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                                                                     Exhibit 4.1


                    SECOND AMENDMENT TO THE RIGHTS AGREEMENT

      THIS SECOND AMENDMENT TO RIGHTS AGREEMENT (this "Amendment"), is dated as
of May 19, 2004, between Advanced Fibre Communications, Inc., a Delaware
corporation (the "Company"), and Equiserve Trust Company, N.A. (the "Rights
Agent").

      WHEREAS, the Company and the Rights Agent (as the successor in interest to
BankBoston, N.A.) are parties to a Rights Agreement, dated as of May 13, 1998,
as amended by the First Amendment thereto, dated as of October 19, 1998 (the
"Rights Agreement"); and

      WHEREAS, pursuant to Section 27 of the Rights Agreement, the Company and
the Rights Agent desire to amend the Rights Agreement as set forth below;

      NOW, THEREFORE, the Rights Agreement is hereby amended as follows:

      1.    Amendment of Section 1.

      Section 1 of the Rights Agreement is amended by adding thereto a new
definition immediately after the definition of "Final Expiration Date" and
immediately before the definition of "NASDAQ" which new definition shall read as
follows:

      "'Merger Agreement' shall mean the Agreement and Plan of Merger, dated as
      of May 19, 2004, among Tellabs, Inc., Chardonnay Merger Corp. and the
      Company, as the same may be amended from time to time."

      2.    Amendment of Section 7.

      Section 7(a) of the Rights Agreement is amended by amending the definition
of "Expiration Date" by deleting the phrase which begins with the word "or"
immediately preceding clause (iii) thereof and ends with the "." at the end of
Section 7(a) and replacing it with the phrase:

      "(iii) the time at which such Rights are exchanged as provided in Section
      24 hereof, or (iv) immediately prior to the Effective Time (as defined in
      the Merger Agreement) (the earliest of (i), (ii), (iii) and (iv) being the
      "Expiration Date")."

      3.    Addition of New Section 35.

      The Rights Agreement is amended by adding a Section 35 immediately after
Section 34 which shall read as follows:
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      "Section 35. Exception for Merger Agreement. Notwithstanding any provision
      of this Agreement to the contrary, (i) no Distribution Date, Shares
      Acquisition Date or Triggering Event shall be deemed to have occurred,
      (ii) none of Parent, Sub, or Sister Subsidiary (each as defined in the
      Merger Agreement), nor any of their Subsidiaries, Affiliates or Associates
      (collectively, the "Acquisition Group"), shall be deemed to have become an
      Acquiring Person, and (iii) no holder of Rights shall be entitled to
      exercise such Rights under, or be entitled to any rights or benefits
      pursuant to, any of Sections 3, 7, 11 or 13, or any other provision of
      this Agreement, in each case solely by reason of (a) the approval,
      execution, delivery and performance of the Merger Agreement, (b)
      consummation of any of the transactions contemplated by the Merger
      Agreement, including, without limitation, the Merger and the Subsequent
      Merger (each as defined in the Merger Agreement), in accordance with the
      provisions of the Merger Agreement, or (c) the announcement of any of
      foregoing; provided that in the event that one or more members of the
      Acquisition Group collectively become the Beneficial Owner of 15% or more
      of the shares of Common Stock then outstanding in any manner other than as
      set forth in the Merger Agreement, the provisions of this Section 35
      (other than this proviso) shall terminate."

      4.    Effectiveness.

            This Amendment shall be deemed effective as of May 19, 2004 as if
executed by both parties hereto on such date. Except as amended hereby, the
Rights Agreement shall remain in full force and effect and shall be otherwise
unaffected hereby.

      5.    Miscellaneous.

            This Amendment shall be deemed to be a contract made under the laws
of the State of Delaware and for all purposes shall be governed by and construed
in accordance with the laws of such state applicable to contracts to be made and
performed entirely within such state. This Amendment may be executed in any
number of counterparts, each of such counterparts shall for all purposes be
deemed to be an original, and all such counterparts shall together constitute
but one and the same instrument. If any term, provision, covenant or restriction
of this Amendment is held by a court of competent jurisdiction or other
authority to be invalid, illegal, or unenforceable, the remainder of the terms,
provisions, covenants and restrictions of this Amendment shall remain in full
force and effect and shall in no way be affected, impaired or invalidated.

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      IN WITNESS WHEREOF, the parties hereto have caused this Amendment to be
duly executed as of the date set forth above.

                         ADVANCED FIBRE COMMUNICATIONS, INC.



                         By:        /s/ John A. Schofield
                             ----------------------------------
                             Name:  John A. Schofield
                             Title: Chief Executive Officer and President


                         EQUISERVE TRUST COMPANY, N.A.



                         By:        /s/ Dennis V. Moccia
                             ----------------------------------
                             Name:  Dennis V. Moccia
                             Title: Managing Director