26), 27359–27370 (2010). [CrossRef] [PubMed]
- D. A. I. Marpaung, L. Chevalier, M. Burla, and C. G. H. Roeloffzen, “Impulse radio ultrawideband pulse shaper based on a programmable photonic chip frequency discriminator,” Opt. Express 19(25), 24838–24848 (2011). [CrossRef] [PubMed]
- M. Burla, D. A. I. Marpaung, L. Zhuang, C. G. H. Roeloffzen, M. R. Khan, A. Leinse, M. Hoekman, and R. G. Heideman, “On-chip CMOS compatible reconfigurable optical delay line with separate carrier tuning for microwave photonic signal processing,” Opt. Express 19(22), 21475–21484 (2011). [CrossRef] [PubMed]
- S. L. Hahn, Transforms and Applications Handbook (CRC Press, 2010).
- R. Ashrafi and J. Azaña, “Terahertz bandwidth all-optical Hilbert transformers based on long-period gratings,” Opt. Lett. 37(13), 2604–2606 (2012). [CrossRef] [PubMed]
- M. H. Asghari and J. Azaña, “All-optical hilbert transformer based on a single phase-shifted fiber bragg grating: design and analysis,” Opt. Lett. 34(3), 334–336 (2009). [CrossRef] [PubMed]
- M. Li and J. Yao, “All-fiber temporal photonic fractional hilbert transformer based on a directly designed fiber bragg grating,” Opt. Lett. 35(2), 223–225 (2010). [CrossRef] [PubMed]
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