<PAGE>   1
                                    FORM 10Q
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549



              [X] QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d)
                     OF THE SECURITIES EXCHANGE ACT OF 1934
                  For the quarterly period ended June 27, 1997
                                                 -------------

                                       OR

              [ ] TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(d)
                     OF THE SECURITIES EXCHANGE ACT OF 1934
                        For the transition period from     to
                                                       ----   ----     
 
                          Commission File Number 1-6549
                                                 ------

                     American Science and Engineering, Inc.
             ------------------------------------------------------
             (Exact name of Registrant as specified in its charter)


          Massachusetts                                        04-2240991
------------------------------                            -------------------
(State or other jurisdiction of                           (I.R.S. Employer
incorporation or organization)                            Identification No.)


   829 Middlesex Turnpike
   Billerica, Massachusetts                                       01821
----------------------------------------                        ----------
(Address of principal executive offices)                        (Zip Code)


                                 (508) 262-8700
              ----------------------------------------------------
              (Registrant's telephone number, including area code)


       -------------------------------------------------------------------
              (Former name, former address and former fiscal year,
                         if changed since last report)

Indicate by check mark whether the Registrant (1) has filed all reports required
to be filed by Section 13 or 15(d) of the Securities Exchange Act of 1934 during
the preceding 12 months, and (2) has been subject to such filing requirements
for the past 90 days.

                                  Yes   X        No
                                      -----         -----     

Indicate the number of shares outstanding of each of the issuer's classes of
common stock, as of the latest practicable date

                                                      Outstanding at
      Class of Common Stock                           June 27, 1997
      ---------------------                           -------------
      $.66 2/3 par value                              4,645,210
                                     

                               Page 1 of 21 Pages
                     The Exhibit Index is Located at Page 9

                                       -1-

<PAGE>   2

                     AMERICAN SCIENCE AND ENGINEERING, INC.
                         PART I - FINANCIAL INFORMATION
                 CONDENSED CONSOLIDATED STATEMENTS OF OPERATIONS
                                   (Unaudited)



<TABLE>
<CAPTION>
                                                                   For The Three Months Ended
                                                                 -------------------------------
Dollars and shares in thousands, except per share amounts        June 27, 1997     June 28, 1996
                                                                 -------------     -------------

<S>                                                                  <C>              <C>    
NET SALES AND CONTRACT REVENUE                                       $7,532           $6,699 
                                                                     ------           ------
                                                                                   
COSTS AND EXPENSES:                                                                
   Cost of sales and contracts                                        4,782            4,608
   Selling, general and administrative expenses                       1,670            1,449
   Research and development                                             464              244
                                                                     ------           ------
   Total costs and expenses                                           6,916            6,301
                                                                     ------           ------
                                                                                   
OPERATING INCOME                                                        616              398
                                                                     ------           ------
                                                                                   
OTHER INCOME (EXPENSE):                                                            
   Interest, net                                                         27               47
   Other, net                                                            (3)             (19)
                                                                     ------           ------
   Total other income                                                    24               28
                                                                     ------           ------
                                                                                   
INCOME BEFORE PROVISION FOR                                                        
   INCOME TAXES                                                         640              426
                                                                                   
PROVISION FOR INCOME TAXES                                               30               20
                                                                     ------           ------
                                                                                   
NET INCOME                                                           $  610           $  406
                                                                     ======           ======
                                                                                   
INCOME PER SHARE-PRIMARY/FULLY                                                     
DILUTED                                                              $  .13           $  .09
                                                                     ======           ======
                                                                                   
DIVIDENDS PAID PER SHARE                                               None             None
                                                                     ======           ======
                                                                                   
WEIGHTED AVERAGE SHARES  - PRIMARY                                    4,805            4,712
                                                                     ======           ======
                         - FULLY DILUTED                              4,805            4,712
                                                                     ======           ======
</TABLE>
                                                                              

         The accompanying notes are an integral part of these condensed
                       consolidated financial statements.

                                       -2-

<PAGE>   3



                    AMERICAN SCIENCE AND ENGINEERING, INC.
                    CONDENSED CONSOLIDATED BALANCE SHEETS

<TABLE>
<CAPTION>
Dollars in thousands

                                                     June 27, 1997   Mar. 28, 1997
                                                     -------------   -------------
                                                     (Unaudited)
<S>                                                       <C>            <C>    
ASSETS   CURRENT ASSETS:                           
         Cash and cash equivalents                      $ 2,319        $ 3,202
         Accounts receivable, net                         6,717          5,019
         Unbilled costs and fees, net                     1,905            981
         Inventories                                      5,018          4,736
         Prepaid expenses and other current assets          149            157
                                                        -------        -------
         TOTAL CURRENT ASSETS                            16,108         14,095
                                                        -------        -------
                                                   
         NONCURRENT ASSETS:                        
         Deposits                                            65            115
         Property and equipment, net               
            of accumulated depreciation of         
            $9,012 at June 27, 1997 and            
            $8,860 at March 28, 1997                      1,492          1,304
                                                        -------        -------
                                                   
                                                        $17,665        $15,514
                                                        =======        =======
</TABLE>
         
      
         The accompanying notes are an integral part of these condensed
                        consolidated financial statements.

                                       -3-

<PAGE>   4



                     AMERICAN SCIENCE AND ENGINEERING, INC.
                     CONDENSED CONSOLIDATED BALANCE SHEETS
                                   (CONTINUED)
<TABLE>
<CAPTION>
Dollars in thousands

                                                                   June 27, 1997       Mar. 28, 1997
                                                                   -------------       -------------
                                                                    (Unaudited)
<S>                                                                   <C>                 <C>    
LIABILITIES &       CURRENT LIABILITIES:
STOCKHOLDERS'       Current maturities of obligations
INVESTMENT             under capital leases                           $    18             $    18
                    Accounts payable                                    2,685               2,253
                    Accrued salaries and benefits                         682                 573
                    Accrued warranty costs                                328                 292
                    Deferred revenue                                      818                 526
                    Customer deposits                                     137                  --
                    Other current liabilities                             686                 577
                                                                      -------             -------
                    TOTAL CURRENT LIABILITIES                           5,354               4,239
                                                                      -------             -------

                    NONCURRENT LIABILITIES:
                    Obligations under capital leases, net
                       of current maturities                               37                  42
                    Deferred revenue                                      715                 660
                    Deferred compensation                                 167                 174
                    Deferred rent                                         254                 249
                                                                      -------             -------
                    TOTAL NONCURRENT LIABILITIES                        1,173               1,125
                                                                      -------             -------

                    STOCKHOLDERS' INVESTMENT: 
                    Preferred stock, no par value
                    Authorized - 100,000 shares 
                    Issued - None 
                    Common stock, $.66-2/3 par value 
                    Authorized -20,000,000 shares 
                    Issued 4,645,210 shares at June 27, 1997
                     and 4,585,209 shares at Mar. 28, 1997              3,098               3,058
                    Capital in excess of par value                     15,611              15,273
                    Accumulated deficit                                (6,755)             (7,365)
                                                                      -------             -------
                                                                       11,954              10,966
                    Note receivable-Officer                              (640)               (640)
                    Less: treasury stock -
                    62,841 shares at June 27, 1997
                      and Mar. 28, 1997 at cost                          (176)               (176)
                                                                      -------             -------

                    TOTAL STOCKHOLDERS' INVESTMENT                     11,138              10,150
                                                                      -------             -------
                                                                      $17,665             $15,514
                                                                      =======             =======
</TABLE>


         The accompanying notes are an integral part of these condensed
                       consolidated financial statements.

                                       -4-

<PAGE>   5



                     AMERICAN SCIENCE AND ENGINEERING, INC.
                 CONDENSED CONSOLIDATED STATEMENTS OF CASH FLOWS
                                   (Unaudited)
<TABLE>
<CAPTION>
Dollars in thousands

                                                                        For the Three Months Ended
                                                                    ---------------------------------
                                                                    June 27, 1997       June 28, 1996
                                                                    -------------       -------------
<S>                                                                    <C>                 <C>    
CASH FLOWS FROM OPERATING ACTIVITIES:
Net income                                                             $   610             $   406
Adjustments to reconcile net income to net cash provided by
(used for) operating activities:
   Depreciation and amortization                                           152                  81
   Provisions for contract, inventory and warranty reserves                208                 213
Changes in assets and liabilities:
   Accounts receivable                                                  (1,698)             (1,078)
   Unbilled costs and fees                                                (974)                500
   Inventories                                                            (332)                101
   Prepaid expenses, other current assets, and deposits                     58                (156)
   Accounts payable                                                        432                 232
   Customer deposits                                                       137                (165)
   Accrued expenses and other current liabilities                          438                 367
   Noncurrent liabilities                                                   53                   2
                                                                       -------             -------
Total adjustments                                                       (1,526)                 97
                                                                       -------             -------

Net cash provided by (used for) operating activities                      (916)                503
                                                                       -------             -------

CASH FLOWS FROM INVESTING ACTIVITIES:
   Purchase of property and equipment, net                                (340)               (148)
                                                                       -------             -------

CASH FLOWS FROM FINANCING ACTIVITIES:
   Proceeds from exercise of stock options                                 378                 127
   Principal payments of capital lease obligations                          (5)                 (4)
                                                                       -------             -------
   Cash (used for) provided by financing activities                        373                 123
                                                                       -------             -------

NET INCREASE IN CASH AND CASH EQUIVALENTS                                 (883)                478
CASH AND CASH EQUIVALENTS AT BEGINNING OF PERIOD                         3,202               3,377
                                                                       -------             -------
CASH AND CASH EQUIVALENTS AT END OF PERIOD                             $ 2,319             $ 3,855
                                                                       =======             =======

SUPPLEMENTAL DISCLOSURES OF CASH FLOW INFORMATION:
   Interest paid                                                       $     2             $     3
   Income taxes paid                                                   $    --             $    30

</TABLE>

         The accompanying notes are an integral part of these condensed
                       consolidated financial statements.

                                       -5-

<PAGE>   6



                     AMERICAN SCIENCE AND ENGINEERING, INC.
           PREPARATION OF CONDENSED CONSOLIDATED FINANCIAL STATEMENTS


The condensed consolidated financial statements included herein have been
prepared by American Science and Engineering, Inc. (the Company) pursuant to the
rules and regulations of the Securities and Exchange Commission, and the annual
condensed consolidated financial statements are subject to year end audit by
independent public accountants. Certain information and footnote disclosures
normally included in financial statements prepared in accordance with generally
accepted accounting principles have been condensed or omitted pursuant to such
rules and regulations. The Company believes, however, that the disclosures are
adequate to make the information presented not misleading. It is suggested that
these condensed consolidated financial statements be read in conjunction with
the consolidated financial statements and the notes thereto included in the
Company's latest annual report on Form 10-K.

The condensed consolidated financial statements, in the opinion of management,
include all adjustments necessary to present fairly the Company's financial
position and the results of operations. These results are not necessarily to be
considered indicative of the results for the entire year.


        NOTES TO CONDENSED CONSOLIDATED FINANCIAL STATEMENTS (Unaudited)

1.   Nature of Operations and Significant Accounting Policies
     --------------------------------------------------------
     American Science and Engineering, Inc., is engaged in the development and
     manufacture of sophisticated X-ray inspection systems for critical
     detection and security screening solutions for sale primarily to U.S. and
     foreign government agencies.

     The significant accounting policies followed by the Company and its
     subsidiary in preparing its consolidated financial statements are set forth
     in Note 1 to the consolidated financial statements included in Form 10-K
     for the year ended March 28, 1997. The Company has made no change in these
     policies during this quarter.


2.   Inventories
     -----------
     Inventories consisted of:

<TABLE>
<CAPTION>
                                     June 27, 1997        Mar. 28, 1997
                                     -------------        -------------
    <S>                                <C>                  <C>       
    Raw materials and completed
       sub-assemblies                  $2,653,000           $3,106,000
    Work in process                     2,265,000            1,334,000
    Finished goods                        100,000              296,000
                                       ----------           ==========
    Total                              $5,018,000           $4,736,000
                                       ==========           ==========
</TABLE>


3.   Income per Common and Common Equivalent Share
     ---------------------------------------------
     Income per common share for the three month period ended June 27, 1997 was
     computed by dividing net income by the weighted average number of common
     stock and common stock equivalents outstanding during the year. If
     dilutive, stock options and stock warrants were considered in the
     computation of earnings per common and common equivalent shares. Under the
     provisions of SFAS No. 128 there would be no material change from primary
     and fully diluted earnings per common share to basic and diluted earnings
     per share for the quarters ended June 27, 1997 and June 28, 1996.
        
                                       -6-

<PAGE>   7


4.   Income Taxes
     ------------
     At March 28, 1997, the Company had approximately $6,986,000 of federal net
     operating loss carry forwards which can be used, subject to certain
     limitations, to offset any future federal taxable income. The carry
     forwards expire through the year 2010. The Company also has unused
     investment tax and other credits of approximately $213,000 expiring through
     2001. The provision for income taxes in the accompanying consolidated
     statement of operations is substantially a current provision and differs
     from the provision calculated by applying the statutory federal income tax
     rate of 34% to income before provision for income taxes due to the
     following:

<TABLE>
<CAPTION>
                                                               Three Months End
                                                                 June 27, 1997
                                                               ----------------
             <S>                                                   <C>        
             Provision for income taxes at statutory rate          $ 217,600  
             Benefit from net operating loss carry forward          (217,600)
             Other tax liabilities                                    30,000
                                                                   ---------
                                                                   $  30,000
                                                                   =========
</TABLE>
                                                              

5.   Presentation
     ------------
     Certain amounts in the quarter ended June 28, 1996 have been reclassified
     to conform to the quarter ended June 27, 1997 financial statements
     presentation.




                                       -7-

<PAGE>   8



                     AMERICAN SCIENCE AND ENGINEERING, INC.
           ITEM 2 - MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL
                       CONDITION AND RESULTS OF OPERATIONS

Overview
--------
In the first quarter of fiscal 1998, the Company produced its ninth consecutive
quarter of increasing profitability. Net sales and contract revenues were
$7,532,000 during the quarter, a 12% increase from the comparable year ago
period and a 5% decrease as compared to the fourth quarter of fiscal 1997. The
company earned net profits of $610,000 in the current quarter, a 50% increase
over the first quarter of fiscal 1997 and an 8% increase over the previous
quarter.

Results of Operations
---------------------
Net sales and contract revenues in the first quarter increased by $833,000 (12%)
in comparison to the corresponding period a year ago and decreased by $436,000
(5%) compared to the fourth quarter of fiscal 1997. Compared to the previous
quarter, security systems and related field service revenues were down by
$409,000 (6%) and research and engineering revenues declined by $27,000 (3%).
The decline in security systems and related field service revenues was due to
customer-requested delays in orders for, and installations of, the Company's
X-ray inspection systems.

For the first quarter, costs of sales and contracts increased to $4,782,000 from
$4,608,000 in the corresponding period a year ago due primarily to increased
sales volume. Costs of sales and contracts represented 63% of revenues versus
69% for the corresponding period last year and 65% for the fourth quarter of
fiscal year 1997. The costs of sales percentage of revenues in the current
quarter decreased from the previous quarter primarily due to the sales mix of
higher margin products.

Selling, general and administrative expenses of $1,670,000 for the first quarter
were higher by 15% compared to the corresponding year-ago period and lower by 3%
compared to the fourth quarter of fiscal 1997. As a percent of sales, selling,
general and administrative expenses were 22% of revenues in the current quarter
compared to 22% of revenues for the corresponding year ago period and 22% for
the previous quarter.

Company-funded research and development expenses of $464,000 for the first
quarter were higher by $220,000 (90%) compared to the first quarter of fiscal
1997 and lower by $7,000 (2%) compared to the fourth quarter of fiscal year
1997. The increase compared to the year ago period is an intentional result of
the Company's growing commitment to research and development.

The Company produced a net profit of $610,000 during the first quarter of fiscal
1998. This represents an improvement of 50% over net profit in the year-ago
quarter and an increase of 8% over the net profit reported in the previous
quarter. The improved profitability results primarily from higher revenue
compared to the year ago quarter and a more profitable product mix compared to
both the year ago and prior quarters.

Liquidity and Capital Resources
-------------------------------
Net cash used by operating activities during the first three months of fiscal
1998 was $916,000, compared to $503,000 cash provided by operating activities
during the corresponding year-ago period. This decrease in cash is primarily due
to an increase in accounts and unbilled receivables of $1,698,000 and $974,000,
respectively, partially offset by increases in accounts payable of $432,000 and
accrued expenses and other current liabilities of $438,000. Cash and cash
equivalents decreased by $883,000 to $2,319,000 as of June 27, 1997, compared to
$3,202,000 on March 28, 1997. Working capital increased by $898,000 (9%) since
March 28, 1997, growing from $9,856,000 to $10,754,000 at the end of the first
quarter.


No external borrowings were outstanding at the end of either the current or
previous quarter. Management believes the Company has access to sufficient
capital resources to support operations over the next several quarters.

                                       -8-

<PAGE>   9



                     AMERICAN SCIENCE AND ENGINEERING, INC.


Part II - Other Information

Item 6. Exhibits and Reports on Form 8-K
        --------------------------------

(a)             Exhibit Index
                -------------

                         Exhibit                                Page
                         -------                                ----
                
         (10)   Statement re: Computation of                      6
                Income per Common
                and Common Equivalent Share
                
         (11)   Employment Agreement between the
                Company and Ralph Sheridan dated
                September 26, 1996                               10



(b)             Reports on Form 8-K
                -------------------

          No reports on Form 8-K were filed during the quarter.


                SIGNATURES
                ----------

Pursuant to the requirements of the Securities Exchange Act of 1934, the
Registrant has duly caused this Report to be signed on its behalf by the
undersigned hereunto duly authorized.


                                      AMERICAN SCIENCE AND ENGINEERING, INC.
                                      (Registrant)


Date: 4 August 1997                   /s/  Lee C. Steele
                                      ------------------------------------------
                                      Lee C. Steele
                                      Vice President and Chief Financial Officer


Safe Harbor Statement
---------------------
The foregoing 10-Q contains statements concerning the Company's financial
performance and business operations which may be considered "forward-looking"
under applicable securities laws.

The Company wishes to caution readers of this Form 10-Q that actual results
might differ materially from those projected in any forward-looking statements.

Factors which might cause actual results to differ materially from those
projected in the forward-looking statements contained herein include the
following: Significant reductions or delays in procurements of the Company's
systems by the United States Government; disruption in the supply of any
sole-source component incorporated into the Company's products (of which there
are several); litigation seeking to restrict the use of intellectual property
used by the Company; potential product liability claims against the Company;
global political trends and events which affect public perception of the threat
presented by drugs, explosives and other contraband; the ability of governments
and private organizations to fund purchases of the Company's products to
address such threats; and the potential insufficiency of Company resources,
including human resources, capital, plant and equipment and management systems,
to accommodate any future growth. These and certain other factors which might
cause actual results to differ materially from those projected are more fully
set forth under the caption "Risk Factors" in the Company's Registration
Statement on Form S-3 (SEC File No. 333-9151).

                                       -9-